head 1.1; branch 1.1.1; access; symbols netbsd-9-5-RELEASE:1.1.1.3 netbsd-11-0-RELEASE:1.1.1.6 netbsd-11-0-RC7:1.1.1.6 netbsd-11-0-RC6:1.1.1.6 netbsd-11-0-RC5:1.1.1.6 netbsd-11-0-RC4:1.1.1.6 netbsd-11-0-RC3:1.1.1.6 netbsd-11-0-RC2:1.1.1.6 netbsd-11-0-RC1:1.1.1.6 perseant-exfatfs-base-20250801:1.1.1.6 netbsd-11:1.1.1.6.0.2 netbsd-11-base:1.1.1.6 netbsd-10-1-RELEASE:1.1.1.4 mpfr-4-2-1:1.1.1.6 perseant-exfatfs-base-20240630:1.1.1.5 perseant-exfatfs:1.1.1.5.0.2 perseant-exfatfs-base:1.1.1.5 netbsd-8-3-RELEASE:1.1.1.1 netbsd-9-4-RELEASE:1.1.1.3 netbsd-10-0-RELEASE:1.1.1.4 netbsd-10-0-RC6:1.1.1.4 netbsd-10-0-RC5:1.1.1.4 netbsd-10-0-RC4:1.1.1.4 netbsd-10-0-RC3:1.1.1.4 netbsd-10-0-RC2:1.1.1.4 netbsd-10-0-RC1:1.1.1.4 mpfr-4-2-0:1.1.1.5 netbsd-10:1.1.1.4.0.6 netbsd-10-base:1.1.1.4 netbsd-9-3-RELEASE:1.1.1.3 cjep_sun2x-base1:1.1.1.4 cjep_sun2x:1.1.1.4.0.4 cjep_sun2x-base:1.1.1.4 cjep_staticlib_x-base1:1.1.1.4 netbsd-9-2-RELEASE:1.1.1.3 cjep_staticlib_x:1.1.1.4.0.2 cjep_staticlib_x-base:1.1.1.4 netbsd-9-1-RELEASE:1.1.1.3 mpfr-4-1-0:1.1.1.4 phil-wifi-20200421:1.1.1.3 phil-wifi-20200411:1.1.1.3 is-mlppp:1.1.1.3.0.4 is-mlppp-base:1.1.1.3 phil-wifi-20200406:1.1.1.3 netbsd-8-2-RELEASE:1.1.1.1 netbsd-9-0-RELEASE:1.1.1.3 netbsd-9-0-RC2:1.1.1.3 netbsd-9-0-RC1:1.1.1.3 phil-wifi-20191119:1.1.1.3 netbsd-9:1.1.1.3.0.2 netbsd-9-base:1.1.1.3 phil-wifi-20190609:1.1.1.3 netbsd-8-1-RELEASE:1.1.1.1 netbsd-8-1-RC1:1.1.1.1 pgoyette-compat-merge-20190127:1.1.1.2.2.1 pgoyette-compat-20190127:1.1.1.3 pgoyette-compat-20190118:1.1.1.3 pgoyette-compat-1226:1.1.1.3 pgoyette-compat-1126:1.1.1.3 pgoyette-compat-1020:1.1.1.3 pgoyette-compat-0930:1.1.1.3 pgoyette-compat-0906:1.1.1.3 mpfr-4-0-1:1.1.1.3 netbsd-7-2-RELEASE:1.1.1.1 pgoyette-compat-0728:1.1.1.2 netbsd-8-0-RELEASE:1.1.1.1 phil-wifi:1.1.1.2.0.4 phil-wifi-base:1.1.1.2 pgoyette-compat-0625:1.1.1.2 netbsd-8-0-RC2:1.1.1.1 pgoyette-compat-0521:1.1.1.2 pgoyette-compat-0502:1.1.1.2 pgoyette-compat-0422:1.1.1.2 netbsd-8-0-RC1:1.1.1.1 pgoyette-compat-0415:1.1.1.2 pgoyette-compat-0407:1.1.1.2 pgoyette-compat-0330:1.1.1.2 pgoyette-compat-0322:1.1.1.2 pgoyette-compat-0315:1.1.1.2 netbsd-7-1-2-RELEASE:1.1.1.1 pgoyette-compat:1.1.1.2.0.2 pgoyette-compat-base:1.1.1.2 netbsd-7-1-1-RELEASE:1.1.1.1 matt-nb8-mediatek:1.1.1.1.0.26 matt-nb8-mediatek-base:1.1.1.1 mpfr-3-1-5:1.1.1.2 perseant-stdc-iso10646:1.1.1.1.0.24 perseant-stdc-iso10646-base:1.1.1.1 netbsd-8:1.1.1.1.0.22 netbsd-8-base:1.1.1.1 prg-localcount2-base3:1.1.1.1 prg-localcount2-base2:1.1.1.1 prg-localcount2-base1:1.1.1.1 prg-localcount2:1.1.1.1.0.20 prg-localcount2-base:1.1.1.1 pgoyette-localcount-20170426:1.1.1.1 bouyer-socketcan-base1:1.1.1.1 pgoyette-localcount-20170320:1.1.1.1 netbsd-7-1:1.1.1.1.0.18 netbsd-7-1-RELEASE:1.1.1.1 netbsd-7-1-RC2:1.1.1.1 netbsd-7-nhusb-base-20170116:1.1.1.1 bouyer-socketcan:1.1.1.1.0.16 bouyer-socketcan-base:1.1.1.1 pgoyette-localcount-20170107:1.1.1.1 netbsd-7-1-RC1:1.1.1.1 pgoyette-localcount-20161104:1.1.1.1 netbsd-7-0-2-RELEASE:1.1.1.1 localcount-20160914:1.1.1.1 netbsd-7-nhusb:1.1.1.1.0.14 netbsd-7-nhusb-base:1.1.1.1 pgoyette-localcount-20160806:1.1.1.1 pgoyette-localcount-20160726:1.1.1.1 pgoyette-localcount:1.1.1.1.0.12 pgoyette-localcount-base:1.1.1.1 netbsd-7-0-1-RELEASE:1.1.1.1 netbsd-7-0:1.1.1.1.0.10 netbsd-7-0-RELEASE:1.1.1.1 netbsd-7-0-RC3:1.1.1.1 netbsd-7-0-RC2:1.1.1.1 netbsd-7-0-RC1:1.1.1.1 tls-maxphys-base:1.1.1.1 tls-maxphys:1.1.1.1.0.8 netbsd-7:1.1.1.1.0.6 netbsd-7-base:1.1.1.1 yamt-pagecache:1.1.1.1.0.4 yamt-pagecache-base9:1.1.1.1 tls-earlyentropy:1.1.1.1.0.2 tls-earlyentropy-base:1.1.1.1 riastradh-xf86-video-intel-2-7-1-pre-2-21-15:1.1.1.1 riastradh-drm2-base3:1.1.1.1 mpfr-3-1-2:1.1.1.1 mpfr:1.1.1; locks; strict; comment @# @; 1.1 date 2013.11.28.12.30.54; author mrg; state Exp; branches 1.1.1.1; next ; commitid suMNN8knGTUjE2fx; 1.1.1.1 date 2013.11.28.12.30.54; author mrg; state Exp; branches 1.1.1.1.4.1 1.1.1.1.8.1; next 1.1.1.2; commitid suMNN8knGTUjE2fx; 1.1.1.2 date 2017.08.17.01.09.24; author mrg; state Exp; branches 1.1.1.2.2.1 1.1.1.2.4.1; next 1.1.1.3; commitid byVWLjxDrcX5fv3A; 1.1.1.3 date 2018.09.04.05.02.02; author mrg; state Exp; branches; next 1.1.1.4; commitid aNyHel12V2dWaKQA; 1.1.1.4 date 2020.09.26.07.25.42; author mrg; state Exp; branches; next 1.1.1.5; commitid uDXWuNicgnpBNwpC; 1.1.1.5 date 2023.03.05.22.08.38; author mrg; state Exp; branches 1.1.1.5.2.1; next 1.1.1.6; commitid AcBV1jKMu2z25ZfE; 1.1.1.6 date 2024.07.01.02.24.32; author mrg; state Exp; branches; next ; commitid E5dBeg2NyM7QV4gF; 1.1.1.1.4.1 date 2013.11.28.12.30.54; author yamt; state dead; branches; next 1.1.1.1.4.2; commitid nx2BSsHy0NPeAxBx; 1.1.1.1.4.2 date 2014.05.22.14.09.15; author yamt; state Exp; branches; next ; commitid nx2BSsHy0NPeAxBx; 1.1.1.1.8.1 date 2013.11.28.12.30.54; author tls; state dead; branches; next 1.1.1.1.8.2; commitid jTnpym9Qu0o4R1Nx; 1.1.1.1.8.2 date 2014.08.20.00.00.04; author tls; state Exp; branches; next ; commitid jTnpym9Qu0o4R1Nx; 1.1.1.2.2.1 date 2018.09.06.06.53.46; author pgoyette; state Exp; branches; next ; commitid HCi1bXD317XIK0RA; 1.1.1.2.4.1 date 2019.06.10.22.02.27; author christos; state Exp; branches; next ; commitid jtc8rnCzWiEEHGqB; 1.1.1.5.2.1 date 2025.08.02.05.50.24; author perseant; state Exp; branches; next ; commitid 23j6GFaDws3O875G; desc @@ 1.1 log @Initial revision @ text @This is mpfr.info, produced by makeinfo version 4.13 from mpfr.texi. This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 3.1.2. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in *note GNU Free Documentation License::. INFO-DIR-SECTION Software libraries START-INFO-DIR-ENTRY * mpfr: (mpfr). Multiple Precision Floating-Point Reliable Library. END-INFO-DIR-ENTRY  File: mpfr.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir) GNU MPFR ******** This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 3.1.2. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in *note GNU Free Documentation License::. * Menu: * Copying:: MPFR Copying Conditions (LGPL). * Introduction to MPFR:: Brief introduction to GNU MPFR. * Installing MPFR:: How to configure and compile the MPFR library. * Reporting Bugs:: How to usefully report bugs. * MPFR Basics:: What every MPFR user should now. * MPFR Interface:: MPFR functions and macros. * API Compatibility:: API compatibility with previous MPFR versions. * Contributors:: * References:: * GNU Free Documentation License:: * Concept Index:: * Function and Type Index::  File: mpfr.info, Node: Copying, Next: Introduction to MPFR, Prev: Top, Up: Top MPFR Copying Conditions *********************** The GNU MPFR library (or MPFR for short) is "free"; this means that everyone is free to use it and free to redistribute it on a free basis. The library is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you. Specifically, we want to make sure that you have the right to give away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things. To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of the GNU MPFR library, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights. Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU MPFR library. If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation. The precise conditions of the license for the GNU MPFR library are found in the Lesser General Public License that accompanies the source code. See the file COPYING.LESSER.  File: mpfr.info, Node: Introduction to MPFR, Next: Installing MPFR, Prev: Copying, Up: Top 1 Introduction to MPFR ********************** MPFR is a portable library written in C for arbitrary precision arithmetic on floating-point numbers. It is based on the GNU MP library. It aims to provide a class of floating-point numbers with precise semantics. The main characteristics of MPFR, which make it differ from most arbitrary precision floating-point software tools, are: * the MPFR code is portable, i.e., the result of any operation does not depend on the machine word size `mp_bits_per_limb' (64 on most current processors); * the precision in bits can be set _exactly_ to any valid value for each variable (including very small precision); * MPFR provides the four rounding modes from the IEEE 754-1985 standard, plus away-from-zero, as well as for basic operations as for other mathematical functions. In particular, with a precision of 53 bits, MPFR is able to exactly reproduce all computations with double-precision machine floating-point numbers (e.g., `double' type in C, with a C implementation that rigorously follows Annex F of the ISO C99 standard and `FP_CONTRACT' pragma set to `OFF') on the four arithmetic operations and the square root, except the default exponent range is much wider and subnormal numbers are not implemented (but can be emulated). This version of MPFR is released under the GNU Lesser General Public License, version 3 or any later version. It is permitted to link MPFR to most non-free programs, as long as when distributing them the MPFR source code and a means to re-link with a modified MPFR library is provided. 1.1 How to Use This Manual ========================== Everyone should read *note MPFR Basics::. If you need to install the library yourself, you need to read *note Installing MPFR::, too. To use the library you will need to refer to *note MPFR Interface::. The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.  File: mpfr.info, Node: Installing MPFR, Next: Reporting Bugs, Prev: Introduction to MPFR, Up: Top 2 Installing MPFR ***************** The MPFR library is already installed on some GNU/Linux distributions, but the development files necessary to the compilation such as `mpfr.h' are not always present. To check that MPFR is fully installed on your computer, you can check the presence of the file `mpfr.h' in `/usr/include', or try to compile a small program having `#include ' (since `mpfr.h' may be installed somewhere else). For instance, you can try to compile: #include #include int main (void) { printf ("MPFR library: %-12s\nMPFR header: %s (based on %d.%d.%d)\n", mpfr_get_version (), MPFR_VERSION_STRING, MPFR_VERSION_MAJOR, MPFR_VERSION_MINOR, MPFR_VERSION_PATCHLEVEL); return 0; } with cc -o version version.c -lmpfr -lgmp and if you get errors whose first line looks like version.c:2:19: error: mpfr.h: No such file or directory then MPFR is probably not installed. Running this program will give you the MPFR version. If MPFR is not installed on your computer, or if you want to install a different version, please follow the steps below. 2.1 How to Install ================== Here are the steps needed to install the library on Unix systems (more details are provided in the `INSTALL' file): 1. To build MPFR, you first have to install GNU MP (version 4.1 or higher) on your computer. You need a C compiler, preferably GCC, but any reasonable compiler should work. And you need the standard Unix `make' command, plus some other standard Unix utility commands. Then, in the MPFR build directory, type the following commands. 2. `./configure' This will prepare the build and setup the options according to your system. You can give options to specify the install directories (instead of the default `/usr/local'), threading support, and so on. See the `INSTALL' file and/or the output of `./configure --help' for more information, in particular if you get error messages. 3. `make' This will compile MPFR, and create a library archive file `libmpfr.a'. On most platforms, a dynamic library will be produced too. 4. `make check' This will make sure MPFR was built correctly. If you get error messages, please report this to the MPFR mailing-list `mpfr@@inria.fr'. (*Note Reporting Bugs::, for information on what to include in useful bug reports.) 5. `make install' This will copy the files `mpfr.h' and `mpf2mpfr.h' to the directory `/usr/local/include', the library files (`libmpfr.a' and possibly others) to the directory `/usr/local/lib', the file `mpfr.info' to the directory `/usr/local/share/info', and some other documentation files to the directory `/usr/local/share/doc/mpfr' (or if you passed the `--prefix' option to `configure', using the prefix directory given as argument to `--prefix' instead of `/usr/local'). 2.2 Other `make' Targets ======================== There are some other useful make targets: * `mpfr.info' or `info' Create or update an info version of the manual, in `mpfr.info'. This file is already provided in the MPFR archives. * `mpfr.pdf' or `pdf' Create a PDF version of the manual, in `mpfr.pdf'. * `mpfr.dvi' or `dvi' Create a DVI version of the manual, in `mpfr.dvi'. * `mpfr.ps' or `ps' Create a Postscript version of the manual, in `mpfr.ps'. * `mpfr.html' or `html' Create a HTML version of the manual, in several pages in the directory `doc/mpfr.html'; if you want only one output HTML file, then type `makeinfo --html --no-split mpfr.texi' from the `doc' directory instead. * `clean' Delete all object files and archive files, but not the configuration files. * `distclean' Delete all generated files not included in the distribution. * `uninstall' Delete all files copied by `make install'. 2.3 Build Problems ================== In case of problem, please read the `INSTALL' file carefully before reporting a bug, in particular section "In case of problem". Some problems are due to bad configuration on the user side (not specific to MPFR). Problems are also mentioned in the FAQ `http://www.mpfr.org/faq.html'. Please report problems to the MPFR mailing-list `mpfr@@inria.fr'. *Note Reporting Bugs::. Some bug fixes are available on the MPFR 3.1.2 web page `http://www.mpfr.org/mpfr-3.1.2/'. 2.4 Getting the Latest Version of MPFR ====================================== The latest version of MPFR is available from `ftp://ftp.gnu.org/gnu/mpfr/' or `http://www.mpfr.org/'.  File: mpfr.info, Node: Reporting Bugs, Next: MPFR Basics, Prev: Installing MPFR, Up: Top 3 Reporting Bugs **************** If you think you have found a bug in the MPFR library, first have a look on the MPFR 3.1.2 web page `http://www.mpfr.org/mpfr-3.1.2/' and the FAQ `http://www.mpfr.org/faq.html': perhaps this bug is already known, in which case you may find there a workaround for it. You might also look in the archives of the MPFR mailing-list: `https://sympa.inria.fr/sympa/arc/mpfr'. Otherwise, please investigate and report it. We have made this library available to you, and it is not to ask too much from you, to ask you to report the bugs that you find. There are a few things you should think about when you put your bug report together. You have to send us a test case that makes it possible for us to reproduce the bug, i.e., a small self-content program, using no other library than MPFR. Include instructions on how to run the test case. You also have to explain what is wrong; if you get a crash, or if the results you get are incorrect and in that case, in what way. Please include compiler version information in your bug report. This can be extracted using `cc -V' on some machines, or, if you're using GCC, `gcc -v'. Also, include the output from `uname -a' and the MPFR version (the GMP version may be useful too). If you get a failure while running `make' or `make check', please include the `config.log' file in your bug report. If your bug report is good, we will do our best to help you to get a corrected version of the library; if the bug report is poor, we will not do anything about it (aside of chiding you to send better bug reports). Send your bug report to the MPFR mailing-list `mpfr@@inria.fr'. If you think something in this manual is unclear, or downright incorrect, or if the language needs to be improved, please send a note to the same address.  File: mpfr.info, Node: MPFR Basics, Next: MPFR Interface, Prev: Reporting Bugs, Up: Top 4 MPFR Basics ************* * Menu: * Headers and Libraries:: * Nomenclature and Types:: * MPFR Variable Conventions:: * Rounding Modes:: * Floating-Point Values on Special Numbers:: * Exceptions:: * Memory Handling::  File: mpfr.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: MPFR Basics, Up: MPFR Basics 4.1 Headers and Libraries ========================= All declarations needed to use MPFR are collected in the include file `mpfr.h'. It is designed to work with both C and C++ compilers. You should include that file in any program using the MPFR library: #include Note however that prototypes for MPFR functions with `FILE *' parameters are provided only if `' is included too (before `mpfr.h'): #include #include Likewise `' (or `') is required for prototypes with `va_list' parameters, such as `mpfr_vprintf'. And for any functions using `intmax_t', you must include `' or `' before `mpfr.h', to allow `mpfr.h' to define prototypes for these functions. Moreover, users of C++ compilers under some platforms may need to define `MPFR_USE_INTMAX_T' (and should do it for portability) before `mpfr.h' has been included; of course, it is possible to do that on the command line, e.g., with `-DMPFR_USE_INTMAX_T'. Note: If `mpfr.h' and/or `gmp.h' (used by `mpfr.h') are included several times (possibly from another header file), `' and/or `' (or `') should be included *before the first inclusion* of `mpfr.h' or `gmp.h'. Alternatively, you can define `MPFR_USE_FILE' (for MPFR I/O functions) and/or `MPFR_USE_VA_LIST' (for MPFR functions with `va_list' parameters) anywhere before the last inclusion of `mpfr.h'. As a consequence, if your file is a public header that includes `mpfr.h', you need to use the latter method. When calling a MPFR macro, it is not allowed to have previously defined a macro with the same name as some keywords (currently `do', `while' and `sizeof'). You can avoid the use of MPFR macros encapsulating functions by defining the `MPFR_USE_NO_MACRO' macro before `mpfr.h' is included. In general this should not be necessary, but this can be useful when debugging user code: with some macros, the compiler may emit spurious warnings with some warning options, and macros can prevent some prototype checking. All programs using MPFR must link against both `libmpfr' and `libgmp' libraries. On a typical Unix-like system this can be done with `-lmpfr -lgmp' (in that order), for example: gcc myprogram.c -lmpfr -lgmp MPFR is built using Libtool and an application can use that to link if desired, *note GNU Libtool: (libtool.info)Top. If MPFR has been installed to a non-standard location, then it may be necessary to set up environment variables such as `C_INCLUDE_PATH' and `LIBRARY_PATH', or use `-I' and `-L' compiler options, in order to point to the right directories. For a shared library, it may also be necessary to set up some sort of run-time library path (e.g., `LD_LIBRARY_PATH') on some systems. Please read the `INSTALL' file for additional information.  File: mpfr.info, Node: Nomenclature and Types, Next: MPFR Variable Conventions, Prev: Headers and Libraries, Up: MPFR Basics 4.2 Nomenclature and Types ========================== A "floating-point number", or "float" for short, is an arbitrary precision significand (also called mantissa) with a limited precision exponent. The C data type for such objects is `mpfr_t' (internally defined as a one-element array of a structure, and `mpfr_ptr' is the C data type representing a pointer to this structure). A floating-point number can have three special values: Not-a-Number (NaN) or plus or minus Infinity. NaN represents an uninitialized object, the result of an invalid operation (like 0 divided by 0), or a value that cannot be determined (like +Infinity minus +Infinity). Moreover, like in the IEEE 754 standard, zero is signed, i.e., there are both +0 and -0; the behavior is the same as in the IEEE 754 standard and it is generalized to the other functions supported by MPFR. Unless documented otherwise, the sign bit of a NaN is unspecified. The "precision" is the number of bits used to represent the significand of a floating-point number; the corresponding C data type is `mpfr_prec_t'. The precision can be any integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. In the current implementation, `MPFR_PREC_MIN' is equal to 2. Warning! MPFR needs to increase the precision internally, in order to provide accurate results (and in particular, correct rounding). Do not attempt to set the precision to any value near `MPFR_PREC_MAX', otherwise MPFR will abort due to an assertion failure. Moreover, you may reach some memory limit on your platform, in which case the program may abort, crash or have undefined behavior (depending on your C implementation). The "rounding mode" specifies the way to round the result of a floating-point operation, in case the exact result can not be represented exactly in the destination significand; the corresponding C data type is `mpfr_rnd_t'.  File: mpfr.info, Node: MPFR Variable Conventions, Next: Rounding Modes, Prev: Nomenclature and Types, Up: MPFR Basics 4.3 MPFR Variable Conventions ============================= Before you can assign to an MPFR variable, you need to initialize it by calling one of the special initialization functions. When you're done with a variable, you need to clear it out, using one of the functions for that purpose. A variable should only be initialized once, or at least cleared out between each initialization. After a variable has been initialized, it may be assigned to any number of times. For efficiency reasons, avoid to initialize and clear out a variable in loops. Instead, initialize it before entering the loop, and clear it out after the loop has exited. You do not need to be concerned about allocating additional space for MPFR variables, since any variable has a significand of fixed size. Hence unless you change its precision, or clear and reinitialize it, a floating-point variable will have the same allocated space during all its life. As a general rule, all MPFR functions expect output arguments before input arguments. This notation is based on an analogy with the assignment operator. MPFR allows you to use the same variable for both input and output in the same expression. For example, the main function for floating-point multiplication, `mpfr_mul', can be used like this: `mpfr_mul (x, x, x, rnd)'. This computes the square of X with rounding mode `rnd' and puts the result back in X.  File: mpfr.info, Node: Rounding Modes, Next: Floating-Point Values on Special Numbers, Prev: MPFR Variable Conventions, Up: MPFR Basics 4.4 Rounding Modes ================== The following five rounding modes are supported: * `MPFR_RNDN': round to nearest (roundTiesToEven in IEEE 754-2008), * `MPFR_RNDZ': round toward zero (roundTowardZero in IEEE 754-2008), * `MPFR_RNDU': round toward plus infinity (roundTowardPositive in IEEE 754-2008), * `MPFR_RNDD': round toward minus infinity (roundTowardNegative in IEEE 754-2008), * `MPFR_RNDA': round away from zero. The `round to nearest' mode works as in the IEEE 754 standard: in case the number to be rounded lies exactly in the middle of two representable numbers, it is rounded to the one with the least significant bit set to zero. For example, the number 2.5, which is represented by (10.1) in binary, is rounded to (10.0)=2 with a precision of two bits, and not to (11.0)=3. This rule avoids the "drift" phenomenon mentioned by Knuth in volume 2 of The Art of Computer Programming (Section 4.2.2). Most MPFR functions take as first argument the destination variable, as second and following arguments the input variables, as last argument a rounding mode, and have a return value of type `int', called the "ternary value". The value stored in the destination variable is correctly rounded, i.e., MPFR behaves as if it computed the result with an infinite precision, then rounded it to the precision of this variable. The input variables are regarded as exact (in particular, their precision does not affect the result). As a consequence, in case of a non-zero real rounded result, the error on the result is less or equal to 1/2 ulp (unit in the last place) of that result in the rounding to nearest mode, and less than 1 ulp of that result in the directed rounding modes (a ulp is the weight of the least significant represented bit of the result after rounding). Unless documented otherwise, functions returning an `int' return a ternary value. If the ternary value is zero, it means that the value stored in the destination variable is the exact result of the corresponding mathematical function. If the ternary value is positive (resp. negative), it means the value stored in the destination variable is greater (resp. lower) than the exact result. For example with the `MPFR_RNDU' rounding mode, the ternary value is usually positive, except when the result is exact, in which case it is zero. In the case of an infinite result, it is considered as inexact when it was obtained by overflow, and exact otherwise. A NaN result (Not-a-Number) always corresponds to an exact return value. The opposite of a returned ternary value is guaranteed to be representable in an `int'. Unless documented otherwise, functions returning as result the value `1' (or any other value specified in this manual) for special cases (like `acos(0)') yield an overflow or an underflow if that value is not representable in the current exponent range.  File: mpfr.info, Node: Floating-Point Values on Special Numbers, Next: Exceptions, Prev: Rounding Modes, Up: MPFR Basics 4.5 Floating-Point Values on Special Numbers ============================================ This section specifies the floating-point values (of type `mpfr_t') returned by MPFR functions (where by "returned" we mean here the modified value of the destination object, which should not be mixed with the ternary return value of type `int' of those functions). For functions returning several values (like `mpfr_sin_cos'), the rules apply to each result separately. Functions can have one or several input arguments. An input point is a mapping from these input arguments to the set of the MPFR numbers. When none of its components are NaN, an input point can also be seen as a tuple in the extended real numbers (the set of the real numbers with both infinities). When the input point is in the domain of the mathematical function, the result is rounded as described in Section "Rounding Modes" (but see below for the specification of the sign of an exact zero). Otherwise the general rules from this section apply unless stated otherwise in the description of the MPFR function (*note MPFR Interface::). When the input point is not in the domain of the mathematical function but is in its closure in the extended real numbers and the function can be extended by continuity, the result is the obtained limit. Examples: `mpfr_hypot' on (+Inf,0) gives +Inf. But `mpfr_pow' cannot be defined on (1,+Inf) using this rule, as one can find sequences (X_N,Y_N) such that X_N goes to 1, Y_N goes to +Inf and X_N to the Y_N goes to any positive value when N goes to the infinity. When the input point is in the closure of the domain of the mathematical function and an input argument is +0 (resp. -0), one considers the limit when the corresponding argument approaches 0 from above (resp. below). If the limit is not defined (e.g., `mpfr_log' on -0), the behavior is specified in the description of the MPFR function. When the result is equal to 0, its sign is determined by considering the limit as if the input point were not in the domain: If one approaches 0 from above (resp. below), the result is +0 (resp. -0); for example, `mpfr_sin' on +0 gives +0. In the other cases, the sign is specified in the description of the MPFR function; for example `mpfr_max' on -0 and +0 gives +0. When the input point is not in the closure of the domain of the function, the result is NaN. Example: `mpfr_sqrt' on -17 gives NaN. When an input argument is NaN, the result is NaN, possibly except when a partial function is constant on the finite floating-point numbers; such a case is always explicitly specified in *note MPFR Interface::. Example: `mpfr_hypot' on (NaN,0) gives NaN, but `mpfr_hypot' on (NaN,+Inf) gives +Inf (as specified in *note Special Functions::), since for any finite input X, `mpfr_hypot' on (X,+Inf) gives +Inf.  File: mpfr.info, Node: Exceptions, Next: Memory Handling, Prev: Floating-Point Values on Special Numbers, Up: MPFR Basics 4.6 Exceptions ============== MPFR supports 6 exception types: * Underflow: An underflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent smaller than the minimum value of the current exponent range. (In the round-to-nearest mode, the halfway case is rounded toward zero.) Note: This is not the single possible definition of the underflow. MPFR chooses to consider the underflow _after_ rounding. The underflow before rounding can also be defined. For instance, consider a function that has the exact result 7 multiplied by two to the power E-4, where E is the smallest exponent (for a significand between 1/2 and 1), with a 2-bit target precision and rounding toward plus infinity. The exact result has the exponent E-1. With the underflow before rounding, such a function call would yield an underflow, as E-1 is outside the current exponent range. However, MPFR first considers the rounded result assuming an unbounded exponent range. The exact result cannot be represented exactly in precision 2, and here, it is rounded to 0.5 times 2 to E, which is representable in the current exponent range. As a consequence, this will not yield an underflow in MPFR. * Overflow: An overflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent larger than the maximum value of the current exponent range. In the round-to-nearest mode, the result is infinite. Note: unlike the underflow case, there is only one possible definition of overflow here. * Divide-by-zero: An exact infinite result is obtained from finite inputs. * NaN: A NaN exception occurs when the result of a function is NaN. * Inexact: An inexact exception occurs when the result of a function cannot be represented exactly and must be rounded. * Range error: A range exception occurs when a function that does not return a MPFR number (such as comparisons and conversions to an integer) has an invalid result (e.g., an argument is NaN in `mpfr_cmp', or a conversion to an integer cannot be represented in the target type). MPFR has a global flag for each exception, which can be cleared, set or tested by functions described in *note Exception Related Functions::. Differences with the ISO C99 standard: * In C, only quiet NaNs are specified, and a NaN propagation does not raise an invalid exception. Unless explicitly stated otherwise, MPFR sets the NaN flag whenever a NaN is generated, even when a NaN is propagated (e.g., in NaN + NaN), as if all NaNs were signaling. * An invalid exception in C corresponds to either a NaN exception or a range error in MPFR.  File: mpfr.info, Node: Memory Handling, Prev: Exceptions, Up: MPFR Basics 4.7 Memory Handling =================== MPFR functions may create caches, e.g., when computing constants such as Pi, either because the user has called a function like `mpfr_const_pi' directly or because such a function was called internally by the MPFR library itself to compute some other function. At any time, the user can free the various caches with `mpfr_free_cache'. It is strongly advised to do that before terminating a thread, or before exiting when using tools like `valgrind' (to avoid memory leaks being reported). MPFR internal data such as flags, the exponent range, the default precision and rounding mode, and caches (i.e., data that are not accessed via parameters) are either global (if MPFR has not been compiled as thread safe) or per-thread (thread local storage, TLS). The initial values of TLS data after a thread is created entirely depend on the compiler and thread implementation (MPFR simply does a conventional variable initialization, the variables being declared with an implementation-defined TLS specifier).  File: mpfr.info, Node: MPFR Interface, Next: API Compatibility, Prev: MPFR Basics, Up: Top 5 MPFR Interface **************** The floating-point functions expect arguments of type `mpfr_t'. The MPFR floating-point functions have an interface that is similar to the GNU MP functions. The function prefix for floating-point operations is `mpfr_'. The user has to specify the precision of each variable. A computation that assigns a variable will take place with the precision of the assigned variable; the cost of that computation should not depend on the precision of variables used as input (on average). The semantics of a calculation in MPFR is specified as follows: Compute the requested operation exactly (with "infinite accuracy"), and round the result to the precision of the destination variable, with the given rounding mode. The MPFR floating-point functions are intended to be a smooth extension of the IEEE 754 arithmetic. The results obtained on a given computer are identical to those obtained on a computer with a different word size, or with a different compiler or operating system. MPFR _does not keep track_ of the accuracy of a computation. This is left to the user or to a higher layer (for example the MPFI library for interval arithmetic). As a consequence, if two variables are used to store only a few significant bits, and their product is stored in a variable with large precision, then MPFR will still compute the result with full precision. The value of the standard C macro `errno' may be set to non-zero by any MPFR function or macro, whether or not there is an error. * Menu: * Initialization Functions:: * Assignment Functions:: * Combined Initialization and Assignment Functions:: * Conversion Functions:: * Basic Arithmetic Functions:: * Comparison Functions:: * Special Functions:: * Input and Output Functions:: * Formatted Output Functions:: * Integer Related Functions:: * Rounding Related Functions:: * Miscellaneous Functions:: * Exception Related Functions:: * Compatibility with MPF:: * Custom Interface:: * Internals::  File: mpfr.info, Node: Initialization Functions, Next: Assignment Functions, Prev: MPFR Interface, Up: MPFR Interface 5.1 Initialization Functions ============================ An `mpfr_t' object must be initialized before storing the first value in it. The functions `mpfr_init' and `mpfr_init2' are used for that purpose. -- Function: void mpfr_init2 (mpfr_t X, mpfr_prec_t PREC) Initialize X, set its precision to be *exactly* PREC bits and its value to NaN. (Warning: the corresponding MPF function initializes to zero instead.) Normally, a variable should be initialized once only or at least be cleared, using `mpfr_clear', between initializations. To change the precision of a variable which has already been initialized, use `mpfr_set_prec'. The precision PREC must be an integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX' (otherwise the behavior is undefined). -- Function: void mpfr_inits2 (mpfr_prec_t PREC, mpfr_t X, ...) Initialize all the `mpfr_t' variables of the given variable argument `va_list', set their precision to be *exactly* PREC bits and their value to NaN. See `mpfr_init2' for more details. The `va_list' is assumed to be composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It begins from X, and ends when it encounters a null pointer (whose type must also be `mpfr_ptr'). -- Function: void mpfr_clear (mpfr_t X) Free the space occupied by the significand of X. Make sure to call this function for all `mpfr_t' variables when you are done with them. -- Function: void mpfr_clears (mpfr_t X, ...) Free the space occupied by all the `mpfr_t' variables of the given `va_list'. See `mpfr_clear' for more details. The `va_list' is assumed to be composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It begins from X, and ends when it encounters a null pointer (whose type must also be `mpfr_ptr'). Here is an example of how to use multiple initialization functions (since `NULL' is not necessarily defined in this context, we use `(mpfr_ptr) 0' instead, but `(mpfr_ptr) NULL' is also correct). { mpfr_t x, y, z, t; mpfr_inits2 (256, x, y, z, t, (mpfr_ptr) 0); ... mpfr_clears (x, y, z, t, (mpfr_ptr) 0); } -- Function: void mpfr_init (mpfr_t X) Initialize X, set its precision to the default precision, and set its value to NaN. The default precision can be changed by a call to `mpfr_set_default_prec'. Warning! In a given program, some other libraries might change the default precision and not restore it. Thus it is safer to use `mpfr_init2'. -- Function: void mpfr_inits (mpfr_t X, ...) Initialize all the `mpfr_t' variables of the given `va_list', set their precision to the default precision and their value to NaN. See `mpfr_init' for more details. The `va_list' is assumed to be composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It begins from X, and ends when it encounters a null pointer (whose type must also be `mpfr_ptr'). Warning! In a given program, some other libraries might change the default precision and not restore it. Thus it is safer to use `mpfr_inits2'. -- Macro: MPFR_DECL_INIT (NAME, PREC) This macro declares NAME as an automatic variable of type `mpfr_t', initializes it and sets its precision to be *exactly* PREC bits and its value to NaN. NAME must be a valid identifier. You must use this macro in the declaration section. This macro is much faster than using `mpfr_init2' but has some drawbacks: * You *must not* call `mpfr_clear' with variables created with this macro (the storage is allocated at the point of declaration and deallocated when the brace-level is exited). * You *cannot* change their precision. * You *should not* create variables with huge precision with this macro. * Your compiler must support `Non-Constant Initializers' (standard in C++ and ISO C99) and `Token Pasting' (standard in ISO C89). If PREC is not a constant expression, your compiler must support `variable-length automatic arrays' (standard in ISO C99). GCC 2.95.3 and above supports all these features. If you compile your program with GCC in C89 mode and with `-pedantic', you may want to define the `MPFR_USE_EXTENSION' macro to avoid warnings due to the `MPFR_DECL_INIT' implementation. -- Function: void mpfr_set_default_prec (mpfr_prec_t PREC) Set the default precision to be *exactly* PREC bits, where PREC can be any integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. The precision of a variable means the number of bits used to store its significand. All subsequent calls to `mpfr_init' or `mpfr_inits' will use this precision, but previously initialized variables are unaffected. The default precision is set to 53 bits initially. Note: when MPFR is built with the `--enable-thread-safe' configure option, the default precision is local to each thread. *Note Memory Handling::, for more information. -- Function: mpfr_prec_t mpfr_get_default_prec (void) Return the current default MPFR precision in bits. See the documentation of `mpfr_set_default_prec'. Here is an example on how to initialize floating-point variables: { mpfr_t x, y; mpfr_init (x); /* use default precision */ mpfr_init2 (y, 256); /* precision _exactly_ 256 bits */ ... /* When the program is about to exit, do ... */ mpfr_clear (x); mpfr_clear (y); mpfr_free_cache (); /* free the cache for constants like pi */ } The following functions are useful for changing the precision during a calculation. A typical use would be for adjusting the precision gradually in iterative algorithms like Newton-Raphson, making the computation precision closely match the actual accurate part of the numbers. -- Function: void mpfr_set_prec (mpfr_t X, mpfr_prec_t PREC) Reset the precision of X to be *exactly* PREC bits, and set its value to NaN. The previous value stored in X is lost. It is equivalent to a call to `mpfr_clear(x)' followed by a call to `mpfr_init2(x, prec)', but more efficient as no allocation is done in case the current allocated space for the significand of X is enough. The precision PREC can be any integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. In case you want to keep the previous value stored in X, use `mpfr_prec_round' instead. -- Function: mpfr_prec_t mpfr_get_prec (mpfr_t X) Return the precision of X, i.e., the number of bits used to store its significand.  File: mpfr.info, Node: Assignment Functions, Next: Combined Initialization and Assignment Functions, Prev: Initialization Functions, Up: MPFR Interface 5.2 Assignment Functions ======================== These functions assign new values to already initialized floats (*note Initialization Functions::). -- Function: int mpfr_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) -- Function: int mpfr_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t RND) -- Function: int mpfr_set_uj (mpfr_t ROP, uintmax_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_sj (mpfr_t ROP, intmax_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_flt (mpfr_t ROP, float OP, mpfr_rnd_t RND) -- Function: int mpfr_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND) -- Function: int mpfr_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t RND) -- Function: int mpfr_set_decimal64 (mpfr_t ROP, _Decimal64 OP, mpfr_rnd_t RND) -- Function: int mpfr_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND) Set the value of ROP from OP, rounded toward the given direction RND. Note that the input 0 is converted to +0 by `mpfr_set_ui', `mpfr_set_si', `mpfr_set_uj', `mpfr_set_sj', `mpfr_set_z', `mpfr_set_q' and `mpfr_set_f', regardless of the rounding mode. If the system does not support the IEEE 754 standard, `mpfr_set_flt', `mpfr_set_d', `mpfr_set_ld' and `mpfr_set_decimal64' might not preserve the signed zeros. The `mpfr_set_decimal64' function is built only with the configure option `--enable-decimal-float', which also requires `--with-gmp-build', and when the compiler or system provides the `_Decimal64' data type (recent versions of GCC support this data type); to use `mpfr_set_decimal64', one should define the macro `MPFR_WANT_DECIMAL_FLOATS' before including `mpfr.h'. `mpfr_set_q' might fail if the numerator (or the denominator) can not be represented as a `mpfr_t'. Note: If you want to store a floating-point constant to a `mpfr_t', you should use `mpfr_set_str' (or one of the MPFR constant functions, such as `mpfr_const_pi' for Pi) instead of `mpfr_set_flt', `mpfr_set_d', `mpfr_set_ld' or `mpfr_set_decimal64'. Otherwise the floating-point constant will be first converted into a reduced-precision (e.g., 53-bit) binary (or decimal, for `mpfr_set_decimal64') number before MPFR can work with it. -- Function: int mpfr_set_ui_2exp (mpfr_t ROP, unsigned long int OP, mpfr_exp_t E, mpfr_rnd_t RND) -- Function: int mpfr_set_si_2exp (mpfr_t ROP, long int OP, mpfr_exp_t E, mpfr_rnd_t RND) -- Function: int mpfr_set_uj_2exp (mpfr_t ROP, uintmax_t OP, intmax_t E, mpfr_rnd_t RND) -- Function: int mpfr_set_sj_2exp (mpfr_t ROP, intmax_t OP, intmax_t E, mpfr_rnd_t RND) -- Function: int mpfr_set_z_2exp (mpfr_t ROP, mpz_t OP, mpfr_exp_t E, mpfr_rnd_t RND) Set the value of ROP from OP multiplied by two to the power E, rounded toward the given direction RND. Note that the input 0 is converted to +0. -- Function: int mpfr_set_str (mpfr_t ROP, const char *S, int BASE, mpfr_rnd_t RND) Set ROP to the value of the string S in base BASE, rounded in the direction RND. See the documentation of `mpfr_strtofr' for a detailed description of the valid string formats. Contrary to `mpfr_strtofr', `mpfr_set_str' requires the _whole_ string to represent a valid floating-point number. The meaning of the return value differs from other MPFR functions: it is 0 if the entire string up to the final null character is a valid number in base BASE; otherwise it is -1, and ROP may have changed (users interested in the *note ternary value:: should use `mpfr_strtofr' instead). Note: it is preferable to use `mpfr_set_str' if one wants to distinguish between an infinite ROP value coming from an infinite S or from an overflow. -- Function: int mpfr_strtofr (mpfr_t ROP, const char *NPTR, char **ENDPTR, int BASE, mpfr_rnd_t RND) Read a floating-point number from a string NPTR in base BASE, rounded in the direction RND; BASE must be either 0 (to detect the base, as described below) or a number from 2 to 62 (otherwise the behavior is undefined). If NPTR starts with valid data, the result is stored in ROP and `*ENDPTR' points to the character just after the valid data (if ENDPTR is not a null pointer); otherwise ROP is set to zero (for consistency with `strtod') and the value of NPTR is stored in the location referenced by ENDPTR (if ENDPTR is not a null pointer). The usual ternary value is returned. Parsing follows the standard C `strtod' function with some extensions. After optional leading whitespace, one has a subject sequence consisting of an optional sign (`+' or `-'), and either numeric data or special data. The subject sequence is defined as the longest initial subsequence of the input string, starting with the first non-whitespace character, that is of the expected form. The form of numeric data is a non-empty sequence of significand digits with an optional decimal point, and an optional exponent consisting of an exponent prefix followed by an optional sign and a non-empty sequence of decimal digits. A significand digit is either a decimal digit or a Latin letter (62 possible characters), with `A' = 10, `B' = 11, ..., `Z' = 35; case is ignored in bases less or equal to 36, in bases larger than 36, `a' = 36, `b' = 37, ..., `z' = 61. The value of a significand digit must be strictly less than the base. The decimal point can be either the one defined by the current locale or the period (the first one is accepted for consistency with the C standard and the practice, the second one is accepted to allow the programmer to provide MPFR numbers from strings in a way that does not depend on the current locale). The exponent prefix can be `e' or `E' for bases up to 10, or `@@' in any base; it indicates a multiplication by a power of the base. In bases 2 and 16, the exponent prefix can also be `p' or `P', in which case the exponent, called _binary exponent_, indicates a multiplication by a power of 2 instead of the base (there is a difference only for base 16); in base 16 for example `1p2' represents 4 whereas `1@@2' represents 256. The value of an exponent is always written in base 10. If the argument BASE is 0, then the base is automatically detected as follows. If the significand starts with `0b' or `0B', base 2 is assumed. If the significand starts with `0x' or `0X', base 16 is assumed. Otherwise base 10 is assumed. Note: The exponent (if present) must contain at least a digit. Otherwise the possible exponent prefix and sign are not part of the number (which ends with the significand). Similarly, if `0b', `0B', `0x' or `0X' is not followed by a binary/hexadecimal digit, then the subject sequence stops at the character `0', thus 0 is read. Special data (for infinities and NaN) can be `@@inf@@' or `@@nan@@(n-char-sequence-opt)', and if BASE <= 16, it can also be `infinity', `inf', `nan' or `nan(n-char-sequence-opt)', all case insensitive. A `n-char-sequence-opt' is a possibly empty string containing only digits, Latin letters and the underscore (0, 1, 2, ..., 9, a, b, ..., z, A, B, ..., Z, _). Note: one has an optional sign for all data, even NaN. For example, `-@@nAn@@(This_Is_Not_17)' is a valid representation for NaN in base 17. -- Function: void mpfr_set_nan (mpfr_t X) -- Function: void mpfr_set_inf (mpfr_t X, int SIGN) -- Function: void mpfr_set_zero (mpfr_t X, int SIGN) Set the variable X to NaN (Not-a-Number), infinity or zero respectively. In `mpfr_set_inf' or `mpfr_set_zero', X is set to plus infinity or plus zero iff SIGN is nonnegative; in `mpfr_set_nan', the sign bit of the result is unspecified. -- Function: void mpfr_swap (mpfr_t X, mpfr_t Y) Swap the values X and Y efficiently. Warning: the precisions are exchanged too; in case the precisions are different, `mpfr_swap' is thus not equivalent to three `mpfr_set' calls using a third auxiliary variable.  File: mpfr.info, Node: Combined Initialization and Assignment Functions, Next: Conversion Functions, Prev: Assignment Functions, Up: MPFR Interface 5.3 Combined Initialization and Assignment Functions ==================================================== -- Macro: int mpfr_init_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND) Initialize ROP and set its value from OP, rounded in the direction RND. The precision of ROP will be taken from the active default precision, as set by `mpfr_set_default_prec'. -- Function: int mpfr_init_set_str (mpfr_t X, const char *S, int BASE, mpfr_rnd_t RND) Initialize X and set its value from the string S in base BASE, rounded in the direction RND. See `mpfr_set_str'.  File: mpfr.info, Node: Conversion Functions, Next: Basic Arithmetic Functions, Prev: Combined Initialization and Assignment Functions, Up: MPFR Interface 5.4 Conversion Functions ======================== -- Function: float mpfr_get_flt (mpfr_t OP, mpfr_rnd_t RND) -- Function: double mpfr_get_d (mpfr_t OP, mpfr_rnd_t RND) -- Function: long double mpfr_get_ld (mpfr_t OP, mpfr_rnd_t RND) -- Function: _Decimal64 mpfr_get_decimal64 (mpfr_t OP, mpfr_rnd_t RND) Convert OP to a `float' (respectively `double', `long double' or `_Decimal64'), using the rounding mode RND. If OP is NaN, some fixed NaN (either quiet or signaling) or the result of 0.0/0.0 is returned. If OP is ±Inf, an infinity of the same sign or the result of ±1.0/0.0 is returned. If OP is zero, these functions return a zero, trying to preserve its sign, if possible. The `mpfr_get_decimal64' function is built only under some conditions: see the documentation of `mpfr_set_decimal64'. -- Function: long mpfr_get_si (mpfr_t OP, mpfr_rnd_t RND) -- Function: unsigned long mpfr_get_ui (mpfr_t OP, mpfr_rnd_t RND) -- Function: intmax_t mpfr_get_sj (mpfr_t OP, mpfr_rnd_t RND) -- Function: uintmax_t mpfr_get_uj (mpfr_t OP, mpfr_rnd_t RND) Convert OP to a `long', an `unsigned long', an `intmax_t' or an `uintmax_t' (respectively) after rounding it with respect to RND. If OP is NaN, 0 is returned and the _erange_ flag is set. If OP is too big for the return type, the function returns the maximum or the minimum of the corresponding C type, depending on the direction of the overflow; the _erange_ flag is set too. See also `mpfr_fits_slong_p', `mpfr_fits_ulong_p', `mpfr_fits_intmax_p' and `mpfr_fits_uintmax_p'. -- Function: double mpfr_get_d_2exp (long *EXP, mpfr_t OP, mpfr_rnd_t RND) -- Function: long double mpfr_get_ld_2exp (long *EXP, mpfr_t OP, mpfr_rnd_t RND) Return D and set EXP (formally, the value pointed to by EXP) such that 0.5<=abs(D)<1 and D times 2 raised to EXP equals OP rounded to double (resp. long double) precision, using the given rounding mode. If OP is zero, then a zero of the same sign (or an unsigned zero, if the implementation does not have signed zeros) is returned, and EXP is set to 0. If OP is NaN or an infinity, then the corresponding double precision (resp. long-double precision) value is returned, and EXP is undefined. -- Function: int mpfr_frexp (mpfr_exp_t *EXP, mpfr_t Y, mpfr_t X, mpfr_rnd_t RND) Set EXP (formally, the value pointed to by EXP) and Y such that 0.5<=abs(Y)<1 and Y times 2 raised to EXP equals X rounded to the precision of Y, using the given rounding mode. If X is zero, then Y is set to a zero of the same sign and EXP is set to 0. If X is NaN or an infinity, then Y is set to the same value and EXP is undefined. -- Function: mpfr_exp_t mpfr_get_z_2exp (mpz_t ROP, mpfr_t OP) Put the scaled significand of OP (regarded as an integer, with the precision of OP) into ROP, and return the exponent EXP (which may be outside the current exponent range) such that OP exactly equals ROP times 2 raised to the power EXP. If OP is zero, the minimal exponent `emin' is returned. If OP is NaN or an infinity, the _erange_ flag is set, ROP is set to 0, and the the minimal exponent `emin' is returned. The returned exponent may be less than the minimal exponent `emin' of MPFR numbers in the current exponent range; in case the exponent is not representable in the `mpfr_exp_t' type, the _erange_ flag is set and the minimal value of the `mpfr_exp_t' type is returned. -- Function: int mpfr_get_z (mpz_t ROP, mpfr_t OP, mpfr_rnd_t RND) Convert OP to a `mpz_t', after rounding it with respect to RND. If OP is NaN or an infinity, the _erange_ flag is set, ROP is set to 0, and 0 is returned. -- Function: int mpfr_get_f (mpf_t ROP, mpfr_t OP, mpfr_rnd_t RND) Convert OP to a `mpf_t', after rounding it with respect to RND. The _erange_ flag is set if OP is NaN or an infinity, which do not exist in MPF. If OP is NaN, then ROP is undefined. If OP is an +Inf (resp. -Inf), then ROP is set to the maximum (resp. minimum) value in the precision of the MPF number; if a future MPF version supports infinities, this behavior will be considered incorrect and will change (portable programs should assume that ROP is set either to this finite number or to an infinite number). Note that since MPFR currently has the same exponent type as MPF (but not with the same radix), the range of values is much larger in MPF than in MPFR, so that an overflow or underflow is not possible. -- Function: char * mpfr_get_str (char *STR, mpfr_exp_t *EXPPTR, int B, size_t N, mpfr_t OP, mpfr_rnd_t RND) Convert OP to a string of digits in base B, with rounding in the direction RND, where N is either zero (see below) or the number of significant digits output in the string; in the latter case, N must be greater or equal to 2. The base may vary from 2 to 62. If the input number is an ordinary number, the exponent is written through the pointer EXPPTR (for input 0, the current minimal exponent is written). The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. For example, the number -3.1416 would be returned as "-31416" in the string and 1 written at EXPPTR. If RND is to nearest, and OP is exactly in the middle of two consecutive possible outputs, the one with an even significand is chosen, where both significands are considered with the exponent of OP. Note that for an odd base, this may not correspond to an even last digit: for example with 2 digits in base 7, (14) and a half is rounded to (15) which is 12 in decimal, (16) and a half is rounded to (20) which is 14 in decimal, and (26) and a half is rounded to (26) which is 20 in decimal. If N is zero, the number of digits of the significand is chosen large enough so that re-reading the printed value with the same precision, assuming both output and input use rounding to nearest, will recover the original value of OP. More precisely, in most cases, the chosen precision of STR is the minimal precision m depending only on P = PREC(OP) and B that satisfies the above property, i.e., m = 1 + ceil(P*log(2)/log(B)), with P replaced by P-1 if B is a power of 2, but in some very rare cases, it might be m+1 (the smallest case for bases up to 62 is when P equals 186564318007 for bases 7 and 49). If STR is a null pointer, space for the significand is allocated using the current allocation function, and a pointer to the string is returned. To free the returned string, you must use `mpfr_free_str'. If STR is not a null pointer, it should point to a block of storage large enough for the significand, i.e., at least `max(N + 2, 7)'. The extra two bytes are for a possible minus sign, and for the terminating null character, and the value 7 accounts for `-@@Inf@@' plus the terminating null character. A pointer to the string is returned, unless there is an error, in which case a null pointer is returned. -- Function: void mpfr_free_str (char *STR) Free a string allocated by `mpfr_get_str' using the current unallocation function. The block is assumed to be `strlen(STR)+1' bytes. For more information about how it is done: *note Custom Allocation: (gmp.info)Custom Allocation. -- Function: int mpfr_fits_ulong_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_slong_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_uint_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_sint_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_ushort_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_sshort_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_uintmax_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_intmax_p (mpfr_t OP, mpfr_rnd_t RND) Return non-zero if OP would fit in the respective C data type, respectively `unsigned long', `long', `unsigned int', `int', `unsigned short', `short', `uintmax_t', `intmax_t', when rounded to an integer in the direction RND.  File: mpfr.info, Node: Basic Arithmetic Functions, Next: Comparison Functions, Prev: Conversion Functions, Up: MPFR Interface 5.5 Basic Arithmetic Functions ============================== -- Function: int mpfr_add (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_d (mpfr_t ROP, mpfr_t OP1, double OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, mpfr_rnd_t RND) Set ROP to OP1 + OP2 rounded in the direction RND. For types having no signed zero, it is considered unsigned (i.e., (+0) + 0 = (+0) and (-0) + 0 = (-0)). The `mpfr_add_d' function assumes that the radix of the `double' type is a power of 2, with a precision at most that declared by the C implementation (macro `IEEE_DBL_MANT_DIG', and if not defined 53 bits). -- Function: int mpfr_sub (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_sub (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_si_sub (mpfr_t ROP, long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_d_sub (mpfr_t ROP, double OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_d (mpfr_t ROP, mpfr_t OP1, double OP2, mpfr_rnd_t RND) -- Function: int mpfr_z_sub (mpfr_t ROP, mpz_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, mpfr_rnd_t RND) Set ROP to OP1 - OP2 rounded in the direction RND. For types having no signed zero, it is considered unsigned (i.e., (+0) - 0 = (+0), (-0) - 0 = (-0), 0 - (+0) = (-0) and 0 - (-0) = (+0)). The same restrictions than for `mpfr_add_d' apply to `mpfr_d_sub' and `mpfr_sub_d'. -- Function: int mpfr_mul (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_d (mpfr_t ROP, mpfr_t OP1, double OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, mpfr_rnd_t RND) Set ROP to OP1 times OP2 rounded in the direction RND. When a result is zero, its sign is the product of the signs of the operands (for types having no signed zero, it is considered positive). The same restrictions than for `mpfr_add_d' apply to `mpfr_mul_d'. -- Function: int mpfr_sqr (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the square of OP rounded in the direction RND. -- Function: int mpfr_div (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_div (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_si_div (mpfr_t ROP, long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_d_div (mpfr_t ROP, double OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_d (mpfr_t ROP, mpfr_t OP1, double OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, mpfr_rnd_t RND) Set ROP to OP1/OP2 rounded in the direction RND. When a result is zero, its sign is the product of the signs of the operands (for types having no signed zero, it is considered positive). The same restrictions than for `mpfr_add_d' apply to `mpfr_d_div' and `mpfr_div_d'. -- Function: int mpfr_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_sqrt_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) Set ROP to the square root of OP rounded in the direction RND (set ROP to -0 if OP is -0, to be consistent with the IEEE 754 standard). Set ROP to NaN if OP is negative. -- Function: int mpfr_rec_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the reciprocal square root of OP rounded in the direction RND. Set ROP to +Inf if OP is ±0, +0 if OP is +Inf, and NaN if OP is negative. -- Function: int mpfr_cbrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_root (mpfr_t ROP, mpfr_t OP, unsigned long int K, mpfr_rnd_t RND) Set ROP to the cubic root (resp. the Kth root) of OP rounded in the direction RND. For K odd (resp. even) and OP negative (including -Inf), set ROP to a negative number (resp. NaN). The Kth root of -0 is defined to be -0, whatever the parity of K. -- Function: int mpfr_pow (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_pow_ui (mpfr_t ROP, unsigned long int OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_pow (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to OP1 raised to OP2, rounded in the direction RND. Special values are handled as described in the ISO C99 and IEEE 754-2008 standards for the `pow' function: * `pow(±0, Y)' returns plus or minus infinity for Y a negative odd integer. * `pow(±0, Y)' returns plus infinity for Y negative and not an odd integer. * `pow(±0, Y)' returns plus or minus zero for Y a positive odd integer. * `pow(±0, Y)' returns plus zero for Y positive and not an odd integer. * `pow(-1, ±Inf)' returns 1. * `pow(+1, Y)' returns 1 for any Y, even a NaN. * `pow(X, ±0)' returns 1 for any X, even a NaN. * `pow(X, Y)' returns NaN for finite negative X and finite non-integer Y. * `pow(X, -Inf)' returns plus infinity for 0 < abs(x) < 1, and plus zero for abs(x) > 1. * `pow(X, +Inf)' returns plus zero for 0 < abs(x) < 1, and plus infinity for abs(x) > 1. * `pow(-Inf, Y)' returns minus zero for Y a negative odd integer. * `pow(-Inf, Y)' returns plus zero for Y negative and not an odd integer. * `pow(-Inf, Y)' returns minus infinity for Y a positive odd integer. * `pow(-Inf, Y)' returns plus infinity for Y positive and not an odd integer. * `pow(+Inf, Y)' returns plus zero for Y negative, and plus infinity for Y positive. -- Function: int mpfr_neg (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_abs (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to -OP and the absolute value of OP respectively, rounded in the direction RND. Just changes or adjusts the sign if ROP and OP are the same variable, otherwise a rounding might occur if the precision of ROP is less than that of OP. -- Function: int mpfr_dim (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the positive difference of OP1 and OP2, i.e., OP1 - OP2 rounded in the direction RND if OP1 > OP2, +0 if OP1 <= OP2, and NaN if OP1 or OP2 is NaN. -- Function: int mpfr_mul_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_2si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) Set ROP to OP1 times 2 raised to OP2 rounded in the direction RND. Just increases the exponent by OP2 when ROP and OP1 are identical. -- Function: int mpfr_div_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_2si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) Set ROP to OP1 divided by 2 raised to OP2 rounded in the direction RND. Just decreases the exponent by OP2 when ROP and OP1 are identical.  File: mpfr.info, Node: Comparison Functions, Next: Special Functions, Prev: Basic Arithmetic Functions, Up: MPFR Interface 5.6 Comparison Functions ======================== -- Function: int mpfr_cmp (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_cmp_ui (mpfr_t OP1, unsigned long int OP2) -- Function: int mpfr_cmp_si (mpfr_t OP1, long int OP2) -- Function: int mpfr_cmp_d (mpfr_t OP1, double OP2) -- Function: int mpfr_cmp_ld (mpfr_t OP1, long double OP2) -- Function: int mpfr_cmp_z (mpfr_t OP1, mpz_t OP2) -- Function: int mpfr_cmp_q (mpfr_t OP1, mpq_t OP2) -- Function: int mpfr_cmp_f (mpfr_t OP1, mpf_t OP2) Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1 < OP2. Both OP1 and OP2 are considered to their full own precision, which may differ. If one of the operands is NaN, set the _erange_ flag and return zero. Note: These functions may be useful to distinguish the three possible cases. If you need to distinguish two cases only, it is recommended to use the predicate functions (e.g., `mpfr_equal_p' for the equality) described below; they behave like the IEEE 754 comparisons, in particular when one or both arguments are NaN. But only floating-point numbers can be compared (you may need to do a conversion first). -- Function: int mpfr_cmp_ui_2exp (mpfr_t OP1, unsigned long int OP2, mpfr_exp_t E) -- Function: int mpfr_cmp_si_2exp (mpfr_t OP1, long int OP2, mpfr_exp_t E) Compare OP1 and OP2 multiplied by two to the power E. Similar as above. -- Function: int mpfr_cmpabs (mpfr_t OP1, mpfr_t OP2) Compare |OP1| and |OP2|. Return a positive value if |OP1| > |OP2|, zero if |OP1| = |OP2|, and a negative value if |OP1| < |OP2|. If one of the operands is NaN, set the _erange_ flag and return zero. -- Function: int mpfr_nan_p (mpfr_t OP) -- Function: int mpfr_inf_p (mpfr_t OP) -- Function: int mpfr_number_p (mpfr_t OP) -- Function: int mpfr_zero_p (mpfr_t OP) -- Function: int mpfr_regular_p (mpfr_t OP) Return non-zero if OP is respectively NaN, an infinity, an ordinary number (i.e., neither NaN nor an infinity), zero, or a regular number (i.e., neither NaN, nor an infinity nor zero). Return zero otherwise. -- Macro: int mpfr_sgn (mpfr_t OP) Return a positive value if OP > 0, zero if OP = 0, and a negative value if OP < 0. If the operand is NaN, set the _erange_ flag and return zero. This is equivalent to `mpfr_cmp_ui (op, 0)', but more efficient. -- Function: int mpfr_greater_p (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_greaterequal_p (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_less_p (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_lessequal_p (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_equal_p (mpfr_t OP1, mpfr_t OP2) Return non-zero if OP1 > OP2, OP1 >= OP2, OP1 < OP2, OP1 <= OP2, OP1 = OP2 respectively, and zero otherwise. Those functions return zero whenever OP1 and/or OP2 is NaN. -- Function: int mpfr_lessgreater_p (mpfr_t OP1, mpfr_t OP2) Return non-zero if OP1 < OP2 or OP1 > OP2 (i.e., neither OP1, nor OP2 is NaN, and OP1 <> OP2), zero otherwise (i.e., OP1 and/or OP2 is NaN, or OP1 = OP2). -- Function: int mpfr_unordered_p (mpfr_t OP1, mpfr_t OP2) Return non-zero if OP1 or OP2 is a NaN (i.e., they cannot be compared), zero otherwise.  File: mpfr.info, Node: Special Functions, Next: Input and Output Functions, Prev: Comparison Functions, Up: MPFR Interface 5.7 Special Functions ===================== All those functions, except explicitly stated (for example `mpfr_sin_cos'), return a *note ternary value::, i.e., zero for an exact return value, a positive value for a return value larger than the exact result, and a negative value otherwise. Important note: in some domains, computing special functions (either with correct or incorrect rounding) is expensive, even for small precision, for example the trigonometric and Bessel functions for large argument. -- Function: int mpfr_log (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_log2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_log10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the natural logarithm of OP, log2(OP) or log10(OP), respectively, rounded in the direction RND. Set ROP to -Inf if OP is -0 (i.e., the sign of the zero has no influence on the result). -- Function: int mpfr_exp (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_exp2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_exp10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential of OP, to 2 power of OP or to 10 power of OP, respectively, rounded in the direction RND. -- Function: int mpfr_cos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_sin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_tan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the cosine of OP, sine of OP, tangent of OP, rounded in the direction RND. -- Function: int mpfr_sin_cos (mpfr_t SOP, mpfr_t COP, mpfr_t OP, mpfr_rnd_t RND) Set simultaneously SOP to the sine of OP and COP to the cosine of OP, rounded in the direction RND with the corresponding precisions of SOP and COP, which must be different variables. Return 0 iff both results are exact, more precisely it returns s+4c where s=0 if SOP is exact, s=1 if SOP is larger than the sine of OP, s=2 if SOP is smaller than the sine of OP, and similarly for c and the cosine of OP. -- Function: int mpfr_sec (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_csc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_cot (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the secant of OP, cosecant of OP, cotangent of OP, rounded in the direction RND. -- Function: int mpfr_acos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_asin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_atan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the arc-cosine, arc-sine or arc-tangent of OP, rounded in the direction RND. Note that since `acos(-1)' returns the floating-point number closest to Pi according to the given rounding mode, this number might not be in the output range 0 <= ROP < \pi of the arc-cosine function; still, the result lies in the image of the output range by the rounding function. The same holds for `asin(-1)', `asin(1)', `atan(-Inf)', `atan(+Inf)' or for `atan(op)' with large OP and small precision of ROP. -- Function: int mpfr_atan2 (mpfr_t ROP, mpfr_t Y, mpfr_t X, mpfr_rnd_t RND) Set ROP to the arc-tangent2 of Y and X, rounded in the direction RND: if `x > 0', `atan2(y, x) = atan (y/x)'; if `x < 0', `atan2(y, x) = sign(y)*(Pi - atan (abs(y/x)))', thus a number from -Pi to Pi. As for `atan', in case the exact mathematical result is +Pi or -Pi, its rounded result might be outside the function output range. `atan2(y, 0)' does not raise any floating-point exception. Special values are handled as described in the ISO C99 and IEEE 754-2008 standards for the `atan2' function: * `atan2(+0, -0)' returns +Pi. * `atan2(-0, -0)' returns -Pi. * `atan2(+0, +0)' returns +0. * `atan2(-0, +0)' returns -0. * `atan2(+0, x)' returns +Pi for x < 0. * `atan2(-0, x)' returns -Pi for x < 0. * `atan2(+0, x)' returns +0 for x > 0. * `atan2(-0, x)' returns -0 for x > 0. * `atan2(y, 0)' returns -Pi/2 for y < 0. * `atan2(y, 0)' returns +Pi/2 for y > 0. * `atan2(+Inf, -Inf)' returns +3*Pi/4. * `atan2(-Inf, -Inf)' returns -3*Pi/4. * `atan2(+Inf, +Inf)' returns +Pi/4. * `atan2(-Inf, +Inf)' returns -Pi/4. * `atan2(+Inf, x)' returns +Pi/2 for finite x. * `atan2(-Inf, x)' returns -Pi/2 for finite x. * `atan2(y, -Inf)' returns +Pi for finite y > 0. * `atan2(y, -Inf)' returns -Pi for finite y < 0. * `atan2(y, +Inf)' returns +0 for finite y > 0. * `atan2(y, +Inf)' returns -0 for finite y < 0. -- Function: int mpfr_cosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_sinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_tanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the hyperbolic cosine, sine or tangent of OP, rounded in the direction RND. -- Function: int mpfr_sinh_cosh (mpfr_t SOP, mpfr_t COP, mpfr_t OP, mpfr_rnd_t RND) Set simultaneously SOP to the hyperbolic sine of OP and COP to the hyperbolic cosine of OP, rounded in the direction RND with the corresponding precision of SOP and COP, which must be different variables. Return 0 iff both results are exact (see `mpfr_sin_cos' for a more detailed description of the return value). -- Function: int mpfr_sech (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_csch (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_coth (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the hyperbolic secant of OP, cosecant of OP, cotangent of OP, rounded in the direction RND. -- Function: int mpfr_acosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_asinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_atanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the inverse hyperbolic cosine, sine or tangent of OP, rounded in the direction RND. -- Function: int mpfr_fac_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) Set ROP to the factorial of OP, rounded in the direction RND. -- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the logarithm of one plus OP, rounded in the direction RND. -- Function: int mpfr_expm1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential of OP followed by a subtraction by one, rounded in the direction RND. -- Function: int mpfr_eint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential integral of OP, rounded in the direction RND. For positive OP, the exponential integral is the sum of Euler's constant, of the logarithm of OP, and of the sum for k from 1 to infinity of OP to the power k, divided by k and factorial(k). For negative OP, ROP is set to NaN (this definition for negative argument follows formula 5.1.2 from the Handbook of Mathematical Functions from Abramowitz and Stegun, a future version might use another definition). -- Function: int mpfr_li2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to real part of the dilogarithm of OP, rounded in the direction RND. MPFR defines the dilogarithm function as the integral of -log(1-t)/t from 0 to OP. -- Function: int mpfr_gamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the Gamma function on OP, rounded in the direction RND. When OP is a negative integer, ROP is set to NaN. -- Function: int mpfr_lngamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the logarithm of the Gamma function on OP, rounded in the direction RND. When -2K-1 <= OP <= -2K, K being a non-negative integer, ROP is set to NaN. See also `mpfr_lgamma'. -- Function: int mpfr_lgamma (mpfr_t ROP, int *SIGNP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the logarithm of the absolute value of the Gamma function on OP, rounded in the direction RND. The sign (1 or -1) of Gamma(OP) is returned in the object pointed to by SIGNP. When OP is an infinity or a non-positive integer, set ROP to +Inf. When OP is NaN, -Inf or a negative integer, *SIGNP is undefined, and when OP is ±0, *SIGNP is the sign of the zero. -- Function: int mpfr_digamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the Digamma (sometimes also called Psi) function on OP, rounded in the direction RND. When OP is a negative integer, set ROP to NaN. -- Function: int mpfr_zeta (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_zeta_ui (mpfr_t ROP, unsigned long OP, mpfr_rnd_t RND) Set ROP to the value of the Riemann Zeta function on OP, rounded in the direction RND. -- Function: int mpfr_erf (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_erfc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the error function on OP (resp. the complementary error function on OP) rounded in the direction RND. -- Function: int mpfr_j0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_j1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_jn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the first kind Bessel function of order 0, (resp. 1 and N) on OP, rounded in the direction RND. When OP is NaN, ROP is always set to NaN. When OP is plus or minus Infinity, ROP is set to +0. When OP is zero, and N is not zero, ROP is set to +0 or -0 depending on the parity and sign of N, and the sign of OP. -- Function: int mpfr_y0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_y1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_yn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the second kind Bessel function of order 0 (resp. 1 and N) on OP, rounded in the direction RND. When OP is NaN or negative, ROP is always set to NaN. When OP is +Inf, ROP is set to +0. When OP is zero, ROP is set to +Inf or -Inf depending on the parity and sign of N. -- Function: int mpfr_fma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_rnd_t RND) -- Function: int mpfr_fms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_rnd_t RND) Set ROP to (OP1 times OP2) + OP3 (resp. (OP1 times OP2) - OP3) rounded in the direction RND. -- Function: int mpfr_agm (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the arithmetic-geometric mean of OP1 and OP2, rounded in the direction RND. The arithmetic-geometric mean is the common limit of the sequences U_N and V_N, where U_0=OP1, V_0=OP2, U_(N+1) is the arithmetic mean of U_N and V_N, and V_(N+1) is the geometric mean of U_N and V_N. If any operand is negative, set ROP to NaN. -- Function: int mpfr_hypot (mpfr_t ROP, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) Set ROP to the Euclidean norm of X and Y, i.e., the square root of the sum of the squares of X and Y, rounded in the direction RND. Special values are handled as described in Section F.9.4.3 of the ISO C99 and IEEE 754-2008 standards: If X or Y is an infinity, then +Inf is returned in ROP, even if the other number is NaN. -- Function: int mpfr_ai (mpfr_t ROP, mpfr_t X, mpfr_rnd_t RND) Set ROP to the value of the Airy function Ai on X, rounded in the direction RND. When X is NaN, ROP is always set to NaN. When X is +Inf or -Inf, ROP is +0. The current implementation is not intended to be used with large arguments. It works with abs(X) typically smaller than 500. For larger arguments, other methods should be used and will be implemented in a future version. -- Function: int mpfr_const_log2 (mpfr_t ROP, mpfr_rnd_t RND) -- Function: int mpfr_const_pi (mpfr_t ROP, mpfr_rnd_t RND) -- Function: int mpfr_const_euler (mpfr_t ROP, mpfr_rnd_t RND) -- Function: int mpfr_const_catalan (mpfr_t ROP, mpfr_rnd_t RND) Set ROP to the logarithm of 2, the value of Pi, of Euler's constant 0.577..., of Catalan's constant 0.915..., respectively, rounded in the direction RND. These functions cache the computed values to avoid other calculations if a lower or equal precision is requested. To free these caches, use `mpfr_free_cache'. -- Function: void mpfr_free_cache (void) Free various caches used by MPFR internally, in particular the caches used by the functions computing constants (`mpfr_const_log2', `mpfr_const_pi', `mpfr_const_euler' and `mpfr_const_catalan'). You should call this function before terminating a thread, even if you did not call these functions directly (they could have been called internally). -- Function: int mpfr_sum (mpfr_t ROP, mpfr_ptr const TAB[], unsigned long int N, mpfr_rnd_t RND) Set ROP to the sum of all elements of TAB, whose size is N, rounded in the direction RND. Warning: for efficiency reasons, TAB is an array of pointers to `mpfr_t', not an array of `mpfr_t'. If the returned `int' value is zero, ROP is guaranteed to be the exact sum; otherwise ROP might be smaller than, equal to, or larger than the exact sum (in accordance to the rounding mode). However, `mpfr_sum' does guarantee the result is correctly rounded.  File: mpfr.info, Node: Input and Output Functions, Next: Formatted Output Functions, Prev: Special Functions, Up: MPFR Interface 5.8 Input and Output Functions ============================== This section describes functions that perform input from an input/output stream, and functions that output to an input/output stream. Passing a null pointer for a `stream' to any of these functions will make them read from `stdin' and write to `stdout', respectively. When using any of these functions, you must include the `' standard header before `mpfr.h', to allow `mpfr.h' to define prototypes for these functions. -- Function: size_t mpfr_out_str (FILE *STREAM, int BASE, size_t N, mpfr_t OP, mpfr_rnd_t RND) Output OP on stream STREAM, as a string of digits in base BASE, rounded in the direction RND. The base may vary from 2 to 62. Print N significant digits exactly, or if N is 0, enough digits so that OP can be read back exactly (see `mpfr_get_str'). In addition to the significant digits, a decimal point (defined by the current locale) at the right of the first digit and a trailing exponent in base 10, in the form `eNNN', are printed. If BASE is greater than 10, `@@' will be used instead of `e' as exponent delimiter. Return the number of characters written, or if an error occurred, return 0. -- Function: size_t mpfr_inp_str (mpfr_t ROP, FILE *STREAM, int BASE, mpfr_rnd_t RND) Input a string in base BASE from stream STREAM, rounded in the direction RND, and put the read float in ROP. This function reads a word (defined as a sequence of characters between whitespace) and parses it using `mpfr_set_str'. See the documentation of `mpfr_strtofr' for a detailed description of the valid string formats. Return the number of bytes read, or if an error occurred, return 0.  File: mpfr.info, Node: Formatted Output Functions, Next: Integer Related Functions, Prev: Input and Output Functions, Up: MPFR Interface 5.9 Formatted Output Functions ============================== 5.9.1 Requirements ------------------ The class of `mpfr_printf' functions provides formatted output in a similar manner as the standard C `printf'. These functions are defined only if your system supports ISO C variadic functions and the corresponding argument access macros. When using any of these functions, you must include the `' standard header before `mpfr.h', to allow `mpfr.h' to define prototypes for these functions. 5.9.2 Format String ------------------- The format specification accepted by `mpfr_printf' is an extension of the `printf' one. The conversion specification is of the form: % [flags] [width] [.[precision]] [type] [rounding] conv `flags', `width', and `precision' have the same meaning as for the standard `printf' (in particular, notice that the `precision' is related to the number of digits displayed in the base chosen by `conv' and not related to the internal precision of the `mpfr_t' variable). `mpfr_printf' accepts the same `type' specifiers as GMP (except the non-standard and deprecated `q', use `ll' instead), namely the length modifiers defined in the C standard: `h' `short' `hh' `char' `j' `intmax_t' or `uintmax_t' `l' `long' or `wchar_t' `ll' `long long' `L' `long double' `t' `ptrdiff_t' `z' `size_t' and the `type' specifiers defined in GMP plus `R' and `P' specific to MPFR (the second column in the table below shows the type of the argument read in the argument list and the kind of `conv' specifier to use after the `type' specifier): `F' `mpf_t', float conversions `Q' `mpq_t', integer conversions `M' `mp_limb_t', integer conversions `N' `mp_limb_t' array, integer conversions `Z' `mpz_t', integer conversions `P' `mpfr_prec_t', integer conversions `R' `mpfr_t', float conversions The `type' specifiers have the same restrictions as those mentioned in the GMP documentation: *note Formatted Output Strings: (gmp.info)Formatted Output Strings. In particular, the `type' specifiers (except `R' and `P') are supported only if they are supported by `gmp_printf' in your GMP build; this implies that the standard specifiers, such as `t', must _also_ be supported by your C library if you want to use them. The `rounding' field is specific to `mpfr_t' arguments and should not be used with other types. With conversion specification not involving `P' and `R' types, `mpfr_printf' behaves exactly as `gmp_printf'. The `P' type specifies that a following `o', `u', `x', or `X' conversion specifier applies to a `mpfr_prec_t' argument. It is needed because the `mpfr_prec_t' type does not necessarily correspond to an `unsigned int' or any fixed standard type. The `precision' field specifies the minimum number of digits to appear. The default `precision' is 1. For example: mpfr_t x; mpfr_prec_t p; mpfr_init (x); ... p = mpfr_get_prec (x); mpfr_printf ("variable x with %Pu bits", p); The `R' type specifies that a following `a', `A', `b', `e', `E', `f', `F', `g', `G', or `n' conversion specifier applies to a `mpfr_t' argument. The `R' type can be followed by a `rounding' specifier denoted by one of the following characters: `U' round toward plus infinity `D' round toward minus infinity `Y' round away from zero `Z' round toward zero `N' round to nearest (with ties to even) `*' rounding mode indicated by the `mpfr_rnd_t' argument just before the corresponding `mpfr_t' variable. The default rounding mode is rounding to nearest. The following three examples are equivalent: mpfr_t x; mpfr_init (x); ... mpfr_printf ("%.128Rf", x); mpfr_printf ("%.128RNf", x); mpfr_printf ("%.128R*f", MPFR_RNDN, x); Note that the rounding away from zero mode is specified with `Y' because ISO C reserves the `A' specifier for hexadecimal output (see below). The output `conv' specifiers allowed with `mpfr_t' parameter are: `a' `A' hex float, C99 style `b' binary output `e' `E' scientific format float `f' `F' fixed point float `g' `G' fixed or scientific float The conversion specifier `b' which displays the argument in binary is specific to `mpfr_t' arguments and should not be used with other types. Other conversion specifiers have the same meaning as for a `double' argument. In case of non-decimal output, only the significand is written in the specified base, the exponent is always displayed in decimal. Special values are always displayed as `nan', `-inf', and `inf' for `a', `b', `e', `f', and `g' specifiers and `NAN', `-INF', and `INF' for `A', `E', `F', and `G' specifiers. If the `precision' field is not empty, the `mpfr_t' number is rounded to the given precision in the direction specified by the rounding mode. If the precision is zero with rounding to nearest mode and one of the following `conv' specifiers: `a', `A', `b', `e', `E', tie case is rounded to even when it lies between two consecutive values at the wanted precision which have the same exponent, otherwise, it is rounded away from zero. For instance, 85 is displayed as "8e+1" and 95 is displayed as "1e+2" with the format specification `"%.0RNe"'. This also applies when the `g' (resp. `G') conversion specifier uses the `e' (resp. `E') style. If the precision is set to a value greater than the maximum value for an `int', it will be silently reduced down to `INT_MAX'. If the `precision' field is empty (as in `%Re' or `%.RE') with `conv' specifier `e' and `E', the number is displayed with enough digits so that it can be read back exactly, assuming that the input and output variables have the same precision and that the input and output rounding modes are both rounding to nearest (as for `mpfr_get_str'). The default precision for an empty `precision' field with `conv' specifiers `f', `F', `g', and `G' is 6. 5.9.3 Functions --------------- For all the following functions, if the number of characters which ought to be written appears to exceed the maximum limit for an `int', nothing is written in the stream (resp. to `stdout', to BUF, to STR), the function returns -1, sets the _erange_ flag, and (in POSIX system only) `errno' is set to `EOVERFLOW'. -- Function: int mpfr_fprintf (FILE *STREAM, const char *TEMPLATE, ...) -- Function: int mpfr_vfprintf (FILE *STREAM, const char *TEMPLATE, va_list AP) Print to the stream STREAM the optional arguments under the control of the template string TEMPLATE. Return the number of characters written or a negative value if an error occurred. -- Function: int mpfr_printf (const char *TEMPLATE, ...) -- Function: int mpfr_vprintf (const char *TEMPLATE, va_list AP) Print to `stdout' the optional arguments under the control of the template string TEMPLATE. Return the number of characters written or a negative value if an error occurred. -- Function: int mpfr_sprintf (char *BUF, const char *TEMPLATE, ...) -- Function: int mpfr_vsprintf (char *BUF, const char *TEMPLATE, va_list AP) Form a null-terminated string corresponding to the optional arguments under the control of the template string TEMPLATE, and print it in BUF. No overlap is permitted between BUF and the other arguments. Return the number of characters written in the array BUF _not counting_ the terminating null character or a negative value if an error occurred. -- Function: int mpfr_snprintf (char *BUF, size_t N, const char *TEMPLATE, ...) -- Function: int mpfr_vsnprintf (char *BUF, size_t N, const char *TEMPLATE, va_list AP) Form a null-terminated string corresponding to the optional arguments under the control of the template string TEMPLATE, and print it in BUF. If N is zero, nothing is written and BUF may be a null pointer, otherwise, the N-1 first characters are written in BUF and the N-th is a null character. Return the number of characters that would have been written had N be sufficiently large, _not counting_ the terminating null character, or a negative value if an error occurred. -- Function: int mpfr_asprintf (char **STR, const char *TEMPLATE, ...) -- Function: int mpfr_vasprintf (char **STR, const char *TEMPLATE, va_list AP) Write their output as a null terminated string in a block of memory allocated using the current allocation function. A pointer to the block is stored in STR. The block of memory must be freed using `mpfr_free_str'. The return value is the number of characters written in the string, excluding the null-terminator, or a negative value if an error occurred.  File: mpfr.info, Node: Integer Related Functions, Next: Rounding Related Functions, Prev: Formatted Output Functions, Up: MPFR Interface 5.10 Integer and Remainder Related Functions ============================================ -- Function: int mpfr_rint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_ceil (mpfr_t ROP, mpfr_t OP) -- Function: int mpfr_floor (mpfr_t ROP, mpfr_t OP) -- Function: int mpfr_round (mpfr_t ROP, mpfr_t OP) -- Function: int mpfr_trunc (mpfr_t ROP, mpfr_t OP) Set ROP to OP rounded to an integer. `mpfr_rint' rounds to the nearest representable integer in the given direction RND, `mpfr_ceil' rounds to the next higher or equal representable integer, `mpfr_floor' to the next lower or equal representable integer, `mpfr_round' to the nearest representable integer, rounding halfway cases away from zero (as in the roundTiesToAway mode of IEEE 754-2008), and `mpfr_trunc' to the next representable integer toward zero. The returned value is zero when the result is exact, positive when it is greater than the original value of OP, and negative when it is smaller. More precisely, the returned value is 0 when OP is an integer representable in ROP, 1 or -1 when OP is an integer that is not representable in ROP, 2 or -2 when OP is not an integer. Note that `mpfr_round' is different from `mpfr_rint' called with the rounding to nearest mode (where halfway cases are rounded to an even integer or significand). Note also that no double rounding is performed; for instance, 10.5 (1010.1 in binary) is rounded by `mpfr_rint' with rounding to nearest to 12 (1100 in binary) in 2-bit precision, because the two enclosing numbers representable on two bits are 8 and 12, and the closest is 12. (If one first rounded to an integer, one would round 10.5 to 10 with even rounding, and then 10 would be rounded to 8 again with even rounding.) -- Function: int mpfr_rint_ceil (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_rint_floor (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_rint_round (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_rint_trunc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to OP rounded to an integer. `mpfr_rint_ceil' rounds to the next higher or equal integer, `mpfr_rint_floor' to the next lower or equal integer, `mpfr_rint_round' to the nearest integer, rounding halfway cases away from zero, and `mpfr_rint_trunc' to the next integer toward zero. If the result is not representable, it is rounded in the direction RND. The returned value is the ternary value associated with the considered round-to-integer function (regarded in the same way as any other mathematical function). Contrary to `mpfr_rint', those functions do perform a double rounding: first OP is rounded to the nearest integer in the direction given by the function name, then this nearest integer (if not representable) is rounded in the given direction RND. For example, `mpfr_rint_round' with rounding to nearest and a precision of two bits rounds 6.5 to 7 (halfway cases away from zero), then 7 is rounded to 8 by the round-even rule, despite the fact that 6 is also representable on two bits, and is closer to 6.5 than 8. -- Function: int mpfr_frac (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the fractional part of OP, having the same sign as OP, rounded in the direction RND (unlike in `mpfr_rint', RND affects only how the exact fractional part is rounded, not how the fractional part is generated). -- Function: int mpfr_modf (mpfr_t IOP, mpfr_t FOP, mpfr_t OP, mpfr_rnd_t RND) Set simultaneously IOP to the integral part of OP and FOP to the fractional part of OP, rounded in the direction RND with the corresponding precision of IOP and FOP (equivalent to `mpfr_trunc(IOP, OP, RND)' and `mpfr_frac(FOP, OP, RND)'). The variables IOP and FOP must be different. Return 0 iff both results are exact (see `mpfr_sin_cos' for a more detailed description of the return value). -- Function: int mpfr_fmod (mpfr_t R, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) -- Function: int mpfr_remainder (mpfr_t R, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) -- Function: int mpfr_remquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) Set R to the value of X - NY, rounded according to the direction RND, where N is the integer quotient of X divided by Y, defined as follows: N is rounded toward zero for `mpfr_fmod', and to the nearest integer (ties rounded to even) for `mpfr_remainder' and `mpfr_remquo'. Special values are handled as described in Section F.9.7.1 of the ISO C99 standard: If X is infinite or Y is zero, R is NaN. If Y is infinite and X is finite, R is X rounded to the precision of R. If R is zero, it has the sign of X. The return value is the ternary value corresponding to R. Additionally, `mpfr_remquo' stores the low significant bits from the quotient N in *Q (more precisely the number of bits in a `long' minus one), with the sign of X divided by Y (except if those low bits are all zero, in which case zero is returned). Note that X may be so large in magnitude relative to Y that an exact representation of the quotient is not practical. The `mpfr_remainder' and `mpfr_remquo' functions are useful for additive argument reduction. -- Function: int mpfr_integer_p (mpfr_t OP) Return non-zero iff OP is an integer.  File: mpfr.info, Node: Rounding Related Functions, Next: Miscellaneous Functions, Prev: Integer Related Functions, Up: MPFR Interface 5.11 Rounding Related Functions =============================== -- Function: void mpfr_set_default_rounding_mode (mpfr_rnd_t RND) Set the default rounding mode to RND. The default rounding mode is to nearest initially. -- Function: mpfr_rnd_t mpfr_get_default_rounding_mode (void) Get the default rounding mode. -- Function: int mpfr_prec_round (mpfr_t X, mpfr_prec_t PREC, mpfr_rnd_t RND) Round X according to RND with precision PREC, which must be an integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX' (otherwise the behavior is undefined). If PREC is greater or equal to the precision of X, then new space is allocated for the significand, and it is filled with zeros. Otherwise, the significand is rounded to precision PREC with the given direction. In both cases, the precision of X is changed to PREC. Here is an example of how to use `mpfr_prec_round' to implement Newton's algorithm to compute the inverse of A, assuming X is already an approximation to N bits: mpfr_set_prec (t, 2 * n); mpfr_set (t, a, MPFR_RNDN); /* round a to 2n bits */ mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to 2n bits */ mpfr_ui_sub (t, 1, t, MPFR_RNDN); /* high n bits cancel with 1 */ mpfr_prec_round (t, n, MPFR_RNDN); /* t is correct to n bits */ mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to n bits */ mpfr_prec_round (x, 2 * n, MPFR_RNDN); /* exact */ mpfr_add (x, x, t, MPFR_RNDN); /* x is correct to 2n bits */ -- Function: int mpfr_can_round (mpfr_t B, mpfr_exp_t ERR, mpfr_rnd_t RND1, mpfr_rnd_t RND2, mpfr_prec_t PREC) Assuming B is an approximation of an unknown number X in the direction RND1 with error at most two to the power E(b)-ERR where E(b) is the exponent of B, return a non-zero value if one is able to round correctly X to precision PREC with the direction RND2, and 0 otherwise (including for NaN and Inf). This function *does not modify* its arguments. If RND1 is `MPFR_RNDN', then the sign of the error is unknown, but its absolute value is the same, so that the possible range is twice as large as with a directed rounding for RND1. Note: if one wants to also determine the correct *note ternary value:: when rounding B to precision PREC with rounding mode RND, a useful trick is the following: if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ, prec + (rnd == MPFR_RNDN))) ... Indeed, if RND is `MPFR_RNDN', this will check if one can round to PREC+1 bits with a directed rounding: if so, one can surely round to nearest to PREC bits, and in addition one can determine the correct ternary value, which would not be the case when B is near from a value exactly representable on PREC bits. -- Function: mpfr_prec_t mpfr_min_prec (mpfr_t X) Return the minimal number of bits required to store the significand of X, and 0 for special values, including 0. (Warning: the returned value can be less than `MPFR_PREC_MIN'.) The function name is subject to change. -- Function: const char * mpfr_print_rnd_mode (mpfr_rnd_t RND) Return a string ("MPFR_RNDD", "MPFR_RNDU", "MPFR_RNDN", "MPFR_RNDZ", "MPFR_RNDA") corresponding to the rounding mode RND, or a null pointer if RND is an invalid rounding mode.  File: mpfr.info, Node: Miscellaneous Functions, Next: Exception Related Functions, Prev: Rounding Related Functions, Up: MPFR Interface 5.12 Miscellaneous Functions ============================ -- Function: void mpfr_nexttoward (mpfr_t X, mpfr_t Y) If X or Y is NaN, set X to NaN. If X and Y are equal, X is unchanged. Otherwise, if X is different from Y, replace X by the next floating-point number (with the precision of X and the current exponent range) in the direction of Y (the infinite values are seen as the smallest and largest floating-point numbers). If the result is zero, it keeps the same sign. No underflow or overflow is generated. -- Function: void mpfr_nextabove (mpfr_t X) -- Function: void mpfr_nextbelow (mpfr_t X) Equivalent to `mpfr_nexttoward' where Y is plus infinity (resp. minus infinity). -- Function: int mpfr_min (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_max (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the minimum (resp. maximum) of OP1 and OP2. If OP1 and OP2 are both NaN, then ROP is set to NaN. If OP1 or OP2 is NaN, then ROP is set to the numeric value. If OP1 and OP2 are zeros of different signs, then ROP is set to -0 (resp. +0). -- Function: int mpfr_urandomb (mpfr_t ROP, gmp_randstate_t STATE) Generate a uniformly distributed random float in the interval 0 <= ROP < 1. More precisely, the number can be seen as a float with a random non-normalized significand and exponent 0, which is then normalized (thus if E denotes the exponent after normalization, then the least -E significant bits of the significand are always 0). Return 0, unless the exponent is not in the current exponent range, in which case ROP is set to NaN and a non-zero value is returned (this should never happen in practice, except in very specific cases). The second argument is a `gmp_randstate_t' structure which should be created using the GMP `gmp_randinit' function (see the GMP manual). Note: for a given version of MPFR, the returned value of ROP and the new value of STATE (which controls further random values) do not depend on the machine word size. -- Function: int mpfr_urandom (mpfr_t ROP, gmp_randstate_t STATE, mpfr_rnd_t RND) Generate a uniformly distributed random float. The floating-point number ROP can be seen as if a random real number is generated according to the continuous uniform distribution on the interval [0, 1] and then rounded in the direction RND. The second argument is a `gmp_randstate_t' structure which should be created using the GMP `gmp_randinit' function (see the GMP manual). Note: the note for `mpfr_urandomb' holds too. In addition, the exponent range and the rounding mode might have a side effect on the next random state. -- Function: int mpfr_grandom (mpfr_t ROP1, mpfr_t ROP2, gmp_randstate_t STATE, mpfr_rnd_t RND) Generate two random floats according to a standard normal gaussian distribution. If ROP2 is a null pointer, then only one value is generated and stored in ROP1. The floating-point number ROP1 (and ROP2) can be seen as if a random real number were generated according to the standard normal gaussian distribution and then rounded in the direction RND. The third argument is a `gmp_randstate_t' structure, which should be created using the GMP `gmp_randinit' function (see the GMP manual). The combination of the ternary values is returned like with `mpfr_sin_cos'. If ROP2 is a null pointer, the second ternary value is assumed to be 0 (note that the encoding of the only ternary value is not the same as the usual encoding for functions that return only one result). Otherwise the ternary value of a random number is always non-zero. Note: the note for `mpfr_urandomb' holds too. In addition, the exponent range and the rounding mode might have a side effect on the next random state. -- Function: mpfr_exp_t mpfr_get_exp (mpfr_t X) Return the exponent of X, assuming that X is a non-zero ordinary number and the significand is considered in [1/2,1). The behavior for NaN, infinity or zero is undefined. -- Function: int mpfr_set_exp (mpfr_t X, mpfr_exp_t E) Set the exponent of X if E is in the current exponent range, and return 0 (even if X is not a non-zero ordinary number); otherwise, return a non-zero value. The significand is assumed to be in [1/2,1). -- Function: int mpfr_signbit (mpfr_t OP) Return a non-zero value iff OP has its sign bit set (i.e., if it is negative, -0, or a NaN whose representation has its sign bit set). -- Function: int mpfr_setsign (mpfr_t ROP, mpfr_t OP, int S, mpfr_rnd_t RND) Set the value of ROP from OP, rounded toward the given direction RND, then set (resp. clear) its sign bit if S is non-zero (resp. zero), even when OP is a NaN. -- Function: int mpfr_copysign (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set the value of ROP from OP1, rounded toward the given direction RND, then set its sign bit to that of OP2 (even when OP1 or OP2 is a NaN). This function is equivalent to `mpfr_setsign (ROP, OP1, mpfr_signbit (OP2), RND)'. -- Function: const char * mpfr_get_version (void) Return the MPFR version, as a null-terminated string. -- Macro: MPFR_VERSION -- Macro: MPFR_VERSION_MAJOR -- Macro: MPFR_VERSION_MINOR -- Macro: MPFR_VERSION_PATCHLEVEL -- Macro: MPFR_VERSION_STRING `MPFR_VERSION' is the version of MPFR as a preprocessing constant. `MPFR_VERSION_MAJOR', `MPFR_VERSION_MINOR' and `MPFR_VERSION_PATCHLEVEL' are respectively the major, minor and patch level of MPFR version, as preprocessing constants. `MPFR_VERSION_STRING' is the version (with an optional suffix, used in development and pre-release versions) as a string constant, which can be compared to the result of `mpfr_get_version' to check at run time the header file and library used match: if (strcmp (mpfr_get_version (), MPFR_VERSION_STRING)) fprintf (stderr, "Warning: header and library do not match\n"); Note: Obtaining different strings is not necessarily an error, as in general, a program compiled with some old MPFR version can be dynamically linked with a newer MPFR library version (if allowed by the library versioning system). -- Macro: long MPFR_VERSION_NUM (MAJOR, MINOR, PATCHLEVEL) Create an integer in the same format as used by `MPFR_VERSION' from the given MAJOR, MINOR and PATCHLEVEL. Here is an example of how to check the MPFR version at compile time: #if (!defined(MPFR_VERSION) || (MPFR_VERSION' line, #include #include any program written for MPF can be compiled directly with MPFR without any changes (except the `gmp_printf' functions will not work for arguments of type `mpfr_t'). All operations are then performed with the default MPFR rounding mode, which can be reset with `mpfr_set_default_rounding_mode'. Warning: the `mpf_init' and `mpf_init2' functions initialize to zero, whereas the corresponding MPFR functions initialize to NaN: this is useful to detect uninitialized values, but is slightly incompatible with MPF. -- Function: void mpfr_set_prec_raw (mpfr_t X, mpfr_prec_t PREC) Reset the precision of X to be *exactly* PREC bits. The only difference with `mpfr_set_prec' is that PREC is assumed to be small enough so that the significand fits into the current allocated memory space for X. Otherwise the behavior is undefined. -- Function: int mpfr_eq (mpfr_t OP1, mpfr_t OP2, unsigned long int OP3) Return non-zero if OP1 and OP2 are both non-zero ordinary numbers with the same exponent and the same first OP3 bits, both zero, or both infinities of the same sign. Return zero otherwise. This function is defined for compatibility with MPF, we do not recommend to use it otherwise. Do not use it either if you want to know whether two numbers are close to each other; for instance, 1.011111 and 1.100000 are regarded as different for any value of OP3 larger than 1. -- Function: void mpfr_reldiff (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Compute the relative difference between OP1 and OP2 and store the result in ROP. This function does not guarantee the correct rounding on the relative difference; it just computes |OP1-OP2|/OP1, using the precision of ROP and the rounding mode RND for all operations. -- Function: int mpfr_mul_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) These functions are identical to `mpfr_mul_2ui' and `mpfr_div_2ui' respectively. These functions are only kept for compatibility with MPF, one should prefer `mpfr_mul_2ui' and `mpfr_div_2ui' otherwise.  File: mpfr.info, Node: Custom Interface, Next: Internals, Prev: Compatibility with MPF, Up: MPFR Interface 5.15 Custom Interface ===================== Some applications use a stack to handle the memory and their objects. However, the MPFR memory design is not well suited for such a thing. So that such applications are able to use MPFR, an auxiliary memory interface has been created: the Custom Interface. The following interface allows one to use MPFR in two ways: * Either directly store a floating-point number as a `mpfr_t' on the stack. * Either store its own representation on the stack and construct a new temporary `mpfr_t' each time it is needed. Nothing has to be done to destroy the floating-point numbers except garbaging the used memory: all the memory management (allocating, destroying, garbaging) is left to the application. Each function in this interface is also implemented as a macro for efficiency reasons: for example `mpfr_custom_init (s, p)' uses the macro, while `(mpfr_custom_init) (s, p)' uses the function. Note 1: MPFR functions may still initialize temporary floating-point numbers using `mpfr_init' and similar functions. See Custom Allocation (GNU MP). Note 2: MPFR functions may use the cached functions (`mpfr_const_pi' for example), even if they are not explicitly called. You have to call `mpfr_free_cache' each time you garbage the memory iff `mpfr_init', through GMP Custom Allocation, allocates its memory on the application stack. -- Function: size_t mpfr_custom_get_size (mpfr_prec_t PREC) Return the needed size in bytes to store the significand of a floating-point number of precision PREC. -- Function: void mpfr_custom_init (void *SIGNIFICAND, mpfr_prec_t PREC) Initialize a significand of precision PREC, where SIGNIFICAND must be an area of `mpfr_custom_get_size (prec)' bytes at least and be suitably aligned for an array of `mp_limb_t' (GMP type, *note Internals::). -- Function: void mpfr_custom_init_set (mpfr_t X, int KIND, mpfr_exp_t EXP, mpfr_prec_t PREC, void *SIGNIFICAND) Perform a dummy initialization of a `mpfr_t' and set it to: * if `ABS(kind) == MPFR_NAN_KIND', X is set to NaN; * if `ABS(kind) == MPFR_INF_KIND', X is set to the infinity of sign `sign(kind)'; * if `ABS(kind) == MPFR_ZERO_KIND', X is set to the zero of sign `sign(kind)'; * if `ABS(kind) == MPFR_REGULAR_KIND', X is set to a regular number: `x = sign(kind)*significand*2^exp'. In all cases, it uses SIGNIFICAND directly for further computing involving X. It will not allocate anything. A floating-point number initialized with this function cannot be resized using `mpfr_set_prec' or `mpfr_prec_round', or cleared using `mpfr_clear'! The SIGNIFICAND must have been initialized with `mpfr_custom_init' using the same precision PREC. -- Function: int mpfr_custom_get_kind (mpfr_t X) Return the current kind of a `mpfr_t' as created by `mpfr_custom_init_set'. The behavior of this function for any `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined. -- Function: void * mpfr_custom_get_significand (mpfr_t X) Return a pointer to the significand used by a `mpfr_t' initialized with `mpfr_custom_init_set'. The behavior of this function for any `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined. -- Function: mpfr_exp_t mpfr_custom_get_exp (mpfr_t X) Return the exponent of X, assuming that X is a non-zero ordinary number. The return value for NaN, Infinity or zero is unspecified but does not produce any trap. The behavior of this function for any `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined. -- Function: void mpfr_custom_move (mpfr_t X, void *NEW_POSITION) Inform MPFR that the significand of X has moved due to a garbage collect and update its new position to `new_position'. However the application has to move the significand and the `mpfr_t' itself. The behavior of this function for any `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined.  File: mpfr.info, Node: Internals, Prev: Custom Interface, Up: MPFR Interface 5.16 Internals ============== A "limb" means the part of a multi-precision number that fits in a single word. Usually a limb contains 32 or 64 bits. The C data type for a limb is `mp_limb_t'. The `mpfr_t' type is internally defined as a one-element array of a structure, and `mpfr_ptr' is the C data type representing a pointer to this structure. The `mpfr_t' type consists of four fields: * The `_mpfr_prec' field is used to store the precision of the variable (in bits); this is not less than `MPFR_PREC_MIN'. * The `_mpfr_sign' field is used to store the sign of the variable. * The `_mpfr_exp' field stores the exponent. An exponent of 0 means a radix point just above the most significant limb. Non-zero values n are a multiplier 2^n relative to that point. A NaN, an infinity and a zero are indicated by special values of the exponent field. * Finally, the `_mpfr_d' field is a pointer to the limbs, least significant limbs stored first. The number of limbs in use is controlled by `_mpfr_prec', namely ceil(`_mpfr_prec'/`mp_bits_per_limb'). Non-singular (i.e., different from NaN, Infinity or zero) values always have the most significant bit of the most significant limb set to 1. When the precision does not correspond to a whole number of limbs, the excess bits at the low end of the data are zeros.  File: mpfr.info, Node: API Compatibility, Next: Contributors, Prev: MPFR Interface, Up: Top 6 API Compatibility ******************* The goal of this section is to describe some API changes that occurred from one version of MPFR to another, and how to write code that can be compiled and run with older MPFR versions. The minimum MPFR version that is considered here is 2.2.0 (released on 20 September 2005). API changes can only occur between major or minor versions. Thus the patchlevel (the third number in the MPFR version) will be ignored in the following. If a program does not use MPFR internals, changes in the behavior between two versions differing only by the patchlevel should only result from what was regarded as a bug or unspecified behavior. As a general rule, a program written for some MPFR version should work with later versions, possibly except at a new major version, where some features (described as obsolete for some time) can be removed. In such a case, a failure should occur during compilation or linking. If a result becomes incorrect because of such a change, please look at the various changes below (they are minimal, and most software should be unaffected), at the FAQ and at the MPFR web page for your version (a bug could have been introduced and be already fixed); and if the problem is not mentioned, please send us a bug report (*note Reporting Bugs::). However, a program written for the current MPFR version (as documented by this manual) may not necessarily work with previous versions of MPFR. This section should help developers to write portable code. Note: Information given here may be incomplete. API changes are also described in the NEWS file (for each version, instead of being classified like here), together with other changes. * Menu: * Type and Macro Changes:: * Added Functions:: * Changed Functions:: * Removed Functions:: * Other Changes::  File: mpfr.info, Node: Type and Macro Changes, Next: Added Functions, Prev: API Compatibility, Up: API Compatibility 6.1 Type and Macro Changes ========================== The official type for exponent values changed from `mp_exp_t' to `mpfr_exp_t' in MPFR 3.0. The type `mp_exp_t' will remain available as it comes from GMP (with a different meaning). These types are currently the same (`mpfr_exp_t' is defined as `mp_exp_t' with `typedef'), so that programs can still use `mp_exp_t'; but this may change in the future. Alternatively, using the following code after including `mpfr.h' will work with official MPFR versions, as `mpfr_exp_t' was never defined in MPFR 2.x: #if MPFR_VERSION_MAJOR < 3 typedef mp_exp_t mpfr_exp_t; #endif The official types for precision values and for rounding modes respectively changed from `mp_prec_t' and `mp_rnd_t' to `mpfr_prec_t' and `mpfr_rnd_t' in MPFR 3.0. This change was actually done a long time ago in MPFR, at least since MPFR 2.2.0, with the following code in `mpfr.h': #ifndef mp_rnd_t # define mp_rnd_t mpfr_rnd_t #endif #ifndef mp_prec_t # define mp_prec_t mpfr_prec_t #endif This means that it is safe to use the new official types `mpfr_prec_t' and `mpfr_rnd_t' in your programs. The types `mp_prec_t' and `mp_rnd_t' (defined in MPFR only) may be removed in the future, as the prefix `mp_' is reserved by GMP. The precision type `mpfr_prec_t' (`mp_prec_t') was unsigned before MPFR 3.0; it is now signed. `MPFR_PREC_MAX' has not changed, though. Indeed the MPFR code requires that `MPFR_PREC_MAX' be representable in the exponent type, which may have the same size as `mpfr_prec_t' but has always been signed. The consequence is that valid code that does not assume anything about the signedness of `mpfr_prec_t' should work with past and new MPFR versions. This change was useful as the use of unsigned types tends to convert signed values to unsigned ones in expressions due to the usual arithmetic conversions, which can yield incorrect results if a negative value is converted in such a way. Warning! A program assuming (intentionally or not) that `mpfr_prec_t' is signed may be affected by this problem when it is built and run against MPFR 2.x. The rounding modes `GMP_RNDx' were renamed to `MPFR_RNDx' in MPFR 3.0. However the old names `GMP_RNDx' have been kept for compatibility (this might change in future versions), using: #define GMP_RNDN MPFR_RNDN #define GMP_RNDZ MPFR_RNDZ #define GMP_RNDU MPFR_RNDU #define GMP_RNDD MPFR_RNDD The rounding mode "round away from zero" (`MPFR_RNDA') was added in MPFR 3.0 (however no rounding mode `GMP_RNDA' exists).  File: mpfr.info, Node: Added Functions, Next: Changed Functions, Prev: Type and Macro Changes, Up: API Compatibility 6.2 Added Functions =================== We give here in alphabetical order the functions that were added after MPFR 2.2, and in which MPFR version. * `mpfr_add_d' in MPFR 2.4. * `mpfr_ai' in MPFR 3.0 (incomplete, experimental). * `mpfr_asprintf' in MPFR 2.4. * `mpfr_buildopt_decimal_p' and `mpfr_buildopt_tls_p' in MPFR 3.0. * `mpfr_buildopt_gmpinternals_p' and `mpfr_buildopt_tune_case' in MPFR 3.1. * `mpfr_clear_divby0' in MPFR 3.1 (new divide-by-zero exception). * `mpfr_copysign' in MPFR 2.3. Note: MPFR 2.2 had a `mpfr_copysign' function that was available, but not documented, and with a slight difference in the semantics (when the second input operand is a NaN). * `mpfr_custom_get_significand' in MPFR 3.0. This function was named `mpfr_custom_get_mantissa' in previous versions; `mpfr_custom_get_mantissa' is still available via a macro in `mpfr.h': #define mpfr_custom_get_mantissa mpfr_custom_get_significand Thus code that needs to work with both MPFR 2.x and MPFR 3.x should use `mpfr_custom_get_mantissa'. * `mpfr_d_div' and `mpfr_d_sub' in MPFR 2.4. * `mpfr_digamma' in MPFR 3.0. * `mpfr_divby0_p' in MPFR 3.1 (new divide-by-zero exception). * `mpfr_div_d' in MPFR 2.4. * `mpfr_fmod' in MPFR 2.4. * `mpfr_fms' in MPFR 2.3. * `mpfr_fprintf' in MPFR 2.4. * `mpfr_frexp' in MPFR 3.1. * `mpfr_get_flt' in MPFR 3.0. * `mpfr_get_patches' in MPFR 2.3. * `mpfr_get_z_2exp' in MPFR 3.0. This function was named `mpfr_get_z_exp' in previous versions; `mpfr_get_z_exp' is still available via a macro in `mpfr.h': #define mpfr_get_z_exp mpfr_get_z_2exp Thus code that needs to work with both MPFR 2.x and MPFR 3.x should use `mpfr_get_z_exp'. * `mpfr_grandom' in MPFR 3.1. * `mpfr_j0', `mpfr_j1' and `mpfr_jn' in MPFR 2.3. * `mpfr_lgamma' in MPFR 2.3. * `mpfr_li2' in MPFR 2.4. * `mpfr_min_prec' in MPFR 3.0. * `mpfr_modf' in MPFR 2.4. * `mpfr_mul_d' in MPFR 2.4. * `mpfr_printf' in MPFR 2.4. * `mpfr_rec_sqrt' in MPFR 2.4. * `mpfr_regular_p' in MPFR 3.0. * `mpfr_remainder' and `mpfr_remquo' in MPFR 2.3. * `mpfr_set_divby0' in MPFR 3.1 (new divide-by-zero exception). * `mpfr_set_flt' in MPFR 3.0. * `mpfr_set_z_2exp' in MPFR 3.0. * `mpfr_set_zero' in MPFR 3.0. * `mpfr_setsign' in MPFR 2.3. * `mpfr_signbit' in MPFR 2.3. * `mpfr_sinh_cosh' in MPFR 2.4. * `mpfr_snprintf' and `mpfr_sprintf' in MPFR 2.4. * `mpfr_sub_d' in MPFR 2.4. * `mpfr_urandom' in MPFR 3.0. * `mpfr_vasprintf', `mpfr_vfprintf', `mpfr_vprintf', `mpfr_vsprintf' and `mpfr_vsnprintf' in MPFR 2.4. * `mpfr_y0', `mpfr_y1' and `mpfr_yn' in MPFR 2.3. * `mpfr_z_sub' in MPFR 3.1.  File: mpfr.info, Node: Changed Functions, Next: Removed Functions, Prev: Added Functions, Up: API Compatibility 6.3 Changed Functions ===================== The following functions have changed after MPFR 2.2. Changes can affect the behavior of code written for some MPFR version when built and run against another MPFR version (older or newer), as described below. * `mpfr_check_range' changed in MPFR 2.3.2 and MPFR 2.4. If the value is an inexact infinity, the overflow flag is now set (in case it was lost), while it was previously left unchanged. This is really what is expected in practice (and what the MPFR code was expecting), so that the previous behavior was regarded as a bug. Hence the change in MPFR 2.3.2. * `mpfr_get_f' changed in MPFR 3.0. This function was returning zero, except for NaN and Inf, which do not exist in MPF. The _erange_ flag is now set in these cases, and `mpfr_get_f' now returns the usual ternary value. * `mpfr_get_si', `mpfr_get_sj', `mpfr_get_ui' and `mpfr_get_uj' changed in MPFR 3.0. In previous MPFR versions, the cases where the _erange_ flag is set were unspecified. * `mpfr_get_z' changed in MPFR 3.0. The return type was `void'; it is now `int', and the usual ternary value is returned. Thus programs that need to work with both MPFR 2.x and 3.x must not use the return value. Even in this case, C code using `mpfr_get_z' as the second or third term of a conditional operator may also be affected. For instance, the following is correct with MPFR 3.0, but not with MPFR 2.x: bool ? mpfr_get_z(...) : mpfr_add(...); On the other hand, the following is correct with MPFR 2.x, but not with MPFR 3.0: bool ? mpfr_get_z(...) : (void) mpfr_add(...); Portable code should cast `mpfr_get_z(...)' to `void' to use the type `void' for both terms of the conditional operator, as in: bool ? (void) mpfr_get_z(...) : (void) mpfr_add(...); Alternatively, `if ... else' can be used instead of the conditional operator. Moreover the cases where the _erange_ flag is set were unspecified in MPFR 2.x. * `mpfr_get_z_exp' changed in MPFR 3.0. In previous MPFR versions, the cases where the _erange_ flag is set were unspecified. Note: this function has been renamed to `mpfr_get_z_2exp' in MPFR 3.0, but `mpfr_get_z_exp' is still available for compatibility reasons. * `mpfr_strtofr' changed in MPFR 2.3.1 and MPFR 2.4. This was actually a bug fix since the code and the documentation did not match. But both were changed in order to have a more consistent and useful behavior. The main changes in the code are as follows. The binary exponent is now accepted even without the `0b' or `0x' prefix. Data corresponding to NaN can now have an optional sign (such data were previously invalid). * `mpfr_strtofr' changed in MPFR 3.0. This function now accepts bases from 37 to 62 (no changes for the other bases). Note: if an unsupported base is provided to this function, the behavior is undefined; more precisely, in MPFR 2.3.1 and later, providing an unsupported base yields an assertion failure (this behavior may change in the future). * `mpfr_subnormalize' changed in MPFR 3.1. This was actually regarded as a bug fix. The `mpfr_subnormalize' implementation up to MPFR 3.0.0 did not change the flags. In particular, it did not follow the generic rule concerning the inexact flag (and no special behavior was specified). The case of the underflow flag was more a lack of specification. * `mpfr_urandom' and `mpfr_urandomb' changed in MPFR 3.1. Their behavior no longer depends on the platform (assuming this is also true for GMP's random generator, which is not the case between GMP 4.1 and 4.2 if `gmp_randinit_default' is used). As a consequence, the returned values can be different between MPFR 3.1 and previous MPFR versions. Note: as the reproducibility of these functions was not specified before MPFR 3.1, the MPFR 3.1 behavior is _not_ regarded as backward incompatible with previous versions.  File: mpfr.info, Node: Removed Functions, Next: Other Changes, Prev: Changed Functions, Up: API Compatibility 6.4 Removed Functions ===================== Functions `mpfr_random' and `mpfr_random2' have been removed in MPFR 3.0 (this only affects old code built against MPFR 3.0 or later). (The function `mpfr_random' had been deprecated since at least MPFR 2.2.0, and `mpfr_random2' since MPFR 2.4.0.)  File: mpfr.info, Node: Other Changes, Prev: Removed Functions, Up: API Compatibility 6.5 Other Changes ================= For users of a C++ compiler, the way how the availability of `intmax_t' is detected has changed in MPFR 3.0. In MPFR 2.x, if a macro `INTMAX_C' or `UINTMAX_C' was defined (e.g. when the `__STDC_CONSTANT_MACROS' macro had been defined before `' or `' has been included), `intmax_t' was assumed to be defined. However this was not always the case (more precisely, `intmax_t' can be defined only in the namespace `std', as with Boost), so that compilations could fail. Thus the check for `INTMAX_C' or `UINTMAX_C' is now disabled for C++ compilers, with the following consequences: * Programs written for MPFR 2.x that need `intmax_t' may no longer be compiled against MPFR 3.0: a `#define MPFR_USE_INTMAX_T' may be necessary before `mpfr.h' is included. * The compilation of programs that work with MPFR 3.0 may fail with MPFR 2.x due to the problem described above. Workarounds are possible, such as defining `intmax_t' and `uintmax_t' in the global namespace, though this is not clean. The divide-by-zero exception is new in MPFR 3.1. However it should not introduce incompatible changes for programs that strictly follow the MPFR API since the exception can only be seen via new functions. As of MPFR 3.1, the `mpfr.h' header can be included several times, while still supporting optional functions (*note Headers and Libraries::).  File: mpfr.info, Node: Contributors, Next: References, Prev: API Compatibility, Up: Top Contributors ************ The main developers of MPFR are Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, Philippe Théveny and Paul Zimmermann. Sylvie Boldo from ENS-Lyon, France, contributed the functions `mpfr_agm' and `mpfr_log'. Sylvain Chevillard contributed the `mpfr_ai' function. David Daney contributed the hyperbolic and inverse hyperbolic functions, the base-2 exponential, and the factorial function. Alain Delplanque contributed the new version of the `mpfr_get_str' function. Mathieu Dutour contributed the functions `mpfr_acos', `mpfr_asin' and `mpfr_atan', and a previous version of `mpfr_gamma'. Laurent Fousse contributed the `mpfr_sum' function. Emmanuel Jeandel, from ENS-Lyon too, contributed the generic hypergeometric code, as well as the internal function `mpfr_exp3', a first implementation of the sine and cosine, and improved versions of `mpfr_const_log2' and `mpfr_const_pi'. Ludovic Meunier helped in the design of the `mpfr_erf' code. Jean-Luc Rémy contributed the `mpfr_zeta' code. Fabrice Rouillier contributed the `mpfr_xxx_z' and `mpfr_xxx_q' functions, and helped to the Microsoft Windows porting. Damien Stehlé contributed the `mpfr_get_ld_2exp' function. We would like to thank Jean-Michel Muller and Joris van der Hoeven for very fruitful discussions at the beginning of that project, Torbjörn Granlund and Kevin Ryde for their help about design issues, and Nathalie Revol for her careful reading of a previous version of this documentation. In particular Kevin Ryde did a tremendous job for the portability of MPFR in 2002-2004. The development of the MPFR library would not have been possible without the continuous support of INRIA, and of the LORIA (Nancy, France) and LIP (Lyon, France) laboratories. In particular the main authors were or are members of the PolKA, Spaces, Cacao and Caramel project-teams at LORIA and of the Arénaire and AriC project-teams at LIP. This project was started during the Fiable (reliable in French) action supported by INRIA, and continued during the AOC action. The development of MPFR was also supported by a grant (202F0659 00 MPN 121) from the Conseil Régional de Lorraine in 2002, from INRIA by an "associate engineer" grant (2003-2005), an "opération de développement logiciel" grant (2007-2009), and the post-doctoral grant of Sylvain Chevillard in 2009-2010. The MPFR-MPC workshop in June 2012 was partly supported by the ERC grant ANTICS of Andreas Enge.  File: mpfr.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top References ********** * Richard Brent and Paul Zimmermann, "Modern Computer Arithmetic", Cambridge University Press (to appear), also available from the authors' web pages. * Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier and Paul Zimmermann, "MPFR: A Multiple-Precision Binary Floating-Point Library With Correct Rounding", ACM Transactions on Mathematical Software, volume 33, issue 2, article 13, 15 pages, 2007, `http://doi.acm.org/10.1145/1236463.1236468'. * Torbjörn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic Library", version 5.0.1, 2010, `http://gmplib.org'. * IEEE standard for binary floating-point arithmetic, Technical Report ANSI-IEEE Standard 754-1985, New York, 1985. Approved March 21, 1985: IEEE Standards Board; approved July 26, 1985: American National Standards Institute, 18 pages. * IEEE Standard for Floating-Point Arithmetic, ANSI-IEEE Standard 754-2008, 2008. Revision of ANSI-IEEE Standard 754-1985, approved June 12, 2008: IEEE Standards Board, 70 pages. * Donald E. Knuth, "The Art of Computer Programming", vol 2, "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981. * Jean-Michel Muller, "Elementary Functions, Algorithms and Implementation", Birkhäuser, Boston, 2nd edition, 2006. * Jean-Michel Muller, Nicolas Brisebarre, Florent de Dinechin, Claude-Pierre Jeannerod, Vincent Lefèvre, Guillaume Melquiond, Nathalie Revol, Damien Stehlé and Serge Torrès, "Handbook of Floating-Point Arithmetic", Birkhäuser, Boston, 2009.  File: mpfr.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top Appendix A GNU Free Documentation License ***************************************** Version 1.2, November 2002 Copyright (C) 2000,2001,2002 Free Software Foundation, Inc. 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 0. PREAMBLE The purpose of this License is to make a manual, textbook, or other functional and useful document "free" in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. 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File: mpfr.info, Node: Concept Index, Next: Function and Type Index, Prev: GNU Free Documentation License, Up: Top Concept Index ************* [index] * Menu: * Accuracy: MPFR Interface. (line 25) * Arithmetic functions: Basic Arithmetic Functions. (line 3) * Assignment functions: Assignment Functions. (line 3) * Basic arithmetic functions: Basic Arithmetic Functions. (line 3) * Combined initialization and assignment functions: Combined Initialization and Assignment Functions. (line 3) * Comparison functions: Comparison Functions. (line 3) * Compatibility with MPF: Compatibility with MPF. (line 3) * Conditions for copying MPFR: Copying. (line 6) * Conversion functions: Conversion Functions. (line 3) * Copying conditions: Copying. (line 6) * Custom interface: Custom Interface. (line 3) * Exception related functions: Exception Related Functions. (line 3) * Float arithmetic functions: Basic Arithmetic Functions. (line 3) * Float comparisons functions: Comparison Functions. (line 3) * Float functions: MPFR Interface. (line 6) * Float input and output functions: Input and Output Functions. (line 3) * Float output functions: Formatted Output Functions. (line 3) * Floating-point functions: MPFR Interface. (line 6) * Floating-point number: Nomenclature and Types. (line 6) * GNU Free Documentation License: GNU Free Documentation License. (line 6) * I/O functions <1>: Formatted Output Functions. (line 3) * I/O functions: Input and Output Functions. (line 3) * Initialization functions: Initialization Functions. (line 3) * Input functions: Input and Output Functions. (line 3) * Installation: Installing MPFR. (line 6) * Integer related functions: Integer Related Functions. (line 3) * Internals: Internals. (line 3) * intmax_t: Headers and Libraries. (line 22) * inttypes.h: Headers and Libraries. (line 22) * libmpfr: Headers and Libraries. (line 50) * Libraries: Headers and Libraries. (line 50) * Libtool: Headers and Libraries. (line 56) * Limb: Internals. (line 6) * Linking: Headers and Libraries. (line 50) * Miscellaneous float functions: Miscellaneous Functions. (line 3) * mpfr.h: Headers and Libraries. (line 6) * Output functions <1>: Formatted Output Functions. (line 3) * Output functions: Input and Output Functions. (line 3) * Precision <1>: MPFR Interface. (line 17) * Precision: Nomenclature and Types. (line 20) * Reporting bugs: Reporting Bugs. (line 6) * Rounding mode related functions: Rounding Related Functions. (line 3) * Rounding Modes: Nomenclature and Types. (line 34) * Special functions: Special Functions. (line 3) * stdarg.h: Headers and Libraries. (line 19) * stdint.h: Headers and Libraries. (line 22) * stdio.h: Headers and Libraries. (line 12) * Ternary value: Rounding Modes. (line 29) * uintmax_t: Headers and Libraries. (line 22)  File: mpfr.info, Node: Function and Type Index, Prev: Concept Index, Up: Top Function and Type Index *********************** [index] * Menu: * mpfr_abs: Basic Arithmetic Functions. (line 175) * mpfr_acos: Special Functions. (line 52) * mpfr_acosh: Special Functions. (line 136) * mpfr_add: Basic Arithmetic Functions. (line 8) * mpfr_add_d: Basic Arithmetic Functions. (line 14) * mpfr_add_q: Basic Arithmetic Functions. (line 18) * mpfr_add_si: Basic Arithmetic Functions. (line 12) * mpfr_add_ui: Basic Arithmetic Functions. (line 10) * mpfr_add_z: Basic Arithmetic Functions. (line 16) * mpfr_agm: Special Functions. (line 232) * mpfr_ai: Special Functions. (line 248) * mpfr_asin: Special Functions. (line 53) * mpfr_asinh: Special Functions. (line 137) * mpfr_asprintf: Formatted Output Functions. (line 194) * mpfr_atan: Special Functions. (line 54) * mpfr_atan2: Special Functions. (line 65) * mpfr_atanh: Special Functions. (line 138) * mpfr_buildopt_decimal_p: Miscellaneous Functions. (line 163) * mpfr_buildopt_gmpinternals_p: Miscellaneous Functions. (line 168) * mpfr_buildopt_tls_p: Miscellaneous Functions. (line 157) * mpfr_buildopt_tune_case: Miscellaneous Functions. (line 173) * mpfr_can_round: Rounding Related Functions. (line 37) * mpfr_cbrt: Basic Arithmetic Functions. (line 109) * mpfr_ceil: Integer Related Functions. (line 8) * mpfr_check_range: Exception Related Functions. (line 38) * mpfr_clear: Initialization Functions. (line 31) * mpfr_clear_divby0: Exception Related Functions. (line 113) * mpfr_clear_erangeflag: Exception Related Functions. (line 116) * mpfr_clear_flags: Exception Related Functions. (line 129) * mpfr_clear_inexflag: Exception Related Functions. (line 115) * mpfr_clear_nanflag: Exception Related Functions. (line 114) * mpfr_clear_overflow: Exception Related Functions. (line 112) * mpfr_clear_underflow: Exception Related Functions. (line 111) * mpfr_clears: Initialization Functions. (line 36) * mpfr_cmp: Comparison Functions. (line 7) * mpfr_cmp_d: Comparison Functions. (line 10) * mpfr_cmp_f: Comparison Functions. (line 14) * mpfr_cmp_ld: Comparison Functions. (line 11) * mpfr_cmp_q: Comparison Functions. (line 13) * mpfr_cmp_si: Comparison Functions. (line 9) * mpfr_cmp_si_2exp: Comparison Functions. (line 31) * mpfr_cmp_ui: Comparison Functions. (line 8) * mpfr_cmp_ui_2exp: Comparison Functions. (line 29) * mpfr_cmp_z: Comparison Functions. (line 12) * mpfr_cmpabs: Comparison Functions. (line 35) * mpfr_const_catalan: Special Functions. (line 259) * mpfr_const_euler: Special Functions. (line 258) * mpfr_const_log2: Special Functions. (line 256) * mpfr_const_pi: Special Functions. (line 257) * mpfr_copysign: Miscellaneous Functions. (line 111) * mpfr_cos: Special Functions. (line 30) * mpfr_cosh: Special Functions. (line 115) * mpfr_cot: Special Functions. (line 48) * mpfr_coth: Special Functions. (line 132) * mpfr_csc: Special Functions. (line 47) * mpfr_csch: Special Functions. (line 131) * mpfr_custom_get_exp: Custom Interface. (line 78) * mpfr_custom_get_kind: Custom Interface. (line 67) * mpfr_custom_get_significand: Custom Interface. (line 72) * mpfr_custom_get_size: Custom Interface. (line 36) * mpfr_custom_init: Custom Interface. (line 41) * mpfr_custom_init_set: Custom Interface. (line 48) * mpfr_custom_move: Custom Interface. (line 85) * mpfr_d_div: Basic Arithmetic Functions. (line 84) * mpfr_d_sub: Basic Arithmetic Functions. (line 37) * MPFR_DECL_INIT: Initialization Functions. (line 75) * mpfr_digamma: Special Functions. (line 187) * mpfr_dim: Basic Arithmetic Functions. (line 182) * mpfr_div: Basic Arithmetic Functions. (line 74) * mpfr_div_2exp: Compatibility with MPF. (line 51) * mpfr_div_2si: Basic Arithmetic Functions. (line 197) * mpfr_div_2ui: Basic Arithmetic Functions. (line 195) * mpfr_div_d: Basic Arithmetic Functions. (line 86) * mpfr_div_q: Basic Arithmetic Functions. (line 90) * mpfr_div_si: Basic Arithmetic Functions. (line 82) * mpfr_div_ui: Basic Arithmetic Functions. (line 78) * mpfr_div_z: Basic Arithmetic Functions. (line 88) * mpfr_divby0_p: Exception Related Functions. (line 135) * mpfr_eint: Special Functions. (line 154) * mpfr_eq: Compatibility with MPF. (line 30) * mpfr_equal_p: Comparison Functions. (line 61) * mpfr_erangeflag_p: Exception Related Functions. (line 138) * mpfr_erf: Special Functions. (line 198) * mpfr_erfc: Special Functions. (line 199) * mpfr_exp: Special Functions. (line 24) * mpfr_exp10: Special Functions. (line 26) * mpfr_exp2: Special Functions. (line 25) * mpfr_expm1: Special Functions. (line 150) * mpfr_fac_ui: Special Functions. (line 143) * mpfr_fits_intmax_p: Conversion Functions. (line 146) * mpfr_fits_sint_p: Conversion Functions. (line 142) * mpfr_fits_slong_p: Conversion Functions. (line 140) * mpfr_fits_sshort_p: Conversion Functions. (line 144) * mpfr_fits_uint_p: Conversion Functions. (line 141) * mpfr_fits_uintmax_p: Conversion Functions. (line 145) * mpfr_fits_ulong_p: Conversion Functions. (line 139) * mpfr_fits_ushort_p: Conversion Functions. (line 143) * mpfr_floor: Integer Related Functions. (line 9) * mpfr_fma: Special Functions. (line 225) * mpfr_fmod: Integer Related Functions. (line 79) * mpfr_fms: Special Functions. (line 227) * mpfr_fprintf: Formatted Output Functions. (line 158) * mpfr_frac: Integer Related Functions. (line 62) * mpfr_free_cache: Special Functions. (line 266) * mpfr_free_str: Conversion Functions. (line 133) * mpfr_frexp: Conversion Functions. (line 47) * mpfr_gamma: Special Functions. (line 169) * mpfr_get_d: Conversion Functions. (line 8) * mpfr_get_d_2exp: Conversion Functions. (line 34) * mpfr_get_decimal64: Conversion Functions. (line 10) * mpfr_get_default_prec: Initialization Functions. (line 114) * mpfr_get_default_rounding_mode: Rounding Related Functions. (line 11) * mpfr_get_emax: Exception Related Functions. (line 8) * mpfr_get_emax_max: Exception Related Functions. (line 31) * mpfr_get_emax_min: Exception Related Functions. (line 30) * mpfr_get_emin: Exception Related Functions. (line 7) * mpfr_get_emin_max: Exception Related Functions. (line 29) * mpfr_get_emin_min: Exception Related Functions. (line 28) * mpfr_get_exp: Miscellaneous Functions. (line 89) * mpfr_get_f: Conversion Functions. (line 73) * mpfr_get_flt: Conversion Functions. (line 7) * mpfr_get_ld: Conversion Functions. (line 9) * mpfr_get_ld_2exp: Conversion Functions. (line 36) * mpfr_get_patches: Miscellaneous Functions. (line 148) * mpfr_get_prec: Initialization Functions. (line 147) * mpfr_get_si: Conversion Functions. (line 20) * mpfr_get_sj: Conversion Functions. (line 22) * mpfr_get_str: Conversion Functions. (line 87) * mpfr_get_ui: Conversion Functions. (line 21) * mpfr_get_uj: Conversion Functions. (line 23) * mpfr_get_version: Miscellaneous Functions. (line 117) * mpfr_get_z: Conversion Functions. (line 68) * mpfr_get_z_2exp: Conversion Functions. (line 55) * mpfr_grandom: Miscellaneous Functions. (line 65) * mpfr_greater_p: Comparison Functions. (line 57) * mpfr_greaterequal_p: Comparison Functions. (line 58) * mpfr_hypot: Special Functions. (line 241) * mpfr_inexflag_p: Exception Related Functions. (line 137) * mpfr_inf_p: Comparison Functions. (line 42) * mpfr_init: Initialization Functions. (line 54) * mpfr_init2: Initialization Functions. (line 11) * mpfr_init_set: Combined Initialization and Assignment Functions. (line 7) * mpfr_init_set_d: Combined Initialization and Assignment Functions. (line 12) * mpfr_init_set_f: Combined Initialization and Assignment Functions. (line 17) * mpfr_init_set_ld: Combined Initialization and Assignment Functions. (line 14) * mpfr_init_set_q: Combined Initialization and Assignment Functions. (line 16) * mpfr_init_set_si: Combined Initialization and Assignment Functions. (line 11) * mpfr_init_set_str: Combined Initialization and Assignment Functions. (line 23) * mpfr_init_set_ui: Combined Initialization and Assignment Functions. (line 9) * mpfr_init_set_z: Combined Initialization and Assignment Functions. (line 15) * mpfr_inits: Initialization Functions. (line 63) * mpfr_inits2: Initialization Functions. (line 23) * mpfr_inp_str: Input and Output Functions. (line 33) * mpfr_integer_p: Integer Related Functions. (line 105) * mpfr_j0: Special Functions. (line 203) * mpfr_j1: Special Functions. (line 204) * mpfr_jn: Special Functions. (line 206) * mpfr_less_p: Comparison Functions. (line 59) * mpfr_lessequal_p: Comparison Functions. (line 60) * mpfr_lessgreater_p: Comparison Functions. (line 66) * mpfr_lgamma: Special Functions. (line 179) * mpfr_li2: Special Functions. (line 164) * mpfr_lngamma: Special Functions. (line 173) * mpfr_log: Special Functions. (line 17) * mpfr_log10: Special Functions. (line 19) * mpfr_log1p: Special Functions. (line 146) * mpfr_log2: Special Functions. (line 18) * mpfr_max: Miscellaneous Functions. (line 24) * mpfr_min: Miscellaneous Functions. (line 22) * mpfr_min_prec: Rounding Related Functions. (line 59) * mpfr_modf: Integer Related Functions. (line 69) * mpfr_mul: Basic Arithmetic Functions. (line 53) * mpfr_mul_2exp: Compatibility with MPF. (line 49) * mpfr_mul_2si: Basic Arithmetic Functions. (line 190) * mpfr_mul_2ui: Basic Arithmetic Functions. (line 188) * mpfr_mul_d: Basic Arithmetic Functions. (line 59) * mpfr_mul_q: Basic Arithmetic Functions. (line 63) * mpfr_mul_si: Basic Arithmetic Functions. (line 57) * mpfr_mul_ui: Basic Arithmetic Functions. (line 55) * mpfr_mul_z: Basic Arithmetic Functions. (line 61) * mpfr_nan_p: Comparison Functions. (line 41) * mpfr_nanflag_p: Exception Related Functions. (line 136) * mpfr_neg: Basic Arithmetic Functions. (line 174) * mpfr_nextabove: Miscellaneous Functions. (line 16) * mpfr_nextbelow: Miscellaneous Functions. (line 17) * mpfr_nexttoward: Miscellaneous Functions. (line 7) * mpfr_number_p: Comparison Functions. (line 43) * mpfr_out_str: Input and Output Functions. (line 17) * mpfr_overflow_p: Exception Related Functions. (line 134) * mpfr_pow: Basic Arithmetic Functions. (line 118) * mpfr_pow_si: Basic Arithmetic Functions. (line 122) * mpfr_pow_ui: Basic Arithmetic Functions. (line 120) * mpfr_pow_z: Basic Arithmetic Functions. (line 124) * mpfr_prec_round: Rounding Related Functions. (line 15) * mpfr_prec_t: Nomenclature and Types. (line 20) * mpfr_print_rnd_mode: Rounding Related Functions. (line 66) * mpfr_printf: Formatted Output Functions. (line 165) * mpfr_rec_sqrt: Basic Arithmetic Functions. (line 104) * mpfr_regular_p: Comparison Functions. (line 45) * mpfr_reldiff: Compatibility with MPF. (line 41) * mpfr_remainder: Integer Related Functions. (line 81) * mpfr_remquo: Integer Related Functions. (line 83) * mpfr_rint: Integer Related Functions. (line 7) * mpfr_rint_ceil: Integer Related Functions. (line 38) * mpfr_rint_floor: Integer Related Functions. (line 40) * mpfr_rint_round: Integer Related Functions. (line 42) * mpfr_rint_trunc: Integer Related Functions. (line 44) * mpfr_rnd_t: Nomenclature and Types. (line 34) * mpfr_root: Basic Arithmetic Functions. (line 111) * mpfr_round: Integer Related Functions. (line 10) * mpfr_sec: Special Functions. (line 46) * mpfr_sech: Special Functions. (line 130) * mpfr_set: Assignment Functions. (line 10) * mpfr_set_d: Assignment Functions. (line 17) * mpfr_set_decimal64: Assignment Functions. (line 21) * mpfr_set_default_prec: Initialization Functions. (line 101) * mpfr_set_default_rounding_mode: Rounding Related Functions. (line 7) * mpfr_set_divby0: Exception Related Functions. (line 122) * mpfr_set_emax: Exception Related Functions. (line 17) * mpfr_set_emin: Exception Related Functions. (line 16) * mpfr_set_erangeflag: Exception Related Functions. (line 125) * mpfr_set_exp: Miscellaneous Functions. (line 94) * mpfr_set_f: Assignment Functions. (line 24) * mpfr_set_flt: Assignment Functions. (line 16) * mpfr_set_inexflag: Exception Related Functions. (line 124) * mpfr_set_inf: Assignment Functions. (line 147) * mpfr_set_ld: Assignment Functions. (line 19) * mpfr_set_nan: Assignment Functions. (line 146) * mpfr_set_nanflag: Exception Related Functions. (line 123) * mpfr_set_overflow: Exception Related Functions. (line 121) * mpfr_set_prec: Initialization Functions. (line 137) * mpfr_set_prec_raw: Compatibility with MPF. (line 23) * mpfr_set_q: Assignment Functions. (line 23) * mpfr_set_si: Assignment Functions. (line 13) * mpfr_set_si_2exp: Assignment Functions. (line 53) * mpfr_set_sj: Assignment Functions. (line 15) * mpfr_set_sj_2exp: Assignment Functions. (line 57) * mpfr_set_str: Assignment Functions. (line 65) * mpfr_set_ui: Assignment Functions. (line 12) * mpfr_set_ui_2exp: Assignment Functions. (line 51) * mpfr_set_uj: Assignment Functions. (line 14) * mpfr_set_uj_2exp: Assignment Functions. (line 55) * mpfr_set_underflow: Exception Related Functions. (line 120) * mpfr_set_z: Assignment Functions. (line 22) * mpfr_set_z_2exp: Assignment Functions. (line 59) * mpfr_set_zero: Assignment Functions. (line 148) * mpfr_setsign: Miscellaneous Functions. (line 105) * mpfr_sgn: Comparison Functions. (line 51) * mpfr_si_div: Basic Arithmetic Functions. (line 80) * mpfr_si_sub: Basic Arithmetic Functions. (line 33) * mpfr_signbit: Miscellaneous Functions. (line 100) * mpfr_sin: Special Functions. (line 31) * mpfr_sin_cos: Special Functions. (line 37) * mpfr_sinh: Special Functions. (line 116) * mpfr_sinh_cosh: Special Functions. (line 122) * mpfr_snprintf: Formatted Output Functions. (line 182) * mpfr_sprintf: Formatted Output Functions. (line 171) * mpfr_sqr: Basic Arithmetic Functions. (line 70) * mpfr_sqrt: Basic Arithmetic Functions. (line 97) * mpfr_sqrt_ui: Basic Arithmetic Functions. (line 99) * mpfr_strtofr: Assignment Functions. (line 83) * mpfr_sub: Basic Arithmetic Functions. (line 27) * mpfr_sub_d: Basic Arithmetic Functions. (line 39) * mpfr_sub_q: Basic Arithmetic Functions. (line 45) * mpfr_sub_si: Basic Arithmetic Functions. (line 35) * mpfr_sub_ui: Basic Arithmetic Functions. (line 31) * mpfr_sub_z: Basic Arithmetic Functions. (line 43) * mpfr_subnormalize: Exception Related Functions. (line 61) * mpfr_sum: Special Functions. (line 275) * mpfr_swap: Assignment Functions. (line 154) * mpfr_t: Nomenclature and Types. (line 6) * mpfr_tan: Special Functions. (line 32) * mpfr_tanh: Special Functions. (line 117) * mpfr_trunc: Integer Related Functions. (line 11) * mpfr_ui_div: Basic Arithmetic Functions. (line 76) * mpfr_ui_pow: Basic Arithmetic Functions. (line 128) * mpfr_ui_pow_ui: Basic Arithmetic Functions. (line 126) * mpfr_ui_sub: Basic Arithmetic Functions. (line 29) * mpfr_underflow_p: Exception Related Functions. (line 133) * mpfr_unordered_p: Comparison Functions. (line 71) * mpfr_urandom: Miscellaneous Functions. (line 50) * mpfr_urandomb: Miscellaneous Functions. (line 30) * mpfr_vasprintf: Formatted Output Functions. (line 196) * MPFR_VERSION: Miscellaneous Functions. (line 120) * MPFR_VERSION_MAJOR: Miscellaneous Functions. (line 121) * MPFR_VERSION_MINOR: Miscellaneous Functions. (line 122) * MPFR_VERSION_NUM: Miscellaneous Functions. (line 140) * MPFR_VERSION_PATCHLEVEL: Miscellaneous Functions. (line 123) * MPFR_VERSION_STRING: Miscellaneous Functions. (line 124) * mpfr_vfprintf: Formatted Output Functions. (line 160) * mpfr_vprintf: Formatted Output Functions. (line 166) * mpfr_vsnprintf: Formatted Output Functions. (line 184) * mpfr_vsprintf: Formatted Output Functions. (line 173) * mpfr_y0: Special Functions. (line 214) * mpfr_y1: Special Functions. (line 215) * mpfr_yn: Special Functions. (line 217) * mpfr_z_sub: Basic Arithmetic Functions. (line 41) * mpfr_zero_p: Comparison Functions. (line 44) * mpfr_zeta: Special Functions. (line 192) * mpfr_zeta_ui: Special Functions. (line 194)  Tag Table: Node: Top892 Node: Copying2243 Node: Introduction to MPFR4003 Node: Installing MPFR6092 Node: Reporting Bugs10914 Node: MPFR Basics12843 Node: Headers and Libraries13159 Node: Nomenclature and Types16143 Node: MPFR Variable Conventions18147 Node: Rounding Modes19677 Ref: ternary value20774 Node: Floating-Point Values on Special Numbers22727 Node: Exceptions25703 Node: Memory Handling28855 Node: MPFR Interface29987 Node: Initialization Functions32083 Node: Assignment Functions38997 Node: Combined Initialization and Assignment Functions47651 Node: Conversion Functions48944 Node: Basic Arithmetic Functions57496 Node: Comparison Functions66504 Node: Special Functions69986 Node: Input and Output Functions83739 Node: Formatted Output Functions85662 Node: Integer Related Functions94781 Node: Rounding Related Functions100543 Node: Miscellaneous Functions104157 Node: Exception Related Functions112724 Node: Compatibility with MPF119478 Node: Custom Interface122166 Node: Internals126411 Node: API Compatibility127895 Node: Type and Macro Changes129825 Node: Added Functions132546 Node: Changed Functions135489 Node: Removed Functions139770 Node: Other Changes140182 Node: Contributors141711 Node: References144285 Node: GNU Free Documentation License146026 Node: Concept Index168469 Node: Function and Type Index174388  End Tag Table  Local Variables: coding: utf-8 End: @ 1.1.1.1 log @initial import of MPFR 3.1.2. changes since 3.0.1: - Bug fixes (see or ChangeLog file). - Bug fixes (see or ChangeLog file). - TLS support is now detected automatically. If TLS is supported, MPFR is built as thread safe by default. To disable TLS explicitly, configure MPFR with --disable-thread-safe. - The mpfr_urandom and mpfr_urandomb functions now return identical values on processors with different word size (assuming the same random seed, and since the GMP random generator does not depend itself on the word size, cf http://gmplib.org/list-archives/gmp-devel/2010-September/001642.html). - The mpfr_add_one_ulp and mpfr_sub_one_ulp macros (which are obsolete and no more documented) will be removed in a future release. - Speed improvement for the mpfr_sqr and mpfr_div functions using Mulders' algorithm. As a consequence, other functions using those routines are also faster. - Much faster formatted output (mpfr_printf, etc.) with %Rg and similar. - New functions mpfr_buildopt_gmpinternals_p, mpfr_buildopt_tune_case, mpfr_frexp, mpfr_grandom and mpfr_z_sub. - New divide-by-zero exception (flag) and associated functions. - Internal change: the logging mechanism has been improved. - Bug fixes, in particular a huge inefficiency in mpfr_exp (when the target precision is less than MPFR_EXP_THRESHOLD) on hard-to-round cases, which can take several minutes. @ text @@ 1.1.1.2 log @initial import of MPFR 3.1.5 package. changes since 3.1.2: Changes from version 3.1.4 to version 3.1.5: - C++11 compatibility. - Bug fixes (see and ChangeLog file). - More tests. Changes from version 3.1.3 to version 3.1.4: - Improved MPFR manual. - Bug fixes (see and ChangeLog file). - MinGW (MS Windows): Added support for thread-safe DLL (shared library). Changes from version 3.1.2 to version 3.1.3: - Better support for Automake 1.13+ (now used to generate the tarball). - Improved MPFR manual. - Bug fixes (see and ChangeLog file). @ text @d1 1 a1 1 This is mpfr.info, produced by makeinfo version 6.3 from mpfr.texi. d4 1 a4 1 Floating-Point Reliable Library, version 3.1.5. d6 10 a15 1 Copyright 1991, 1993-2016 Free Software Foundation, Inc. a16 6 Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in *note GNU Free Documentation License::. d28 2 a29 2 This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 3.1.5. d31 10 a40 1 Copyright 1991, 1993-2016 Free Software Foundation, Inc. a41 6 Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in *note GNU Free Documentation License::. d64 1 a64 1 The GNU MPFR library (or MPFR for short) is “free”; this means that d66 6 a71 5 The library is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you. d74 3 a76 3 away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things. d81 2 a82 2 rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights. d86 3 a88 3 modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation. d101 4 a104 4 arithmetic on floating-point numbers. It is based on the GNU MP library. It aims to provide a class of floating-point numbers with precise semantics. The main characteristics of MPFR, which make it differ from most arbitrary precision floating-point software tools, are: d106 2 a107 2 • the MPFR code is portable, i.e., the result of any operation does not depend on the machine word size ‘mp_bits_per_limb’ (64 on most d110 1 a110 1 • the precision in bits can be set _exactly_ to any valid value for d113 1 a113 1 • MPFR provides the four rounding modes from the IEEE 754-1985 d119 3 a121 3 numbers (e.g., ‘double’ type in C, with a C implementation that rigorously follows Annex F of the ISO C99 standard and ‘FP_CONTRACT’ pragma set to ‘OFF’) on the four arithmetic operations and the square d135 2 a136 2 library yourself, you need to read *note Installing MPFR::, too. To use the library you will need to refer to *note MPFR Interface::. d148 5 a152 5 but the development files necessary to the compilation such as ‘mpfr.h’ are not always present. To check that MPFR is fully installed on your computer, you can check the presence of the file ‘mpfr.h’ in ‘/usr/include’, or try to compile a small program having ‘#include ’ (since ‘mpfr.h’ may be installed somewhere else). For d173 1 a173 1 then MPFR is probably not installed. Running this program will give you d183 1 a183 1 details are provided in the ‘INSTALL’ file): d187 3 a189 3 but any reasonable compiler should work. And you need the standard Unix ‘make’ command, plus some other standard Unix utility commands. d193 1 a193 1 2. ‘./configure’ d195 6 a200 6 This will prepare the build and setup the options according to your system. You can give options to specify the install directories (instead of the default ‘/usr/local’), threading support, and so on. See the ‘INSTALL’ file and/or the output of ‘./configure --help’ for more information, in particular if you get error messages. d202 1 a202 1 3. ‘make’ d205 2 a206 2 ‘libmpfr.a’. On most platforms, a dynamic library will be produced too. d208 1 a208 1 4. ‘make check’ d210 14 a223 22 This will make sure that MPFR was built correctly. If any test fails, information about this failure can be found in the ‘tests/test-suite.log’ file. If you want the contents of this file to be automatically output in case of failure, you can set the ‘VERBOSE’ environment variable to 1 before running ‘make check’, for instance by typing: ‘VERBOSE=1 make check’ In case of failure, you may want to check whether the problem is already known. If not, please report this failure to the MPFR mailing-list ‘mpfr@@inria.fr’. For details, *Note Reporting Bugs::. 5. ‘make install’ This will copy the files ‘mpfr.h’ and ‘mpf2mpfr.h’ to the directory ‘/usr/local/include’, the library files (‘libmpfr.a’ and possibly others) to the directory ‘/usr/local/lib’, the file ‘mpfr.info’ to the directory ‘/usr/local/share/info’, and some other documentation files to the directory ‘/usr/local/share/doc/mpfr’ (or if you passed the ‘--prefix’ option to ‘configure’, using the prefix directory given as argument to ‘--prefix’ instead of ‘/usr/local’). d225 1 a225 1 2.2 Other ‘make’ Targets d230 1 a230 1 • ‘mpfr.info’ or ‘info’ d232 1 a232 1 Create or update an info version of the manual, in ‘mpfr.info’. d236 1 a236 1 • ‘mpfr.pdf’ or ‘pdf’ d238 1 a238 1 Create a PDF version of the manual, in ‘mpfr.pdf’. d240 1 a240 1 • ‘mpfr.dvi’ or ‘dvi’ d242 1 a242 1 Create a DVI version of the manual, in ‘mpfr.dvi’. d244 1 a244 1 • ‘mpfr.ps’ or ‘ps’ d246 1 a246 1 Create a Postscript version of the manual, in ‘mpfr.ps’. d248 1 a248 1 • ‘mpfr.html’ or ‘html’ d251 2 a252 2 directory ‘doc/mpfr.html’; if you want only one output HTML file, then type ‘makeinfo --html --no-split mpfr.texi’ from the ‘doc’ d255 1 a255 1 • ‘clean’ d260 1 a260 1 • ‘distclean’ d264 1 a264 1 • ‘uninstall’ d266 1 a266 1 Delete all files copied by ‘make install’. d271 2 a272 2 In case of problem, please read the ‘INSTALL’ file carefully before reporting a bug, in particular section “In case of problem”. Some d274 2 a275 2 MPFR). Problems are also mentioned in the FAQ . d277 3 a279 3 Please report problems to the MPFR mailing-list ‘mpfr@@inria.fr’. *Note Reporting Bugs::. Some bug fixes are available on the MPFR 3.1.5 web page . d285 1 a285 1 or . d294 5 a298 5 on the MPFR 3.1.5 web page and the FAQ : perhaps this bug is already known, in which case you may find there a workaround for it. You might also look in the archives of the MPFR mailing-list: . Otherwise, please investigate d310 2 a311 2 You also have to explain what is wrong; if you get a crash, or if the results you get are incorrect and in that case, in what way. d313 6 a318 7 Please include compiler version information in your bug report. This can be extracted using ‘cc -V’ on some machines, or, if you’re using GCC, ‘gcc -v’. Also, include the output from ‘uname -a’ and the MPFR version (the GMP version may be useful too). If you get a failure while running ‘make’ or ‘make check’, please include the ‘config.log’ file in your bug report, and in case of test failure, the ‘tests/test-suite.log’ file too. d321 3 a323 2 corrected version of the library; if the bug report is poor, we will not do anything about it (aside of chiding you to send better bug reports). d325 1 a325 1 Send your bug report to the MPFR mailing-list ‘mpfr@@inria.fr’. d354 1 a354 1 ‘mpfr.h’. It is designed to work with both C and C++ compilers. You d359 3 a361 3 Note however that prototypes for MPFR functions with ‘FILE *’ parameters are provided only if ‘’ is included too (before ‘mpfr.h’): d366 2 a367 2 Likewise ‘’ (or ‘’) is required for prototypes with ‘va_list’ parameters, such as ‘mpfr_vprintf’. d369 16 a384 16 And for any functions using ‘intmax_t’, you must include ‘’ or ‘’ before ‘mpfr.h’, to allow ‘mpfr.h’ to define prototypes for these functions. Moreover, users of C++ compilers under some platforms may need to define ‘MPFR_USE_INTMAX_T’ (and should do it for portability) before ‘mpfr.h’ has been included; of course, it is possible to do that on the command line, e.g., with ‘-DMPFR_USE_INTMAX_T’. Note: If ‘mpfr.h’ and/or ‘gmp.h’ (used by ‘mpfr.h’) are included several times (possibly from another header file), ‘’ and/or ‘’ (or ‘’) should be included *before the first inclusion* of ‘mpfr.h’ or ‘gmp.h’. Alternatively, you can define ‘MPFR_USE_FILE’ (for MPFR I/O functions) and/or ‘MPFR_USE_VA_LIST’ (for MPFR functions with ‘va_list’ parameters) anywhere before the last inclusion of ‘mpfr.h’. As a consequence, if your file is a public header that includes ‘mpfr.h’, you need to use the latter method. d387 2 a388 2 defined a macro with the same name as some keywords (currently ‘do’, ‘while’ and ‘sizeof’). d391 1 a391 1 defining the ‘MPFR_USE_NO_MACRO’ macro before ‘mpfr.h’ is included. In d397 3 a399 3 All programs using MPFR must link against both ‘libmpfr’ and ‘libgmp’ libraries. On a typical Unix-like system this can be done with ‘-lmpfr -lgmp’ (in that order), for example: d404 1 a404 1 if desired, *note GNU Libtool: (libtool)Top. d407 3 a409 3 necessary to set up environment variables such as ‘C_INCLUDE_PATH’ and ‘LIBRARY_PATH’, or use ‘-I’ and ‘-L’ compiler options, in order to point to the right directories. For a shared library, it may also be d411 1 a411 1 ‘LD_LIBRARY_PATH’) on some systems. Please read the ‘INSTALL’ file for d420 1 a420 1 A “floating-point number”, or “float” for short, is an arbitrary d422 3 a424 3 exponent. The C data type for such objects is ‘mpfr_t’ (internally defined as a one-element array of a structure, and ‘mpfr_ptr’ is the C data type representing a pointer to this structure). A floating-point d426 1 a426 1 minus Infinity. NaN represents an uninitialized object, the result of d428 2 a429 2 determined (like +Infinity minus +Infinity). Moreover, like in the IEEE 754 standard, zero is signed, i.e., there are both +0 and −0; the d431 1 a431 1 to the other functions supported by MPFR. Unless documented otherwise, d434 1 a434 1 The “precision” is the number of bits used to represent the significand d436 8 a443 8 ‘mpfr_prec_t’. The precision can be any integer between ‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’. In the current implementation, ‘MPFR_PREC_MIN’ is equal to 2. Warning! MPFR needs to increase the precision internally, in order to provide accurate results (and in particular, correct rounding). Do not attempt to set the precision to any value near ‘MPFR_PREC_MAX’, otherwise MPFR will abort due to an assertion failure. Moreover, you d448 1 a448 1 The “rounding mode” specifies the way to round the result of a d451 1 a451 1 data type is ‘mpfr_rnd_t’. d460 1 a460 1 calling one of the special initialization functions. When you’re done d468 2 a469 2 allocating additional space for MPFR variables, since any variable has a significand of fixed size. Hence unless you change its precision, or d476 4 a479 4 input and output in the same expression. For example, the main function for floating-point multiplication, ‘mpfr_mul’, can be used like this: ‘mpfr_mul (x, x, x, rnd)’. This computes the square of X with rounding mode ‘rnd’ and puts the result back in X. d488 6 a493 3 • ‘MPFR_RNDN’: round to nearest (roundTiesToEven in IEEE 754-2008), • ‘MPFR_RNDZ’: round toward zero (roundTowardZero in IEEE 754-2008), • ‘MPFR_RNDU’: round toward plus infinity (roundTowardPositive in d495 2 a496 1 • ‘MPFR_RNDD’: round toward minus infinity (roundTowardNegative in a497 1 • ‘MPFR_RNDA’: round away from zero. d499 3 a501 1 The ‘round to nearest’ mode works as in the IEEE 754 standard: in d505 4 a508 4 represented by (10.1) in binary, is rounded to (10.0)=2 with a precision of two bits, and not to (11.0)=3. This rule avoids the “drift” phenomenon mentioned by Knuth in volume 2 of The Art of Computer Programming (Section 4.2.2). d512 2 a513 2 a rounding mode, and have a return value of type ‘int’, called the “ternary value”. The value stored in the destination variable is d520 4 a523 4 error on the result is less or equal to 1/2 ulp (unit in the last place) of that result in the rounding to nearest mode, and less than 1 ulp of that result in the directed rounding modes (a ulp is the weight of the least significant represented bit of the result after rounding). d525 1 a525 1 Unless documented otherwise, functions returning an ‘int’ return a d528 1 a528 1 corresponding mathematical function. If the ternary value is positive d530 5 a534 5 is greater (resp. lower) than the exact result. For example with the ‘MPFR_RNDU’ rounding mode, the ternary value is usually positive, except when the result is exact, in which case it is zero. In the case of an infinite result, it is considered as inexact when it was obtained by overflow, and exact otherwise. A NaN result (Not-a-Number) always d536 1 a536 1 ternary value is guaranteed to be representable in an ‘int’. d539 2 a540 2 ‘1’ (or any other value specified in this manual) for special cases (like ‘acos(0)’) yield an overflow or an underflow if that value is not d549 5 a553 5 This section specifies the floating-point values (of type ‘mpfr_t’) returned by MPFR functions (where by “returned” we mean here the modified value of the destination object, which should not be mixed with the ternary return value of type ‘int’ of those functions). For functions returning several values (like ‘mpfr_sin_cos’), the rules d556 1 a556 1 Functions can have one or several input arguments. An input point is d563 4 a566 4 the result is rounded as described in Section “Rounding Modes” (but see below for the specification of the sign of an exact zero). Otherwise the general rules from this section apply unless stated otherwise in the description of the MPFR function (*note MPFR Interface::). d571 4 a574 4 limit. Examples: ‘mpfr_hypot’ on (+Inf,0) gives +Inf. But ‘mpfr_pow’ cannot be defined on (1,+Inf) using this rule, as one can find sequences (X_N,Y_N) such that X_N goes to 1, Y_N goes to +Inf and X_N to the Y_N goes to any positive value when N goes to the infinity. d577 1 a577 1 mathematical function and an input argument is +0 (resp. −0), one d579 2 a580 4 above (resp. below), if possible. If the limit is not defined (e.g., ‘mpfr_sqrt’ and ‘mpfr_log’ on −0), the behavior is specified in the description of the MPFR function, but must be consistent with the rule from the above paragraph (e.g., ‘mpfr_log’ on ±0 gives −Inf). d584 4 a587 5 approaches 0 from above (resp. below), the result is +0 (resp. −0); for example, ‘mpfr_sin’ on −0 gives −0 and ‘mpfr_acos’ on 1 gives +0 (in all rounding modes). In the other cases, the sign is specified in the description of the MPFR function; for example ‘mpfr_max’ on −0 and +0 gives +0. d590 1 a590 1 function, the result is NaN. Example: ‘mpfr_sqrt’ on −17 gives NaN. d595 4 a598 4 Interface::. Example: ‘mpfr_hypot’ on (NaN,0) gives NaN, but ‘mpfr_hypot’ on (NaN,+Inf) gives +Inf (as specified in *note Special Functions::), since for any finite or infinite input X, ‘mpfr_hypot’ on (X,+Inf) gives +Inf. d608 1 a608 1 • Underflow: An underflow occurs when the exact result of a function d612 1 a612 1 exponent range. (In the round-to-nearest mode, the halfway case is d616 2 a617 2 MPFR chooses to consider the underflow _after_ rounding. The underflow before rounding can also be defined. For instance, d619 1 a619 1 to the power E−4, where E is the smallest exponent (for a d622 3 a624 3 E−1. With the underflow before rounding, such a function call would yield an underflow, as E−1 is outside the current exponent range. However, MPFR first considers the rounded result assuming d627 2 a628 2 times 2 to E, which is representable in the current exponent range. As a consequence, this will not yield an underflow in MPFR. d630 7 a636 7 • Overflow: An overflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent larger than the maximum value of the current exponent range. In the round-to-nearest mode, the result is infinite. Note: unlike the underflow case, there is only one possible definition of overflow here. d638 1 a638 1 • Divide-by-zero: An exact infinite result is obtained from finite d641 1 a641 1 • NaN: A NaN exception occurs when the result of a function is NaN. d643 1 a643 1 • Inexact: An inexact exception occurs when the result of a function d646 4 a649 4 • Range error: A range exception occurs when a function that does not return a MPFR number (such as comparisons and conversions to an integer) has an invalid result (e.g., an argument is NaN in ‘mpfr_cmp’, or a conversion to an integer cannot be represented in d652 1 d658 5 a662 4 • In C, only quiet NaNs are specified, and a NaN propagation does not raise an invalid exception. Unless explicitly stated otherwise, MPFR sets the NaN flag whenever a NaN is generated, even when a NaN is propagated (e.g., in NaN + NaN), as if all NaNs were signaling. d664 1 a664 1 • An invalid exception in C corresponds to either a NaN exception or d667 1 d674 4 a677 4 MPFR functions may create caches, e.g., when computing constants such as Pi, either because the user has called a function like ‘mpfr_const_pi’ directly or because such a function was called internally by the MPFR library itself to compute some other function. d680 2 a681 2 ‘mpfr_free_cache’. It is strongly advised to do that before terminating a thread, or before exiting when using tools like ‘valgrind’ (to avoid d687 5 a691 5 compiled as thread safe) or per-thread (thread local storage, TLS). The initial values of TLS data after a thread is created entirely depend on the compiler and thread implementation (MPFR simply does a conventional variable initialization, the variables being declared with an implementation-defined TLS specifier). d699 1 a699 1 The floating-point functions expect arguments of type ‘mpfr_t’. d703 1 a703 1 operations is ‘mpfr_’. d707 2 a708 2 of the assigned variable; the cost of that computation should not depend on the precision of variables used as input (on average). d711 1 a711 1 Compute the requested operation exactly (with “infinite accuracy”), and d714 3 a716 3 be a smooth extension of the IEEE 754 arithmetic. The results obtained on a given computer are identical to those obtained on a computer with a different word size, or with a different compiler or operating system. d718 1 a718 1 MPFR _does not keep track_ of the accuracy of a computation. This is d725 1 a725 1 The value of the standard C macro ‘errno’ may be set to non-zero by d753 2 a754 2 An ‘mpfr_t’ object must be initialized before storing the first value in it. The functions ‘mpfr_init’ and ‘mpfr_init2’ are used for that d759 1 a759 1 value to NaN. (Warning: the corresponding MPF function initializes d762 6 a767 6 Normally, a variable should be initialized once only or at least be cleared, using ‘mpfr_clear’, between initializations. To change the precision of a variable which has already been initialized, use ‘mpfr_set_prec’. The precision PREC must be an integer between ‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’ (otherwise the behavior is undefined). d770 6 a775 6 Initialize all the ‘mpfr_t’ variables of the given variable argument ‘va_list’, set their precision to be *exactly* PREC bits and their value to NaN. See ‘mpfr_init2’ for more details. The ‘va_list’ is assumed to be composed only of type ‘mpfr_t’ (or equivalently ‘mpfr_ptr’). It begins from X, and ends when it encounters a null pointer (whose type must also be ‘mpfr_ptr’). d778 3 a780 3 Free the space occupied by the significand of X. Make sure to call this function for all ‘mpfr_t’ variables when you are done with them. d783 5 a787 5 Free the space occupied by all the ‘mpfr_t’ variables of the given ‘va_list’. See ‘mpfr_clear’ for more details. The ‘va_list’ is assumed to be composed only of type ‘mpfr_t’ (or equivalently ‘mpfr_ptr’). It begins from X, and ends when it encounters a null pointer (whose type must also be ‘mpfr_ptr’). d790 2 a791 2 (since ‘NULL’ is not necessarily defined in this context, we use ‘(mpfr_ptr) 0’ instead, but ‘(mpfr_ptr) NULL’ is also correct). d803 1 a803 1 to ‘mpfr_set_default_prec’. d805 3 a807 3 Warning! In a given program, some other libraries might change the default precision and not restore it. Thus it is safer to use ‘mpfr_init2’. d810 1 a810 1 Initialize all the ‘mpfr_t’ variables of the given ‘va_list’, set d812 2 a813 2 See ‘mpfr_init’ for more details. The ‘va_list’ is assumed to be composed only of type ‘mpfr_t’ (or equivalently ‘mpfr_ptr’). It d815 1 a815 1 type must also be ‘mpfr_ptr’). d817 3 a819 3 Warning! In a given program, some other libraries might change the default precision and not restore it. Thus it is safer to use ‘mpfr_inits2’. d822 5 a826 5 This macro declares NAME as an automatic variable of type ‘mpfr_t’, initializes it and sets its precision to be *exactly* PREC bits and its value to NaN. NAME must be a valid identifier. You must use this macro in the declaration section. This macro is much faster than using ‘mpfr_init2’ but has some drawbacks: d828 1 a828 1 • You *must not* call ‘mpfr_clear’ with variables created with d832 1 a832 1 • You *cannot* change their precision. d834 1 a834 1 • You *should not* create variables with huge precision with d837 9 a845 9 • Your compiler must support ‘Non-Constant Initializers’ (standard in C++ and ISO C99) and ‘Token Pasting’ (standard in ISO C89). If PREC is not a constant expression, your compiler must support ‘variable-length automatic arrays’ (standard in ISO C99). GCC 2.95.3 and above supports all these features. If you compile your program with GCC in C89 mode and with ‘-pedantic’, you may want to define the ‘MPFR_USE_EXTENSION’ macro to avoid warnings due to the ‘MPFR_DECL_INIT’ implementation. d848 7 a854 6 Set the default precision to be *exactly* PREC bits, where PREC can be any integer between ‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’. The precision of a variable means the number of bits used to store its significand. All subsequent calls to ‘mpfr_init’ or ‘mpfr_inits’ will use this precision, but previously initialized variables are unaffected. The default precision is set to 53 bits initially. d856 2 a857 2 Note: when MPFR is built with the ‘--enable-thread-safe’ configure option, the default precision is local to each thread. *Note d862 1 a862 1 documentation of ‘mpfr_set_default_prec’. d885 3 a887 3 value to NaN. The previous value stored in X is lost. It is equivalent to a call to ‘mpfr_clear(x)’ followed by a call to ‘mpfr_init2(x, prec)’, but more efficient as no allocation is done d890 2 a891 6 ‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’. In case you want to keep the previous value stored in X, use ‘mpfr_prec_round’ instead. Warning! You must not use this function if X was initialized with ‘MPFR_DECL_INIT’ or with ‘mpfr_custom_init_set’ (*note Custom Interface::). d922 20 a941 19 RND. Note that the input 0 is converted to +0 by ‘mpfr_set_ui’, ‘mpfr_set_si’, ‘mpfr_set_uj’, ‘mpfr_set_sj’, ‘mpfr_set_z’, ‘mpfr_set_q’ and ‘mpfr_set_f’, regardless of the rounding mode. If the system does not support the IEEE 754 standard, ‘mpfr_set_flt’, ‘mpfr_set_d’, ‘mpfr_set_ld’ and ‘mpfr_set_decimal64’ might not preserve the signed zeros. The ‘mpfr_set_decimal64’ function is built only with the configure option ‘--enable-decimal-float’, which also requires ‘--with-gmp-build’, and when the compiler or system provides the ‘_Decimal64’ data type (recent versions of GCC support this data type); to use ‘mpfr_set_decimal64’, one should define the macro ‘MPFR_WANT_DECIMAL_FLOATS’ before including ‘mpfr.h’. ‘mpfr_set_q’ might fail if the numerator (or the denominator) can not be represented as a ‘mpfr_t’. Note: If you want to store a floating-point constant to a ‘mpfr_t’, you should use ‘mpfr_set_str’ (or one of the MPFR constant functions, such as ‘mpfr_const_pi’ for Pi) instead of ‘mpfr_set_flt’, ‘mpfr_set_d’, ‘mpfr_set_ld’ or ‘mpfr_set_decimal64’. Otherwise the floating-point constant will d943 1 a943 1 (or decimal, for ‘mpfr_set_decimal64’) number before MPFR can work d952 2 a953 2 -- Function: int mpfr_set_sj_2exp (mpfr_t ROP, intmax_t OP, intmax_t E, mpfr_rnd_t RND) d963 1 a963 1 direction RND. See the documentation of ‘mpfr_strtofr’ for a d965 1 a965 1 ‘mpfr_strtofr’, ‘mpfr_set_str’ requires the _whole_ string to d970 1 a970 1 valid number in base BASE; otherwise it is −1, and ROP may have d972 1 a972 1 ‘mpfr_strtofr’ instead). d974 3 a976 3 Note: it is preferable to use ‘mpfr_strtofr’ if one wants to distinguish between an infinite ROP value coming from an infinite S or from an overflow. d983 2 a984 2 behavior is undefined). If NPTR starts with valid data, the result is stored in ROP and ‘*ENDPTR’ points to the character just after d986 1 a986 1 set to zero (for consistency with ‘strtod’) and the value of NPTR d988 1 a988 1 null pointer). The usual ternary value is returned. d990 1 a990 1 Parsing follows the standard C ‘strtod’ function with some d992 2 a993 2 sequence consisting of an optional sign (‘+’ or ‘-’), and either numeric data or special data. The subject sequence is defined as d999 2 a1000 2 consisting of an exponent prefix followed by an optional sign and a non-empty sequence of decimal digits. A significand digit is d1002 3 a1004 3 with ‘A’ = 10, ‘B’ = 11, ..., ‘Z’ = 35; case is ignored in bases less or equal to 36, in bases larger than 36, ‘a’ = 36, ‘b’ = 37, ..., ‘z’ = 61. The value of a significand digit must be strictly d1010 4 a1013 4 locale). The exponent prefix can be ‘e’ or ‘E’ for bases up to 10, or ‘@@’ in any base; it indicates a multiplication by a power of the base. In bases 2 and 16, the exponent prefix can also be ‘p’ or ‘P’, in which case the exponent, called _binary exponent_, d1016 1 a1016 1 ‘1p2’ represents 4 whereas ‘1@@2’ represents 256. The value of an d1020 3 a1022 3 as follows. If the significand starts with ‘0b’ or ‘0B’, base 2 is assumed. If the significand starts with ‘0x’ or ‘0X’, base 16 is assumed. Otherwise base 10 is assumed. d1025 4 a1028 4 Otherwise the possible exponent prefix and sign are not part of the number (which ends with the significand). Similarly, if ‘0b’, ‘0B’, ‘0x’ or ‘0X’ is not followed by a binary/hexadecimal digit, then the subject sequence stops at the character ‘0’, thus 0 is d1031 4 a1034 4 Special data (for infinities and NaN) can be ‘@@inf@@’ or ‘@@nan@@(n-char-sequence-opt)’, and if BASE <= 16, it can also be ‘infinity’, ‘inf’, ‘nan’ or ‘nan(n-char-sequence-opt)’, all case insensitive. A ‘n-char-sequence-opt’ is a possibly empty string d1036 5 a1040 3 ..., 9, a, b, ..., z, A, B, ..., Z, _). Note: one has an optional sign for all data, even NaN. For example, ‘-@@nAn@@(This_Is_Not_17)’ is a valid representation for NaN in base 17. d1046 1 a1046 1 respectively. In ‘mpfr_set_inf’ or ‘mpfr_set_zero’, X is set to d1048 1 a1048 1 ‘mpfr_set_nan’, the sign bit of the result is unspecified. d1051 4 a1054 11 Swap the structures pointed to by X and Y. In particular, the values are exchanged without rounding (this may be different from three ‘mpfr_set’ calls using a third auxiliary variable). Warning! Since the precisions are exchanged, this will affect future assignments. Moreover, since the significand pointers are also exchanged, you must not use this function if the allocation method used for X and/or Y does not permit it. This is the case when X and/or Y were declared and initialized with ‘MPFR_DECL_INIT’, and possibly with ‘mpfr_custom_init_set’ (*note Custom Interface::). d1075 1 a1075 1 precision, as set by ‘mpfr_set_default_prec’. d1080 1 a1080 1 rounded in the direction RND. See ‘mpfr_set_str’. d1092 2 a1093 2 Convert OP to a ‘float’ (respectively ‘double’, ‘long double’ or ‘_Decimal64’), using the rounding mode RND. If OP is NaN, some d1095 2 a1096 2 returned. If OP is ±Inf, an infinity of the same sign or the result of ±1.0/0.0 is returned. If OP is zero, these functions d1098 2 a1099 2 ‘mpfr_get_decimal64’ function is built only under some conditions: see the documentation of ‘mpfr_set_decimal64’. d1105 8 a1112 8 Convert OP to a ‘long’, an ‘unsigned long’, an ‘intmax_t’ or an ‘uintmax_t’ (respectively) after rounding it with respect to RND. If OP is NaN, 0 is returned and the _erange_ flag is set. If OP is too big for the return type, the function returns the maximum or the minimum of the corresponding C type, depending on the direction of the overflow; the _erange_ flag is set too. See also ‘mpfr_fits_slong_p’, ‘mpfr_fits_ulong_p’, ‘mpfr_fits_intmax_p’ and ‘mpfr_fits_uintmax_p’. d1119 2 a1120 2 that 0.5<=abs(D)<1 and D times 2 raised to EXP equals OP rounded to double (resp. long double) precision, using the given rounding d1141 7 a1147 7 exponent ‘emin’ is returned. If OP is NaN or an infinity, the _erange_ flag is set, ROP is set to 0, and the the minimal exponent ‘emin’ is returned. The returned exponent may be less than the minimal exponent ‘emin’ of MPFR numbers in the current exponent range; in case the exponent is not representable in the ‘mpfr_exp_t’ type, the _erange_ flag is set and the minimal value of the ‘mpfr_exp_t’ type is returned. d1150 1 a1150 1 Convert OP to a ‘mpz_t’, after rounding it with respect to RND. If d1155 1 a1155 1 Convert OP to a ‘mpf_t’, after rounding it with respect to RND. d1157 9 a1165 9 exist in MPF. If OP is NaN, then ROP is undefined. If OP is +Inf (resp. −Inf), then ROP is set to the maximum (resp. minimum) value in the precision of the MPF number; if a future MPF version supports infinities, this behavior will be considered incorrect and will change (portable programs should assume that ROP is set either to this finite number or to an infinite number). Note that since MPFR currently has the same exponent type as MPF (but not with the same radix), the range of values is much larger in MPF than in MPFR, so that an overflow or underflow is not possible. d1167 2 a1168 2 -- Function: char * mpfr_get_str (char *STR, mpfr_exp_t *EXPPTR, int B, size_t N, mpfr_t OP, mpfr_rnd_t RND) d1171 5 a1175 7 significant digits output in the string; in the latter case, N must be greater or equal to 2. The base may vary from 2 to 62; otherwise the function does nothing and immediately returns a null pointer. If the input number is an ordinary number, the exponent is written through the pointer EXPPTR (for input 0, the current minimal exponent is written); the type ‘mpfr_exp_t’ is large enough to hold the exponent in all cases. d1179 1 a1179 1 number −3.1416 would be returned as "−31416" in the string and 1 d1184 4 a1187 4 correspond to an even last digit: for example with 2 digits in base 7, (14) and a half is rounded to (15) which is 12 in decimal, (16) and a half is rounded to (20) which is 14 in decimal, and (26) and a half is rounded to (26) which is 20 in decimal. d1196 1 a1196 1 P−1 if B is a power of 2, but in some very rare cases, it might be d1201 3 a1203 3 using the current allocation function and a pointer to the string is returned (unless the base is invalid). To free the returned string, you must use ‘mpfr_free_str’. d1206 1 a1206 1 large enough for the significand, i.e., at least ‘max(N + 2, 7)’. d1208 5 a1212 8 terminating null character, and the value 7 accounts for ‘-@@Inf@@’ plus the terminating null character. The pointer to the string STR is returned (unless the base is invalid). Note: The NaN and inexact flags are currently not set when need be; this will be fixed in future versions. Programmers should currently assume that whether the flags are set by this function is unspecified. d1215 4 a1218 4 Free a string allocated by ‘mpfr_get_str’ using the current unallocation function. The block is assumed to be ‘strlen(STR)+1’ bytes. For more information about how it is done: *note (gmp.info)Custom Allocation::. d1229 3 a1231 3 respectively ‘unsigned long’, ‘long’, ‘unsigned int’, ‘int’, ‘unsigned short’, ‘short’, ‘uintmax_t’, ‘intmax_t’, when rounded to an integer in the direction RND. d1241 2 a1242 2 -- Function: int mpfr_add_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) d1251 6 a1256 7 Set ROP to OP1 + OP2 rounded in the direction RND. The IEEE-754 rules are used, in particular for signed zeros. But for types having no signed zeros, 0 is considered unsigned (i.e., (+0) + 0 = (+0) and (−0) + 0 = (−0)). The ‘mpfr_add_d’ function assumes that the radix of the ‘double’ type is a power of 2, with a precision at most that declared by the C implementation (macro ‘IEEE_DBL_MANT_DIG’, and if not defined 53 bits). d1260 4 a1263 4 -- Function: int mpfr_ui_sub (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) d1278 5 a1282 6 Set ROP to OP1 - OP2 rounded in the direction RND. The IEEE-754 rules are used, in particular for signed zeros. But for types having no signed zeros, 0 is considered unsigned (i.e., (+0) − 0 = (+0), (−0) − 0 = (−0), 0 − (+0) = (−0) and 0 − (−0) = (+0)). The same restrictions than for ‘mpfr_add_d’ apply to ‘mpfr_d_sub’ and ‘mpfr_sub_d’. d1286 2 a1287 2 -- Function: int mpfr_mul_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) d1298 3 a1300 3 operands (for types having no signed zeros, 0 is considered positive). The same restrictions than for ‘mpfr_add_d’ apply to ‘mpfr_mul_d’. d1307 4 a1310 4 -- Function: int mpfr_ui_div (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) d1325 3 a1327 3 types having no signed zeros, 0 is considered positive). The same restrictions than for ‘mpfr_add_d’ apply to ‘mpfr_d_div’ and ‘mpfr_div_d’. d1332 3 a1334 3 Set ROP to the square root of OP rounded in the direction RND. Set ROP to −0 if OP is −0, to be consistent with the IEEE 754 standard. Set ROP to NaN if OP is negative. d1338 2 a1339 5 direction RND. Set ROP to +Inf if OP is ±0, +0 if OP is +Inf, and NaN if OP is negative. Warning! Therefore the result on −0 is different from the one of the rSqrt function recommended by the IEEE 754-2008 standard (Section 9.2.1), which is −Inf instead of +Inf. d1342 6 a1347 6 -- Function: int mpfr_root (mpfr_t ROP, mpfr_t OP, unsigned long int K, mpfr_rnd_t RND) Set ROP to the cubic root (resp. the Kth root) of OP rounded in the direction RND. For K odd (resp. even) and OP negative (including −Inf), set ROP to a negative number (resp. NaN). The Kth root of −0 is defined to be −0, whatever the parity of K. d1351 2 a1352 2 -- Function: int mpfr_pow_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) d1359 2 a1360 2 -- Function: int mpfr_ui_pow (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) d1363 2 a1364 2 754-2008 standards for the ‘pow’ function: • ‘pow(±0, Y)’ returns plus or minus infinity for Y a negative d1366 2 a1367 1 • ‘pow(±0, Y)’ returns plus infinity for Y negative and not an d1369 2 a1370 1 • ‘pow(±0, Y)’ returns plus or minus zero for Y a positive odd d1372 2 a1373 1 • ‘pow(±0, Y)’ returns plus zero for Y positive and not an odd d1375 8 a1382 4 • ‘pow(-1, ±Inf)’ returns 1. • ‘pow(+1, Y)’ returns 1 for any Y, even a NaN. • ‘pow(X, ±0)’ returns 1 for any X, even a NaN. • ‘pow(X, Y)’ returns NaN for finite negative X and finite d1384 2 a1385 1 • ‘pow(X, -Inf)’ returns plus infinity for 0 < abs(x) < 1, and d1387 2 a1388 1 • ‘pow(X, +Inf)’ returns plus zero for 0 < abs(x) < 1, and plus d1390 2 a1391 1 • ‘pow(-Inf, Y)’ returns minus zero for Y a negative odd d1393 5 a1397 1 • ‘pow(-Inf, Y)’ returns plus zero for Y negative and not an odd d1399 5 a1403 5 • ‘pow(-Inf, Y)’ returns minus infinity for Y a positive odd integer. • ‘pow(-Inf, Y)’ returns plus infinity for Y positive and not an odd integer. • ‘pow(+Inf, Y)’ returns plus zero for Y negative, and plus d1431 1 a1431 1 RND. Just decreases the exponent by OP2 when ROP and OP1 are d1448 4 a1451 4 Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1 < OP2. Both OP1 and OP2 are considered to their full own precision, which may differ. If one of the operands is NaN, set the _erange_ flag and return zero. d1455 1 a1455 1 recommended to use the predicate functions (e.g., ‘mpfr_equal_p’ d1457 1 a1457 1 comparisons, in particular when one or both arguments are NaN. But d1463 3 a1465 3 -- Function: int mpfr_cmp_si_2exp (mpfr_t OP1, long int OP2, mpfr_exp_t E) Compare OP1 and OP2 multiplied by two to the power E. Similar as d1469 4 a1472 3 Compare |OP1| and |OP2|. Return a positive value if |OP1| > |OP2|, zero if |OP1| = |OP2|, and a negative value if |OP1| < |OP2|. If one of the operands is NaN, set the _erange_ flag and return zero. d1481 1 a1481 1 number (i.e., neither NaN, nor an infinity nor zero). Return zero d1487 2 a1488 2 return zero. This is equivalent to ‘mpfr_cmp_ui (op, 0)’, but more efficient. d1496 2 a1497 2 OP1 = OP2 respectively, and zero otherwise. Those functions return zero whenever OP1 and/or OP2 is NaN. d1515 3 a1517 3 ‘mpfr_sin_cos’), return a *note ternary value::, i.e., zero for an exact return value, a positive value for a return value larger than the exact result, and a negative value otherwise. d1528 2 a1529 4 respectively, rounded in the direction RND. Set ROP to +0 if OP is 1 (in all rounding modes), for consistency with the ISO C99 and IEEE 754-2008 standards. Set ROP to −Inf if OP is ±0 (i.e., the sign of the zero has no influence on the result). d1534 1 a1534 1 Set ROP to the exponential of OP, to 2 power of OP or to 10 power d1548 4 a1551 4 both results are exact, more precisely it returns s+4c where s=0 if SOP is exact, s=1 if SOP is larger than the sine of OP, s=2 if SOP is smaller than the sine of OP, and similarly for c and the cosine of OP. d1563 7 a1569 7 in the direction RND. Note that since ‘acos(-1)’ returns the floating-point number closest to Pi according to the given rounding mode, this number might not be in the output range 0 <= ROP < \pi of the arc-cosine function; still, the result lies in the image of the output range by the rounding function. The same holds for ‘asin(-1)’, ‘asin(1)’, ‘atan(-Inf)’, ‘atan(+Inf)’ or for ‘atan(op)’ with large OP and small precision of ROP. d1571 2 a1572 2 -- Function: int mpfr_atan2 (mpfr_t ROP, mpfr_t Y, mpfr_t X, mpfr_rnd_t RND) d1574 3 a1576 3 RND: if ‘x > 0’, ‘atan2(y, x) = atan (y/x)’; if ‘x < 0’, ‘atan2(y, x) = sign(y)*(Pi - atan (abs(y/x)))’, thus a number from -Pi to Pi. As for ‘atan’, in case the exact mathematical result is +Pi or -Pi, d1579 42 a1620 23 ‘atan2(y, 0)’ does not raise any floating-point exception. Special values are handled as described in the ISO C99 and IEEE 754-2008 standards for the ‘atan2’ function: • ‘atan2(+0, -0)’ returns +Pi. • ‘atan2(-0, -0)’ returns -Pi. • ‘atan2(+0, +0)’ returns +0. • ‘atan2(-0, +0)’ returns −0. • ‘atan2(+0, x)’ returns +Pi for x < 0. • ‘atan2(-0, x)’ returns -Pi for x < 0. • ‘atan2(+0, x)’ returns +0 for x > 0. • ‘atan2(-0, x)’ returns −0 for x > 0. • ‘atan2(y, 0)’ returns -Pi/2 for y < 0. • ‘atan2(y, 0)’ returns +Pi/2 for y > 0. • ‘atan2(+Inf, -Inf)’ returns +3*Pi/4. • ‘atan2(-Inf, -Inf)’ returns -3*Pi/4. • ‘atan2(+Inf, +Inf)’ returns +Pi/4. • ‘atan2(-Inf, +Inf)’ returns -Pi/4. • ‘atan2(+Inf, x)’ returns +Pi/2 for finite x. • ‘atan2(-Inf, x)’ returns -Pi/2 for finite x. • ‘atan2(y, -Inf)’ returns +Pi for finite y > 0. • ‘atan2(y, -Inf)’ returns -Pi for finite y < 0. • ‘atan2(y, +Inf)’ returns +0 for finite y > 0. • ‘atan2(y, +Inf)’ returns −0 for finite y < 0. d1625 2 a1626 2 Set ROP to the hyperbolic cosine, sine or tangent of OP, rounded in the direction RND. d1633 3 a1635 2 variables. Return 0 iff both results are exact (see ‘mpfr_sin_cos’ for a more detailed description of the return value). d1662 8 a1669 8 Set ROP to the exponential integral of OP, rounded in the direction RND. For positive OP, the exponential integral is the sum of Euler’s constant, of the logarithm of OP, and of the sum for k from 1 to infinity of OP to the power k, divided by k and factorial(k). For negative OP, ROP is set to NaN (this definition for negative argument follows formula 5.1.2 from the Handbook of Mathematical Functions from Abramowitz and Stegun, a future version might use another definition). d1673 1 a1673 1 direction RND. MPFR defines the dilogarithm function as the d1678 1 a1678 1 direction RND. When OP is a negative integer, ROP is set to NaN. d1682 2 a1683 5 rounded in the direction RND. When OP is 1 or 2, set ROP to +0 (in all rounding modes). When OP is an infinity or a nonpositive integer, set ROP to +Inf, following the general rules on special values. When −2K−1 < OP < −2K, K being a nonnegative integer, set ROP to NaN. See also ‘mpfr_lgamma’. d1688 5 a1692 6 Gamma function on OP, rounded in the direction RND. The sign (1 or −1) of Gamma(OP) is returned in the object pointed to by SIGNP. When OP is 1 or 2, set ROP to +0 (in all rounding modes). When OP is an infinity or a nonpositive integer, set ROP to +Inf. When OP is NaN, −Inf or a negative integer, *SIGNP is undefined, and when OP is ±0, *SIGNP is the sign of the zero. d1700 4 a1703 4 -- Function: int mpfr_zeta_ui (mpfr_t ROP, unsigned long OP, mpfr_rnd_t RND) Set ROP to the value of the Riemann Zeta function on OP, rounded in the direction RND. d1715 4 a1718 4 (resp. 1 and N) on OP, rounded in the direction RND. When OP is NaN, ROP is always set to NaN. When OP is plus or minus Infinity, ROP is set to +0. When OP is zero, and N is not zero, ROP is set to +0 or −0 depending on the parity and sign of N, and the sign of d1726 3 a1728 3 (resp. 1 and N) on OP, rounded in the direction RND. When OP is NaN or negative, ROP is always set to NaN. When OP is +Inf, ROP is set to +0. When OP is zero, ROP is set to +Inf or −Inf depending d1736 1 a1736 4 rounded in the direction RND. Concerning special values (signed zeros, infinities, NaN), these functions behave like a multiplication followed by a separate addition or subtraction. That is, the fused operation matters only for rounding. d1740 6 a1745 5 Set ROP to the arithmetic-geometric mean of OP1 and OP2, rounded in the direction RND. The arithmetic-geometric mean is the common limit of the sequences U_N and V_N, where U_0=OP1, V_0=OP2, U_(N+1) is the arithmetic mean of U_N and V_N, and V_(N+1) is the geometric mean of U_N and V_N. If any operand is negative, set ROP to NaN. d1747 2 a1748 2 -- Function: int mpfr_hypot (mpfr_t ROP, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) d1751 3 a1753 4 Special values are handled as described in the ISO C99 (Section F.9.4.3) and IEEE 754-2008 (Section 9.2.1) standards: If X or Y is an infinity, then +Inf is returned in ROP, even if the other number is NaN. d1756 3 a1758 3 Set ROP to the value of the Airy function Ai on X, rounded in the direction RND. When X is NaN, ROP is always set to NaN. When X is +Inf or −Inf, ROP is +0. The current implementation is not d1760 1 a1760 1 typically smaller than 500. For larger arguments, other methods d1767 5 a1771 5 Set ROP to the logarithm of 2, the value of Pi, of Euler’s constant 0.577..., of Catalan’s constant 0.915..., respectively, rounded in the direction RND. These functions cache the computed values to avoid other calculations if a lower or equal precision is requested. To free these caches, use ‘mpfr_free_cache’. d1776 2 a1777 2 (‘mpfr_const_log2’, ‘mpfr_const_pi’, ‘mpfr_const_euler’ and ‘mpfr_const_catalan’). You should call this function before d1783 7 a1789 7 Set ROP to the sum of all elements of TAB, whose size is N, rounded in the direction RND. Warning: for efficiency reasons, TAB is an array of pointers to ‘mpfr_t’, not an array of ‘mpfr_t’. If the returned ‘int’ value is zero, ROP is guaranteed to be the exact sum; otherwise ROP might be smaller than, equal to, or larger than the exact sum (in accordance to the rounding mode). However, ‘mpfr_sum’ does guarantee the result is correctly rounded. d1799 2 a1800 2 null pointer for a ‘stream’ to any of these functions will make them read from ‘stdin’ and write to ‘stdout’, respectively. d1802 2 a1803 2 When using any of these functions, you must include the ‘’ standard header before ‘mpfr.h’, to allow ‘mpfr.h’ to define prototypes d1811 1 a1811 1 that OP can be read back exactly (see ‘mpfr_get_str’). d1815 2 a1816 2 exponent in base 10, in the form ‘eNNN’, are printed. If BASE is greater than 10, ‘@@’ will be used instead of ‘e’ as exponent d1828 2 a1829 2 between whitespace) and parses it using ‘mpfr_set_str’. See the documentation of ‘mpfr_strtofr’ for a detailed description of the d1843 2 a1844 2 The class of ‘mpfr_printf’ functions provides formatted output in a similar manner as the standard C ‘printf’. These functions are defined d1848 2 a1849 2 When using any of these functions, you must include the ‘’ standard header before ‘mpfr.h’, to allow ‘mpfr.h’ to define prototypes d1855 2 a1856 2 The format specification accepted by ‘mpfr_printf’ is an extension of the ‘printf’ one. The conversion specification is of the form: d1858 6 a1863 6 ‘flags’, ‘width’, and ‘precision’ have the same meaning as for the standard ‘printf’ (in particular, notice that the ‘precision’ is related to the number of digits displayed in the base chosen by ‘conv’ and not related to the internal precision of the ‘mpfr_t’ variable). ‘mpfr_printf’ accepts the same ‘type’ specifiers as GMP (except the non-standard and deprecated ‘q’, use ‘ll’ instead), namely the length d1866 42 a1907 42 ‘h’ ‘short’ ‘hh’ ‘char’ ‘j’ ‘intmax_t’ or ‘uintmax_t’ ‘l’ ‘long’ or ‘wchar_t’ ‘ll’ ‘long long’ ‘L’ ‘long double’ ‘t’ ‘ptrdiff_t’ ‘z’ ‘size_t’ and the ‘type’ specifiers defined in GMP plus ‘R’ and ‘P’ specific to MPFR (the second column in the table below shows the type of the argument read in the argument list and the kind of ‘conv’ specifier to use after the ‘type’ specifier): ‘F’ ‘mpf_t’, float conversions ‘Q’ ‘mpq_t’, integer conversions ‘M’ ‘mp_limb_t’, integer conversions ‘N’ ‘mp_limb_t’ array, integer conversions ‘Z’ ‘mpz_t’, integer conversions ‘P’ ‘mpfr_prec_t’, integer conversions ‘R’ ‘mpfr_t’, float conversions The ‘type’ specifiers have the same restrictions as those mentioned in the GMP documentation: *note (gmp.info)Formatted Output Strings::. In particular, the ‘type’ specifiers (except ‘R’ and ‘P’) are supported only if they are supported by ‘gmp_printf’ in your GMP build; this implies that the standard specifiers, such as ‘t’, must _also_ be supported by your C library if you want to use them. The ‘rounding’ field is specific to ‘mpfr_t’ arguments and should not be used with other types. With conversion specification not involving ‘P’ and ‘R’ types, ‘mpfr_printf’ behaves exactly as ‘gmp_printf’. The ‘P’ type specifies that a following ‘d’, ‘i’, ‘o’, ‘u’, ‘x’, or ‘X’ conversion specifier applies to a ‘mpfr_prec_t’ argument. It is needed because the ‘mpfr_prec_t’ type does not necessarily correspond to an ‘int’ or any fixed standard type. The ‘precision’ field specifies the minimum number of digits to appear. The default ‘precision’ is 1. For example: d1915 3 a1917 3 The ‘R’ type specifies that a following ‘a’, ‘A’, ‘b’, ‘e’, ‘E’, ‘f’, ‘F’, ‘g’, ‘G’, or ‘n’ conversion specifier applies to a ‘mpfr_t’ argument. The ‘R’ type can be followed by a ‘rounding’ specifier d1920 8 a1927 8 ‘U’ round toward plus infinity ‘D’ round toward minus infinity ‘Y’ round away from zero ‘Z’ round toward zero ‘N’ round to nearest (with ties to even) ‘*’ rounding mode indicated by the ‘mpfr_rnd_t’ argument just before the corresponding ‘mpfr_t’ variable. d1938 2 a1939 2 Note that the rounding away from zero mode is specified with ‘Y’ because ISO C reserves the ‘A’ specifier for hexadecimal output (see d1942 1 a1942 1 The output ‘conv’ specifiers allowed with ‘mpfr_t’ parameter are: d1944 9 a1952 9 ‘a’ ‘A’ hex float, C99 style ‘b’ binary output ‘e’ ‘E’ scientific format float ‘f’ ‘F’ fixed point float ‘g’ ‘G’ fixed or scientific float The conversion specifier ‘b’ which displays the argument in binary is specific to ‘mpfr_t’ arguments and should not be used with other types. Other conversion specifiers have the same meaning as for a ‘double’ d1957 24 a1980 24 values are always displayed as ‘nan’, ‘-inf’, and ‘inf’ for ‘a’, ‘b’, ‘e’, ‘f’, and ‘g’ specifiers and ‘NAN’, ‘-INF’, and ‘INF’ for ‘A’, ‘E’, ‘F’, and ‘G’ specifiers. If the ‘precision’ field is not empty, the ‘mpfr_t’ number is rounded to the given precision in the direction specified by the rounding mode. If the precision is zero with rounding to nearest mode and one of the following ‘conv’ specifiers: ‘a’, ‘A’, ‘b’, ‘e’, ‘E’, tie case is rounded to even when it lies between two consecutive values at the wanted precision which have the same exponent, otherwise, it is rounded away from zero. For instance, 85 is displayed as "8e+1" and 95 is displayed as "1e+2" with the format specification ‘"%.0RNe"’. This also applies when the ‘g’ (resp. ‘G’) conversion specifier uses the ‘e’ (resp. ‘E’) style. If the precision is set to a value greater than the maximum value for an ‘int’, it will be silently reduced down to ‘INT_MAX’. If the ‘precision’ field is empty (as in ‘%Re’ or ‘%.RE’) with ‘conv’ specifier ‘e’ and ‘E’, the number is displayed with enough digits so that it can be read back exactly, assuming that the input and output variables have the same precision and that the input and output rounding modes are both rounding to nearest (as for ‘mpfr_get_str’). The default precision for an empty ‘precision’ field with ‘conv’ specifiers ‘f’, ‘F’, ‘g’, and ‘G’ is 6. d1985 5 a1989 5 For all the following functions, if the number of characters which ought to be written appears to exceed the maximum limit for an ‘int’, nothing is written in the stream (resp. to ‘stdout’, to BUF, to STR), the function returns −1, sets the _erange_ flag, and (in POSIX system only) ‘errno’ is set to ‘EOVERFLOW’. d1994 3 a1996 3 Print to the stream STREAM the optional arguments under the control of the template string TEMPLATE. Return the number of characters written or a negative value if an error occurred. d2000 1 a2000 1 Print to ‘stdout’ the optional arguments under the control of the d2009 1 a2009 1 print it in BUF. No overlap is permitted between BUF and the other d2020 2 a2021 2 print it in BUF. If N is zero, nothing is written and BUF may be a null pointer, otherwise, the N−1 first characters are written in d2024 2 a2025 2 large, _not counting_ the terminating null character, or a negative value if an error occurred. d2030 6 a2035 6 Write their output as a null terminated string in a block of memory allocated using the current allocation function. A pointer to the block is stored in STR. The block of memory must be freed using ‘mpfr_free_str’. The return value is the number of characters written in the string, excluding the null-terminator, or a negative value if an error occurred. d2048 1 a2048 1 Set ROP to OP rounded to an integer. ‘mpfr_rint’ rounds to the d2050 3 a2052 3 ‘mpfr_ceil’ rounds to the next higher or equal representable integer, ‘mpfr_floor’ to the next lower or equal representable integer, ‘mpfr_round’ to the nearest representable integer, d2054 1 a2054 1 mode of IEEE 754-2008), and ‘mpfr_trunc’ to the next representable d2060 2 a2061 2 integer representable in ROP, 1 or −1 when OP is an integer that is not representable in ROP, 2 or −2 when OP is not an integer. d2063 7 a2069 16 When OP is NaN, the NaN flag is set as usual. In the other cases, the inexact flag is set when ROP differs from OP, following the ISO C99 rule for the ‘rint’ function. If you want the behavior to be more like IEEE 754 / ISO TS 18661-1, i.e., the usual behavior where the round-to-integer function is regarded as any other mathematical function, you should use one the ‘mpfr_rint_*’ functions instead (however it is not possible to round to nearest with the even rounding rule yet). Note that ‘mpfr_round’ is different from ‘mpfr_rint’ called with the rounding to nearest mode (where halfway cases are rounded to an even integer or significand). Note also that no double rounding is performed; for instance, 10.5 (1010.1 in binary) is rounded by ‘mpfr_rint’ with rounding to nearest to 12 (1100 in binary) in 2-bit precision, because the two enclosing numbers representable on two bits are 8 and 12, and the closest is 12. (If one first d2081 16 a2096 22 Set ROP to OP rounded to an integer. ‘mpfr_rint_ceil’ rounds to the next higher or equal integer, ‘mpfr_rint_floor’ to the next lower or equal integer, ‘mpfr_rint_round’ to the nearest integer, rounding halfway cases away from zero, and ‘mpfr_rint_trunc’ to the next integer toward zero. If the result is not representable, it is rounded in the direction RND. The returned value is the ternary value associated with the considered round-to-integer function (regarded in the same way as any other mathematical function). Contrary to ‘mpfr_rint’, those functions do perform a double rounding: first OP is rounded to the nearest integer in the direction given by the function name, then this nearest integer (if not representable) is rounded in the given direction RND. Thus these round-to-integer functions behave more like the other mathematical functions, i.e., the returned result is the correct rounding of the exact result of the function in the real numbers. For example, ‘mpfr_rint_round’ with rounding to nearest and a precision of two bits rounds 6.5 to 7 (halfway cases away from zero), then 7 is rounded to 8 by the round-even rule, despite the fact that 6 is also representable on two bits, and is closer to 6.5 than 8. d2100 1 a2100 1 rounded in the direction RND (unlike in ‘mpfr_rint’, RND affects d2109 3 a2111 3 ‘mpfr_trunc(IOP, OP, RND)’ and ‘mpfr_frac(FOP, OP, RND)’). The variables IOP and FOP must be different. Return 0 iff both results are exact (see ‘mpfr_sin_cos’ for a more detailed description of d2122 3 a2124 3 follows: N is rounded toward zero for ‘mpfr_fmod’, and to the nearest integer (ties rounded to even) for ‘mpfr_remainder’ and ‘mpfr_remquo’. d2127 12 a2138 12 ISO C99 standard: If X is infinite or Y is zero, R is NaN. If Y is infinite and X is finite, R is X rounded to the precision of R. If R is zero, it has the sign of X. The return value is the ternary value corresponding to R. Additionally, ‘mpfr_remquo’ stores the low significant bits from the quotient N in *Q (more precisely the number of bits in a ‘long’ minus one), with the sign of X divided by Y (except if those low bits are all zero, in which case zero is returned). Note that X may be so large in magnitude relative to Y that an exact representation of the quotient is not practical. The ‘mpfr_remainder’ and ‘mpfr_remquo’ functions are useful for d2151 2 a2152 2 Set the default rounding mode to RND. The default rounding mode is to nearest initially. d2160 1 a2160 1 integer between ‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’ (otherwise the d2163 3 a2165 3 and it is filled with zeros. Otherwise, the significand is rounded to precision PREC with the given direction. In both cases, the precision of X is changed to PREC. d2167 2 a2168 2 Here is an example of how to use ‘mpfr_prec_round’ to implement Newton’s algorithm to compute the inverse of A, assuming X is a2178 4 Warning! You must not use this function if X was initialized with ‘MPFR_DECL_INIT’ or with ‘mpfr_custom_init_set’ (*note Custom Interface::). d2184 7 a2190 7 to round correctly X to precision PREC with the direction RND2, and 0 otherwise (including for NaN and Inf). This function *does not modify* its arguments. If RND1 is ‘MPFR_RNDN’, then the sign of the error is unknown, but its absolute value is the same, so that the possible range is twice as large as with a directed rounding for RND1. d2193 8 a2200 10 value:: when rounding B to precision PREC with rounding mode RND, a useful trick is the following: if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ, prec + (rnd == MPFR_RNDN))) ... Indeed, if RND is ‘MPFR_RNDN’, this will check if one can round to PREC+1 bits with a directed rounding: if so, one can surely round to nearest to PREC bits, and in addition one can determine the correct ternary value, which would not be the case when B is near from a value exactly representable on PREC bits. d2203 3 a2205 3 Return the minimal number of bits required to store the significand of X, and 0 for special values, including 0. (Warning: the returned value can be less than ‘MPFR_PREC_MIN’.) d2221 7 a2227 7 If X or Y is NaN, set X to NaN. If X and Y are equal, X is unchanged. Otherwise, if X is different from Y, replace X by the next floating-point number (with the precision of X and the current exponent range) in the direction of Y (the infinite values are seen as the smallest and largest floating-point numbers). If the result is zero, it keeps the same sign. No underflow or overflow is generated. d2231 1 a2231 1 Equivalent to ‘mpfr_nexttoward’ where Y is plus infinity (resp. d2238 4 a2241 4 Set ROP to the minimum (resp. maximum) of OP1 and OP2. If OP1 and OP2 are both NaN, then ROP is set to NaN. If OP1 or OP2 is NaN, then ROP is set to the numeric value. If OP1 and OP2 are zeros of different signs, then ROP is set to −0 (resp. +0). d2245 1 a2245 1 ROP < 1. More precisely, the number can be seen as a float with a d2251 6 a2256 6 Return 0, unless the exponent is not in the current exponent range, in which case ROP is set to NaN and a non-zero value is returned (this should never happen in practice, except in very specific cases). The second argument is a ‘gmp_randstate_t’ structure which should be created using the GMP ‘gmp_randinit’ function (see the GMP manual). d2269 2 a2270 2 The second argument is a ‘gmp_randstate_t’ structure which should be created using the GMP ‘gmp_randinit’ function (see the GMP d2273 1 a2273 1 Note: the note for ‘mpfr_urandomb’ holds too. In addition, the d2280 1 a2280 1 distribution. If ROP2 is a null pointer, then only one value is d2287 2 a2288 2 The third argument is a ‘gmp_randstate_t’ structure, which should be created using the GMP ‘gmp_randinit’ function (see the GMP d2292 1 a2292 1 ‘mpfr_sin_cos’. If ROP2 is a null pointer, the second ternary d2295 1 a2295 1 that return only one result). Otherwise the ternary value of a d2298 1 a2298 1 Note: the note for ‘mpfr_urandomb’ holds too. In addition, the d2304 1 a2304 1 number and the significand is considered in [1/2,1). The behavior d2315 1 a2315 1 negative, −0, or a NaN whose representation has its sign bit set). d2317 2 a2318 2 -- Function: int mpfr_setsign (mpfr_t ROP, mpfr_t OP, int S, mpfr_rnd_t RND) d2327 2 a2328 2 a NaN). This function is equivalent to ‘mpfr_setsign (ROP, OP1, mpfr_signbit (OP2), RND)’. d2338 3 a2340 3 ‘MPFR_VERSION’ is the version of MPFR as a preprocessing constant. ‘MPFR_VERSION_MAJOR’, ‘MPFR_VERSION_MINOR’ and ‘MPFR_VERSION_PATCHLEVEL’ are respectively the major, minor and d2342 1 a2342 1 ‘MPFR_VERSION_STRING’ is the version (with an optional suffix, used d2344 1 a2344 1 which can be compared to the result of ‘mpfr_get_version’ to check d2350 2 a2351 2 dynamically linked with a newer MPFR library version (if allowed by the library versioning system). d2354 3 a2356 3 Create an integer in the same format as used by ‘MPFR_VERSION’ from the given MAJOR, MINOR and PATCHLEVEL. Here is an example of how to check the MPFR version at compile time: d2363 1 a2363 1 applied to the MPFR library (contents of the ‘PATCHES’ file), d2373 1 a2373 1 the ‘--enable-thread-safe’ configure option, see ‘INSTALL’ file), d2378 1 a2378 1 support (that is, MPFR was built with the ‘--enable-decimal-float’ d2383 2 a2384 2 (that is, MPFR was built with either ‘--with-gmp-build’ or ‘--enable-gmp-internals’ configure option), return zero otherwise. d2400 1 a2400 1 floating-point variable. The smallest positive value of a d2408 9 a2416 9 Set the smallest and largest exponents allowed for a floating-point variable. Return a non-zero value when EXP is not in the range accepted by the implementation (in that case the smallest or largest exponent is not changed), and zero otherwise. If the user changes the exponent range, it is her/his responsibility to check that all current floating-point variables are in the new allowed range (for example using ‘mpfr_check_range’), otherwise the subsequent behavior will be undefined, in the sense of the ISO C standard. d2423 4 a2426 4 ‘mpfr_set_emin’ and ‘mpfr_set_emax’ respectively. These values are implementation dependent, thus a program using ‘mpfr_set_emax(mpfr_get_emax_max())’ or ‘mpfr_set_emin(mpfr_get_emin_min())’ may not be portable. d2430 3 a2432 3 real value Y in the direction RND and some extended exponent range, and that T is the corresponding *note ternary value::. For example, one performed ‘t = mpfr_log (x, u, rnd)’, and Y is the d2438 1 a2438 1 may be used to avoid a double rounding. This function returns zero d2447 1 a2447 1 is set. This is useful because ‘mpfr_check_range’ is typically d2455 7 a2461 7 ‘EXP(x)-emin+1’ according to rounding mode RND and previous ternary value T, avoiding double rounding problems. More precisely in the subnormal domain, denoting by E the value of ‘emin’, X is rounded in fixed-point arithmetic to an integer multiple of two to the power E−1; as a consequence, 1.5 multiplied by two to the power E−1 when T is zero is rounded to two to the power E with rounding to nearest. d2463 1 a2463 1 ‘PREC(x)’ is not modified by this function. RND and T must be the d2465 6 a2470 6 (as in ‘mpfr_check_range’). The subnormal exponent range is from ‘emin’ to ‘emin+PREC(x)-1’. If the result cannot be represented in the current exponent range (due to a too small ‘emax’), the behavior is undefined. Note that unlike most functions, the result is compared to the exact one, not the input value X, i.e., the ternary value is propagated. d2507 2 a2508 2 Clear the underflow, overflow, divide-by-zero, invalid, inexact and _erange_ flags. d2530 2 a2531 2 invalid, inexact, _erange_) flag, which is non-zero iff the flag is set. d2539 15 a2553 15 A header file ‘mpf2mpfr.h’ is included in the distribution of MPFR for compatibility with the GNU MP class MPF. By inserting the following two lines after the ‘#include ’ line, #include #include any program written for MPF can be compiled directly with MPFR without any changes (except the ‘gmp_printf’ functions will not work for arguments of type ‘mpfr_t’). All operations are then performed with the default MPFR rounding mode, which can be reset with ‘mpfr_set_default_rounding_mode’. Warning: the ‘mpf_init’ and ‘mpf_init2’ functions initialize to zero, whereas the corresponding MPFR functions initialize to NaN: this is useful to detect uninitialized values, but is slightly incompatible with MPF. d2557 3 a2559 3 difference with ‘mpfr_set_prec’ is that PREC is assumed to be small enough so that the significand fits into the current allocated memory space for X. Otherwise the behavior is undefined. d2565 1 a2565 1 both infinities of the same sign. Return zero otherwise. This d2568 3 a2570 3 whether two numbers are close to each other; for instance, 1.011111 and 1.100000 are regarded as different for any value of OP3 larger than 1. d2577 2 a2578 2 |OP1-OP2|/OP1, using the precision of ROP and the rounding mode RND for all operations. d2584 4 a2587 3 These functions are identical to ‘mpfr_mul_2ui’ and ‘mpfr_div_2ui’ respectively. These functions are only kept for compatibility with MPF, one should prefer ‘mpfr_mul_2ui’ and ‘mpfr_div_2ui’ otherwise. d2596 1 a2596 1 However, the MPFR memory design is not well suited for such a thing. So d2601 1 a2601 2 • Either directly store a floating-point number as a ‘mpfr_t’ on the d2604 2 a2605 3 • Either store its own representation on the stack and construct a new temporary ‘mpfr_t’ each time it is needed. d2611 2 a2612 2 efficiency reasons: for example ‘mpfr_custom_init (s, p)’ uses the macro, while ‘(mpfr_custom_init) (s, p)’ uses the function. d2615 1 a2615 1 numbers using ‘mpfr_init’ and similar functions. See Custom Allocation d2618 3 a2620 3 Note 2: MPFR functions may use the cached functions (‘mpfr_const_pi’ for example), even if they are not explicitly called. You have to call ‘mpfr_free_cache’ each time you garbage the memory iff ‘mpfr_init’, d2631 2 a2632 2 be an area of ‘mpfr_custom_get_size (prec)’ bytes at least and be suitably aligned for an array of ‘mp_limb_t’ (GMP type, *note d2637 11 a2647 8 Perform a dummy initialization of a ‘mpfr_t’ and set it to: • if ‘ABS(kind) == MPFR_NAN_KIND’, X is set to NaN; • if ‘ABS(kind) == MPFR_INF_KIND’, X is set to the infinity of sign ‘sign(kind)’; • if ‘ABS(kind) == MPFR_ZERO_KIND’, X is set to the zero of sign ‘sign(kind)’; • if ‘ABS(kind) == MPFR_REGULAR_KIND’, X is set to a regular number: ‘x = sign(kind)*significand*2^exp’. d2649 1 a2649 1 involving X. It will not allocate anything. A floating-point d2651 3 a2653 3 ‘mpfr_set_prec’ or ‘mpfr_prec_round’, or cleared using ‘mpfr_clear’! The SIGNIFICAND must have been initialized with ‘mpfr_custom_init’ using the same precision PREC. d2656 3 a2658 3 Return the current kind of a ‘mpfr_t’ as created by ‘mpfr_custom_init_set’. The behavior of this function for any ‘mpfr_t’ not initialized with ‘mpfr_custom_init_set’ is undefined. d2661 4 a2664 3 Return a pointer to the significand used by a ‘mpfr_t’ initialized with ‘mpfr_custom_init_set’. The behavior of this function for any ‘mpfr_t’ not initialized with ‘mpfr_custom_init_set’ is undefined. d2668 1 a2668 1 number. The return value for NaN, Infinity or zero is unspecified d2670 1 a2670 1 any ‘mpfr_t’ not initialized with ‘mpfr_custom_init_set’ is d2675 4 a2678 4 collect and update its new position to ‘new_position’. However the application has to move the significand and the ‘mpfr_t’ itself. The behavior of this function for any ‘mpfr_t’ not initialized with ‘mpfr_custom_init_set’ is undefined. d2686 3 a2688 3 A “limb” means the part of a multi-precision number that fits in a single word. Usually a limb contains 32 or 64 bits. The C data type for a limb is ‘mp_limb_t’. d2690 3 a2692 3 The ‘mpfr_t’ type is internally defined as a one-element array of a structure, and ‘mpfr_ptr’ is the C data type representing a pointer to this structure. The ‘mpfr_t’ type consists of four fields: d2694 2 a2695 2 • The ‘_mpfr_prec’ field is used to store the precision of the variable (in bits); this is not less than ‘MPFR_PREC_MIN’. d2697 1 a2697 1 • The ‘_mpfr_sign’ field is used to store the sign of the variable. d2699 1 a2699 1 • The ‘_mpfr_exp’ field stores the exponent. An exponent of 0 means d2705 1 a2705 1 • Finally, the ‘_mpfr_d’ field is a pointer to the limbs, least d2707 2 a2708 2 controlled by ‘_mpfr_prec’, namely ceil(‘_mpfr_prec’/‘mp_bits_per_limb’). Non-singular (i.e., d2714 1 d2727 5 a2731 4 patchlevel (the third number in the MPFR version) will be ignored in the following. If a program does not use MPFR internals, changes in the behavior between two versions differing only by the patchlevel should only result from what was regarded as a bug or unspecified behavior. d2736 2 a2737 2 such a case, a failure should occur during compilation or linking. If a result becomes incorrect because of such a change, please look at the d2739 4 a2742 3 unaffected), at the FAQ and at the MPFR web page for your version (a bug could have been introduced and be already fixed); and if the problem is not mentioned, please send us a bug report (*note Reporting Bugs::). d2746 2 a2747 2 versions of MPFR. This section should help developers to write portable code. d2749 2 a2750 2 Note: Information given here may be incomplete. API changes are also described in the NEWS file (for each version, instead of being d2767 8 a2774 8 The official type for exponent values changed from ‘mp_exp_t’ to ‘mpfr_exp_t’ in MPFR 3.0. The type ‘mp_exp_t’ will remain available as it comes from GMP (with a different meaning). These types are currently the same (‘mpfr_exp_t’ is defined as ‘mp_exp_t’ with ‘typedef’), so that programs can still use ‘mp_exp_t’; but this may change in the future. Alternatively, using the following code after including ‘mpfr.h’ will work with official MPFR versions, as ‘mpfr_exp_t’ was never defined in MPFR 2.x: d2780 4 a2783 4 respectively changed from ‘mp_prec_t’ and ‘mp_rnd_t’ to ‘mpfr_prec_t’ and ‘mpfr_rnd_t’ in MPFR 3.0. This change was actually done a long time ago in MPFR, at least since MPFR 2.2.0, with the following code in ‘mpfr.h’: d2791 11 a2801 11 ‘mpfr_prec_t’ and ‘mpfr_rnd_t’ in your programs. The types ‘mp_prec_t’ and ‘mp_rnd_t’ (defined in MPFR only) may be removed in the future, as the prefix ‘mp_’ is reserved by GMP. The precision type ‘mpfr_prec_t’ (‘mp_prec_t’) was unsigned before MPFR 3.0; it is now signed. ‘MPFR_PREC_MAX’ has not changed, though. Indeed the MPFR code requires that ‘MPFR_PREC_MAX’ be representable in the exponent type, which may have the same size as ‘mpfr_prec_t’ but has always been signed. The consequence is that valid code that does not assume anything about the signedness of ‘mpfr_prec_t’ should work with past and new MPFR versions. This change was useful as the use of d2805 1 a2805 1 Warning! A program assuming (intentionally or not) that ‘mpfr_prec_t’ d2809 2 a2810 2 The rounding modes ‘GMP_RNDx’ were renamed to ‘MPFR_RNDx’ in MPFR 3.0. However the old names ‘GMP_RNDx’ have been kept for compatibility d2816 2 a2817 2 The rounding mode “round away from zero” (‘MPFR_RNDA’) was added in MPFR 3.0 (however no rounding mode ‘GMP_RNDA’ exists). d2828 1 a2828 1 • ‘mpfr_add_d’ in MPFR 2.4. d2830 1 a2830 1 • ‘mpfr_ai’ in MPFR 3.0 (incomplete, experimental). d2832 1 a2832 1 • ‘mpfr_asprintf’ in MPFR 2.4. d2834 1 a2834 1 • ‘mpfr_buildopt_decimal_p’ and ‘mpfr_buildopt_tls_p’ in MPFR 3.0. d2836 1 a2836 1 • ‘mpfr_buildopt_gmpinternals_p’ and ‘mpfr_buildopt_tune_case’ in d2839 1 a2839 1 • ‘mpfr_clear_divby0’ in MPFR 3.1 (new divide-by-zero exception). d2841 1 a2841 1 • ‘mpfr_copysign’ in MPFR 2.3. Note: MPFR 2.2 had a ‘mpfr_copysign’ d2846 4 a2849 4 • ‘mpfr_custom_get_significand’ in MPFR 3.0. This function was named ‘mpfr_custom_get_mantissa’ in previous versions; ‘mpfr_custom_get_mantissa’ is still available via a macro in ‘mpfr.h’: d2852 1 a2852 1 use ‘mpfr_custom_get_mantissa’. d2854 1 a2854 1 • ‘mpfr_d_div’ and ‘mpfr_d_sub’ in MPFR 2.4. d2856 1 a2856 1 • ‘mpfr_digamma’ in MPFR 3.0. d2858 1 a2858 1 • ‘mpfr_divby0_p’ in MPFR 3.1 (new divide-by-zero exception). d2860 1 a2860 1 • ‘mpfr_div_d’ in MPFR 2.4. d2862 1 a2862 1 • ‘mpfr_fmod’ in MPFR 2.4. d2864 1 a2864 1 • ‘mpfr_fms’ in MPFR 2.3. d2866 1 a2866 1 • ‘mpfr_fprintf’ in MPFR 2.4. d2868 1 a2868 1 • ‘mpfr_frexp’ in MPFR 3.1. d2870 1 a2870 1 • ‘mpfr_get_flt’ in MPFR 3.0. d2872 1 a2872 1 • ‘mpfr_get_patches’ in MPFR 2.3. d2874 3 a2876 3 • ‘mpfr_get_z_2exp’ in MPFR 3.0. This function was named ‘mpfr_get_z_exp’ in previous versions; ‘mpfr_get_z_exp’ is still available via a macro in ‘mpfr.h’: d2879 1 a2879 1 use ‘mpfr_get_z_exp’. d2881 1 a2881 1 • ‘mpfr_grandom’ in MPFR 3.1. d2883 1 a2883 1 • ‘mpfr_j0’, ‘mpfr_j1’ and ‘mpfr_jn’ in MPFR 2.3. d2885 1 a2885 1 • ‘mpfr_lgamma’ in MPFR 2.3. d2887 1 a2887 1 • ‘mpfr_li2’ in MPFR 2.4. d2889 1 a2889 1 • ‘mpfr_min_prec’ in MPFR 3.0. d2891 1 a2891 1 • ‘mpfr_modf’ in MPFR 2.4. d2893 1 a2893 1 • ‘mpfr_mul_d’ in MPFR 2.4. d2895 1 a2895 1 • ‘mpfr_printf’ in MPFR 2.4. d2897 1 a2897 1 • ‘mpfr_rec_sqrt’ in MPFR 2.4. d2899 1 a2899 1 • ‘mpfr_regular_p’ in MPFR 3.0. d2901 1 a2901 1 • ‘mpfr_remainder’ and ‘mpfr_remquo’ in MPFR 2.3. d2903 1 a2903 1 • ‘mpfr_set_divby0’ in MPFR 3.1 (new divide-by-zero exception). d2905 1 a2905 1 • ‘mpfr_set_flt’ in MPFR 3.0. d2907 1 a2907 1 • ‘mpfr_set_z_2exp’ in MPFR 3.0. d2909 1 a2909 1 • ‘mpfr_set_zero’ in MPFR 3.0. d2911 1 a2911 1 • ‘mpfr_setsign’ in MPFR 2.3. d2913 1 a2913 1 • ‘mpfr_signbit’ in MPFR 2.3. d2915 1 a2915 1 • ‘mpfr_sinh_cosh’ in MPFR 2.4. d2917 1 a2917 1 • ‘mpfr_snprintf’ and ‘mpfr_sprintf’ in MPFR 2.4. d2919 1 a2919 1 • ‘mpfr_sub_d’ in MPFR 2.4. d2921 1 a2921 1 • ‘mpfr_urandom’ in MPFR 3.0. d2923 2 a2924 2 • ‘mpfr_vasprintf’, ‘mpfr_vfprintf’, ‘mpfr_vprintf’, ‘mpfr_vsprintf’ and ‘mpfr_vsnprintf’ in MPFR 2.4. d2926 3 a2928 1 • ‘mpfr_y0’, ‘mpfr_y1’ and ‘mpfr_yn’ in MPFR 2.3. a2929 1 • ‘mpfr_z_sub’ in MPFR 3.1. d2937 1 a2937 1 The following functions have changed after MPFR 2.2. Changes can affect d2941 4 a2944 4 • ‘mpfr_check_range’ changed in MPFR 2.3.2 and MPFR 2.4. If the value is an inexact infinity, the overflow flag is now set (in case it was lost), while it was previously left unchanged. This is really what is expected in practice (and what the MPFR code was d2948 3 a2950 3 • ‘mpfr_get_f’ changed in MPFR 3.0. This function was returning zero, except for NaN and Inf, which do not exist in MPF. The _erange_ flag is now set in these cases, and ‘mpfr_get_f’ now d2953 1 a2953 1 • ‘mpfr_get_si’, ‘mpfr_get_sj’, ‘mpfr_get_ui’ and ‘mpfr_get_uj’ d2957 2 a2958 2 • ‘mpfr_get_z’ changed in MPFR 3.0. The return type was ‘void’; it is now ‘int’, and the usual ternary value is returned. Thus d2960 1 a2960 1 the return value. Even in this case, C code using ‘mpfr_get_z’ as d2962 1 a2962 1 affected. For instance, the following is correct with MPFR 3.0, d2968 2 a2969 2 Portable code should cast ‘mpfr_get_z(...)’ to ‘void’ to use the type ‘void’ for both terms of the conditional operator, as in: d2971 2 a2972 2 Alternatively, ‘if ... else’ can be used instead of the conditional operator. d2977 1 a2977 1 • ‘mpfr_get_z_exp’ changed in MPFR 3.0. In previous MPFR versions, d2979 2 a2980 2 this function has been renamed to ‘mpfr_get_z_2exp’ in MPFR 3.0, but ‘mpfr_get_z_exp’ is still available for compatibility reasons. d2982 1 a2982 1 • ‘mpfr_strtofr’ changed in MPFR 2.3.1 and MPFR 2.4. This was d2986 1 a2986 1 The binary exponent is now accepted even without the ‘0b’ or ‘0x’ d2990 1 a2990 1 • ‘mpfr_strtofr’ changed in MPFR 3.0. This function now accepts d2997 6 a3002 6 • ‘mpfr_subnormalize’ changed in MPFR 3.1. This was actually regarded as a bug fix. The ‘mpfr_subnormalize’ implementation up to MPFR 3.0.0 did not change the flags. In particular, it did not follow the generic rule concerning the inexact flag (and no special behavior was specified). The case of the underflow flag was more a lack of specification. d3004 1 a3004 1 • ‘mpfr_urandom’ and ‘mpfr_urandomb’ changed in MPFR 3.1. Their d3006 2 a3007 2 true for GMP’s random generator, which is not the case between GMP 4.1 and 4.2 if ‘gmp_randinit_default’ is used). As a consequence, d3009 2 a3010 2 MPFR versions. Note: as the reproducibility of these functions was not specified before MPFR 3.1, the MPFR 3.1 behavior is _not_ d3013 1 d3020 4 a3023 4 Functions ‘mpfr_random’ and ‘mpfr_random2’ have been removed in MPFR 3.0 (this only affects old code built against MPFR 3.0 or later). (The function ‘mpfr_random’ had been deprecated since at least MPFR 2.2.0, and ‘mpfr_random2’ since MPFR 2.4.0.) d3031 13 a3043 13 For users of a C++ compiler, the way how the availability of ‘intmax_t’ is detected has changed in MPFR 3.0. In MPFR 2.x, if a macro ‘INTMAX_C’ or ‘UINTMAX_C’ was defined (e.g. when the ‘__STDC_CONSTANT_MACROS’ macro had been defined before ‘’ or ‘’ has been included), ‘intmax_t’ was assumed to be defined. However this was not always the case (more precisely, ‘intmax_t’ can be defined only in the namespace ‘std’, as with Boost), so that compilations could fail. Thus the check for ‘INTMAX_C’ or ‘UINTMAX_C’ is now disabled for C++ compilers, with the following consequences: • Programs written for MPFR 2.x that need ‘intmax_t’ may no longer be compiled against MPFR 3.0: a ‘#define MPFR_USE_INTMAX_T’ may be necessary before ‘mpfr.h’ is included. d3045 1 a3045 1 • The compilation of programs that work with MPFR 3.0 may fail with d3047 1 a3047 1 possible, such as defining ‘intmax_t’ and ‘uintmax_t’ in the global a3049 3 The divide-by-zero exception is new in MPFR 3.1. However it should not introduce incompatible changes for programs that strictly follow the MPFR API since the exception can only be seen via new functions. d3051 5 a3055 1 As of MPFR 3.1, the ‘mpfr.h’ header can be included several times, d3069 15 a3083 15 ‘mpfr_agm’ and ‘mpfr_log’. Sylvain Chevillard contributed the ‘mpfr_ai’ function. David Daney contributed the hyperbolic and inverse hyperbolic functions, the base-2 exponential, and the factorial function. Alain Delplanque contributed the new version of the ‘mpfr_get_str’ function. Mathieu Dutour contributed the functions ‘mpfr_acos’, ‘mpfr_asin’ and ‘mpfr_atan’, and a previous version of ‘mpfr_gamma’. Laurent Fousse contributed the ‘mpfr_sum’ function. Emmanuel Jeandel, from ENS-Lyon too, contributed the generic hypergeometric code, as well as the internal function ‘mpfr_exp3’, a first implementation of the sine and cosine, and improved versions of ‘mpfr_const_log2’ and ‘mpfr_const_pi’. Ludovic Meunier helped in the design of the ‘mpfr_erf’ code. Jean-Luc Rémy contributed the ‘mpfr_zeta’ code. Fabrice Rouillier contributed the ‘mpfr_xxx_z’ and ‘mpfr_xxx_q’ functions, and helped to the Microsoft Windows porting. Damien Stehlé contributed the ‘mpfr_get_ld_2exp’ function. d3086 5 a3090 5 for very fruitful discussions at the beginning of that project, Torbjörn Granlund and Kevin Ryde for their help about design issues, and Nathalie Revol for her careful reading of a previous version of this documentation. In particular Kevin Ryde did a tremendous job for the portability of MPFR in 2002-2004. d3094 11 a3104 11 France) and LIP (Lyon, France) laboratories. In particular the main authors were or are members of the PolKA, Spaces, Cacao, Caramel and Caramba project-teams at LORIA and of the Arénaire and AriC project-teams at LIP. This project was started during the Fiable (reliable in French) action supported by INRIA, and continued during the AOC action. The development of MPFR was also supported by a grant (202F0659 00 MPN 121) from the Conseil Régional de Lorraine in 2002, from INRIA by an "associate engineer" grant (2003-2005), an "opération de développement logiciel" grant (2007-2009), and the post-doctoral grant of Sylvain Chevillard in 2009-2010. The MPFR-MPC workshop in June 2012 was partly supported by the ERC grant ANTICS of Andreas Enge. d3112 1 a3112 1 • Richard Brent and Paul Zimmermann, "Modern Computer Arithmetic", d3114 1 a3114 1 authors’ web pages. d3116 1 a3116 1 • Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick d3120 1 a3120 1 2007, . d3122 2 a3123 2 • Torbjörn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic Library", version 5.0.1, 2010, . d3125 4 a3128 4 • IEEE standard for binary floating-point arithmetic, Technical Report ANSI-IEEE Standard 754-1985, New York, 1985. Approved March 21, 1985: IEEE Standards Board; approved July 26, 1985: American National Standards Institute, 18 pages. d3130 1 a3130 1 • IEEE Standard for Floating-Point Arithmetic, ANSI-IEEE Standard d3134 1 a3134 1 • Donald E. Knuth, "The Art of Computer Programming", vol 2, d3137 1 a3137 1 • Jean-Michel Muller, "Elementary Functions, Algorithms and d3140 1 a3140 1 • Jean-Michel Muller, Nicolas Brisebarre, Florent de Dinechin, d3145 1 d3154 1 a3154 1 Copyright © 2000,2001,2002 Free Software Foundation, Inc. d3163 1 a3163 1 functional and useful document “free” in the sense of freedom: to d3170 1 a3170 1 This License is a kind of “copyleft”, which means that derivative d3180 2 a3181 2 of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is d3187 2 a3188 2 that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice d3191 4 a3194 4 “Document”, below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as “you”. You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law. d3196 1 a3196 1 A “Modified Version” of the Document means any work containing the d3200 1 a3200 1 A “Secondary Section” is a named appendix or a front-matter section d3202 1 a3202 1 publishers or authors of the Document to the Document’s overall d3211 7 a3217 7 The “Invariant Sections” are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none. d3219 1 a3219 1 The “Cover Texts” are certain short passages of text that are d3225 1 a3225 1 A “Transparent” copy of the Document means a machine-readable copy, d3228 10 a3237 10 straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. 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FUTURE REVISIONS OF THIS LICENSE d3535 1 a3535 1 . d3539 1 a3539 1 version of this License “or any later version” applies to it, you d3542 4 a3545 4 published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation. d3563 1 a3563 1 Texts, replace the “with...Texts.” line with this: d3574 3 a3576 3 recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software. d3617 2 a3618 2 * GNU Free Documentation License <1>: GNU Free Documentation License. (line 6) a3620 2 * I/O functions <1>: Formatted Output Functions. (line 3) d3646 2 d3650 1 a3650 2 * Output functions <1>: Formatted Output Functions. (line 3) a3652 1 * Precision <1>: MPFR Interface. (line 17) d3665 1 a3665 1 * Ternary value: Rounding Modes. (line 24) d3679 3 a3681 3 (line 165) * mpfr_acos: Special Functions. (line 53) * mpfr_acosh: Special Functions. (line 117) d3683 1 a3683 1 (line 6) d3685 1 a3685 1 (line 12) d3687 1 a3687 1 (line 16) d3689 2 a3691 2 * mpfr_add_ui: Basic Arithmetic Functions. (line 8) d3693 5 a3697 5 (line 14) * mpfr_agm: Special Functions. (line 219) * mpfr_ai: Special Functions. (line 236) * mpfr_asin: Special Functions. (line 54) * mpfr_asinh: Special Functions. (line 118) d3699 2 a3700 2 (line 193) * mpfr_atan: Special Functions. (line 55) d3702 1 a3702 1 * mpfr_atanh: Special Functions. (line 119) d3704 1 a3704 1 (line 162) d3706 1 a3706 1 (line 167) d3708 1 a3708 1 (line 156) d3710 1 a3710 1 (line 172) d3712 1 a3712 1 (line 39) d3714 1 a3714 1 (line 113) d3716 1 a3716 1 (line 7) d3718 1 a3718 1 (line 37) d3720 1 a3720 3 (line 30) * mpfr_clears: Initialization Functions. (line 35) d3722 1 a3722 1 (line 112) d3724 1 a3724 1 (line 115) d3726 1 a3726 1 (line 128) d3728 2 a3730 2 * mpfr_clear_nanflag: Exception Related Functions. (line 113) d3732 2 d3735 2 a3736 2 * mpfr_clear_underflow: Exception Related Functions. (line 110) d3738 1 a3738 3 (line 6) * mpfr_cmpabs: Comparison Functions. (line 34) d3740 1 a3740 1 (line 9) d3742 1 a3742 1 (line 13) d3744 1 a3744 1 (line 10) d3746 1 a3746 1 (line 12) d3748 1 a3748 1 (line 8) d3750 1 a3750 1 (line 29) d3752 1 a3752 1 (line 7) d3754 1 a3754 1 (line 27) d3756 7 a3762 5 (line 11) * mpfr_const_catalan: Special Functions. (line 247) * mpfr_const_euler: Special Functions. (line 246) * mpfr_const_log2: Special Functions. (line 244) * mpfr_const_pi: Special Functions. (line 245) d3764 11 a3774 11 (line 109) * mpfr_cos: Special Functions. (line 31) * mpfr_cosh: Special Functions. (line 97) * mpfr_cot: Special Functions. (line 49) * mpfr_coth: Special Functions. (line 113) * mpfr_csc: Special Functions. (line 48) * mpfr_csch: Special Functions. (line 112) * mpfr_custom_get_exp: Custom Interface. (line 75) * mpfr_custom_get_kind: Custom Interface. (line 65) * mpfr_custom_get_significand: Custom Interface. (line 70) * mpfr_custom_get_size: Custom Interface. (line 37) d3777 5 a3781 1 * mpfr_custom_move: Custom Interface. (line 82) d3783 2 a3784 2 (line 74) * mpfr_digamma: Special Functions. (line 172) d3786 1 a3786 1 (line 171) a3788 2 * mpfr_divby0_p: Exception Related Functions. (line 134) d3790 1 a3790 1 (line 49) d3792 1 a3792 1 (line 186) d3794 1 a3794 1 (line 184) d3805 3 a3807 5 * mpfr_d_div: Basic Arithmetic Functions. (line 84) * mpfr_d_sub: Basic Arithmetic Functions. (line 36) * mpfr_eint: Special Functions. (line 135) d3809 1 a3809 1 (line 28) d3811 1 a3811 1 (line 59) d3813 8 a3820 8 (line 137) * mpfr_erf: Special Functions. (line 183) * mpfr_erfc: Special Functions. (line 184) * mpfr_exp: Special Functions. (line 25) * mpfr_exp10: Special Functions. (line 27) * mpfr_exp2: Special Functions. (line 26) * mpfr_expm1: Special Functions. (line 131) * mpfr_fac_ui: Special Functions. (line 123) d3822 1 a3822 1 (line 150) d3824 1 a3824 1 (line 146) d3826 2 d3829 2 a3830 2 * mpfr_fits_sshort_p: Conversion Functions. (line 148) a3831 2 (line 149) * mpfr_fits_uint_p: Conversion Functions. d3834 2 a3836 2 * mpfr_fits_ushort_p: Conversion Functions. (line 147) d3838 2 a3839 2 (line 8) * mpfr_fma: Special Functions. (line 209) d3841 2 a3842 2 (line 92) * mpfr_fms: Special Functions. (line 211) d3844 1 a3844 1 (line 157) d3846 2 a3847 2 (line 76) * mpfr_free_cache: Special Functions. (line 254) d3849 1 a3849 1 (line 137) d3851 2 a3852 2 (line 45) * mpfr_gamma: Special Functions. (line 150) d3854 3 a3856 1 (line 7) d3858 1 a3858 1 (line 9) d3860 1 a3860 1 (line 112) d3862 1 a3862 3 (line 10) * mpfr_get_d_2exp: Conversion Functions. (line 32) d3864 1 a3864 1 (line 7) d3866 2 a3868 2 * mpfr_get_emax_min: Exception Related Functions. (line 29) d3870 1 a3870 1 (line 6) d3872 2 a3874 2 * mpfr_get_emin_min: Exception Related Functions. (line 27) d3876 1 a3876 1 (line 88) d3878 1 a3878 1 (line 72) d3880 1 a3880 1 (line 6) d3882 1 a3882 1 (line 8) d3884 1 a3884 1 (line 34) d3886 2 a3888 2 * mpfr_get_prec: Initialization Functions. (line 149) d3890 1 a3890 1 (line 19) d3892 1 a3892 1 (line 21) d3894 1 a3894 1 (line 85) d3896 1 a3896 1 (line 20) d3898 1 a3898 1 (line 22) d3900 1 a3900 1 (line 116) d3902 1 a3902 1 (line 67) d3904 1 a3904 1 (line 54) d3906 3 a3908 1 (line 63) d3910 2 a3911 4 (line 56) * mpfr_greater_p: Comparison Functions. (line 55) * mpfr_hypot: Special Functions. (line 227) d3913 1 a3913 1 (line 136) d3915 1 a3915 1 (line 40) d3917 1 a3917 1 (line 53) d3919 1 a3919 5 (line 10) * mpfr_inits: Initialization Functions. (line 62) * mpfr_inits2: Initialization Functions. (line 22) d3921 1 a3921 1 (line 6) d3923 1 a3923 1 (line 11) d3925 1 a3925 1 (line 16) d3927 1 a3927 1 (line 12) d3929 1 a3929 1 (line 15) d3931 1 a3931 1 (line 9) d3933 1 a3933 1 (line 21) d3935 1 a3935 1 (line 7) d3937 5 a3941 1 (line 14) d3943 1 a3943 1 (line 31) d3945 6 a3950 4 (line 119) * mpfr_j0: Special Functions. (line 188) * mpfr_j1: Special Functions. (line 189) * mpfr_jn: Special Functions. (line 190) d3952 1 a3952 1 (line 58) d3954 8 a3961 10 (line 64) * mpfr_less_p: Comparison Functions. (line 57) * mpfr_lgamma: Special Functions. (line 162) * mpfr_li2: Special Functions. (line 145) * mpfr_lngamma: Special Functions. (line 154) * mpfr_log: Special Functions. (line 16) * mpfr_log10: Special Functions. (line 18) * mpfr_log1p: Special Functions. (line 127) * mpfr_log2: Special Functions. (line 17) d3963 2 a3965 2 * mpfr_min: Miscellaneous Functions. (line 20) d3967 1 a3967 1 (line 64) d3969 1 a3969 1 (line 82) d3973 1 a3973 1 (line 47) d3975 1 a3975 1 (line 179) d3977 1 a3977 1 (line 177) d3988 2 d3991 1 a3991 3 (line 135) * mpfr_nan_p: Comparison Functions. (line 39) d3993 1 a3993 1 (line 164) d3995 1 a3995 1 (line 15) d3997 1 a3997 1 (line 16) d3999 1 a3999 1 (line 6) d4001 1 a4001 1 (line 41) d4003 1 a4003 1 (line 15) d4005 1 a4005 1 (line 133) d4007 1 a4007 1 (line 121) d4009 1 a4009 1 (line 125) d4011 1 a4011 1 (line 123) d4013 1 a4013 1 (line 127) d4015 1 a4015 1 (line 13) d4018 2 d4021 1 a4021 3 (line 164) * mpfr_print_rnd_mode: Rounding Related Functions. (line 71) d4023 1 a4023 1 (line 105) d4025 1 a4025 1 (line 43) d4027 1 a4027 1 (line 39) d4029 1 a4029 1 (line 94) d4031 1 a4031 1 (line 96) d4033 1 a4033 1 (line 6) d4035 1 a4035 1 (line 46) d4037 1 a4037 1 (line 47) d4039 1 a4039 1 (line 49) d4041 1 a4041 1 (line 51) d4045 1 a4045 1 (line 114) d4047 3 a4049 3 (line 9) * mpfr_sec: Special Functions. (line 47) * mpfr_sech: Special Functions. (line 111) d4051 1 a4051 3 (line 9) * mpfr_setsign: Miscellaneous Functions. (line 103) d4053 1 a4053 1 (line 16) d4055 1 a4055 1 (line 19) d4057 1 a4057 1 (line 100) d4059 1 a4059 1 (line 6) d4061 1 a4061 1 (line 121) d4063 2 a4065 2 * mpfr_set_emin: Exception Related Functions. (line 15) d4067 1 a4067 1 (line 124) d4069 1 a4069 1 (line 93) d4071 1 a4071 1 (line 23) d4073 1 a4073 1 (line 15) d4075 1 a4075 1 (line 123) d4077 1 a4077 1 (line 143) d4079 1 a4079 1 (line 17) d4081 1 a4081 1 (line 142) d4083 1 a4083 1 (line 122) d4085 1 a4085 1 (line 120) d4087 1 a4087 1 (line 135) d4089 1 a4089 1 (line 22) d4091 1 a4091 1 (line 22) d4093 1 a4093 1 (line 12) d4095 1 a4095 1 (line 50) d4097 1 a4097 1 (line 14) d4099 1 a4099 1 (line 54) d4101 1 a4101 1 (line 62) d4103 1 a4103 1 (line 10) d4105 1 a4105 1 (line 48) d4107 1 a4107 1 (line 13) d4109 1 a4109 1 (line 52) d4111 1 a4111 1 (line 119) d4113 3 a4115 1 (line 21) d4117 3 a4119 3 (line 144) * mpfr_set_z_2exp: Assignment Functions. (line 56) d4121 1 a4121 7 (line 49) * mpfr_signbit: Miscellaneous Functions. (line 99) * mpfr_sin: Special Functions. (line 32) * mpfr_sinh: Special Functions. (line 98) * mpfr_sinh_cosh: Special Functions. (line 103) * mpfr_sin_cos: Special Functions. (line 37) d4125 7 a4131 1 (line 32) d4133 1 a4133 1 (line 180) d4135 1 a4135 1 (line 170) d4137 1 a4137 1 (line 71) d4139 1 a4139 1 (line 98) d4143 1 a4143 1 (line 80) d4145 1 a4145 3 (line 26) * mpfr_subnormalize: Exception Related Functions. (line 60) d4147 1 a4147 1 (line 38) d4149 1 a4149 1 (line 44) d4151 1 a4151 1 (line 34) d4153 1 a4153 1 (line 30) d4155 4 a4158 2 (line 42) * mpfr_sum: Special Functions. (line 262) d4160 1 a4160 1 (line 150) d4163 2 a4164 2 * mpfr_tan: Special Functions. (line 33) * mpfr_tanh: Special Functions. (line 99) d4166 1 a4166 1 (line 10) d4170 1 a4170 1 (line 131) d4172 1 a4172 1 (line 129) d4174 1 a4174 1 (line 28) d4176 1 a4176 1 (line 132) d4178 1 a4178 1 (line 69) d4180 1 a4180 1 (line 48) d4182 1 a4182 1 (line 29) d4184 1 a4184 1 (line 194) d4186 1 a4186 1 (line 119) d4188 1 a4188 1 (line 120) d4190 1 a4190 1 (line 121) d4192 1 a4192 1 (line 139) d4194 1 a4194 1 (line 122) d4196 1 a4196 1 (line 123) d4198 1 a4198 1 (line 158) d4200 1 a4200 1 (line 165) d4202 1 a4202 1 (line 182) d4204 6 a4209 4 (line 171) * mpfr_y0: Special Functions. (line 199) * mpfr_y1: Special Functions. (line 200) * mpfr_yn: Special Functions. (line 201) d4211 3 a4213 5 (line 42) * mpfr_zeta: Special Functions. (line 177) * mpfr_zeta_ui: Special Functions. (line 178) * mpfr_z_sub: Basic Arithmetic Functions. (line 40) d4218 42 a4259 42 Node: Top775 Node: Copying2007 Node: Introduction to MPFR3770 Node: Installing MPFR5884 Node: Reporting Bugs11328 Node: MPFR Basics13359 Node: Headers and Libraries13675 Node: Nomenclature and Types16829 Node: MPFR Variable Conventions18892 Node: Rounding Modes20436 Ref: ternary value21566 Node: Floating-Point Values on Special Numbers23552 Node: Exceptions26812 Node: Memory Handling29989 Node: MPFR Interface31135 Node: Initialization Functions33249 Node: Assignment Functions40564 Node: Combined Initialization and Assignment Functions49920 Node: Conversion Functions51221 Node: Basic Arithmetic Functions60283 Node: Comparison Functions69785 Node: Special Functions73273 Node: Input and Output Functions87874 Node: Formatted Output Functions89846 Node: Integer Related Functions99633 Node: Rounding Related Functions106254 Node: Miscellaneous Functions110091 Node: Exception Related Functions118774 Node: Compatibility with MPF125592 Node: Custom Interface128334 Node: Internals132733 Node: API Compatibility134277 Node: Type and Macro Changes136207 Node: Added Functions139056 Node: Changed Functions142344 Node: Removed Functions146758 Node: Other Changes147186 Node: Contributors148789 Node: References151442 Node: GNU Free Documentation License153196 Node: Concept Index175789 Node: Function and Type Index181854 @ 1.1.1.2.4.1 log @Sync with HEAD @ text @d1 1 a1 1 This is mpfr.info, produced by makeinfo version 6.5 from mpfr.texi. d4 1 a4 1 Floating-Point Reliable Library, version 4.0.1. d6 1 a6 1 Copyright 1991, 1993-2018 Free Software Foundation, Inc. d26 1 a26 1 Floating-Point Reliable Library, version 4.0.1. d28 1 a28 1 Copyright 1991, 1993-2018 Free Software Foundation, Inc. a45 1 * MPFR and the IEEE 754 Standard:: d101 1 a101 3 current processors), possibly except in faithful rounding. It does not depend either on the machine rounding mode or rounding precision; d108 9 a116 12 for other mathematical functions. Faithful rounding (partially supported) is provided too, but the results may no longer be reproducible. In particular, with a precision of 53 bits and in any of the four standard rounding modes, MPFR is able to exactly reproduce all computations with double-precision machine floating-point numbers (e.g., ‘double’ type in C, with a C implementation that rigorously follows Annex F of the ISO C99 standard and ‘FP_CONTRACT’ pragma set to ‘OFF’) on the four arithmetic operations and the square root, except the default exponent range is much wider and subnormal numbers are not implemented (but can be emulated). d279 2 a280 2 *Note Reporting Bugs::. Some bug fixes are available on the MPFR 4.0.1 web page . d286 1 a286 1 or . d295 1 a295 1 on the MPFR 4.0.1 web page and the FAQ d315 1 a315 1 can be extracted using ‘cc -V’ on some machines, or, if you are using a346 1 * Getting the Best Efficiency Out of MPFR:: d372 5 a376 5 prototypes for these functions. Moreover, under some platforms (in particular with C++ compilers), users may need to define ‘MPFR_USE_INTMAX_T’ (and should do it for portability) before ‘mpfr.h’ has been included; of course, it is possible to do that on the command line, e.g., with ‘-DMPFR_USE_INTMAX_T’. a414 10 Alternatively, it is possible to use ‘pkg-config’ (a file ‘mpfr.pc’ is provided as of MPFR 4.0): cc myprogram.c $(pkg-config --cflags --libs mpfr) Note that the ‘MPFR_’ and ‘mpfr_’ prefixes are reserved for MPFR. As a general rule, in order to avoid clashes, software using MPFR (directly or indirectly) and system headers/libraries should not define macros and symbols using these prefixes. d439 1 a439 1 equal to 1. a453 4 MPFR has a global (or per-thread) flag for each supported exception and provides operations on flags (*note Exceptions::). This C data type is used to represent a group of flags (or a mask). d460 2 a461 2 Before you can assign to a MPFR variable, you need to initialize it by calling one of the special initialization functions. When you are done d488 1 a488 1 The following rounding modes are supported: a495 10 • ‘MPFR_RNDF’: faithful rounding. This feature is currently experimental. Specific support for this rounding mode has been added to some functions, such as the basic operations (addition, subtraction, multiplication, square, division, square root) or when explicitly documented. It might also work with other functions, as it is possible that they do not need modification in their code; even though a correct behavior is not guaranteed yet (corrections were done when failures occurred in the test suite, but almost nothing has been checked manually), failures should be regarded as bugs and reported, so that they can be fixed. a505 14 The ‘MPFR_RNDF’ mode works as follows: the computed value is either that corresponding to ‘MPFR_RNDD’ or that corresponding to ‘MPFR_RNDU’. In particular when those values are identical, i.e., when the result of the corresponding operation is exactly representable, that exact result is returned. Thus, the computed result can take at most two possible values, and in absence of underflow/overflow, the corresponding error is strictly less than one ulp (unit in the last place) of that result and of the exact result. For ‘MPFR_RNDF’, the ternary value (defined below) and the inexact flag (defined later, as with the other flags) are unspecified, the divide-by-zero flag is as with other roundings, and the underflow and overflow flags match what would be obtained in the case the computed value is the same as with ‘MPFR_RNDD’ or ‘MPFR_RNDU’. The results may not be reproducible. d605 8 a612 17 MPFR defines a global (or per-thread) flag for each supported exception. A macro evaluating to a power of two is associated with each flag and exception, in order to be able to specify a group of flags (or a mask) by OR’ing such macros. Flags can be cleared (lowered), set (raised), and tested by functions described in *note Exception Related Functions::. The supported exceptions are listed below. The macro associated with each exception is in parentheses. • Underflow (‘MPFR_FLAGS_UNDERFLOW’): An underflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent smaller than the minimum value of the current exponent range. (In the round-to-nearest mode, the halfway case is rounded toward zero.) d629 24 a652 27 • Overflow (‘MPFR_FLAGS_OVERFLOW’): An overflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent larger than the maximum value of the current exponent range. In the round-to-nearest mode, the result is infinite. Note: unlike the underflow case, there is only one possible definition of overflow here. • Divide-by-zero (‘MPFR_FLAGS_DIVBY0’): An exact infinite result is obtained from finite inputs. • NaN (‘MPFR_FLAGS_NAN’): A NaN exception occurs when the result of a function is NaN. • Inexact (‘MPFR_FLAGS_INEXACT’): An inexact exception occurs when the result of a function cannot be represented exactly and must be rounded. • Range error (‘MPFR_FLAGS_ERANGE’): A range exception occurs when a function that does not return a MPFR number (such as comparisons and conversions to an integer) has an invalid result (e.g., an argument is NaN in ‘mpfr_cmp’, or a conversion to an integer cannot be represented in the target type). Moreover, the group consisting of all the flags is represented by the ‘MPFR_FLAGS_ALL’ macro (if new flags are added in future MPFR versions, they will be added to this macro too). d665 1 a665 1 File: mpfr.info, Node: Memory Handling, Next: Getting the Best Efficiency Out of MPFR, Prev: Exceptions, Up: MPFR Basics d673 6 a678 37 library itself to compute some other function. When more precision is needed, the value is automatically recomputed; a minimum of 10% increase of the precision is guaranteed to avoid too many recomputations. MPFR functions may also create thread-local pools for internal use to avoid the cost of memory allocation. The pools can be freed with ‘mpfr_free_pool’ (but with a default MPFR build, they should not take much memory, as the allocation size is limited). At any time, the user can free various caches and pools with ‘mpfr_free_cache’ and ‘mpfr_free_cache2’. It is strongly advised to free thread-local caches before terminating a thread, and all caches before exiting when using tools like ‘valgrind’ (to avoid memory leaks being reported). MPFR allocates its memory either on the stack (for temporary memory only) or with the same allocator as the one configured for GMP: *note (gmp.info)Custom Allocation::. This means that the application must make sure that data allocated with the current allocator will not be reallocated or freed with a new allocator. So, in practice, if an application needs to change the allocator with ‘mp_set_memory_functions’, it should first free all data allocated with the current allocator: for its own data, with ‘mpfr_clear’, etc.; for the caches and pools, with ‘mpfr_mp_memory_cleanup’ in all threads where MPFR is potentially used. This function is currently equivalent to ‘mpfr_free_cache’, but ‘mpfr_mp_memory_cleanup’ is the recommended way in case the allocation method changes in the future (for instance, one may choose to allocate the caches for floating-point constants with ‘malloc’ to avoid freeing them if the allocator changes). Developers should also be aware that MPFR may also be used indirectly by libraries, so that libraries based on MPFR should provide a clean-up function calling ‘mpfr_mp_memory_cleanup’ and/or warn their users about this issue. Note: For multithreaded applications, the allocator must be valid in all threads where MPFR may be used; data allocated in one thread may be reallocated and/or freed in some other thread. a688 35 Writers of libraries using MPFR should be aware that the application and/or another library used by the application may also use MPFR, so that changing the exponent range, the default precision, or the default rounding mode may have an effect on this other use of MPFR since these data are not duplicated (unless they are in a different thread). Therefore any such value changed in a library function should be restored before the function returns (unless the purpose of the function is to do such a change). Writers of software using MPFR should also be careful when changing such a value if they use a library using MPFR (directly or indirectly), in order to make sure that such a change is compatible with the library.  File: mpfr.info, Node: Getting the Best Efficiency Out of MPFR, Prev: Memory Handling, Up: MPFR Basics 4.8 Getting the Best Efficiency Out of MPFR =========================================== Here are a few hints to get the best efficiency out of MPFR: • you should avoid allocating and clearing variables. Reuse variables whenever possible, allocate or clear outside of loops, pass temporary variables to subroutines instead of allocating them inside the subroutines; • use ‘mpfr_swap’ instead of ‘mpfr_set’ whenever possible. This will avoid copying the significands; • avoid using MPFR from C++, or make sure your C++ interface does not perform unnecessary allocations or copies; • MPFR functions work in-place: to compute ‘a = a + b’ you don’t need an auxiliary variable, you can directly write ‘mpfr_add (a, a, b, ...)’. d721 2 a722 5 The value of the standard C macro ‘errno’ may be set to non-zero after calling any MPFR function or macro, whether or not there is an error. Except when documented, MPFR will not set ‘errno’, but functions called by the MPFR code (libc functions, memory allocator, etc.) may do so. d735 2 a736 2 * Integer and Remainder Related Functions:: * Rounding-Related Functions:: a914 2 -- Function: int mpfr_set_float128 (mpfr_t ROP, __float128 OP, mpfr_rnd_t RND) d922 1 a922 6 ‘mpfr_set_si’, ‘mpfr_set_uj’, ‘mpfr_set_sj’, The ‘mpfr_set_float128’ function is built only with the configure option ‘--enable-float128’, which requires the compiler or system provides the ‘__float128’ data type (GCC 4.3 or later supports this data type); to use ‘mpfr_set_float128’, one should define the macro ‘MPFR_WANT_FLOAT128’ before including ‘mpfr.h’. ‘mpfr_set_z’, d927 7 a933 11 built only with the configure option ‘--enable-decimal-float’, and when the compiler or system provides the ‘_Decimal64’ data type (recent versions of GCC support this data type); to use ‘mpfr_set_decimal64’, one should define the macro ‘MPFR_WANT_DECIMAL_FLOATS’ before including ‘mpfr.h’. ‘mpfr_set_q’ might fail if the numerator (or the denominator) can not be represented as a ‘mpfr_t’. For ‘mpfr_set’, the sign of a NaN is propagated in order to mimic the IEEE 754 ‘copy’ operation. But contrary to IEEE 754, the NaN flag is set as usual. a1093 1 -- Function: __float128 mpfr_get_float128 (mpfr_t OP, mpfr_rnd_t RND) d1101 2 a1102 3 ‘mpfr_get_float128’ and ‘mpfr_get_decimal64’ functions are built only under some conditions: see the documentation of ‘mpfr_set_float128’ and ‘mpfr_set_decimal64’ respectively. d1109 7 a1115 9 ‘uintmax_t’ (respectively) after rounding it to an integer with respect to RND. If OP is NaN, 0 is returned and the _erange_ flag is set. If OP is too big for the return type, the function returns the maximum or the minimum of the corresponding C type, depending on the direction of the overflow; the _erange_ flag is set too. When there is no such range error, if the return value differs from OP, i.e., if OP is not an integer, the inexact flag is set. See also ‘mpfr_fits_slong_p’, ‘mpfr_fits_ulong_p’, ‘mpfr_fits_intmax_p’ and ‘mpfr_fits_uintmax_p’. d1155 1 a1155 10 0, and 0 is returned. Otherwise the return value is zero when ROP is equal to OP (i.e., when OP is an integer), positive when it is greater than OP, and negative when it is smaller than OP; moreover, if ROP differs from OP, i.e., if OP is not an integer, the inexact flag is set. -- Function: void mpfr_get_q (mpq_t ROP, mpfr_t OP) Convert OP to a ‘mpq_t’. If OP is NaN or an infinity, the _erange_ flag is set and ROP is set to 0. Otherwise the conversion is always exact. d1177 4 a1180 10 pointer. If the input is NaN, then the returned string is ‘@@NaN@@’ and the NaN flag is set. If the input is +Inf (resp. −Inf), then the returned string is ‘@@Inf@@’ (resp. ‘-@@Inf@@’). If the input number is a finite number, the exponent is written through the pointer EXPPTR (for input 0, the current minimal exponent is written); the type ‘mpfr_exp_t’ is large enough to hold the exponent in all cases. d1206 3 a1208 3 using the allocation function (*note Memory Handling::) and a pointer to the string is returned (unless the base is invalid). To free the returned string, you must use ‘mpfr_free_str’. d1211 10 a1220 10 large enough for the significand. A safe block size (sufficient for any value) is ‘max(N + 2, 7)’ if N is not zero; if N is zero, replace it by m+1, as discussed above. The extra two bytes are for a possible minus sign, and for the terminating null character, and the value 7 accounts for ‘-@@Inf@@’ plus the terminating null character. The pointer to the string STR is returned (unless the base is invalid). Like in usual functions, the inexact flag is set iff the result is inexact. d1223 4 a1226 3 Free a string allocated by ‘mpfr_get_str’ using the unallocation function (*note Memory Handling::). The block is assumed to be ‘strlen(STR)+1’ bytes. d1239 1 a1239 9 an integer in the direction RND. For instance, with the ‘MPFR_RNDU’ rounding mode on −0.5, the result will be non-zero for all these functions. For ‘MPFR_RNDF’, those functions return non-zero when it is guaranteed that the corresponding conversion function (for example ‘mpfr_get_ui’ for ‘mpfr_fits_ulong_p’), when called with faithful rounding, will always return a number that is representable in the corresponding type. As a consequence, for ‘MPFR_RNDF’, ‘mpfr_fits_ulong_p’ will return non-zero for a non-negative number less or equal to ‘ULONG_MAX’. d1259 1 a1259 1 Set ROP to OP1 + OP2 rounded in the direction RND. The IEEE 754 d1287 1 a1287 1 Set ROP to OP1 - OP2 rounded in the direction RND. The IEEE 754 d1334 4 a1337 6 zero, its sign is the product of the signs of the operands. For types having no signed zeros, 0 is considered positive; but note that if OP1 is non-zero and OP2 is zero, the result might change from ±Inf to NaN in future MPFR versions if there is an opposite decision on the IEEE 754 side. The same restrictions than for ‘mpfr_add_d’ apply to ‘mpfr_d_div’ and ‘mpfr_div_d’. a1354 12 -- Function: int mpfr_rootn_ui (mpfr_t ROP, mpfr_t OP, unsigned long int K, mpfr_rnd_t RND) Set ROP to the cubic root (resp. the Kth root) of OP rounded in the direction RND. For K = 0, set ROP to NaN. For K odd (resp. even) and OP negative (including −Inf), set ROP to a negative number (resp. NaN). If OP is zero, set ROP to zero with the sign obtained by the usual limit rules, i.e., the same sign as OP if K is odd, and positive if K is even. These functions agree with the rootn function of the IEEE 754-2008 standard (Section 9.2). d1357 4 a1360 8 This function is the same as ‘mpfr_rootn_ui’ except when OP is −0 and K is even: the result is −0 instead of +0 (the reason was to be consistent with ‘mpfr_sqrt’). Said otherwise, if OP is zero, set ROP to OP. This function predates the IEEE 754-2008 standard and behaves differently from its rootn function. It is marked as deprecated and will be removed in a future release. a1403 3 Note: When 0 is of integer type, it is regarded as +0 by these functions. We do not use the usual limit rules in this case, as these rules are not used for ‘pow’. a1411 5 The sign rule also applies to NaN in order to mimic the IEEE 754 ‘negate’ and ‘abs’ operations, i.e., for ‘mpfr_neg’, the sign is reversed, and for ‘mpfr_abs’, the sign is set to positive. But contrary to IEEE 754, the NaN flag is set as usual. d1517 4 a1520 7 Important note: in some domains, computing special functions (even more with correct rounding) is expensive, even for small precision, for example the trigonometric and Bessel functions for large argument. For some functions, the memory usage might depend not only on the output precision: it is the case of the ‘mpfr_rootn_ui’ function where the memory usage is also linear in the argument K, and of the incomplete Gamma function (dependence on the precision of OP). a1522 2 -- Function: int mpfr_log_ui (mpfr_t ROP, unsigned long OP, mpfr_rnd_t RND) a1530 4 -- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the logarithm of one plus OP, rounded in the direction RND. Set ROP to −Inf if OP is −1. a1536 4 -- Function: int mpfr_expm1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential of OP followed by a subtraction by one, rounded in the direction RND. d1574 2 a1575 2 RND: if ‘x > 0’, ‘atan2(y, x) = atan(y/x)’; if ‘x < 0’, ‘atan2(y, x) = sign(y)*(Pi - atan(abs(y/x)))’, thus a number from -Pi to Pi. d1633 8 d1643 7 a1649 7 RND. This is the sum of Euler’s constant, of the logarithm of the absolute value of OP, and of the sum for k from 1 to infinity of OP to the power k, divided by k and factorial(k). For positive OP, it corresponds to the Ei function at OP (see formula 5.1.10 from the Handbook of Mathematical Functions from Abramowitz and Stegun), and for negative OP, to the opposite of the E1 function (sometimes called eint1) at −OP (formula 5.1.1 from the same reference). d1657 2 a1658 12 -- Function: int mpfr_gamma_inc (mpfr_t ROP, mpfr_t OP, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the value of the Gamma function on OP, resp. the incomplete Gamma function on OP and OP2, rounded in the direction RND. (In the literature, ‘mpfr_gamma_inc’ is called upper incomplete Gamma function, or sometimes complementary incomplete Gamma function.) For ‘mpfr_gamma’ (and ‘mpfr_gamma_inc’ when OP2 is zero), when OP is a negative integer, ROP is set to NaN. Note: the current implementation of ‘mpfr_gamma_inc’ is slow for large values of ROP or OP, in which case some internal overflow might also occur. a1682 6 -- Function: int mpfr_beta (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the value of the Beta function at arguments OP1 and OP2. Note: the current code does not try to avoid internal overflow or underflow, and might use a huge internal precision in some cases. a1724 10 -- Function: int mpfr_fmma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_t OP4, mpfr_rnd_t RND) -- Function: int mpfr_fmms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_t OP4, mpfr_rnd_t RND) Set ROP to (OP1 times OP2) + (OP3 times OP4) (resp. (OP1 times OP2) - (OP3 times OP4)) rounded in the direction RND. In case the computation of OP1 times OP2 overflows or underflows (or that of OP3 times OP4), the result ROP is computed as if the two intermediate products were computed with rounding toward zero. d1731 1 a1731 3 mean of U_N and V_N. If any operand is negative and the other one is not zero, set ROP to NaN. If any operand is zero and the other one is finite (resp. infinite), set ROP to +0 (resp. NaN). d1758 1 a1758 2 requested. To free these caches, use ‘mpfr_free_cache’ or ‘mpfr_free_cache2’. d1761 6 a1766 31 Free all caches and pools used by MPFR internally (thoses local to the current thread and those shared by all threads). You should call this function before terminating a thread, even if you did not call ‘mpfr_const_*’ functions directly (they could have been called internally). -- Function: void mpfr_free_cache2 (mpfr_free_cache_t WAY) Free various caches and pools used by MPFR internally, as specified by WAY, which is a set of flags: • those local to the current thread if flag ‘MPFR_FREE_LOCAL_CACHE’ is set; • those shared by all threads if flag ‘MPFR_FREE_GLOBAL_CACHE’ is set. The other bits of WAY are currently ignored and are reserved for future use; they should be zero. Note: ‘mpfr_free_cache2(MPFR_FREE_LOCAL_CACHE|MPFR_FREE_GLOBAL_CACHE)’ is currently equivalent to ‘mpfr_free_cache()’. -- Function: void mpfr_free_pool (void) Free the pools used by MPFR internally. Note: This function is automatically called after the thread-local caches are freed (with ‘mpfr_free_cache’ or ‘mpfr_free_cache2’). -- Function: int mpfr_mp_memory_cleanup (void) This function should be called before calling ‘mp_set_memory_functions’. *Note Memory Handling::, for more information. Zero is returned in case of success, non-zero in case of error. Errors are currently not possible, but checking the return value is recommended for future compatibility. d1768 1 a1768 1 -- Function: int mpfr_sum (mpfr_t ROP, const mpfr_ptr TAB[], unsigned d1770 7 a1776 14 Set ROP to the sum of all elements of TAB, whose size is N, correctly rounded in the direction RND. Warning: for efficiency reasons, TAB is an array of pointers to ‘mpfr_t’, not an array of ‘mpfr_t’. If N = 0, then the result is +0, and if N = 1, then the function is equivalent to ‘mpfr_set’. For the special exact cases, the result is the same as the one obtained with a succession of additions (‘mpfr_add’) in infinite precision. In particular, if the result is an exact zero and N >= 1: • if all the inputs have the same sign (i.e., all +0 or all −0), then the result has the same sign as the inputs; • otherwise, either because all inputs are zeros with at least a +0 and a −0, or because some inputs are non-zero (but they globally cancel), the result is +0, except for the ‘MPFR_RNDD’ rounding mode, where it is −0. d1789 3 a1791 3 When using a function that takes a ‘FILE *’ argument, you must include the ‘’ standard header before ‘mpfr.h’, to allow ‘mpfr.h’ to define prototypes for these functions. a1820 49 -- Function: int mpfr_fpif_export (FILE *STREAM, mpfr_t OP) Export the number OP to the stream STREAM in a floating-point interchange format. In particular one can export on a 32-bit computer and import on a 64-bit computer, or export on a little-endian computer and import on a big-endian computer. The precision of OP and the sign bit of a NaN are stored too. Return 0 iff the export was successful. Note: this function is experimental and its interface might change in future versions. -- Function: int mpfr_fpif_import (mpfr_t OP, FILE *STREAM) Import the number OP from the stream STREAM in a floating-point interchange format (see ‘mpfr_fpif_export’). Note that the precision of OP is set to the one read from the stream, and the sign bit is always retrieved (even for NaN). If the stored precision is zero or greater than ‘MPFR_PREC_MAX’, the function fails (it returns non-zero) and OP is unchanged. If the function fails for another reason, OP is set to NaN and it is unspecified whether the precision of OP has changed to the one read from the file. Return 0 iff the import was successful. Note: this function is experimental and its interface might change in future versions. -- Function: void mpfr_dump (mpfr_t OP) Output OP on ‘stdout’ in some unspecified format, then a newline character. This function is mainly for debugging purpose. Thus invalid data may be supported. Everything that is not specified may change without breaking the ABI and may depend on the environment. The current output format is the following: a minus sign if the sign bit is set (even for NaN); ‘@@NaN@@’, ‘@@Inf@@’ or ‘0’ if the argument is NaN, an infinity or zero, respectively; otherwise the remaining of the output is as follows: ‘0.’ then the P bits of the binary significand, where P is the precision of the number; if the trailing bits are not all zeros (which must not occur with valid data), they are output enclosed by square brackets; the character ‘E’ followed by the exponent written in base 10; in case of invalid data or out-of-range exponent, this function outputs three exclamation marks (‘!!!’), followed by flags, followed by three exclamation marks (‘!!!’) again. These flags are: ‘N’ if the most significant bit of the significand is 0 (i.e., the number is not normalized); ‘T’ if there are non-zero trailing bits; ‘U’ if this is a UBF number (internal use only); ‘<’ if the exponent is less than the current minimum exponent; ‘>’ if the exponent is greater than the current maximum exponent. d1822 1 a1822 1 File: mpfr.info, Node: Formatted Output Functions, Next: Integer and Remainder Related Functions, Prev: Input and Output Functions, Up: MPFR Interface d1848 1 a1848 2 related to the internal precision of the ‘mpfr_t’ variable), but note that for ‘Re’, the default precision is not the same as the one for ‘e’. d1933 3 a1935 3 ‘e’ ‘E’ scientific-format float ‘f’ ‘F’ fixed-point float ‘g’ ‘G’ fixed-point or scientific float d1972 2 a1973 2 For all the following functions, if the number of characters that ought to be written exceeds the maximum limit ‘INT_MAX’ for an ‘int’, nothing d1975 2 a1976 5 function returns −1, sets the _erange_ flag, and ‘errno’ is set to ‘EOVERFLOW’ if the ‘EOVERFLOW’ macro is defined (such as on POSIX systems). Note, however, that ‘errno’ might be changed to another value by some internal library call if another error occurs there (currently, this would come from the unallocation function). d2010 1 a2010 1 characters that would have been written had N been sufficiently d2018 5 a2022 6 allocated using the allocation function (*note Memory Handling::). A pointer to the block is stored in STR. The block of memory must be freed using ‘mpfr_free_str’. The return value is the number of characters written in the string, excluding the null-terminator, or a negative value if an error occurred, in which case the contents of STR are undefined. d2025 1 a2025 1 File: mpfr.info, Node: Integer and Remainder Related Functions, Next: Rounding-Related Functions, Prev: Formatted Output Functions, Up: MPFR Interface a2033 1 -- Function: int mpfr_roundeven (mpfr_t ROP, mpfr_t OP) d2036 11 a2046 21 nearest representable integer in the given direction RND, and the other five functions behave in a similar way with some fixed rounding mode: • ‘mpfr_ceil’: to the next higher or equal representable integer (like ‘mpfr_rint’ with ‘MPFR_RNDU’); • ‘mpfr_floor’ to the next lower or equal representable integer (like ‘mpfr_rint’ with ‘MPFR_RNDD’); • ‘mpfr_round’ to the nearest representable integer, rounding halfway cases away from zero (as in the roundTiesToAway mode of IEEE 754-2008); • ‘mpfr_roundeven’ to the nearest representable integer, rounding halfway cases with the even-rounding rule (like ‘mpfr_rint’ with ‘MPFR_RNDN’); • ‘mpfr_trunc’ to the next representable integer toward zero (like ‘mpfr_rint’ with ‘MPFR_RNDZ’). When OP is a zero or an infinity, set ROP to the same value (with the same sign). The return value is zero when the result is exact, positive when it is greater than the original value of OP, and negative when it is smaller. More precisely, the return value is 0 when OP is an d2055 14 a2068 9 function, you should use one the ‘mpfr_rint_*’ functions instead. Note that no double rounding is performed; for instance, 10.5 (1010.1 in binary) is rounded by ‘mpfr_rint’ with rounding to nearest to 12 (1100 in binary) in 2-bit precision, because the two enclosing numbers representable on two bits are 8 and 12, and the closest is 12. (If one first rounded to an integer, one would round 10.5 to 10 with even rounding, and then 10 would be rounded to 8 again with even rounding.) a2074 2 -- Function: int mpfr_rint_roundeven (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) d2077 8 a2084 13 Set ROP to OP rounded to an integer: • ‘mpfr_rint_ceil’: to the next higher or equal integer; • ‘mpfr_rint_floor’: to the next lower or equal integer; • ‘mpfr_rint_round’: to the nearest integer, rounding halfway cases away from zero; • ‘mpfr_rint_roundeven’: to the nearest integer, rounding halfway cases to the nearest even integer; • ‘mpfr_rint_trunc’ to the next integer toward zero. If the result is not representable, it is rounded in the direction RND. When OP is a zero or an infinity, set ROP to the same value (with the same sign). The return value is the ternary value associated with the considered round-to-integer function (regarded in the same way as any other mathematical function). d2104 1 a2104 2 fractional part is generated). When OP is an integer or an infinity, set ROP to zero with the same sign as OP. a2117 2 -- Function: int mpfr_fmodquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) d2124 3 a2126 3 follows: N is rounded toward zero for ‘mpfr_fmod’ and ‘mpfr_fmodquo’, and to the nearest integer (ties rounded to even) for ‘mpfr_remainder’ and ‘mpfr_remquo’. d2134 6 a2139 6 Additionally, ‘mpfr_fmodquo’ and ‘mpfr_remquo’ store the low significant bits from the quotient N in *Q (more precisely the number of bits in a ‘long’ minus one), with the sign of X divided by Y (except if those low bits are all zero, in which case zero is returned). Note that X may be so large in magnitude relative to Y that an exact representation of the quotient is not practical. The d2147 1 a2147 1 File: mpfr.info, Node: Rounding-Related Functions, Next: Miscellaneous Functions, Prev: Integer and Remainder Related Functions, Up: MPFR Interface d2149 1 a2149 1 5.11 Rounding-Related Functions d2190 11 a2200 24 to round correctly X to precision PREC with the direction RND2 assuming an unbounded exponent range, and 0 otherwise (including for NaN and Inf). In other words, if the error on B is bounded by two to the power K ulps, and B has precision PREC, you should give ERR=PREC−K. This function *does not modify* its arguments. If RND1 is ‘MPFR_RNDN’ or ‘MPFR_RNDF’, the error is considered to be either positive or negative, thus the possible range is twice as large as with a directed rounding for RND1 (with the same value of ERR). When RND2 is ‘MPFR_RNDF’, let RND3 be the opposite direction if RND1 is a directed rounding, and ‘MPFR_RNDN’ if RND1 is ‘MPFR_RNDN’ or ‘MPFR_RNDF’. The returned value of ‘mpfr_can_round (b, err, rnd1, MPFR_RNDF, prec)’ is non-zero iff after the call ‘mpfr_set (y, b, rnd3)’ with Y of precision PREC, Y is guaranteed to be a faithful rounding of X. Note: The *note ternary value:: cannot be determined in general with this function. However, if it is known that the exact value is not exactly representable in precision PREC, then one can use the following trick to determine the (non-zero) ternary value in any rounding mode RND2 (note that ‘MPFR_RNDZ’ below can be replaced by any directed rounding mode): d2202 3 a2204 7 prec + (rnd2 == MPFR_RNDN))) { /* round the approximation 'b' to the result 'r' of 'prec' bits with rounding mode 'rnd2' and get the ternary value 'inex' */ inex = mpfr_set (r, b, rnd2); } Indeed, if RND2 is ‘MPFR_RNDN’, this will check if one can round to a2209 3 A detailed example is available in the ‘examples’ subdirectory, file ‘can_round.c’. d2212 4 a2215 1 of X, and 0 for special values, including 0. a2221 28 -- Macro: int mpfr_round_nearest_away (int (FOO)(mpfr_t, type1_t, ..., mpfr_rnd_t), mpfr_t ROP, type1_t OP, ...) Given a function FOO and one or more values OP (which may be a ‘mpfr_t’, a ‘long’, a ‘double’, etc.), put in ROP the round-to-nearest-away rounding of ‘FOO(OP,...)’. This rounding is defined in the same way as round-to-nearest-even, except in case of tie, where the value away from zero is returned. The function FOO takes as input, from second to penultimate argument(s), the argument list given after ROP, a rounding mode as final argument, puts in its first argument the value ‘FOO(OP,...)’ rounded according to this rounding mode, and returns the corresponding ternary value (which is expected to be correct, otherwise ‘mpfr_round_nearest_away’ will not work as desired). Due to implementation constraints, this function must not be called when the minimal exponent ‘emin’ is the smallest possible one. This macro has been made such that the compiler is able to detect mismatch between the argument list OP and the function prototype of FOO. Multiple input arguments OP are supported only with C99 compilers. Otherwise, for C89 compilers, only one such argument is supported. Note: this macro is experimental and its interface might change in future versions. unsigned long ul; mpfr_t f, r; /* Code that inits and sets r, f, and ul, and if needed sets emin */ int i = mpfr_round_nearest_away (mpfr_add_ui, r, f, ul); d2223 1 a2223 1 File: mpfr.info, Node: Miscellaneous Functions, Next: Exception Related Functions, Prev: Rounding-Related Functions, Up: MPFR Interface d2229 7 a2235 7 If X or Y is NaN, set X to NaN; note that the NaN flag is set as usual. If X and Y are equal, X is unchanged. Otherwise, if X is different from Y, replace X by the next floating-point number (with the precision of X and the current exponent range) in the direction of Y (the infinite values are seen as the smallest and largest floating-point numbers). If the result is zero, it keeps the same sign. No underflow, overflow, or inexact exception is raised. d2281 3 a2283 7 Note: the note for ‘mpfr_urandomb’ holds too. Moreover, the exact number (the random value to be rounded) and the next random state do not depend on the current exponent range and the rounding mode. However, they depend on the target precision: from the same state of the random generator, if the precision of the destination is changed, then the value may be completely different (and the state of the random generator is different too). a2284 2 -- Function: int mpfr_nrandom (mpfr_t ROP1, gmp_randstate_t STATE, mpfr_rnd_t RND) d2287 2 a2288 4 Generate one (possibly two for ‘mpfr_grandom’) random floating-point number according to a standard normal Gaussian distribution (with mean zero and variance one). For ‘mpfr_grandom’, if ROP2 is a null pointer, then only one value is d2293 1 a2293 1 Gaussian distribution and then rounded in the direction RND. d2295 3 a2297 2 The ‘gmp_randstate_t’ argument should be created using the GMP ‘gmp_randinit’ function (see the GMP manual). d2299 6 a2304 6 For ‘mpfr_grandom’, the combination of the ternary values is returned like with ‘mpfr_sin_cos’. If ROP2 is a null pointer, the second ternary value is assumed to be 0 (note that the encoding of the only ternary value is not the same as the usual encoding for functions that return only one result). Otherwise the ternary value of a random number is always non-zero. a2309 10 Note: ‘mpfr_nrandom’ is much more efficient than ‘mpfr_grandom’, especially for large precision. Thus ‘mpfr_grandom’ is marked as deprecated and will be removed in a future release. -- Function: int mpfr_erandom (mpfr_t ROP1, gmp_randstate_t STATE, mpfr_rnd_t RND) Generate one random floating-point number according to an exponential distribution, with mean one. Other characteristics are identical to ‘mpfr_nrandom’. d2312 2 a2313 4 number and the significand is considered in [1/2,1). For this function, X is allowed to be outside of the current range of acceptable values. The behavior for NaN, infinity or zero is undefined. d2316 4 a2319 3 Set the exponent of X to E if X is a non-zero ordinary number and E is in the current exponent range, and return 0; otherwise, return a non-zero value (X is not changed). a2383 5 -- Function: int mpfr_buildopt_float128_p (void) Return a non-zero value if MPFR was compiled with ‘__float128’ support (that is, MPFR was built with the ‘--enable-float128’ configure option), return zero otherwise. a2393 8 -- Function: int mpfr_buildopt_sharedcache_p (void) Return a non-zero value if MPFR was compiled so that all threads share the same cache for one MPFR constant, like ‘mpfr_const_pi’ or ‘mpfr_const_log2’ (that is, MPFR was built with the ‘--enable-shared-cache’ configure option), return zero otherwise. If the return value is non-zero, MPFR applications may need to be compiled with the ‘-pthread’ option. d2419 6 a2424 19 largest exponent is not changed), and zero otherwise. For the subsequent operations, it is the user’s responsibility to check that any floating-point value used as an input is in the new exponent range (for example using ‘mpfr_check_range’). If a floating-point value outside the new exponent range is used as an input, the default behavior is undefined, in the sense of the ISO C standard; the behavior may also be explicitly documented, such as for ‘mpfr_check_range’. Note: Caches may still have values outside the current exponent range. This is not an issue as the user cannot use these caches directly via the API (MPFR extends the exponent range internally when need be). If ‘emin’ > ‘emax’ and a floating-point value needs to be produced as output, the behavior is undefined (‘mpfr_set_emin’ and ‘mpfr_set_emax’ do not check this condition as it might occur between successive calls to these two functions). d2437 1 a2437 1 This function assumes that X is the correctly rounded value of some d2461 2 a2462 3 is outside the subnormal exponent range of the emulated floating-point system, this function just propagates the *note ternary value:: T; otherwise, it rounds X to precision d2475 1 a2475 1 the current exponent range of MPFR (due to a too small ‘emax’), the a2483 5 Warning! If you change ‘emin’ (with ‘mpfr_set_emin’) just before calling ‘mpfr_subnormalize’, you need to make sure that the value is in the current exponent range of MPFR. But it is better to change ‘emin’ before any computation, if possible. d2505 3 a2507 25 Note that ‘mpfr_set_emin’ and ‘mpfr_set_emax’ are called early enough in order to make sure that all computed values are in the current exponent range. Warning! This emulates a double IEEE 754 arithmetic with correct rounding in the subnormal range, which may not be the case for your hardware. Below is another example showing how to emulate fixed-point arithmetic in a specific case. Here we compute the sine of the integers 1 to 17 with a result in a fixed-point arithmetic rounded at 2 power -42 (using the fact that the result is at most 1 in absolute value): { mpfr_t x; int i, inex; mpfr_set_emin (-41); mpfr_init2 (x, 42); for (i = 1; i <= 17; i++) { mpfr_set_ui (x, i, MPFR_RNDN); inex = mpfr_sin (x, x, MPFR_RNDZ); mpfr_subnormalize (x, inex, MPFR_RNDZ); mpfr_dump (x); } mpfr_clear (x); } d2515 2 a2516 7 Clear (lower) the underflow, overflow, divide-by-zero, invalid, inexact and _erange_ flags. -- Function: void mpfr_clear_flags (void) Clear (lower) all global flags (underflow, overflow, divide-by-zero, invalid, inexact, _erange_). Note: a group of flags can be cleared by using ‘mpfr_flags_clear’. d2524 6 a2529 2 Set (raise) the underflow, overflow, divide-by-zero, invalid, inexact and _erange_ flags. a2540 35 The ‘mpfr_flags_’ functions below that take an argument MASK can operate on any subset of the exception flags: a flag is part of this subset (or group) if and only if the corresponding bit of the argument MASK is set. The ‘MPFR_FLAGS_’ macros will normally be used to build this argument. *Note Exceptions::. -- Function: void mpfr_flags_clear (mpfr_flags_t MASK) Clear (lower) the group of flags specified by MASK. -- Function: void mpfr_flags_set (mpfr_flags_t MASK) Set (raise) the group of flags specified by MASK. -- Function: mpfr_flags_t mpfr_flags_test (mpfr_flags_t MASK) Return the flags specified by MASK. To test whether any flag from MASK is set, compare the return value to 0. You can also test individual flags by AND’ing the result with ‘MPFR_FLAGS_’ macros. Example: mpfr_flags_t t = mpfr_flags_test (MPFR_FLAGS_UNDERFLOW| MPFR_FLAGS_OVERFLOW) ... if (t) /* underflow and/or overflow (unlikely) */ { if (t & MPFR_FLAGS_UNDERFLOW) { /* handle underflow */ } if (t & MPFR_FLAGS_OVERFLOW) { /* handle overflow */ } } -- Function: mpfr_flags_t mpfr_flags_save (void) Return all the flags. It is equivalent to ‘mpfr_flags_test(MPFR_FLAGS_ALL)’. -- Function: void mpfr_flags_restore (mpfr_flags_t FLAGS, mpfr_flags_t MASK) Restore the flags specified by MASK to their state represented in FLAGS. d2552 4 a2555 3 many programs written for MPF can be compiled directly against MPFR without any changes. All operations are then performed with the default MPFR rounding mode, which can be reset with d2558 4 a2561 8 Warning! There are some differences. In particular: • The precision is different: MPFR rounds to the exact number of bits (zeroing trailing bits in the internal representation). Users may need to increase the precision of their variables. • The exponent range is also different. • The formatted output functions (‘gmp_printf’, etc.) will not work for arguments of arbitrary-precision floating-point type (‘mpf_t’, which ‘mpf2mpfr.h’ redefines as ‘mpfr_t’). d2647 13 a2659 14 • if abs(KIND) = ‘MPFR_NAN_KIND’, X is set to NaN; • if abs(KIND) = ‘MPFR_INF_KIND’, X is set to the infinity of the same sign as KIND; • if abs(KIND) = ‘MPFR_ZERO_KIND’, X is set to the zero of the same sign as KIND; • if abs(KIND) = ‘MPFR_REGULAR_KIND’, X is set to the regular number whose sign is the one of KIND, and whose exponent and significand are given by EXP and SIGNIFICAND. In all cases, SIGNIFICAND will be used directly for further computing involving X. This function does not allocate anything. A floating-point number initialized with this function cannot be resized using ‘mpfr_set_prec’ or ‘mpfr_prec_round’, or cleared using ‘mpfr_clear’! The SIGNIFICAND must have been initialized with ‘mpfr_custom_init’ using the same precision PREC. d2673 4 a2676 6 number and the significand is considered in [1/2,1). But if X is NaN, infinity or zero, contrary to ‘mpfr_get_exp’ (where the behavior is undefined), the return value is here an unspecified, valid value of the ‘mpfr_exp_t’ type. The behavior of this function for any ‘mpfr_t’ not initialized with ‘mpfr_custom_init_set’ is undefined. d2720 1 a2720 1 File: mpfr.info, Node: API Compatibility, Next: MPFR and the IEEE 754 Standard, Prev: MPFR Interface, Up: Top d2819 1 a2819 8 MPFR 3.0 (however no rounding mode ‘GMP_RNDA’ exists). Faithful rounding (‘MPFR_RNDF’) was added in MPFR 4.0, but currently, it is partially supported. The flags-related macros, whose name starts with ‘MPFR_FLAGS_’, were added in MPFR 4.0 (for the new functions ‘mpfr_flags_clear’, ‘mpfr_flags_restore’, ‘mpfr_flags_set’ and ‘mpfr_flags_test’, in particular). d2827 2 a2828 2 We give here in alphabetical order the functions (and function-like macros) that were added after MPFR 2.2, and in which MPFR version. d2836 1 a2836 7 • ‘mpfr_beta’ in MPFR 4.0 (incomplete, experimental). • ‘mpfr_buildopt_decimal_p’ in MPFR 3.0. • ‘mpfr_buildopt_float128_p’ in MPFR 4.0. • ‘mpfr_buildopt_gmpinternals_p’ in MPFR 3.1. d2838 2 a2839 5 • ‘mpfr_buildopt_sharedcache_p’ in MPFR 4.0. • ‘mpfr_buildopt_tls_p’ in MPFR 3.0. • ‘mpfr_buildopt_tune_case’ in MPFR 3.1. a2863 7 • ‘mpfr_erandom’ in MPFR 4.0. • ‘mpfr_flags_clear’, ‘mpfr_flags_restore’, ‘mpfr_flags_save’, ‘mpfr_flags_set’ and ‘mpfr_flags_test’ in MPFR 4.0. • ‘mpfr_fmma’ and ‘mpfr_fmms’ in MPFR 4.0. a2865 2 • ‘mpfr_fmodquo’ in MPFR 4.0. a2867 2 • ‘mpfr_fpif_export’ and ‘mpfr_fpif_import’ in MPFR 4.0. a2869 4 • ‘mpfr_free_cache2’ in MPFR 4.0. • ‘mpfr_free_pool’ in MPFR 4.0. a2871 5 • ‘mpfr_gamma_inc’ in MPFR 4.0. • ‘mpfr_get_float128’ in MPFR 4.0 if configured with ‘--enable-float128’. a2875 2 • ‘mpfr_get_q’ in MPFR 4.0. a2890 2 • ‘mpfr_log_ui’ in MPFR 4.0. a2894 2 • ‘mpfr_mp_memory_cleanup’ in MPFR 4.0. a2896 2 • ‘mpfr_nrandom’ in MPFR 4.0. a2904 6 • ‘mpfr_rint_roundeven’ and ‘mpfr_roundeven’ in MPFR 4.0. • ‘mpfr_round_nearest_away’ in MPFR 4.0. • ‘mpfr_rootn_ui’ in MPFR 4.0. a2906 3 • ‘mpfr_set_float128’ in MPFR 4.0 if configured with ‘--enable-float128’. a2941 5 • ‘mpfr_abs’, ‘mpfr_neg’ and ‘mpfr_set’ changed in MPFR 4.0. In previous MPFR versions, the sign bit of a NaN was unspecified; however, in practice, it was set as now specified except for ‘mpfr_neg’ with a reused argument: ‘mpfr_neg(x,x,rnd)’. a2948 4 • ‘mpfr_eint’ changed in MPFR 4.0. This function now returns the value of the E1/eint1 function for negative argument (before MPFR 4.0, it was returning NaN). a2957 5 • ‘mpfr_get_str’ changed in MPFR 4.0. This function now sets the NaN flag on NaN input (to follow the usual MPFR rules on NaN and IEEE 754-2008 recommendations on string conversions from Subclause 5.12.1) and sets the inexact flag when the conversion is inexact. a2982 10 • ‘mpfr_set_exp’ changed in MPFR 4.0. Before MPFR 4.0, the exponent was set whatever the contents of the MPFR object in argument. In practice, this could be useful as a low-level function when the MPFR number was being constructed by setting the fields of its internal structure, but the API does not provide a way to do this except by using internals. Thus, for the API, this behavior was useless and could quickly lead to undefined behavior due to the fact that the generated value could have an invalid format if the MPFR object contained a special value (NaN, infinity or zero). a3004 7 • ‘mpfr_sum’ changed in MPFR 4.0. The ‘mpfr_sum’ function has completely been rewritten for MPFR 4.0, with an update of the specification: the sign of an exact zero result is now specified, and the return value is now the usual ternary value. The old ‘mpfr_sum’ implementation could also take all the memory and crash on inputs of very different magnitude. a3013 7 • ‘mpfr_urandom’ changed in MPFR 4.0. The next random state no longer depends on the current exponent range and the rounding mode. The exceptions due to the rounding of the random number are now correctly generated, following the uniform distribution. As a consequence, the returned values can be different between MPFR 4.0 and previous MPFR versions. a3024 7 Macros ‘mpfr_add_one_ulp’ and ‘mpfr_sub_one_ulp’ have been removed in MPFR 4.0. They were no longer documented since MPFR 2.1.0 and were announced as deprecated since MPFR 3.1.0. Function ‘mpfr_grandom’ is marked as deprecated in MPFR 4.0. It will be removed in a future release. a3057 3 The way memory is allocated by MPFR should be regarded as well-specified only as of MPFR 4.0. d3059 1 a3059 51 File: mpfr.info, Node: MPFR and the IEEE 754 Standard, Next: Contributors, Prev: API Compatibility, Up: Top 7 MPFR and the IEEE 754 Standard ******************************** This section describes differences between MPFR and the IEEE 754 standard, and behaviors that are not specified yet in IEEE 754. The MPFR numbers do not include subnormals. The reason is that subnormals are less useful than in IEEE 754 as the default exponent range in MPFR is large and they would have made the implementation more complex. However, subnormals can be emulated using ‘mpfr_subnormalize’. MPFR has a single NaN. The behavior is similar either to a signaling NaN or to a quiet NaN, depending on the context. For any function returning a NaN (either produced or propagated), the NaN flag is set, while in IEEE 754, some operations are quiet (even on a signaling NaN). The ‘mpfr_rec_sqrt’ function differs from IEEE 754 on −0, where it gives +Inf (like for +0), following the usual limit rules, instead of −Inf. The ‘mpfr_root’ function predates IEEE 754-2008 and behaves differently from its rootn operation. It is deprecated and ‘mpfr_rootn_ui’ should be used instead. Operations with an unsigned zero: For functions taking an argument of integer or rational type, a zero of such a type is unsigned unlike the floating-point zero (this includes the zero of type ‘unsigned long’, which is a mathematical, exact zero, as opposed to a floating-point zero, which may come from an underflow and whose sign would correspond to the sign of the real non-zero value). Unless documented otherwise, this zero is regarded as +0, as if it were first converted to a MPFR number with ‘mpfr_set_ui’ or ‘mpfr_set_si’ (thus the result may not agree with the usual limit rules applied to a mathematical zero). This is not the case of addition and subtraction (‘mpfr_add_ui’, etc.), but for these functions, only the sign of a zero result would be affected, with +0 and −0 considered equal. Such operations are currently out of the scope of the IEEE 754 standard, and at the time of specification in MPFR, the Floating-Point Working Group in charge of the revision of IEEE 754 did not want to discuss issues with non-floating-point types in general. Note also that some obvious differences may come from the fact that in MPFR, each variable has its own precision. For instance, a subtraction of two numbers of the same sign may yield an overflow; idem for a call to ‘mpfr_set’, ‘mpfr_neg’ or ‘mpfr_abs’, if the destination variable has a smaller precision.  File: mpfr.info, Node: Contributors, Next: References, Prev: MPFR and the IEEE 754 Standard, Up: Top d3074 9 a3082 10 contributed the original version of the ‘mpfr_sum’ function (used up to MPFR 3.1). Emmanuel Jeandel, from ENS-Lyon too, contributed the generic hypergeometric code, as well as the internal function ‘mpfr_exp3’, a first implementation of the sine and cosine, and improved versions of ‘mpfr_const_log2’ and ‘mpfr_const_pi’. Ludovic Meunier helped in the design of the ‘mpfr_erf’ code. Jean-Luc Rémy contributed the ‘mpfr_zeta’ code. Fabrice Rouillier contributed the ‘mpfr_xxx_z’ and ‘mpfr_xxx_q’ functions, and helped to the Microsoft Windows porting. Damien Stehlé contributed the ‘mpfr_get_ld_2exp’ function. Charles Karney contributed the ‘mpfr_nrandom’ and ‘mpfr_erandom’ functions. d3103 1 a3103 5 2012 was partly supported by the ERC grant ANTICS of Andreas Enge. The MPFR-MPC workshop in January 2013 was partly supported by the ERC grant ANTICS, the GDR IM and the Caramel project-team, during which Mickaël Gastineau contributed the MPFRbench program, and Fredrik Johannsson a faster version of ‘mpfr_const_euler’. d3112 2 a3113 4 Cambridge University Press, Cambridge Monographs on Applied and Computational Mathematics, Number 18, 2010. Electronic version freely available at . d3122 1 a3122 1 Library", version 6.1.2, 2016, . d3137 1 a3137 1 Implementation", Birkhäuser, Boston, 3rd edition, 2016. a3613 2 * Group of flags: Nomenclature and Types. (line 39) d3623 1 a3623 1 * Integer related functions: Integer and Remainder Related Functions. a3649 2 * Remainder related functions: Integer and Remainder Related Functions. (line 3) d3651 1 a3651 1 * Rounding mode related functions: Rounding-Related Functions. d3662 1 a3662 1 * Ternary value: Rounding Modes. (line 48) d3676 3 a3678 3 (line 186) * mpfr_acos: Special Functions. (line 66) * mpfr_acosh: Special Functions. (line 130) d3691 4 a3694 4 * mpfr_agm: Special Functions. (line 250) * mpfr_ai: Special Functions. (line 269) * mpfr_asin: Special Functions. (line 67) * mpfr_asinh: Special Functions. (line 131) d3696 4 a3699 5 (line 197) * mpfr_atan: Special Functions. (line 68) * mpfr_atan2: Special Functions. (line 78) * mpfr_atanh: Special Functions. (line 132) * mpfr_beta: Special Functions. (line 192) d3701 1 a3701 3 (line 185) * mpfr_buildopt_float128_p: Miscellaneous Functions. (line 180) d3703 1 a3703 3 (line 190) * mpfr_buildopt_sharedcache_p: Miscellaneous Functions. (line 195) d3705 1 a3705 1 (line 174) d3707 2 a3708 2 (line 203) * mpfr_can_round: Rounding-Related Functions. d3711 2 a3712 2 (line 115) * mpfr_ceil: Integer and Remainder Related Functions. d3715 1 a3715 1 (line 50) d3721 1 a3721 1 (line 153) d3723 1 a3723 1 (line 156) d3725 1 a3725 1 (line 160) d3727 1 a3727 1 (line 155) d3729 1 a3729 1 (line 154) d3731 1 a3731 1 (line 152) d3733 1 a3733 1 (line 151) d3756 4 a3759 4 * mpfr_const_catalan: Special Functions. (line 280) * mpfr_const_euler: Special Functions. (line 279) * mpfr_const_log2: Special Functions. (line 277) * mpfr_const_pi: Special Functions. (line 278) d3761 10 a3770 10 (line 127) * mpfr_cos: Special Functions. (line 44) * mpfr_cosh: Special Functions. (line 110) * mpfr_cot: Special Functions. (line 62) * mpfr_coth: Special Functions. (line 126) * mpfr_csc: Special Functions. (line 61) * mpfr_csch: Special Functions. (line 125) * mpfr_custom_get_exp: Custom Interface. (line 76) * mpfr_custom_get_kind: Custom Interface. (line 66) * mpfr_custom_get_significand: Custom Interface. (line 71) d3774 1 a3774 1 * mpfr_custom_move: Custom Interface. (line 85) d3777 1 a3777 1 * mpfr_digamma: Special Functions. (line 187) d3779 1 a3779 1 (line 197) d3783 1 a3783 1 (line 176) d3785 1 a3785 1 (line 52) d3787 1 a3787 1 (line 212) d3789 1 a3789 1 (line 210) a3799 2 * mpfr_dump: Input and Output Functions. (line 68) d3804 1 a3804 1 * mpfr_eint: Special Functions. (line 140) d3806 1 a3806 1 (line 31) a3808 2 * mpfr_erandom: Miscellaneous Functions. (line 99) d3810 8 a3817 8 (line 179) * mpfr_erf: Special Functions. (line 204) * mpfr_erfc: Special Functions. (line 205) * mpfr_exp: Special Functions. (line 34) * mpfr_exp10: Special Functions. (line 36) * mpfr_exp2: Special Functions. (line 35) * mpfr_expm1: Special Functions. (line 40) * mpfr_fac_ui: Special Functions. (line 136) d3819 1 a3819 1 (line 168) d3821 1 a3821 1 (line 164) d3823 1 a3823 1 (line 162) d3825 1 a3825 1 (line 166) d3827 1 a3827 1 (line 167) d3829 1 a3829 1 (line 163) d3831 1 a3831 1 (line 161) d3833 2 a3834 14 (line 165) * mpfr_flags_clear: Exception Related Functions. (line 190) * mpfr_flags_restore: Exception Related Functions. (line 214) * mpfr_flags_save: Exception Related Functions. (line 210) * mpfr_flags_set: Exception Related Functions. (line 193) * mpfr_flags_t: Nomenclature and Types. (line 39) * mpfr_flags_test: Exception Related Functions. (line 196) * mpfr_floor: Integer and Remainder Related Functions. d3836 4 a3839 12 * mpfr_fma: Special Functions. (line 230) * mpfr_fmma: Special Functions. (line 240) * mpfr_fmms: Special Functions. (line 242) * mpfr_fmod: Integer and Remainder Related Functions. (line 106) * mpfr_fmodquo: Integer and Remainder Related Functions. (line 108) * mpfr_fms: Special Functions. (line 232) * mpfr_fpif_export: Input and Output Functions. (line 43) * mpfr_fpif_import: Input and Output Functions. (line 54) d3841 4 a3844 6 (line 161) * mpfr_frac: Integer and Remainder Related Functions. (line 89) * mpfr_free_cache: Special Functions. (line 288) * mpfr_free_cache2: Special Functions. (line 295) * mpfr_free_pool: Special Functions. (line 309) d3846 1 a3846 1 (line 156) d3848 2 a3849 3 (line 49) * mpfr_gamma: Special Functions. (line 155) * mpfr_gamma_inc: Special Functions. (line 156) d3853 1 a3853 1 (line 10) d3856 1 a3856 1 * mpfr_get_default_rounding_mode: Rounding-Related Functions. d3859 1 a3859 1 (line 36) d3863 1 a3863 1 (line 43) d3865 1 a3865 1 (line 42) d3869 1 a3869 1 (line 41) d3871 1 a3871 1 (line 40) d3873 1 a3873 1 (line 105) d3875 1 a3875 3 (line 85) * mpfr_get_float128: Conversion Functions. (line 9) d3881 1 a3881 1 (line 38) d3883 1 a3883 1 (line 165) a3885 2 * mpfr_get_q: Conversion Functions. (line 80) d3887 2 a3889 2 * mpfr_get_sj: Conversion Functions. (line 23) d3891 1 a3891 1 (line 98) d3893 2 a3895 2 * mpfr_get_uj: Conversion Functions. (line 24) d3897 1 a3897 1 (line 134) d3899 1 a3899 1 (line 71) d3901 1 a3901 1 (line 58) d3903 1 a3903 1 (line 69) d3908 1 a3908 1 * mpfr_hypot: Special Functions. (line 260) d3910 1 a3910 1 (line 178) d3941 5 a3945 5 * mpfr_integer_p: Integer and Remainder Related Functions. (line 135) * mpfr_j0: Special Functions. (line 209) * mpfr_j1: Special Functions. (line 210) * mpfr_jn: Special Functions. (line 211) d3952 7 a3958 8 * mpfr_lgamma: Special Functions. (line 177) * mpfr_li2: Special Functions. (line 150) * mpfr_lngamma: Special Functions. (line 169) * mpfr_log: Special Functions. (line 19) * mpfr_log10: Special Functions. (line 23) * mpfr_log1p: Special Functions. (line 30) * mpfr_log2: Special Functions. (line 22) * mpfr_log_ui: Special Functions. (line 20) d3963 4 a3966 5 * mpfr_min_prec: Rounding-Related Functions. (line 84) * mpfr_modf: Integer and Remainder Related Functions. (line 96) * mpfr_mp_memory_cleanup: Special Functions. (line 314) d3970 1 a3970 1 (line 50) d3972 1 a3972 1 (line 205) d3974 1 a3974 1 (line 203) d3986 1 a3986 1 (line 177) d3990 1 a3990 1 (line 185) a3996 2 * mpfr_nrandom: Miscellaneous Functions. (line 67) d4002 1 a4002 1 (line 175) d4004 1 a4004 1 (line 139) d4006 1 a4006 1 (line 143) d4008 1 a4008 1 (line 141) d4010 2 a4011 2 (line 145) * mpfr_prec_round: Rounding-Related Functions. d4016 3 a4018 3 (line 168) * mpfr_print_rnd_mode: Rounding-Related Functions. (line 88) d4020 1 a4020 1 (line 107) d4024 6 a4029 6 (line 42) * mpfr_remainder: Integer and Remainder Related Functions. (line 110) * mpfr_remquo: Integer and Remainder Related Functions. (line 112) * mpfr_rint: Integer and Remainder Related Functions. d4031 8 a4038 10 * mpfr_rint_ceil: Integer and Remainder Related Functions. (line 52) * mpfr_rint_floor: Integer and Remainder Related Functions. (line 53) * mpfr_rint_round: Integer and Remainder Related Functions. (line 55) * mpfr_rint_roundeven: Integer and Remainder Related Functions. (line 57) * mpfr_rint_trunc: Integer and Remainder Related Functions. (line 59) d4042 2 a4043 4 (line 128) * mpfr_rootn_ui: Basic Arithmetic Functions. (line 116) * mpfr_round: Integer and Remainder Related Functions. d4045 2 a4046 6 * mpfr_roundeven: Integer and Remainder Related Functions. (line 10) * mpfr_round_nearest_away: Rounding-Related Functions. (line 93) * mpfr_sec: Special Functions. (line 60) * mpfr_sech: Special Functions. (line 124) d4050 1 a4050 1 (line 121) d4054 1 a4054 1 (line 21) d4057 1 a4057 1 * mpfr_set_default_rounding_mode: Rounding-Related Functions. d4060 1 a4060 1 (line 167) d4066 1 a4066 1 (line 170) d4068 1 a4068 1 (line 112) d4070 1 a4070 3 (line 25) * mpfr_set_float128: Assignment Functions. (line 19) d4074 1 a4074 1 (line 169) d4076 1 a4076 1 (line 154) d4080 1 a4080 1 (line 153) d4082 1 a4082 1 (line 168) d4084 1 a4084 1 (line 166) d4088 1 a4088 1 (line 25) d4090 1 a4090 1 (line 24) d4094 1 a4094 1 (line 61) d4098 1 a4098 1 (line 65) d4100 1 a4100 1 (line 73) d4104 1 a4104 1 (line 59) d4108 1 a4108 1 (line 63) d4110 1 a4110 1 (line 165) d4112 1 a4112 1 (line 23) d4114 1 a4114 1 (line 155) d4116 1 a4116 1 (line 67) d4120 5 a4124 5 (line 117) * mpfr_sin: Special Functions. (line 45) * mpfr_sinh: Special Functions. (line 111) * mpfr_sinh_cosh: Special Functions. (line 116) * mpfr_sin_cos: Special Functions. (line 50) d4130 1 a4130 1 (line 184) d4132 1 a4132 1 (line 174) d4136 1 a4136 1 (line 100) d4138 1 a4138 1 (line 101) d4140 1 a4140 1 (line 91) d4144 1 a4144 1 (line 73) d4155 1 a4155 1 * mpfr_sum: Special Functions. (line 321) d4157 1 a4157 1 (line 161) d4160 4 a4163 4 * mpfr_tan: Special Functions. (line 46) * mpfr_tanh: Special Functions. (line 112) * mpfr_trunc: Integer and Remainder Related Functions. (line 11) d4167 1 a4167 1 (line 149) d4169 1 a4169 1 (line 147) d4173 1 a4173 1 (line 174) d4181 1 a4181 1 (line 198) d4183 1 a4183 1 (line 137) d4185 1 a4185 1 (line 138) d4187 2 a4189 2 * MPFR_VERSION_NUM: Miscellaneous Functions. (line 157) d4191 1 a4191 1 (line 140) d4193 1 a4193 1 (line 141) d4195 1 a4195 1 (line 162) d4197 1 a4197 1 (line 169) d4199 1 a4199 1 (line 186) d4201 4 a4204 4 (line 175) * mpfr_y0: Special Functions. (line 220) * mpfr_y1: Special Functions. (line 221) * mpfr_yn: Special Functions. (line 222) d4207 2 a4208 2 * mpfr_zeta: Special Functions. (line 198) * mpfr_zeta_ui: Special Functions. (line 199) d4216 41 a4256 43 Node: Copying2042 Node: Introduction to MPFR3805 Node: Installing MPFR6208 Node: Reporting Bugs11654 Node: MPFR Basics13684 Node: Headers and Libraries14044 Node: Nomenclature and Types17640 Node: MPFR Variable Conventions19899 Node: Rounding Modes21441 Ref: ternary value24166 Node: Floating-Point Values on Special Numbers26152 Node: Exceptions29412 Node: Memory Handling33240 Node: Getting the Best Efficiency Out of MPFR37040 Node: MPFR Interface37952 Node: Initialization Functions40238 Node: Assignment Functions47553 Node: Combined Initialization and Assignment Functions57499 Node: Conversion Functions58800 Node: Basic Arithmetic Functions69381 Node: Comparison Functions80277 Node: Special Functions83765 Node: Input and Output Functions101974 Node: Formatted Output Functions106751 Node: Integer and Remainder Related Functions116956 Node: Rounding-Related Functions124484 Node: Miscellaneous Functions131001 Node: Exception Related Functions141493 Node: Compatibility with MPF151733 Node: Custom Interface154679 Node: Internals159310 Node: API Compatibility160854 Node: Type and Macro Changes162802 Node: Added Functions165985 Node: Changed Functions170499 Node: Removed Functions177095 Node: Other Changes177825 Node: MPFR and the IEEE 754 Standard179526 Node: Contributors182143 Node: References185200 Node: GNU Free Documentation License187084 Node: Concept Index209677 Node: Function and Type Index216049 @ 1.1.1.2.2.1 log @Sync with HEAD Resolve a couple of conflicts (result of the uimin/uimax changes) @ text @d1 1 a1 1 This is mpfr.info, produced by makeinfo version 6.5 from mpfr.texi. d4 1 a4 1 Floating-Point Reliable Library, version 4.0.1. d6 1 a6 1 Copyright 1991, 1993-2018 Free Software Foundation, Inc. d26 1 a26 1 Floating-Point Reliable Library, version 4.0.1. d28 1 a28 1 Copyright 1991, 1993-2018 Free Software Foundation, Inc. a45 1 * MPFR and the IEEE 754 Standard:: d101 1 a101 3 current processors), possibly except in faithful rounding. It does not depend either on the machine rounding mode or rounding precision; d108 9 a116 12 for other mathematical functions. Faithful rounding (partially supported) is provided too, but the results may no longer be reproducible. In particular, with a precision of 53 bits and in any of the four standard rounding modes, MPFR is able to exactly reproduce all computations with double-precision machine floating-point numbers (e.g., ‘double’ type in C, with a C implementation that rigorously follows Annex F of the ISO C99 standard and ‘FP_CONTRACT’ pragma set to ‘OFF’) on the four arithmetic operations and the square root, except the default exponent range is much wider and subnormal numbers are not implemented (but can be emulated). d279 2 a280 2 *Note Reporting Bugs::. Some bug fixes are available on the MPFR 4.0.1 web page . d286 1 a286 1 or . d295 1 a295 1 on the MPFR 4.0.1 web page and the FAQ d315 1 a315 1 can be extracted using ‘cc -V’ on some machines, or, if you are using a346 1 * Getting the Best Efficiency Out of MPFR:: d372 5 a376 5 prototypes for these functions. Moreover, under some platforms (in particular with C++ compilers), users may need to define ‘MPFR_USE_INTMAX_T’ (and should do it for portability) before ‘mpfr.h’ has been included; of course, it is possible to do that on the command line, e.g., with ‘-DMPFR_USE_INTMAX_T’. a414 10 Alternatively, it is possible to use ‘pkg-config’ (a file ‘mpfr.pc’ is provided as of MPFR 4.0): cc myprogram.c $(pkg-config --cflags --libs mpfr) Note that the ‘MPFR_’ and ‘mpfr_’ prefixes are reserved for MPFR. As a general rule, in order to avoid clashes, software using MPFR (directly or indirectly) and system headers/libraries should not define macros and symbols using these prefixes. d439 1 a439 1 equal to 1. a453 4 MPFR has a global (or per-thread) flag for each supported exception and provides operations on flags (*note Exceptions::). This C data type is used to represent a group of flags (or a mask). d460 2 a461 2 Before you can assign to a MPFR variable, you need to initialize it by calling one of the special initialization functions. When you are done d488 1 a488 1 The following rounding modes are supported: a495 10 • ‘MPFR_RNDF’: faithful rounding. This feature is currently experimental. Specific support for this rounding mode has been added to some functions, such as the basic operations (addition, subtraction, multiplication, square, division, square root) or when explicitly documented. It might also work with other functions, as it is possible that they do not need modification in their code; even though a correct behavior is not guaranteed yet (corrections were done when failures occurred in the test suite, but almost nothing has been checked manually), failures should be regarded as bugs and reported, so that they can be fixed. a505 14 The ‘MPFR_RNDF’ mode works as follows: the computed value is either that corresponding to ‘MPFR_RNDD’ or that corresponding to ‘MPFR_RNDU’. In particular when those values are identical, i.e., when the result of the corresponding operation is exactly representable, that exact result is returned. Thus, the computed result can take at most two possible values, and in absence of underflow/overflow, the corresponding error is strictly less than one ulp (unit in the last place) of that result and of the exact result. For ‘MPFR_RNDF’, the ternary value (defined below) and the inexact flag (defined later, as with the other flags) are unspecified, the divide-by-zero flag is as with other roundings, and the underflow and overflow flags match what would be obtained in the case the computed value is the same as with ‘MPFR_RNDD’ or ‘MPFR_RNDU’. The results may not be reproducible. d605 8 a612 17 MPFR defines a global (or per-thread) flag for each supported exception. A macro evaluating to a power of two is associated with each flag and exception, in order to be able to specify a group of flags (or a mask) by OR’ing such macros. Flags can be cleared (lowered), set (raised), and tested by functions described in *note Exception Related Functions::. The supported exceptions are listed below. The macro associated with each exception is in parentheses. • Underflow (‘MPFR_FLAGS_UNDERFLOW’): An underflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent smaller than the minimum value of the current exponent range. (In the round-to-nearest mode, the halfway case is rounded toward zero.) d629 24 a652 27 • Overflow (‘MPFR_FLAGS_OVERFLOW’): An overflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent larger than the maximum value of the current exponent range. In the round-to-nearest mode, the result is infinite. Note: unlike the underflow case, there is only one possible definition of overflow here. • Divide-by-zero (‘MPFR_FLAGS_DIVBY0’): An exact infinite result is obtained from finite inputs. • NaN (‘MPFR_FLAGS_NAN’): A NaN exception occurs when the result of a function is NaN. • Inexact (‘MPFR_FLAGS_INEXACT’): An inexact exception occurs when the result of a function cannot be represented exactly and must be rounded. • Range error (‘MPFR_FLAGS_ERANGE’): A range exception occurs when a function that does not return a MPFR number (such as comparisons and conversions to an integer) has an invalid result (e.g., an argument is NaN in ‘mpfr_cmp’, or a conversion to an integer cannot be represented in the target type). Moreover, the group consisting of all the flags is represented by the ‘MPFR_FLAGS_ALL’ macro (if new flags are added in future MPFR versions, they will be added to this macro too). d665 1 a665 1 File: mpfr.info, Node: Memory Handling, Next: Getting the Best Efficiency Out of MPFR, Prev: Exceptions, Up: MPFR Basics d673 6 a678 37 library itself to compute some other function. When more precision is needed, the value is automatically recomputed; a minimum of 10% increase of the precision is guaranteed to avoid too many recomputations. MPFR functions may also create thread-local pools for internal use to avoid the cost of memory allocation. The pools can be freed with ‘mpfr_free_pool’ (but with a default MPFR build, they should not take much memory, as the allocation size is limited). At any time, the user can free various caches and pools with ‘mpfr_free_cache’ and ‘mpfr_free_cache2’. It is strongly advised to free thread-local caches before terminating a thread, and all caches before exiting when using tools like ‘valgrind’ (to avoid memory leaks being reported). MPFR allocates its memory either on the stack (for temporary memory only) or with the same allocator as the one configured for GMP: *note (gmp.info)Custom Allocation::. This means that the application must make sure that data allocated with the current allocator will not be reallocated or freed with a new allocator. So, in practice, if an application needs to change the allocator with ‘mp_set_memory_functions’, it should first free all data allocated with the current allocator: for its own data, with ‘mpfr_clear’, etc.; for the caches and pools, with ‘mpfr_mp_memory_cleanup’ in all threads where MPFR is potentially used. This function is currently equivalent to ‘mpfr_free_cache’, but ‘mpfr_mp_memory_cleanup’ is the recommended way in case the allocation method changes in the future (for instance, one may choose to allocate the caches for floating-point constants with ‘malloc’ to avoid freeing them if the allocator changes). Developers should also be aware that MPFR may also be used indirectly by libraries, so that libraries based on MPFR should provide a clean-up function calling ‘mpfr_mp_memory_cleanup’ and/or warn their users about this issue. Note: For multithreaded applications, the allocator must be valid in all threads where MPFR may be used; data allocated in one thread may be reallocated and/or freed in some other thread. a688 35 Writers of libraries using MPFR should be aware that the application and/or another library used by the application may also use MPFR, so that changing the exponent range, the default precision, or the default rounding mode may have an effect on this other use of MPFR since these data are not duplicated (unless they are in a different thread). Therefore any such value changed in a library function should be restored before the function returns (unless the purpose of the function is to do such a change). Writers of software using MPFR should also be careful when changing such a value if they use a library using MPFR (directly or indirectly), in order to make sure that such a change is compatible with the library.  File: mpfr.info, Node: Getting the Best Efficiency Out of MPFR, Prev: Memory Handling, Up: MPFR Basics 4.8 Getting the Best Efficiency Out of MPFR =========================================== Here are a few hints to get the best efficiency out of MPFR: • you should avoid allocating and clearing variables. Reuse variables whenever possible, allocate or clear outside of loops, pass temporary variables to subroutines instead of allocating them inside the subroutines; • use ‘mpfr_swap’ instead of ‘mpfr_set’ whenever possible. This will avoid copying the significands; • avoid using MPFR from C++, or make sure your C++ interface does not perform unnecessary allocations or copies; • MPFR functions work in-place: to compute ‘a = a + b’ you don’t need an auxiliary variable, you can directly write ‘mpfr_add (a, a, b, ...)’. d721 2 a722 5 The value of the standard C macro ‘errno’ may be set to non-zero after calling any MPFR function or macro, whether or not there is an error. Except when documented, MPFR will not set ‘errno’, but functions called by the MPFR code (libc functions, memory allocator, etc.) may do so. d735 2 a736 2 * Integer and Remainder Related Functions:: * Rounding-Related Functions:: a914 2 -- Function: int mpfr_set_float128 (mpfr_t ROP, __float128 OP, mpfr_rnd_t RND) d922 1 a922 6 ‘mpfr_set_si’, ‘mpfr_set_uj’, ‘mpfr_set_sj’, The ‘mpfr_set_float128’ function is built only with the configure option ‘--enable-float128’, which requires the compiler or system provides the ‘__float128’ data type (GCC 4.3 or later supports this data type); to use ‘mpfr_set_float128’, one should define the macro ‘MPFR_WANT_FLOAT128’ before including ‘mpfr.h’. ‘mpfr_set_z’, d927 7 a933 11 built only with the configure option ‘--enable-decimal-float’, and when the compiler or system provides the ‘_Decimal64’ data type (recent versions of GCC support this data type); to use ‘mpfr_set_decimal64’, one should define the macro ‘MPFR_WANT_DECIMAL_FLOATS’ before including ‘mpfr.h’. ‘mpfr_set_q’ might fail if the numerator (or the denominator) can not be represented as a ‘mpfr_t’. For ‘mpfr_set’, the sign of a NaN is propagated in order to mimic the IEEE 754 ‘copy’ operation. But contrary to IEEE 754, the NaN flag is set as usual. a1093 1 -- Function: __float128 mpfr_get_float128 (mpfr_t OP, mpfr_rnd_t RND) d1101 2 a1102 3 ‘mpfr_get_float128’ and ‘mpfr_get_decimal64’ functions are built only under some conditions: see the documentation of ‘mpfr_set_float128’ and ‘mpfr_set_decimal64’ respectively. d1109 7 a1115 9 ‘uintmax_t’ (respectively) after rounding it to an integer with respect to RND. If OP is NaN, 0 is returned and the _erange_ flag is set. If OP is too big for the return type, the function returns the maximum or the minimum of the corresponding C type, depending on the direction of the overflow; the _erange_ flag is set too. When there is no such range error, if the return value differs from OP, i.e., if OP is not an integer, the inexact flag is set. See also ‘mpfr_fits_slong_p’, ‘mpfr_fits_ulong_p’, ‘mpfr_fits_intmax_p’ and ‘mpfr_fits_uintmax_p’. d1155 1 a1155 10 0, and 0 is returned. Otherwise the return value is zero when ROP is equal to OP (i.e., when OP is an integer), positive when it is greater than OP, and negative when it is smaller than OP; moreover, if ROP differs from OP, i.e., if OP is not an integer, the inexact flag is set. -- Function: void mpfr_get_q (mpq_t ROP, mpfr_t OP) Convert OP to a ‘mpq_t’. If OP is NaN or an infinity, the _erange_ flag is set and ROP is set to 0. Otherwise the conversion is always exact. d1177 4 a1180 10 pointer. If the input is NaN, then the returned string is ‘@@NaN@@’ and the NaN flag is set. If the input is +Inf (resp. −Inf), then the returned string is ‘@@Inf@@’ (resp. ‘-@@Inf@@’). If the input number is a finite number, the exponent is written through the pointer EXPPTR (for input 0, the current minimal exponent is written); the type ‘mpfr_exp_t’ is large enough to hold the exponent in all cases. d1206 3 a1208 3 using the allocation function (*note Memory Handling::) and a pointer to the string is returned (unless the base is invalid). To free the returned string, you must use ‘mpfr_free_str’. d1211 10 a1220 10 large enough for the significand. A safe block size (sufficient for any value) is ‘max(N + 2, 7)’ if N is not zero; if N is zero, replace it by m+1, as discussed above. The extra two bytes are for a possible minus sign, and for the terminating null character, and the value 7 accounts for ‘-@@Inf@@’ plus the terminating null character. The pointer to the string STR is returned (unless the base is invalid). Like in usual functions, the inexact flag is set iff the result is inexact. d1223 4 a1226 3 Free a string allocated by ‘mpfr_get_str’ using the unallocation function (*note Memory Handling::). The block is assumed to be ‘strlen(STR)+1’ bytes. d1239 1 a1239 9 an integer in the direction RND. For instance, with the ‘MPFR_RNDU’ rounding mode on −0.5, the result will be non-zero for all these functions. For ‘MPFR_RNDF’, those functions return non-zero when it is guaranteed that the corresponding conversion function (for example ‘mpfr_get_ui’ for ‘mpfr_fits_ulong_p’), when called with faithful rounding, will always return a number that is representable in the corresponding type. As a consequence, for ‘MPFR_RNDF’, ‘mpfr_fits_ulong_p’ will return non-zero for a non-negative number less or equal to ‘ULONG_MAX’. d1259 1 a1259 1 Set ROP to OP1 + OP2 rounded in the direction RND. The IEEE 754 d1287 1 a1287 1 Set ROP to OP1 - OP2 rounded in the direction RND. The IEEE 754 d1334 4 a1337 6 zero, its sign is the product of the signs of the operands. For types having no signed zeros, 0 is considered positive; but note that if OP1 is non-zero and OP2 is zero, the result might change from ±Inf to NaN in future MPFR versions if there is an opposite decision on the IEEE 754 side. The same restrictions than for ‘mpfr_add_d’ apply to ‘mpfr_d_div’ and ‘mpfr_div_d’. a1354 12 -- Function: int mpfr_rootn_ui (mpfr_t ROP, mpfr_t OP, unsigned long int K, mpfr_rnd_t RND) Set ROP to the cubic root (resp. the Kth root) of OP rounded in the direction RND. For K = 0, set ROP to NaN. For K odd (resp. even) and OP negative (including −Inf), set ROP to a negative number (resp. NaN). If OP is zero, set ROP to zero with the sign obtained by the usual limit rules, i.e., the same sign as OP if K is odd, and positive if K is even. These functions agree with the rootn function of the IEEE 754-2008 standard (Section 9.2). d1357 4 a1360 8 This function is the same as ‘mpfr_rootn_ui’ except when OP is −0 and K is even: the result is −0 instead of +0 (the reason was to be consistent with ‘mpfr_sqrt’). Said otherwise, if OP is zero, set ROP to OP. This function predates the IEEE 754-2008 standard and behaves differently from its rootn function. It is marked as deprecated and will be removed in a future release. a1403 3 Note: When 0 is of integer type, it is regarded as +0 by these functions. We do not use the usual limit rules in this case, as these rules are not used for ‘pow’. a1411 5 The sign rule also applies to NaN in order to mimic the IEEE 754 ‘negate’ and ‘abs’ operations, i.e., for ‘mpfr_neg’, the sign is reversed, and for ‘mpfr_abs’, the sign is set to positive. But contrary to IEEE 754, the NaN flag is set as usual. d1517 4 a1520 7 Important note: in some domains, computing special functions (even more with correct rounding) is expensive, even for small precision, for example the trigonometric and Bessel functions for large argument. For some functions, the memory usage might depend not only on the output precision: it is the case of the ‘mpfr_rootn_ui’ function where the memory usage is also linear in the argument K, and of the incomplete Gamma function (dependence on the precision of OP). a1522 2 -- Function: int mpfr_log_ui (mpfr_t ROP, unsigned long OP, mpfr_rnd_t RND) a1530 4 -- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the logarithm of one plus OP, rounded in the direction RND. Set ROP to −Inf if OP is −1. a1536 4 -- Function: int mpfr_expm1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential of OP followed by a subtraction by one, rounded in the direction RND. d1574 2 a1575 2 RND: if ‘x > 0’, ‘atan2(y, x) = atan(y/x)’; if ‘x < 0’, ‘atan2(y, x) = sign(y)*(Pi - atan(abs(y/x)))’, thus a number from -Pi to Pi. d1633 8 d1643 7 a1649 7 RND. This is the sum of Euler’s constant, of the logarithm of the absolute value of OP, and of the sum for k from 1 to infinity of OP to the power k, divided by k and factorial(k). For positive OP, it corresponds to the Ei function at OP (see formula 5.1.10 from the Handbook of Mathematical Functions from Abramowitz and Stegun), and for negative OP, to the opposite of the E1 function (sometimes called eint1) at −OP (formula 5.1.1 from the same reference). d1657 2 a1658 12 -- Function: int mpfr_gamma_inc (mpfr_t ROP, mpfr_t OP, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the value of the Gamma function on OP, resp. the incomplete Gamma function on OP and OP2, rounded in the direction RND. (In the literature, ‘mpfr_gamma_inc’ is called upper incomplete Gamma function, or sometimes complementary incomplete Gamma function.) For ‘mpfr_gamma’ (and ‘mpfr_gamma_inc’ when OP2 is zero), when OP is a negative integer, ROP is set to NaN. Note: the current implementation of ‘mpfr_gamma_inc’ is slow for large values of ROP or OP, in which case some internal overflow might also occur. a1682 6 -- Function: int mpfr_beta (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the value of the Beta function at arguments OP1 and OP2. Note: the current code does not try to avoid internal overflow or underflow, and might use a huge internal precision in some cases. a1724 10 -- Function: int mpfr_fmma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_t OP4, mpfr_rnd_t RND) -- Function: int mpfr_fmms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_t OP4, mpfr_rnd_t RND) Set ROP to (OP1 times OP2) + (OP3 times OP4) (resp. (OP1 times OP2) - (OP3 times OP4)) rounded in the direction RND. In case the computation of OP1 times OP2 overflows or underflows (or that of OP3 times OP4), the result ROP is computed as if the two intermediate products were computed with rounding toward zero. d1731 1 a1731 3 mean of U_N and V_N. If any operand is negative and the other one is not zero, set ROP to NaN. If any operand is zero and the other one is finite (resp. infinite), set ROP to +0 (resp. NaN). d1758 1 a1758 2 requested. To free these caches, use ‘mpfr_free_cache’ or ‘mpfr_free_cache2’. d1761 6 a1766 31 Free all caches and pools used by MPFR internally (thoses local to the current thread and those shared by all threads). You should call this function before terminating a thread, even if you did not call ‘mpfr_const_*’ functions directly (they could have been called internally). -- Function: void mpfr_free_cache2 (mpfr_free_cache_t WAY) Free various caches and pools used by MPFR internally, as specified by WAY, which is a set of flags: • those local to the current thread if flag ‘MPFR_FREE_LOCAL_CACHE’ is set; • those shared by all threads if flag ‘MPFR_FREE_GLOBAL_CACHE’ is set. The other bits of WAY are currently ignored and are reserved for future use; they should be zero. Note: ‘mpfr_free_cache2(MPFR_FREE_LOCAL_CACHE|MPFR_FREE_GLOBAL_CACHE)’ is currently equivalent to ‘mpfr_free_cache()’. -- Function: void mpfr_free_pool (void) Free the pools used by MPFR internally. Note: This function is automatically called after the thread-local caches are freed (with ‘mpfr_free_cache’ or ‘mpfr_free_cache2’). -- Function: int mpfr_mp_memory_cleanup (void) This function should be called before calling ‘mp_set_memory_functions’. *Note Memory Handling::, for more information. Zero is returned in case of success, non-zero in case of error. Errors are currently not possible, but checking the return value is recommended for future compatibility. d1768 1 a1768 1 -- Function: int mpfr_sum (mpfr_t ROP, const mpfr_ptr TAB[], unsigned d1770 7 a1776 14 Set ROP to the sum of all elements of TAB, whose size is N, correctly rounded in the direction RND. Warning: for efficiency reasons, TAB is an array of pointers to ‘mpfr_t’, not an array of ‘mpfr_t’. If N = 0, then the result is +0, and if N = 1, then the function is equivalent to ‘mpfr_set’. For the special exact cases, the result is the same as the one obtained with a succession of additions (‘mpfr_add’) in infinite precision. In particular, if the result is an exact zero and N >= 1: • if all the inputs have the same sign (i.e., all +0 or all −0), then the result has the same sign as the inputs; • otherwise, either because all inputs are zeros with at least a +0 and a −0, or because some inputs are non-zero (but they globally cancel), the result is +0, except for the ‘MPFR_RNDD’ rounding mode, where it is −0. d1789 3 a1791 3 When using a function that takes a ‘FILE *’ argument, you must include the ‘’ standard header before ‘mpfr.h’, to allow ‘mpfr.h’ to define prototypes for these functions. a1820 49 -- Function: int mpfr_fpif_export (FILE *STREAM, mpfr_t OP) Export the number OP to the stream STREAM in a floating-point interchange format. In particular one can export on a 32-bit computer and import on a 64-bit computer, or export on a little-endian computer and import on a big-endian computer. The precision of OP and the sign bit of a NaN are stored too. Return 0 iff the export was successful. Note: this function is experimental and its interface might change in future versions. -- Function: int mpfr_fpif_import (mpfr_t OP, FILE *STREAM) Import the number OP from the stream STREAM in a floating-point interchange format (see ‘mpfr_fpif_export’). Note that the precision of OP is set to the one read from the stream, and the sign bit is always retrieved (even for NaN). If the stored precision is zero or greater than ‘MPFR_PREC_MAX’, the function fails (it returns non-zero) and OP is unchanged. If the function fails for another reason, OP is set to NaN and it is unspecified whether the precision of OP has changed to the one read from the file. Return 0 iff the import was successful. Note: this function is experimental and its interface might change in future versions. -- Function: void mpfr_dump (mpfr_t OP) Output OP on ‘stdout’ in some unspecified format, then a newline character. This function is mainly for debugging purpose. Thus invalid data may be supported. Everything that is not specified may change without breaking the ABI and may depend on the environment. The current output format is the following: a minus sign if the sign bit is set (even for NaN); ‘@@NaN@@’, ‘@@Inf@@’ or ‘0’ if the argument is NaN, an infinity or zero, respectively; otherwise the remaining of the output is as follows: ‘0.’ then the P bits of the binary significand, where P is the precision of the number; if the trailing bits are not all zeros (which must not occur with valid data), they are output enclosed by square brackets; the character ‘E’ followed by the exponent written in base 10; in case of invalid data or out-of-range exponent, this function outputs three exclamation marks (‘!!!’), followed by flags, followed by three exclamation marks (‘!!!’) again. These flags are: ‘N’ if the most significant bit of the significand is 0 (i.e., the number is not normalized); ‘T’ if there are non-zero trailing bits; ‘U’ if this is a UBF number (internal use only); ‘<’ if the exponent is less than the current minimum exponent; ‘>’ if the exponent is greater than the current maximum exponent. d1822 1 a1822 1 File: mpfr.info, Node: Formatted Output Functions, Next: Integer and Remainder Related Functions, Prev: Input and Output Functions, Up: MPFR Interface d1848 1 a1848 2 related to the internal precision of the ‘mpfr_t’ variable), but note that for ‘Re’, the default precision is not the same as the one for ‘e’. d1933 3 a1935 3 ‘e’ ‘E’ scientific-format float ‘f’ ‘F’ fixed-point float ‘g’ ‘G’ fixed-point or scientific float d1972 2 a1973 2 For all the following functions, if the number of characters that ought to be written exceeds the maximum limit ‘INT_MAX’ for an ‘int’, nothing d1975 2 a1976 5 function returns −1, sets the _erange_ flag, and ‘errno’ is set to ‘EOVERFLOW’ if the ‘EOVERFLOW’ macro is defined (such as on POSIX systems). Note, however, that ‘errno’ might be changed to another value by some internal library call if another error occurs there (currently, this would come from the unallocation function). d2010 1 a2010 1 characters that would have been written had N been sufficiently d2018 5 a2022 6 allocated using the allocation function (*note Memory Handling::). A pointer to the block is stored in STR. The block of memory must be freed using ‘mpfr_free_str’. The return value is the number of characters written in the string, excluding the null-terminator, or a negative value if an error occurred, in which case the contents of STR are undefined. d2025 1 a2025 1 File: mpfr.info, Node: Integer and Remainder Related Functions, Next: Rounding-Related Functions, Prev: Formatted Output Functions, Up: MPFR Interface a2033 1 -- Function: int mpfr_roundeven (mpfr_t ROP, mpfr_t OP) d2036 11 a2046 21 nearest representable integer in the given direction RND, and the other five functions behave in a similar way with some fixed rounding mode: • ‘mpfr_ceil’: to the next higher or equal representable integer (like ‘mpfr_rint’ with ‘MPFR_RNDU’); • ‘mpfr_floor’ to the next lower or equal representable integer (like ‘mpfr_rint’ with ‘MPFR_RNDD’); • ‘mpfr_round’ to the nearest representable integer, rounding halfway cases away from zero (as in the roundTiesToAway mode of IEEE 754-2008); • ‘mpfr_roundeven’ to the nearest representable integer, rounding halfway cases with the even-rounding rule (like ‘mpfr_rint’ with ‘MPFR_RNDN’); • ‘mpfr_trunc’ to the next representable integer toward zero (like ‘mpfr_rint’ with ‘MPFR_RNDZ’). When OP is a zero or an infinity, set ROP to the same value (with the same sign). The return value is zero when the result is exact, positive when it is greater than the original value of OP, and negative when it is smaller. More precisely, the return value is 0 when OP is an d2055 14 a2068 9 function, you should use one the ‘mpfr_rint_*’ functions instead. Note that no double rounding is performed; for instance, 10.5 (1010.1 in binary) is rounded by ‘mpfr_rint’ with rounding to nearest to 12 (1100 in binary) in 2-bit precision, because the two enclosing numbers representable on two bits are 8 and 12, and the closest is 12. (If one first rounded to an integer, one would round 10.5 to 10 with even rounding, and then 10 would be rounded to 8 again with even rounding.) a2074 2 -- Function: int mpfr_rint_roundeven (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) d2077 8 a2084 13 Set ROP to OP rounded to an integer: • ‘mpfr_rint_ceil’: to the next higher or equal integer; • ‘mpfr_rint_floor’: to the next lower or equal integer; • ‘mpfr_rint_round’: to the nearest integer, rounding halfway cases away from zero; • ‘mpfr_rint_roundeven’: to the nearest integer, rounding halfway cases to the nearest even integer; • ‘mpfr_rint_trunc’ to the next integer toward zero. If the result is not representable, it is rounded in the direction RND. When OP is a zero or an infinity, set ROP to the same value (with the same sign). The return value is the ternary value associated with the considered round-to-integer function (regarded in the same way as any other mathematical function). d2104 1 a2104 2 fractional part is generated). When OP is an integer or an infinity, set ROP to zero with the same sign as OP. a2117 2 -- Function: int mpfr_fmodquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) d2124 3 a2126 3 follows: N is rounded toward zero for ‘mpfr_fmod’ and ‘mpfr_fmodquo’, and to the nearest integer (ties rounded to even) for ‘mpfr_remainder’ and ‘mpfr_remquo’. d2134 6 a2139 6 Additionally, ‘mpfr_fmodquo’ and ‘mpfr_remquo’ store the low significant bits from the quotient N in *Q (more precisely the number of bits in a ‘long’ minus one), with the sign of X divided by Y (except if those low bits are all zero, in which case zero is returned). Note that X may be so large in magnitude relative to Y that an exact representation of the quotient is not practical. The d2147 1 a2147 1 File: mpfr.info, Node: Rounding-Related Functions, Next: Miscellaneous Functions, Prev: Integer and Remainder Related Functions, Up: MPFR Interface d2149 1 a2149 1 5.11 Rounding-Related Functions d2190 11 a2200 24 to round correctly X to precision PREC with the direction RND2 assuming an unbounded exponent range, and 0 otherwise (including for NaN and Inf). In other words, if the error on B is bounded by two to the power K ulps, and B has precision PREC, you should give ERR=PREC−K. This function *does not modify* its arguments. If RND1 is ‘MPFR_RNDN’ or ‘MPFR_RNDF’, the error is considered to be either positive or negative, thus the possible range is twice as large as with a directed rounding for RND1 (with the same value of ERR). When RND2 is ‘MPFR_RNDF’, let RND3 be the opposite direction if RND1 is a directed rounding, and ‘MPFR_RNDN’ if RND1 is ‘MPFR_RNDN’ or ‘MPFR_RNDF’. The returned value of ‘mpfr_can_round (b, err, rnd1, MPFR_RNDF, prec)’ is non-zero iff after the call ‘mpfr_set (y, b, rnd3)’ with Y of precision PREC, Y is guaranteed to be a faithful rounding of X. Note: The *note ternary value:: cannot be determined in general with this function. However, if it is known that the exact value is not exactly representable in precision PREC, then one can use the following trick to determine the (non-zero) ternary value in any rounding mode RND2 (note that ‘MPFR_RNDZ’ below can be replaced by any directed rounding mode): d2202 3 a2204 7 prec + (rnd2 == MPFR_RNDN))) { /* round the approximation 'b' to the result 'r' of 'prec' bits with rounding mode 'rnd2' and get the ternary value 'inex' */ inex = mpfr_set (r, b, rnd2); } Indeed, if RND2 is ‘MPFR_RNDN’, this will check if one can round to a2209 3 A detailed example is available in the ‘examples’ subdirectory, file ‘can_round.c’. d2212 4 a2215 1 of X, and 0 for special values, including 0. a2221 28 -- Macro: int mpfr_round_nearest_away (int (FOO)(mpfr_t, type1_t, ..., mpfr_rnd_t), mpfr_t ROP, type1_t OP, ...) Given a function FOO and one or more values OP (which may be a ‘mpfr_t’, a ‘long’, a ‘double’, etc.), put in ROP the round-to-nearest-away rounding of ‘FOO(OP,...)’. This rounding is defined in the same way as round-to-nearest-even, except in case of tie, where the value away from zero is returned. The function FOO takes as input, from second to penultimate argument(s), the argument list given after ROP, a rounding mode as final argument, puts in its first argument the value ‘FOO(OP,...)’ rounded according to this rounding mode, and returns the corresponding ternary value (which is expected to be correct, otherwise ‘mpfr_round_nearest_away’ will not work as desired). Due to implementation constraints, this function must not be called when the minimal exponent ‘emin’ is the smallest possible one. This macro has been made such that the compiler is able to detect mismatch between the argument list OP and the function prototype of FOO. Multiple input arguments OP are supported only with C99 compilers. Otherwise, for C89 compilers, only one such argument is supported. Note: this macro is experimental and its interface might change in future versions. unsigned long ul; mpfr_t f, r; /* Code that inits and sets r, f, and ul, and if needed sets emin */ int i = mpfr_round_nearest_away (mpfr_add_ui, r, f, ul); d2223 1 a2223 1 File: mpfr.info, Node: Miscellaneous Functions, Next: Exception Related Functions, Prev: Rounding-Related Functions, Up: MPFR Interface d2229 7 a2235 7 If X or Y is NaN, set X to NaN; note that the NaN flag is set as usual. If X and Y are equal, X is unchanged. Otherwise, if X is different from Y, replace X by the next floating-point number (with the precision of X and the current exponent range) in the direction of Y (the infinite values are seen as the smallest and largest floating-point numbers). If the result is zero, it keeps the same sign. No underflow, overflow, or inexact exception is raised. d2281 3 a2283 7 Note: the note for ‘mpfr_urandomb’ holds too. Moreover, the exact number (the random value to be rounded) and the next random state do not depend on the current exponent range and the rounding mode. However, they depend on the target precision: from the same state of the random generator, if the precision of the destination is changed, then the value may be completely different (and the state of the random generator is different too). a2284 2 -- Function: int mpfr_nrandom (mpfr_t ROP1, gmp_randstate_t STATE, mpfr_rnd_t RND) d2287 2 a2288 4 Generate one (possibly two for ‘mpfr_grandom’) random floating-point number according to a standard normal Gaussian distribution (with mean zero and variance one). For ‘mpfr_grandom’, if ROP2 is a null pointer, then only one value is d2293 1 a2293 1 Gaussian distribution and then rounded in the direction RND. d2295 3 a2297 2 The ‘gmp_randstate_t’ argument should be created using the GMP ‘gmp_randinit’ function (see the GMP manual). d2299 6 a2304 6 For ‘mpfr_grandom’, the combination of the ternary values is returned like with ‘mpfr_sin_cos’. If ROP2 is a null pointer, the second ternary value is assumed to be 0 (note that the encoding of the only ternary value is not the same as the usual encoding for functions that return only one result). Otherwise the ternary value of a random number is always non-zero. a2309 10 Note: ‘mpfr_nrandom’ is much more efficient than ‘mpfr_grandom’, especially for large precision. Thus ‘mpfr_grandom’ is marked as deprecated and will be removed in a future release. -- Function: int mpfr_erandom (mpfr_t ROP1, gmp_randstate_t STATE, mpfr_rnd_t RND) Generate one random floating-point number according to an exponential distribution, with mean one. Other characteristics are identical to ‘mpfr_nrandom’. d2312 2 a2313 4 number and the significand is considered in [1/2,1). For this function, X is allowed to be outside of the current range of acceptable values. The behavior for NaN, infinity or zero is undefined. d2316 4 a2319 3 Set the exponent of X to E if X is a non-zero ordinary number and E is in the current exponent range, and return 0; otherwise, return a non-zero value (X is not changed). a2383 5 -- Function: int mpfr_buildopt_float128_p (void) Return a non-zero value if MPFR was compiled with ‘__float128’ support (that is, MPFR was built with the ‘--enable-float128’ configure option), return zero otherwise. a2393 8 -- Function: int mpfr_buildopt_sharedcache_p (void) Return a non-zero value if MPFR was compiled so that all threads share the same cache for one MPFR constant, like ‘mpfr_const_pi’ or ‘mpfr_const_log2’ (that is, MPFR was built with the ‘--enable-shared-cache’ configure option), return zero otherwise. If the return value is non-zero, MPFR applications may need to be compiled with the ‘-pthread’ option. d2419 6 a2424 19 largest exponent is not changed), and zero otherwise. For the subsequent operations, it is the user’s responsibility to check that any floating-point value used as an input is in the new exponent range (for example using ‘mpfr_check_range’). If a floating-point value outside the new exponent range is used as an input, the default behavior is undefined, in the sense of the ISO C standard; the behavior may also be explicitly documented, such as for ‘mpfr_check_range’. Note: Caches may still have values outside the current exponent range. This is not an issue as the user cannot use these caches directly via the API (MPFR extends the exponent range internally when need be). If ‘emin’ > ‘emax’ and a floating-point value needs to be produced as output, the behavior is undefined (‘mpfr_set_emin’ and ‘mpfr_set_emax’ do not check this condition as it might occur between successive calls to these two functions). d2437 1 a2437 1 This function assumes that X is the correctly rounded value of some d2461 2 a2462 3 is outside the subnormal exponent range of the emulated floating-point system, this function just propagates the *note ternary value:: T; otherwise, it rounds X to precision d2475 1 a2475 1 the current exponent range of MPFR (due to a too small ‘emax’), the a2483 5 Warning! If you change ‘emin’ (with ‘mpfr_set_emin’) just before calling ‘mpfr_subnormalize’, you need to make sure that the value is in the current exponent range of MPFR. But it is better to change ‘emin’ before any computation, if possible. d2505 3 a2507 25 Note that ‘mpfr_set_emin’ and ‘mpfr_set_emax’ are called early enough in order to make sure that all computed values are in the current exponent range. Warning! This emulates a double IEEE 754 arithmetic with correct rounding in the subnormal range, which may not be the case for your hardware. Below is another example showing how to emulate fixed-point arithmetic in a specific case. Here we compute the sine of the integers 1 to 17 with a result in a fixed-point arithmetic rounded at 2 power -42 (using the fact that the result is at most 1 in absolute value): { mpfr_t x; int i, inex; mpfr_set_emin (-41); mpfr_init2 (x, 42); for (i = 1; i <= 17; i++) { mpfr_set_ui (x, i, MPFR_RNDN); inex = mpfr_sin (x, x, MPFR_RNDZ); mpfr_subnormalize (x, inex, MPFR_RNDZ); mpfr_dump (x); } mpfr_clear (x); } d2515 2 a2516 7 Clear (lower) the underflow, overflow, divide-by-zero, invalid, inexact and _erange_ flags. -- Function: void mpfr_clear_flags (void) Clear (lower) all global flags (underflow, overflow, divide-by-zero, invalid, inexact, _erange_). Note: a group of flags can be cleared by using ‘mpfr_flags_clear’. d2524 6 a2529 2 Set (raise) the underflow, overflow, divide-by-zero, invalid, inexact and _erange_ flags. a2540 35 The ‘mpfr_flags_’ functions below that take an argument MASK can operate on any subset of the exception flags: a flag is part of this subset (or group) if and only if the corresponding bit of the argument MASK is set. The ‘MPFR_FLAGS_’ macros will normally be used to build this argument. *Note Exceptions::. -- Function: void mpfr_flags_clear (mpfr_flags_t MASK) Clear (lower) the group of flags specified by MASK. -- Function: void mpfr_flags_set (mpfr_flags_t MASK) Set (raise) the group of flags specified by MASK. -- Function: mpfr_flags_t mpfr_flags_test (mpfr_flags_t MASK) Return the flags specified by MASK. To test whether any flag from MASK is set, compare the return value to 0. You can also test individual flags by AND’ing the result with ‘MPFR_FLAGS_’ macros. Example: mpfr_flags_t t = mpfr_flags_test (MPFR_FLAGS_UNDERFLOW| MPFR_FLAGS_OVERFLOW) ... if (t) /* underflow and/or overflow (unlikely) */ { if (t & MPFR_FLAGS_UNDERFLOW) { /* handle underflow */ } if (t & MPFR_FLAGS_OVERFLOW) { /* handle overflow */ } } -- Function: mpfr_flags_t mpfr_flags_save (void) Return all the flags. It is equivalent to ‘mpfr_flags_test(MPFR_FLAGS_ALL)’. -- Function: void mpfr_flags_restore (mpfr_flags_t FLAGS, mpfr_flags_t MASK) Restore the flags specified by MASK to their state represented in FLAGS. d2552 4 a2555 3 many programs written for MPF can be compiled directly against MPFR without any changes. All operations are then performed with the default MPFR rounding mode, which can be reset with d2558 4 a2561 8 Warning! There are some differences. In particular: • The precision is different: MPFR rounds to the exact number of bits (zeroing trailing bits in the internal representation). Users may need to increase the precision of their variables. • The exponent range is also different. • The formatted output functions (‘gmp_printf’, etc.) will not work for arguments of arbitrary-precision floating-point type (‘mpf_t’, which ‘mpf2mpfr.h’ redefines as ‘mpfr_t’). d2647 13 a2659 14 • if abs(KIND) = ‘MPFR_NAN_KIND’, X is set to NaN; • if abs(KIND) = ‘MPFR_INF_KIND’, X is set to the infinity of the same sign as KIND; • if abs(KIND) = ‘MPFR_ZERO_KIND’, X is set to the zero of the same sign as KIND; • if abs(KIND) = ‘MPFR_REGULAR_KIND’, X is set to the regular number whose sign is the one of KIND, and whose exponent and significand are given by EXP and SIGNIFICAND. In all cases, SIGNIFICAND will be used directly for further computing involving X. This function does not allocate anything. A floating-point number initialized with this function cannot be resized using ‘mpfr_set_prec’ or ‘mpfr_prec_round’, or cleared using ‘mpfr_clear’! The SIGNIFICAND must have been initialized with ‘mpfr_custom_init’ using the same precision PREC. d2673 4 a2676 6 number and the significand is considered in [1/2,1). But if X is NaN, infinity or zero, contrary to ‘mpfr_get_exp’ (where the behavior is undefined), the return value is here an unspecified, valid value of the ‘mpfr_exp_t’ type. The behavior of this function for any ‘mpfr_t’ not initialized with ‘mpfr_custom_init_set’ is undefined. d2720 1 a2720 1 File: mpfr.info, Node: API Compatibility, Next: MPFR and the IEEE 754 Standard, Prev: MPFR Interface, Up: Top d2819 1 a2819 8 MPFR 3.0 (however no rounding mode ‘GMP_RNDA’ exists). Faithful rounding (‘MPFR_RNDF’) was added in MPFR 4.0, but currently, it is partially supported. The flags-related macros, whose name starts with ‘MPFR_FLAGS_’, were added in MPFR 4.0 (for the new functions ‘mpfr_flags_clear’, ‘mpfr_flags_restore’, ‘mpfr_flags_set’ and ‘mpfr_flags_test’, in particular). d2827 2 a2828 2 We give here in alphabetical order the functions (and function-like macros) that were added after MPFR 2.2, and in which MPFR version. d2836 1 a2836 7 • ‘mpfr_beta’ in MPFR 4.0 (incomplete, experimental). • ‘mpfr_buildopt_decimal_p’ in MPFR 3.0. • ‘mpfr_buildopt_float128_p’ in MPFR 4.0. • ‘mpfr_buildopt_gmpinternals_p’ in MPFR 3.1. d2838 2 a2839 5 • ‘mpfr_buildopt_sharedcache_p’ in MPFR 4.0. • ‘mpfr_buildopt_tls_p’ in MPFR 3.0. • ‘mpfr_buildopt_tune_case’ in MPFR 3.1. a2863 7 • ‘mpfr_erandom’ in MPFR 4.0. • ‘mpfr_flags_clear’, ‘mpfr_flags_restore’, ‘mpfr_flags_save’, ‘mpfr_flags_set’ and ‘mpfr_flags_test’ in MPFR 4.0. • ‘mpfr_fmma’ and ‘mpfr_fmms’ in MPFR 4.0. a2865 2 • ‘mpfr_fmodquo’ in MPFR 4.0. a2867 2 • ‘mpfr_fpif_export’ and ‘mpfr_fpif_import’ in MPFR 4.0. a2869 4 • ‘mpfr_free_cache2’ in MPFR 4.0. • ‘mpfr_free_pool’ in MPFR 4.0. a2871 5 • ‘mpfr_gamma_inc’ in MPFR 4.0. • ‘mpfr_get_float128’ in MPFR 4.0 if configured with ‘--enable-float128’. a2875 2 • ‘mpfr_get_q’ in MPFR 4.0. a2890 2 • ‘mpfr_log_ui’ in MPFR 4.0. a2894 2 • ‘mpfr_mp_memory_cleanup’ in MPFR 4.0. a2896 2 • ‘mpfr_nrandom’ in MPFR 4.0. a2904 6 • ‘mpfr_rint_roundeven’ and ‘mpfr_roundeven’ in MPFR 4.0. • ‘mpfr_round_nearest_away’ in MPFR 4.0. • ‘mpfr_rootn_ui’ in MPFR 4.0. a2906 3 • ‘mpfr_set_float128’ in MPFR 4.0 if configured with ‘--enable-float128’. a2941 5 • ‘mpfr_abs’, ‘mpfr_neg’ and ‘mpfr_set’ changed in MPFR 4.0. In previous MPFR versions, the sign bit of a NaN was unspecified; however, in practice, it was set as now specified except for ‘mpfr_neg’ with a reused argument: ‘mpfr_neg(x,x,rnd)’. a2948 4 • ‘mpfr_eint’ changed in MPFR 4.0. This function now returns the value of the E1/eint1 function for negative argument (before MPFR 4.0, it was returning NaN). a2957 5 • ‘mpfr_get_str’ changed in MPFR 4.0. This function now sets the NaN flag on NaN input (to follow the usual MPFR rules on NaN and IEEE 754-2008 recommendations on string conversions from Subclause 5.12.1) and sets the inexact flag when the conversion is inexact. a2982 10 • ‘mpfr_set_exp’ changed in MPFR 4.0. Before MPFR 4.0, the exponent was set whatever the contents of the MPFR object in argument. In practice, this could be useful as a low-level function when the MPFR number was being constructed by setting the fields of its internal structure, but the API does not provide a way to do this except by using internals. Thus, for the API, this behavior was useless and could quickly lead to undefined behavior due to the fact that the generated value could have an invalid format if the MPFR object contained a special value (NaN, infinity or zero). a3004 7 • ‘mpfr_sum’ changed in MPFR 4.0. The ‘mpfr_sum’ function has completely been rewritten for MPFR 4.0, with an update of the specification: the sign of an exact zero result is now specified, and the return value is now the usual ternary value. The old ‘mpfr_sum’ implementation could also take all the memory and crash on inputs of very different magnitude. a3013 7 • ‘mpfr_urandom’ changed in MPFR 4.0. The next random state no longer depends on the current exponent range and the rounding mode. The exceptions due to the rounding of the random number are now correctly generated, following the uniform distribution. As a consequence, the returned values can be different between MPFR 4.0 and previous MPFR versions. a3024 7 Macros ‘mpfr_add_one_ulp’ and ‘mpfr_sub_one_ulp’ have been removed in MPFR 4.0. They were no longer documented since MPFR 2.1.0 and were announced as deprecated since MPFR 3.1.0. Function ‘mpfr_grandom’ is marked as deprecated in MPFR 4.0. It will be removed in a future release. a3057 3 The way memory is allocated by MPFR should be regarded as well-specified only as of MPFR 4.0. d3059 1 a3059 51 File: mpfr.info, Node: MPFR and the IEEE 754 Standard, Next: Contributors, Prev: API Compatibility, Up: Top 7 MPFR and the IEEE 754 Standard ******************************** This section describes differences between MPFR and the IEEE 754 standard, and behaviors that are not specified yet in IEEE 754. The MPFR numbers do not include subnormals. The reason is that subnormals are less useful than in IEEE 754 as the default exponent range in MPFR is large and they would have made the implementation more complex. However, subnormals can be emulated using ‘mpfr_subnormalize’. MPFR has a single NaN. The behavior is similar either to a signaling NaN or to a quiet NaN, depending on the context. For any function returning a NaN (either produced or propagated), the NaN flag is set, while in IEEE 754, some operations are quiet (even on a signaling NaN). The ‘mpfr_rec_sqrt’ function differs from IEEE 754 on −0, where it gives +Inf (like for +0), following the usual limit rules, instead of −Inf. The ‘mpfr_root’ function predates IEEE 754-2008 and behaves differently from its rootn operation. It is deprecated and ‘mpfr_rootn_ui’ should be used instead. Operations with an unsigned zero: For functions taking an argument of integer or rational type, a zero of such a type is unsigned unlike the floating-point zero (this includes the zero of type ‘unsigned long’, which is a mathematical, exact zero, as opposed to a floating-point zero, which may come from an underflow and whose sign would correspond to the sign of the real non-zero value). Unless documented otherwise, this zero is regarded as +0, as if it were first converted to a MPFR number with ‘mpfr_set_ui’ or ‘mpfr_set_si’ (thus the result may not agree with the usual limit rules applied to a mathematical zero). This is not the case of addition and subtraction (‘mpfr_add_ui’, etc.), but for these functions, only the sign of a zero result would be affected, with +0 and −0 considered equal. Such operations are currently out of the scope of the IEEE 754 standard, and at the time of specification in MPFR, the Floating-Point Working Group in charge of the revision of IEEE 754 did not want to discuss issues with non-floating-point types in general. Note also that some obvious differences may come from the fact that in MPFR, each variable has its own precision. For instance, a subtraction of two numbers of the same sign may yield an overflow; idem for a call to ‘mpfr_set’, ‘mpfr_neg’ or ‘mpfr_abs’, if the destination variable has a smaller precision.  File: mpfr.info, Node: Contributors, Next: References, Prev: MPFR and the IEEE 754 Standard, Up: Top d3074 9 a3082 10 contributed the original version of the ‘mpfr_sum’ function (used up to MPFR 3.1). Emmanuel Jeandel, from ENS-Lyon too, contributed the generic hypergeometric code, as well as the internal function ‘mpfr_exp3’, a first implementation of the sine and cosine, and improved versions of ‘mpfr_const_log2’ and ‘mpfr_const_pi’. Ludovic Meunier helped in the design of the ‘mpfr_erf’ code. Jean-Luc Rémy contributed the ‘mpfr_zeta’ code. Fabrice Rouillier contributed the ‘mpfr_xxx_z’ and ‘mpfr_xxx_q’ functions, and helped to the Microsoft Windows porting. Damien Stehlé contributed the ‘mpfr_get_ld_2exp’ function. Charles Karney contributed the ‘mpfr_nrandom’ and ‘mpfr_erandom’ functions. d3103 1 a3103 5 2012 was partly supported by the ERC grant ANTICS of Andreas Enge. The MPFR-MPC workshop in January 2013 was partly supported by the ERC grant ANTICS, the GDR IM and the Caramel project-team, during which Mickaël Gastineau contributed the MPFRbench program, and Fredrik Johannsson a faster version of ‘mpfr_const_euler’. d3112 2 a3113 4 Cambridge University Press, Cambridge Monographs on Applied and Computational Mathematics, Number 18, 2010. Electronic version freely available at . d3122 1 a3122 1 Library", version 6.1.2, 2016, . d3137 1 a3137 1 Implementation", Birkhäuser, Boston, 3rd edition, 2016. a3613 2 * Group of flags: Nomenclature and Types. (line 39) d3623 1 a3623 1 * Integer related functions: Integer and Remainder Related Functions. a3649 2 * Remainder related functions: Integer and Remainder Related Functions. (line 3) d3651 1 a3651 1 * Rounding mode related functions: Rounding-Related Functions. d3662 1 a3662 1 * Ternary value: Rounding Modes. (line 48) d3676 3 a3678 3 (line 186) * mpfr_acos: Special Functions. (line 66) * mpfr_acosh: Special Functions. (line 130) d3691 4 a3694 4 * mpfr_agm: Special Functions. (line 250) * mpfr_ai: Special Functions. (line 269) * mpfr_asin: Special Functions. (line 67) * mpfr_asinh: Special Functions. (line 131) d3696 4 a3699 5 (line 197) * mpfr_atan: Special Functions. (line 68) * mpfr_atan2: Special Functions. (line 78) * mpfr_atanh: Special Functions. (line 132) * mpfr_beta: Special Functions. (line 192) d3701 1 a3701 3 (line 185) * mpfr_buildopt_float128_p: Miscellaneous Functions. (line 180) d3703 1 a3703 3 (line 190) * mpfr_buildopt_sharedcache_p: Miscellaneous Functions. (line 195) d3705 1 a3705 1 (line 174) d3707 2 a3708 2 (line 203) * mpfr_can_round: Rounding-Related Functions. d3711 2 a3712 2 (line 115) * mpfr_ceil: Integer and Remainder Related Functions. d3715 1 a3715 1 (line 50) d3721 1 a3721 1 (line 153) d3723 1 a3723 1 (line 156) d3725 1 a3725 1 (line 160) d3727 1 a3727 1 (line 155) d3729 1 a3729 1 (line 154) d3731 1 a3731 1 (line 152) d3733 1 a3733 1 (line 151) d3756 4 a3759 4 * mpfr_const_catalan: Special Functions. (line 280) * mpfr_const_euler: Special Functions. (line 279) * mpfr_const_log2: Special Functions. (line 277) * mpfr_const_pi: Special Functions. (line 278) d3761 10 a3770 10 (line 127) * mpfr_cos: Special Functions. (line 44) * mpfr_cosh: Special Functions. (line 110) * mpfr_cot: Special Functions. (line 62) * mpfr_coth: Special Functions. (line 126) * mpfr_csc: Special Functions. (line 61) * mpfr_csch: Special Functions. (line 125) * mpfr_custom_get_exp: Custom Interface. (line 76) * mpfr_custom_get_kind: Custom Interface. (line 66) * mpfr_custom_get_significand: Custom Interface. (line 71) d3774 1 a3774 1 * mpfr_custom_move: Custom Interface. (line 85) d3777 1 a3777 1 * mpfr_digamma: Special Functions. (line 187) d3779 1 a3779 1 (line 197) d3783 1 a3783 1 (line 176) d3785 1 a3785 1 (line 52) d3787 1 a3787 1 (line 212) d3789 1 a3789 1 (line 210) a3799 2 * mpfr_dump: Input and Output Functions. (line 68) d3804 1 a3804 1 * mpfr_eint: Special Functions. (line 140) d3806 1 a3806 1 (line 31) a3808 2 * mpfr_erandom: Miscellaneous Functions. (line 99) d3810 8 a3817 8 (line 179) * mpfr_erf: Special Functions. (line 204) * mpfr_erfc: Special Functions. (line 205) * mpfr_exp: Special Functions. (line 34) * mpfr_exp10: Special Functions. (line 36) * mpfr_exp2: Special Functions. (line 35) * mpfr_expm1: Special Functions. (line 40) * mpfr_fac_ui: Special Functions. (line 136) d3819 1 a3819 1 (line 168) d3821 1 a3821 1 (line 164) d3823 1 a3823 1 (line 162) d3825 1 a3825 1 (line 166) d3827 1 a3827 1 (line 167) d3829 1 a3829 1 (line 163) d3831 1 a3831 1 (line 161) d3833 2 a3834 14 (line 165) * mpfr_flags_clear: Exception Related Functions. (line 190) * mpfr_flags_restore: Exception Related Functions. (line 214) * mpfr_flags_save: Exception Related Functions. (line 210) * mpfr_flags_set: Exception Related Functions. (line 193) * mpfr_flags_t: Nomenclature and Types. (line 39) * mpfr_flags_test: Exception Related Functions. (line 196) * mpfr_floor: Integer and Remainder Related Functions. d3836 4 a3839 12 * mpfr_fma: Special Functions. (line 230) * mpfr_fmma: Special Functions. (line 240) * mpfr_fmms: Special Functions. (line 242) * mpfr_fmod: Integer and Remainder Related Functions. (line 106) * mpfr_fmodquo: Integer and Remainder Related Functions. (line 108) * mpfr_fms: Special Functions. (line 232) * mpfr_fpif_export: Input and Output Functions. (line 43) * mpfr_fpif_import: Input and Output Functions. (line 54) d3841 4 a3844 6 (line 161) * mpfr_frac: Integer and Remainder Related Functions. (line 89) * mpfr_free_cache: Special Functions. (line 288) * mpfr_free_cache2: Special Functions. (line 295) * mpfr_free_pool: Special Functions. (line 309) d3846 1 a3846 1 (line 156) d3848 2 a3849 3 (line 49) * mpfr_gamma: Special Functions. (line 155) * mpfr_gamma_inc: Special Functions. (line 156) d3853 1 a3853 1 (line 10) d3856 1 a3856 1 * mpfr_get_default_rounding_mode: Rounding-Related Functions. d3859 1 a3859 1 (line 36) d3863 1 a3863 1 (line 43) d3865 1 a3865 1 (line 42) d3869 1 a3869 1 (line 41) d3871 1 a3871 1 (line 40) d3873 1 a3873 1 (line 105) d3875 1 a3875 3 (line 85) * mpfr_get_float128: Conversion Functions. (line 9) d3881 1 a3881 1 (line 38) d3883 1 a3883 1 (line 165) a3885 2 * mpfr_get_q: Conversion Functions. (line 80) d3887 2 a3889 2 * mpfr_get_sj: Conversion Functions. (line 23) d3891 1 a3891 1 (line 98) d3893 2 a3895 2 * mpfr_get_uj: Conversion Functions. (line 24) d3897 1 a3897 1 (line 134) d3899 1 a3899 1 (line 71) d3901 1 a3901 1 (line 58) d3903 1 a3903 1 (line 69) d3908 1 a3908 1 * mpfr_hypot: Special Functions. (line 260) d3910 1 a3910 1 (line 178) d3941 5 a3945 5 * mpfr_integer_p: Integer and Remainder Related Functions. (line 135) * mpfr_j0: Special Functions. (line 209) * mpfr_j1: Special Functions. (line 210) * mpfr_jn: Special Functions. (line 211) d3952 7 a3958 8 * mpfr_lgamma: Special Functions. (line 177) * mpfr_li2: Special Functions. (line 150) * mpfr_lngamma: Special Functions. (line 169) * mpfr_log: Special Functions. (line 19) * mpfr_log10: Special Functions. (line 23) * mpfr_log1p: Special Functions. (line 30) * mpfr_log2: Special Functions. (line 22) * mpfr_log_ui: Special Functions. (line 20) d3963 4 a3966 5 * mpfr_min_prec: Rounding-Related Functions. (line 84) * mpfr_modf: Integer and Remainder Related Functions. (line 96) * mpfr_mp_memory_cleanup: Special Functions. (line 314) d3970 1 a3970 1 (line 50) d3972 1 a3972 1 (line 205) d3974 1 a3974 1 (line 203) d3986 1 a3986 1 (line 177) d3990 1 a3990 1 (line 185) a3996 2 * mpfr_nrandom: Miscellaneous Functions. (line 67) d4002 1 a4002 1 (line 175) d4004 1 a4004 1 (line 139) d4006 1 a4006 1 (line 143) d4008 1 a4008 1 (line 141) d4010 2 a4011 2 (line 145) * mpfr_prec_round: Rounding-Related Functions. d4016 3 a4018 3 (line 168) * mpfr_print_rnd_mode: Rounding-Related Functions. (line 88) d4020 1 a4020 1 (line 107) d4024 6 a4029 6 (line 42) * mpfr_remainder: Integer and Remainder Related Functions. (line 110) * mpfr_remquo: Integer and Remainder Related Functions. (line 112) * mpfr_rint: Integer and Remainder Related Functions. d4031 8 a4038 10 * mpfr_rint_ceil: Integer and Remainder Related Functions. (line 52) * mpfr_rint_floor: Integer and Remainder Related Functions. (line 53) * mpfr_rint_round: Integer and Remainder Related Functions. (line 55) * mpfr_rint_roundeven: Integer and Remainder Related Functions. (line 57) * mpfr_rint_trunc: Integer and Remainder Related Functions. (line 59) d4042 2 a4043 4 (line 128) * mpfr_rootn_ui: Basic Arithmetic Functions. (line 116) * mpfr_round: Integer and Remainder Related Functions. d4045 2 a4046 6 * mpfr_roundeven: Integer and Remainder Related Functions. (line 10) * mpfr_round_nearest_away: Rounding-Related Functions. (line 93) * mpfr_sec: Special Functions. (line 60) * mpfr_sech: Special Functions. (line 124) d4050 1 a4050 1 (line 121) d4054 1 a4054 1 (line 21) d4057 1 a4057 1 * mpfr_set_default_rounding_mode: Rounding-Related Functions. d4060 1 a4060 1 (line 167) d4066 1 a4066 1 (line 170) d4068 1 a4068 1 (line 112) d4070 1 a4070 3 (line 25) * mpfr_set_float128: Assignment Functions. (line 19) d4074 1 a4074 1 (line 169) d4076 1 a4076 1 (line 154) d4080 1 a4080 1 (line 153) d4082 1 a4082 1 (line 168) d4084 1 a4084 1 (line 166) d4088 1 a4088 1 (line 25) d4090 1 a4090 1 (line 24) d4094 1 a4094 1 (line 61) d4098 1 a4098 1 (line 65) d4100 1 a4100 1 (line 73) d4104 1 a4104 1 (line 59) d4108 1 a4108 1 (line 63) d4110 1 a4110 1 (line 165) d4112 1 a4112 1 (line 23) d4114 1 a4114 1 (line 155) d4116 1 a4116 1 (line 67) d4120 5 a4124 5 (line 117) * mpfr_sin: Special Functions. (line 45) * mpfr_sinh: Special Functions. (line 111) * mpfr_sinh_cosh: Special Functions. (line 116) * mpfr_sin_cos: Special Functions. (line 50) d4130 1 a4130 1 (line 184) d4132 1 a4132 1 (line 174) d4136 1 a4136 1 (line 100) d4138 1 a4138 1 (line 101) d4140 1 a4140 1 (line 91) d4144 1 a4144 1 (line 73) d4155 1 a4155 1 * mpfr_sum: Special Functions. (line 321) d4157 1 a4157 1 (line 161) d4160 4 a4163 4 * mpfr_tan: Special Functions. (line 46) * mpfr_tanh: Special Functions. (line 112) * mpfr_trunc: Integer and Remainder Related Functions. (line 11) d4167 1 a4167 1 (line 149) d4169 1 a4169 1 (line 147) d4173 1 a4173 1 (line 174) d4181 1 a4181 1 (line 198) d4183 1 a4183 1 (line 137) d4185 1 a4185 1 (line 138) d4187 2 a4189 2 * MPFR_VERSION_NUM: Miscellaneous Functions. (line 157) d4191 1 a4191 1 (line 140) d4193 1 a4193 1 (line 141) d4195 1 a4195 1 (line 162) d4197 1 a4197 1 (line 169) d4199 1 a4199 1 (line 186) d4201 4 a4204 4 (line 175) * mpfr_y0: Special Functions. (line 220) * mpfr_y1: Special Functions. (line 221) * mpfr_yn: Special Functions. (line 222) d4207 2 a4208 2 * mpfr_zeta: Special Functions. (line 198) * mpfr_zeta_ui: Special Functions. (line 199) d4216 41 a4256 43 Node: Copying2042 Node: Introduction to MPFR3805 Node: Installing MPFR6208 Node: Reporting Bugs11654 Node: MPFR Basics13684 Node: Headers and Libraries14044 Node: Nomenclature and Types17640 Node: MPFR Variable Conventions19899 Node: Rounding Modes21441 Ref: ternary value24166 Node: Floating-Point Values on Special Numbers26152 Node: Exceptions29412 Node: Memory Handling33240 Node: Getting the Best Efficiency Out of MPFR37040 Node: MPFR Interface37952 Node: Initialization Functions40238 Node: Assignment Functions47553 Node: Combined Initialization and Assignment Functions57499 Node: Conversion Functions58800 Node: Basic Arithmetic Functions69381 Node: Comparison Functions80277 Node: Special Functions83765 Node: Input and Output Functions101974 Node: Formatted Output Functions106751 Node: Integer and Remainder Related Functions116956 Node: Rounding-Related Functions124484 Node: Miscellaneous Functions131001 Node: Exception Related Functions141493 Node: Compatibility with MPF151733 Node: Custom Interface154679 Node: Internals159310 Node: API Compatibility160854 Node: Type and Macro Changes162802 Node: Added Functions165985 Node: Changed Functions170499 Node: Removed Functions177095 Node: Other Changes177825 Node: MPFR and the IEEE 754 Standard179526 Node: Contributors182143 Node: References185200 Node: GNU Free Documentation License187084 Node: Concept Index209677 Node: Function and Type Index216049 @ 1.1.1.3 log @import mpfr 4.0.1. main changes since 3.1.5 are: Changes from version 4.0.0 to version 4.0.1: - Bug fixes (see ChangeLog file), in particular in mpfr_div_ui, which could yield an incorrectly rounded result to nearest when using different precisions; this bug had been present since the introduction of mpfr_div_ui, and in MPFR 4.0.0, it was affecting mpfr_div too. Changes from versions 3.1.* to version 4.0.0: - Partial support of MPFR_RNDF (faithful rounding). - New functions: mpfr_fpif_export and mpfr_fpif_import to export and import numbers in a floating-point interchange format, independent both on the number of bits per word and on the endianness. - New function mpfr_fmodquo to return the low bits of the quotient corresponding to mpfr_fmod. - New functions mpfr_flags_clear, mpfr_flags_set, mpfr_flags_test, mpfr_flags_save and mpfr_flags_restore to operate on groups of flags. - New functions mpfr_set_float128 and mpfr_get_float128 to convert from/to the __float128 type (requires --enable-float128 and compiler support). - New functions mpfr_buildopt_float128_p and mpfr_buildopt_sharedcache_p. - New functions mpfr_rint_roundeven and mpfr_roundeven, completing the other similar round-to-integer functions for rounding to nearest with the even-rounding rule. - New macro mpfr_round_nearest_away to add partial emulation of the rounding to nearest-away (as defined in IEEE 754-2008). - New functions mpfr_nrandom and mpfr_erandom to generate random numbers following normal and exponential distributions respectively. - New functions mpfr_fmma and mpfr_fmms to compute a*b+c*d and a*b-c*d. - New function mpfr_rootn_ui, similar to mpfr_root, but agreeing with the rootn function of the IEEE 754-2008 standard. - New functions mpfr_log_ui to compute the logarithm of an integer, mpfr_gamma_inc for the incomplete Gamma function. - New function mpfr_beta for the Beta function (incomplete, experimental). - New function mpfr_get_q to convert a floating-point number into rational. - Dropped K&R C compatibility. - Major speedup in mpfr_add, mpfr_sub, mpfr_mul, mpfr_div and mpfr_sqrt when all operands have the same precision and this precision is less than twice the number of bits per word, e.g., less than 128 on a 64-bit computer. - Speedup by a factor of almost 2 in the double <--> mpfr conversions (mpfr_set_d and mpfr_get_d). - Speedup in mpfr_log1p and mpfr_atanh for small arguments. - Speedup in the mpfr_const_euler function (contributed by Fredrik Johansson), in the computation of Bernoulli numbers (used in mpfr_gamma, mpfr_li2, mpfr_digamma, mpfr_lngamma and mpfr_lgamma), in mpfr_div, in mpfr_fma and mpfr_fms. @ text @d1 1 a1 1 This is mpfr.info, produced by makeinfo version 6.5 from mpfr.texi. d4 1 a4 1 Floating-Point Reliable Library, version 4.0.1. d6 1 a6 1 Copyright 1991, 1993-2018 Free Software Foundation, Inc. d26 1 a26 1 Floating-Point Reliable Library, version 4.0.1. d28 1 a28 1 Copyright 1991, 1993-2018 Free Software Foundation, Inc. a45 1 * MPFR and the IEEE 754 Standard:: d101 1 a101 3 current processors), possibly except in faithful rounding. It does not depend either on the machine rounding mode or rounding precision; d108 9 a116 12 for other mathematical functions. Faithful rounding (partially supported) is provided too, but the results may no longer be reproducible. In particular, with a precision of 53 bits and in any of the four standard rounding modes, MPFR is able to exactly reproduce all computations with double-precision machine floating-point numbers (e.g., ‘double’ type in C, with a C implementation that rigorously follows Annex F of the ISO C99 standard and ‘FP_CONTRACT’ pragma set to ‘OFF’) on the four arithmetic operations and the square root, except the default exponent range is much wider and subnormal numbers are not implemented (but can be emulated). d279 2 a280 2 *Note Reporting Bugs::. Some bug fixes are available on the MPFR 4.0.1 web page . d286 1 a286 1 or . d295 1 a295 1 on the MPFR 4.0.1 web page and the FAQ d315 1 a315 1 can be extracted using ‘cc -V’ on some machines, or, if you are using a346 1 * Getting the Best Efficiency Out of MPFR:: d372 5 a376 5 prototypes for these functions. Moreover, under some platforms (in particular with C++ compilers), users may need to define ‘MPFR_USE_INTMAX_T’ (and should do it for portability) before ‘mpfr.h’ has been included; of course, it is possible to do that on the command line, e.g., with ‘-DMPFR_USE_INTMAX_T’. a414 10 Alternatively, it is possible to use ‘pkg-config’ (a file ‘mpfr.pc’ is provided as of MPFR 4.0): cc myprogram.c $(pkg-config --cflags --libs mpfr) Note that the ‘MPFR_’ and ‘mpfr_’ prefixes are reserved for MPFR. As a general rule, in order to avoid clashes, software using MPFR (directly or indirectly) and system headers/libraries should not define macros and symbols using these prefixes. d439 1 a439 1 equal to 1. a453 4 MPFR has a global (or per-thread) flag for each supported exception and provides operations on flags (*note Exceptions::). This C data type is used to represent a group of flags (or a mask). d460 2 a461 2 Before you can assign to a MPFR variable, you need to initialize it by calling one of the special initialization functions. When you are done d488 1 a488 1 The following rounding modes are supported: a495 10 • ‘MPFR_RNDF’: faithful rounding. This feature is currently experimental. Specific support for this rounding mode has been added to some functions, such as the basic operations (addition, subtraction, multiplication, square, division, square root) or when explicitly documented. It might also work with other functions, as it is possible that they do not need modification in their code; even though a correct behavior is not guaranteed yet (corrections were done when failures occurred in the test suite, but almost nothing has been checked manually), failures should be regarded as bugs and reported, so that they can be fixed. a505 14 The ‘MPFR_RNDF’ mode works as follows: the computed value is either that corresponding to ‘MPFR_RNDD’ or that corresponding to ‘MPFR_RNDU’. In particular when those values are identical, i.e., when the result of the corresponding operation is exactly representable, that exact result is returned. Thus, the computed result can take at most two possible values, and in absence of underflow/overflow, the corresponding error is strictly less than one ulp (unit in the last place) of that result and of the exact result. For ‘MPFR_RNDF’, the ternary value (defined below) and the inexact flag (defined later, as with the other flags) are unspecified, the divide-by-zero flag is as with other roundings, and the underflow and overflow flags match what would be obtained in the case the computed value is the same as with ‘MPFR_RNDD’ or ‘MPFR_RNDU’. The results may not be reproducible. d605 8 a612 17 MPFR defines a global (or per-thread) flag for each supported exception. A macro evaluating to a power of two is associated with each flag and exception, in order to be able to specify a group of flags (or a mask) by OR’ing such macros. Flags can be cleared (lowered), set (raised), and tested by functions described in *note Exception Related Functions::. The supported exceptions are listed below. The macro associated with each exception is in parentheses. • Underflow (‘MPFR_FLAGS_UNDERFLOW’): An underflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent smaller than the minimum value of the current exponent range. (In the round-to-nearest mode, the halfway case is rounded toward zero.) d629 24 a652 27 • Overflow (‘MPFR_FLAGS_OVERFLOW’): An overflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent larger than the maximum value of the current exponent range. In the round-to-nearest mode, the result is infinite. Note: unlike the underflow case, there is only one possible definition of overflow here. • Divide-by-zero (‘MPFR_FLAGS_DIVBY0’): An exact infinite result is obtained from finite inputs. • NaN (‘MPFR_FLAGS_NAN’): A NaN exception occurs when the result of a function is NaN. • Inexact (‘MPFR_FLAGS_INEXACT’): An inexact exception occurs when the result of a function cannot be represented exactly and must be rounded. • Range error (‘MPFR_FLAGS_ERANGE’): A range exception occurs when a function that does not return a MPFR number (such as comparisons and conversions to an integer) has an invalid result (e.g., an argument is NaN in ‘mpfr_cmp’, or a conversion to an integer cannot be represented in the target type). Moreover, the group consisting of all the flags is represented by the ‘MPFR_FLAGS_ALL’ macro (if new flags are added in future MPFR versions, they will be added to this macro too). d665 1 a665 1 File: mpfr.info, Node: Memory Handling, Next: Getting the Best Efficiency Out of MPFR, Prev: Exceptions, Up: MPFR Basics d673 6 a678 37 library itself to compute some other function. When more precision is needed, the value is automatically recomputed; a minimum of 10% increase of the precision is guaranteed to avoid too many recomputations. MPFR functions may also create thread-local pools for internal use to avoid the cost of memory allocation. The pools can be freed with ‘mpfr_free_pool’ (but with a default MPFR build, they should not take much memory, as the allocation size is limited). At any time, the user can free various caches and pools with ‘mpfr_free_cache’ and ‘mpfr_free_cache2’. It is strongly advised to free thread-local caches before terminating a thread, and all caches before exiting when using tools like ‘valgrind’ (to avoid memory leaks being reported). MPFR allocates its memory either on the stack (for temporary memory only) or with the same allocator as the one configured for GMP: *note (gmp.info)Custom Allocation::. This means that the application must make sure that data allocated with the current allocator will not be reallocated or freed with a new allocator. So, in practice, if an application needs to change the allocator with ‘mp_set_memory_functions’, it should first free all data allocated with the current allocator: for its own data, with ‘mpfr_clear’, etc.; for the caches and pools, with ‘mpfr_mp_memory_cleanup’ in all threads where MPFR is potentially used. This function is currently equivalent to ‘mpfr_free_cache’, but ‘mpfr_mp_memory_cleanup’ is the recommended way in case the allocation method changes in the future (for instance, one may choose to allocate the caches for floating-point constants with ‘malloc’ to avoid freeing them if the allocator changes). Developers should also be aware that MPFR may also be used indirectly by libraries, so that libraries based on MPFR should provide a clean-up function calling ‘mpfr_mp_memory_cleanup’ and/or warn their users about this issue. Note: For multithreaded applications, the allocator must be valid in all threads where MPFR may be used; data allocated in one thread may be reallocated and/or freed in some other thread. a688 35 Writers of libraries using MPFR should be aware that the application and/or another library used by the application may also use MPFR, so that changing the exponent range, the default precision, or the default rounding mode may have an effect on this other use of MPFR since these data are not duplicated (unless they are in a different thread). Therefore any such value changed in a library function should be restored before the function returns (unless the purpose of the function is to do such a change). Writers of software using MPFR should also be careful when changing such a value if they use a library using MPFR (directly or indirectly), in order to make sure that such a change is compatible with the library.  File: mpfr.info, Node: Getting the Best Efficiency Out of MPFR, Prev: Memory Handling, Up: MPFR Basics 4.8 Getting the Best Efficiency Out of MPFR =========================================== Here are a few hints to get the best efficiency out of MPFR: • you should avoid allocating and clearing variables. Reuse variables whenever possible, allocate or clear outside of loops, pass temporary variables to subroutines instead of allocating them inside the subroutines; • use ‘mpfr_swap’ instead of ‘mpfr_set’ whenever possible. This will avoid copying the significands; • avoid using MPFR from C++, or make sure your C++ interface does not perform unnecessary allocations or copies; • MPFR functions work in-place: to compute ‘a = a + b’ you don’t need an auxiliary variable, you can directly write ‘mpfr_add (a, a, b, ...)’. d721 2 a722 5 The value of the standard C macro ‘errno’ may be set to non-zero after calling any MPFR function or macro, whether or not there is an error. Except when documented, MPFR will not set ‘errno’, but functions called by the MPFR code (libc functions, memory allocator, etc.) may do so. d735 2 a736 2 * Integer and Remainder Related Functions:: * Rounding-Related Functions:: a914 2 -- Function: int mpfr_set_float128 (mpfr_t ROP, __float128 OP, mpfr_rnd_t RND) d922 1 a922 6 ‘mpfr_set_si’, ‘mpfr_set_uj’, ‘mpfr_set_sj’, The ‘mpfr_set_float128’ function is built only with the configure option ‘--enable-float128’, which requires the compiler or system provides the ‘__float128’ data type (GCC 4.3 or later supports this data type); to use ‘mpfr_set_float128’, one should define the macro ‘MPFR_WANT_FLOAT128’ before including ‘mpfr.h’. ‘mpfr_set_z’, d927 7 a933 11 built only with the configure option ‘--enable-decimal-float’, and when the compiler or system provides the ‘_Decimal64’ data type (recent versions of GCC support this data type); to use ‘mpfr_set_decimal64’, one should define the macro ‘MPFR_WANT_DECIMAL_FLOATS’ before including ‘mpfr.h’. ‘mpfr_set_q’ might fail if the numerator (or the denominator) can not be represented as a ‘mpfr_t’. For ‘mpfr_set’, the sign of a NaN is propagated in order to mimic the IEEE 754 ‘copy’ operation. But contrary to IEEE 754, the NaN flag is set as usual. a1093 1 -- Function: __float128 mpfr_get_float128 (mpfr_t OP, mpfr_rnd_t RND) d1101 2 a1102 3 ‘mpfr_get_float128’ and ‘mpfr_get_decimal64’ functions are built only under some conditions: see the documentation of ‘mpfr_set_float128’ and ‘mpfr_set_decimal64’ respectively. d1109 7 a1115 9 ‘uintmax_t’ (respectively) after rounding it to an integer with respect to RND. If OP is NaN, 0 is returned and the _erange_ flag is set. If OP is too big for the return type, the function returns the maximum or the minimum of the corresponding C type, depending on the direction of the overflow; the _erange_ flag is set too. When there is no such range error, if the return value differs from OP, i.e., if OP is not an integer, the inexact flag is set. See also ‘mpfr_fits_slong_p’, ‘mpfr_fits_ulong_p’, ‘mpfr_fits_intmax_p’ and ‘mpfr_fits_uintmax_p’. d1155 1 a1155 10 0, and 0 is returned. Otherwise the return value is zero when ROP is equal to OP (i.e., when OP is an integer), positive when it is greater than OP, and negative when it is smaller than OP; moreover, if ROP differs from OP, i.e., if OP is not an integer, the inexact flag is set. -- Function: void mpfr_get_q (mpq_t ROP, mpfr_t OP) Convert OP to a ‘mpq_t’. If OP is NaN or an infinity, the _erange_ flag is set and ROP is set to 0. Otherwise the conversion is always exact. d1177 4 a1180 10 pointer. If the input is NaN, then the returned string is ‘@@NaN@@’ and the NaN flag is set. If the input is +Inf (resp. −Inf), then the returned string is ‘@@Inf@@’ (resp. ‘-@@Inf@@’). If the input number is a finite number, the exponent is written through the pointer EXPPTR (for input 0, the current minimal exponent is written); the type ‘mpfr_exp_t’ is large enough to hold the exponent in all cases. d1206 3 a1208 3 using the allocation function (*note Memory Handling::) and a pointer to the string is returned (unless the base is invalid). To free the returned string, you must use ‘mpfr_free_str’. d1211 10 a1220 10 large enough for the significand. A safe block size (sufficient for any value) is ‘max(N + 2, 7)’ if N is not zero; if N is zero, replace it by m+1, as discussed above. The extra two bytes are for a possible minus sign, and for the terminating null character, and the value 7 accounts for ‘-@@Inf@@’ plus the terminating null character. The pointer to the string STR is returned (unless the base is invalid). Like in usual functions, the inexact flag is set iff the result is inexact. d1223 4 a1226 3 Free a string allocated by ‘mpfr_get_str’ using the unallocation function (*note Memory Handling::). The block is assumed to be ‘strlen(STR)+1’ bytes. d1239 1 a1239 9 an integer in the direction RND. For instance, with the ‘MPFR_RNDU’ rounding mode on −0.5, the result will be non-zero for all these functions. For ‘MPFR_RNDF’, those functions return non-zero when it is guaranteed that the corresponding conversion function (for example ‘mpfr_get_ui’ for ‘mpfr_fits_ulong_p’), when called with faithful rounding, will always return a number that is representable in the corresponding type. As a consequence, for ‘MPFR_RNDF’, ‘mpfr_fits_ulong_p’ will return non-zero for a non-negative number less or equal to ‘ULONG_MAX’. d1259 1 a1259 1 Set ROP to OP1 + OP2 rounded in the direction RND. The IEEE 754 d1287 1 a1287 1 Set ROP to OP1 - OP2 rounded in the direction RND. The IEEE 754 d1334 4 a1337 6 zero, its sign is the product of the signs of the operands. For types having no signed zeros, 0 is considered positive; but note that if OP1 is non-zero and OP2 is zero, the result might change from ±Inf to NaN in future MPFR versions if there is an opposite decision on the IEEE 754 side. The same restrictions than for ‘mpfr_add_d’ apply to ‘mpfr_d_div’ and ‘mpfr_div_d’. a1354 12 -- Function: int mpfr_rootn_ui (mpfr_t ROP, mpfr_t OP, unsigned long int K, mpfr_rnd_t RND) Set ROP to the cubic root (resp. the Kth root) of OP rounded in the direction RND. For K = 0, set ROP to NaN. For K odd (resp. even) and OP negative (including −Inf), set ROP to a negative number (resp. NaN). If OP is zero, set ROP to zero with the sign obtained by the usual limit rules, i.e., the same sign as OP if K is odd, and positive if K is even. These functions agree with the rootn function of the IEEE 754-2008 standard (Section 9.2). d1357 4 a1360 8 This function is the same as ‘mpfr_rootn_ui’ except when OP is −0 and K is even: the result is −0 instead of +0 (the reason was to be consistent with ‘mpfr_sqrt’). Said otherwise, if OP is zero, set ROP to OP. This function predates the IEEE 754-2008 standard and behaves differently from its rootn function. It is marked as deprecated and will be removed in a future release. a1403 3 Note: When 0 is of integer type, it is regarded as +0 by these functions. We do not use the usual limit rules in this case, as these rules are not used for ‘pow’. a1411 5 The sign rule also applies to NaN in order to mimic the IEEE 754 ‘negate’ and ‘abs’ operations, i.e., for ‘mpfr_neg’, the sign is reversed, and for ‘mpfr_abs’, the sign is set to positive. But contrary to IEEE 754, the NaN flag is set as usual. d1517 4 a1520 7 Important note: in some domains, computing special functions (even more with correct rounding) is expensive, even for small precision, for example the trigonometric and Bessel functions for large argument. For some functions, the memory usage might depend not only on the output precision: it is the case of the ‘mpfr_rootn_ui’ function where the memory usage is also linear in the argument K, and of the incomplete Gamma function (dependence on the precision of OP). a1522 2 -- Function: int mpfr_log_ui (mpfr_t ROP, unsigned long OP, mpfr_rnd_t RND) a1530 4 -- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the logarithm of one plus OP, rounded in the direction RND. Set ROP to −Inf if OP is −1. a1536 4 -- Function: int mpfr_expm1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential of OP followed by a subtraction by one, rounded in the direction RND. d1574 2 a1575 2 RND: if ‘x > 0’, ‘atan2(y, x) = atan(y/x)’; if ‘x < 0’, ‘atan2(y, x) = sign(y)*(Pi - atan(abs(y/x)))’, thus a number from -Pi to Pi. d1633 8 d1643 7 a1649 7 RND. This is the sum of Euler’s constant, of the logarithm of the absolute value of OP, and of the sum for k from 1 to infinity of OP to the power k, divided by k and factorial(k). For positive OP, it corresponds to the Ei function at OP (see formula 5.1.10 from the Handbook of Mathematical Functions from Abramowitz and Stegun), and for negative OP, to the opposite of the E1 function (sometimes called eint1) at −OP (formula 5.1.1 from the same reference). d1657 2 a1658 12 -- Function: int mpfr_gamma_inc (mpfr_t ROP, mpfr_t OP, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the value of the Gamma function on OP, resp. the incomplete Gamma function on OP and OP2, rounded in the direction RND. (In the literature, ‘mpfr_gamma_inc’ is called upper incomplete Gamma function, or sometimes complementary incomplete Gamma function.) For ‘mpfr_gamma’ (and ‘mpfr_gamma_inc’ when OP2 is zero), when OP is a negative integer, ROP is set to NaN. Note: the current implementation of ‘mpfr_gamma_inc’ is slow for large values of ROP or OP, in which case some internal overflow might also occur. a1682 6 -- Function: int mpfr_beta (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the value of the Beta function at arguments OP1 and OP2. Note: the current code does not try to avoid internal overflow or underflow, and might use a huge internal precision in some cases. a1724 10 -- Function: int mpfr_fmma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_t OP4, mpfr_rnd_t RND) -- Function: int mpfr_fmms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_t OP4, mpfr_rnd_t RND) Set ROP to (OP1 times OP2) + (OP3 times OP4) (resp. (OP1 times OP2) - (OP3 times OP4)) rounded in the direction RND. In case the computation of OP1 times OP2 overflows or underflows (or that of OP3 times OP4), the result ROP is computed as if the two intermediate products were computed with rounding toward zero. d1731 1 a1731 3 mean of U_N and V_N. If any operand is negative and the other one is not zero, set ROP to NaN. If any operand is zero and the other one is finite (resp. infinite), set ROP to +0 (resp. NaN). d1758 1 a1758 2 requested. To free these caches, use ‘mpfr_free_cache’ or ‘mpfr_free_cache2’. d1761 6 a1766 31 Free all caches and pools used by MPFR internally (thoses local to the current thread and those shared by all threads). You should call this function before terminating a thread, even if you did not call ‘mpfr_const_*’ functions directly (they could have been called internally). -- Function: void mpfr_free_cache2 (mpfr_free_cache_t WAY) Free various caches and pools used by MPFR internally, as specified by WAY, which is a set of flags: • those local to the current thread if flag ‘MPFR_FREE_LOCAL_CACHE’ is set; • those shared by all threads if flag ‘MPFR_FREE_GLOBAL_CACHE’ is set. The other bits of WAY are currently ignored and are reserved for future use; they should be zero. Note: ‘mpfr_free_cache2(MPFR_FREE_LOCAL_CACHE|MPFR_FREE_GLOBAL_CACHE)’ is currently equivalent to ‘mpfr_free_cache()’. -- Function: void mpfr_free_pool (void) Free the pools used by MPFR internally. Note: This function is automatically called after the thread-local caches are freed (with ‘mpfr_free_cache’ or ‘mpfr_free_cache2’). -- Function: int mpfr_mp_memory_cleanup (void) This function should be called before calling ‘mp_set_memory_functions’. *Note Memory Handling::, for more information. Zero is returned in case of success, non-zero in case of error. Errors are currently not possible, but checking the return value is recommended for future compatibility. d1768 1 a1768 1 -- Function: int mpfr_sum (mpfr_t ROP, const mpfr_ptr TAB[], unsigned d1770 7 a1776 14 Set ROP to the sum of all elements of TAB, whose size is N, correctly rounded in the direction RND. Warning: for efficiency reasons, TAB is an array of pointers to ‘mpfr_t’, not an array of ‘mpfr_t’. If N = 0, then the result is +0, and if N = 1, then the function is equivalent to ‘mpfr_set’. For the special exact cases, the result is the same as the one obtained with a succession of additions (‘mpfr_add’) in infinite precision. In particular, if the result is an exact zero and N >= 1: • if all the inputs have the same sign (i.e., all +0 or all −0), then the result has the same sign as the inputs; • otherwise, either because all inputs are zeros with at least a +0 and a −0, or because some inputs are non-zero (but they globally cancel), the result is +0, except for the ‘MPFR_RNDD’ rounding mode, where it is −0. d1789 3 a1791 3 When using a function that takes a ‘FILE *’ argument, you must include the ‘’ standard header before ‘mpfr.h’, to allow ‘mpfr.h’ to define prototypes for these functions. a1820 49 -- Function: int mpfr_fpif_export (FILE *STREAM, mpfr_t OP) Export the number OP to the stream STREAM in a floating-point interchange format. In particular one can export on a 32-bit computer and import on a 64-bit computer, or export on a little-endian computer and import on a big-endian computer. The precision of OP and the sign bit of a NaN are stored too. Return 0 iff the export was successful. Note: this function is experimental and its interface might change in future versions. -- Function: int mpfr_fpif_import (mpfr_t OP, FILE *STREAM) Import the number OP from the stream STREAM in a floating-point interchange format (see ‘mpfr_fpif_export’). Note that the precision of OP is set to the one read from the stream, and the sign bit is always retrieved (even for NaN). If the stored precision is zero or greater than ‘MPFR_PREC_MAX’, the function fails (it returns non-zero) and OP is unchanged. If the function fails for another reason, OP is set to NaN and it is unspecified whether the precision of OP has changed to the one read from the file. Return 0 iff the import was successful. Note: this function is experimental and its interface might change in future versions. -- Function: void mpfr_dump (mpfr_t OP) Output OP on ‘stdout’ in some unspecified format, then a newline character. This function is mainly for debugging purpose. Thus invalid data may be supported. Everything that is not specified may change without breaking the ABI and may depend on the environment. The current output format is the following: a minus sign if the sign bit is set (even for NaN); ‘@@NaN@@’, ‘@@Inf@@’ or ‘0’ if the argument is NaN, an infinity or zero, respectively; otherwise the remaining of the output is as follows: ‘0.’ then the P bits of the binary significand, where P is the precision of the number; if the trailing bits are not all zeros (which must not occur with valid data), they are output enclosed by square brackets; the character ‘E’ followed by the exponent written in base 10; in case of invalid data or out-of-range exponent, this function outputs three exclamation marks (‘!!!’), followed by flags, followed by three exclamation marks (‘!!!’) again. These flags are: ‘N’ if the most significant bit of the significand is 0 (i.e., the number is not normalized); ‘T’ if there are non-zero trailing bits; ‘U’ if this is a UBF number (internal use only); ‘<’ if the exponent is less than the current minimum exponent; ‘>’ if the exponent is greater than the current maximum exponent. d1822 1 a1822 1 File: mpfr.info, Node: Formatted Output Functions, Next: Integer and Remainder Related Functions, Prev: Input and Output Functions, Up: MPFR Interface d1848 1 a1848 2 related to the internal precision of the ‘mpfr_t’ variable), but note that for ‘Re’, the default precision is not the same as the one for ‘e’. d1933 3 a1935 3 ‘e’ ‘E’ scientific-format float ‘f’ ‘F’ fixed-point float ‘g’ ‘G’ fixed-point or scientific float d1972 2 a1973 2 For all the following functions, if the number of characters that ought to be written exceeds the maximum limit ‘INT_MAX’ for an ‘int’, nothing d1975 2 a1976 5 function returns −1, sets the _erange_ flag, and ‘errno’ is set to ‘EOVERFLOW’ if the ‘EOVERFLOW’ macro is defined (such as on POSIX systems). Note, however, that ‘errno’ might be changed to another value by some internal library call if another error occurs there (currently, this would come from the unallocation function). d2010 1 a2010 1 characters that would have been written had N been sufficiently d2018 5 a2022 6 allocated using the allocation function (*note Memory Handling::). A pointer to the block is stored in STR. The block of memory must be freed using ‘mpfr_free_str’. The return value is the number of characters written in the string, excluding the null-terminator, or a negative value if an error occurred, in which case the contents of STR are undefined. d2025 1 a2025 1 File: mpfr.info, Node: Integer and Remainder Related Functions, Next: Rounding-Related Functions, Prev: Formatted Output Functions, Up: MPFR Interface a2033 1 -- Function: int mpfr_roundeven (mpfr_t ROP, mpfr_t OP) d2036 11 a2046 21 nearest representable integer in the given direction RND, and the other five functions behave in a similar way with some fixed rounding mode: • ‘mpfr_ceil’: to the next higher or equal representable integer (like ‘mpfr_rint’ with ‘MPFR_RNDU’); • ‘mpfr_floor’ to the next lower or equal representable integer (like ‘mpfr_rint’ with ‘MPFR_RNDD’); • ‘mpfr_round’ to the nearest representable integer, rounding halfway cases away from zero (as in the roundTiesToAway mode of IEEE 754-2008); • ‘mpfr_roundeven’ to the nearest representable integer, rounding halfway cases with the even-rounding rule (like ‘mpfr_rint’ with ‘MPFR_RNDN’); • ‘mpfr_trunc’ to the next representable integer toward zero (like ‘mpfr_rint’ with ‘MPFR_RNDZ’). When OP is a zero or an infinity, set ROP to the same value (with the same sign). The return value is zero when the result is exact, positive when it is greater than the original value of OP, and negative when it is smaller. More precisely, the return value is 0 when OP is an d2055 14 a2068 9 function, you should use one the ‘mpfr_rint_*’ functions instead. Note that no double rounding is performed; for instance, 10.5 (1010.1 in binary) is rounded by ‘mpfr_rint’ with rounding to nearest to 12 (1100 in binary) in 2-bit precision, because the two enclosing numbers representable on two bits are 8 and 12, and the closest is 12. (If one first rounded to an integer, one would round 10.5 to 10 with even rounding, and then 10 would be rounded to 8 again with even rounding.) a2074 2 -- Function: int mpfr_rint_roundeven (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) d2077 8 a2084 13 Set ROP to OP rounded to an integer: • ‘mpfr_rint_ceil’: to the next higher or equal integer; • ‘mpfr_rint_floor’: to the next lower or equal integer; • ‘mpfr_rint_round’: to the nearest integer, rounding halfway cases away from zero; • ‘mpfr_rint_roundeven’: to the nearest integer, rounding halfway cases to the nearest even integer; • ‘mpfr_rint_trunc’ to the next integer toward zero. If the result is not representable, it is rounded in the direction RND. When OP is a zero or an infinity, set ROP to the same value (with the same sign). The return value is the ternary value associated with the considered round-to-integer function (regarded in the same way as any other mathematical function). d2104 1 a2104 2 fractional part is generated). When OP is an integer or an infinity, set ROP to zero with the same sign as OP. a2117 2 -- Function: int mpfr_fmodquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) d2124 3 a2126 3 follows: N is rounded toward zero for ‘mpfr_fmod’ and ‘mpfr_fmodquo’, and to the nearest integer (ties rounded to even) for ‘mpfr_remainder’ and ‘mpfr_remquo’. d2134 6 a2139 6 Additionally, ‘mpfr_fmodquo’ and ‘mpfr_remquo’ store the low significant bits from the quotient N in *Q (more precisely the number of bits in a ‘long’ minus one), with the sign of X divided by Y (except if those low bits are all zero, in which case zero is returned). Note that X may be so large in magnitude relative to Y that an exact representation of the quotient is not practical. The d2147 1 a2147 1 File: mpfr.info, Node: Rounding-Related Functions, Next: Miscellaneous Functions, Prev: Integer and Remainder Related Functions, Up: MPFR Interface d2149 1 a2149 1 5.11 Rounding-Related Functions d2190 11 a2200 24 to round correctly X to precision PREC with the direction RND2 assuming an unbounded exponent range, and 0 otherwise (including for NaN and Inf). In other words, if the error on B is bounded by two to the power K ulps, and B has precision PREC, you should give ERR=PREC−K. This function *does not modify* its arguments. If RND1 is ‘MPFR_RNDN’ or ‘MPFR_RNDF’, the error is considered to be either positive or negative, thus the possible range is twice as large as with a directed rounding for RND1 (with the same value of ERR). When RND2 is ‘MPFR_RNDF’, let RND3 be the opposite direction if RND1 is a directed rounding, and ‘MPFR_RNDN’ if RND1 is ‘MPFR_RNDN’ or ‘MPFR_RNDF’. The returned value of ‘mpfr_can_round (b, err, rnd1, MPFR_RNDF, prec)’ is non-zero iff after the call ‘mpfr_set (y, b, rnd3)’ with Y of precision PREC, Y is guaranteed to be a faithful rounding of X. Note: The *note ternary value:: cannot be determined in general with this function. However, if it is known that the exact value is not exactly representable in precision PREC, then one can use the following trick to determine the (non-zero) ternary value in any rounding mode RND2 (note that ‘MPFR_RNDZ’ below can be replaced by any directed rounding mode): d2202 3 a2204 7 prec + (rnd2 == MPFR_RNDN))) { /* round the approximation 'b' to the result 'r' of 'prec' bits with rounding mode 'rnd2' and get the ternary value 'inex' */ inex = mpfr_set (r, b, rnd2); } Indeed, if RND2 is ‘MPFR_RNDN’, this will check if one can round to a2209 3 A detailed example is available in the ‘examples’ subdirectory, file ‘can_round.c’. d2212 4 a2215 1 of X, and 0 for special values, including 0. a2221 28 -- Macro: int mpfr_round_nearest_away (int (FOO)(mpfr_t, type1_t, ..., mpfr_rnd_t), mpfr_t ROP, type1_t OP, ...) Given a function FOO and one or more values OP (which may be a ‘mpfr_t’, a ‘long’, a ‘double’, etc.), put in ROP the round-to-nearest-away rounding of ‘FOO(OP,...)’. This rounding is defined in the same way as round-to-nearest-even, except in case of tie, where the value away from zero is returned. The function FOO takes as input, from second to penultimate argument(s), the argument list given after ROP, a rounding mode as final argument, puts in its first argument the value ‘FOO(OP,...)’ rounded according to this rounding mode, and returns the corresponding ternary value (which is expected to be correct, otherwise ‘mpfr_round_nearest_away’ will not work as desired). Due to implementation constraints, this function must not be called when the minimal exponent ‘emin’ is the smallest possible one. This macro has been made such that the compiler is able to detect mismatch between the argument list OP and the function prototype of FOO. Multiple input arguments OP are supported only with C99 compilers. Otherwise, for C89 compilers, only one such argument is supported. Note: this macro is experimental and its interface might change in future versions. unsigned long ul; mpfr_t f, r; /* Code that inits and sets r, f, and ul, and if needed sets emin */ int i = mpfr_round_nearest_away (mpfr_add_ui, r, f, ul); d2223 1 a2223 1 File: mpfr.info, Node: Miscellaneous Functions, Next: Exception Related Functions, Prev: Rounding-Related Functions, Up: MPFR Interface d2229 7 a2235 7 If X or Y is NaN, set X to NaN; note that the NaN flag is set as usual. If X and Y are equal, X is unchanged. Otherwise, if X is different from Y, replace X by the next floating-point number (with the precision of X and the current exponent range) in the direction of Y (the infinite values are seen as the smallest and largest floating-point numbers). If the result is zero, it keeps the same sign. No underflow, overflow, or inexact exception is raised. d2281 3 a2283 7 Note: the note for ‘mpfr_urandomb’ holds too. Moreover, the exact number (the random value to be rounded) and the next random state do not depend on the current exponent range and the rounding mode. However, they depend on the target precision: from the same state of the random generator, if the precision of the destination is changed, then the value may be completely different (and the state of the random generator is different too). a2284 2 -- Function: int mpfr_nrandom (mpfr_t ROP1, gmp_randstate_t STATE, mpfr_rnd_t RND) d2287 2 a2288 4 Generate one (possibly two for ‘mpfr_grandom’) random floating-point number according to a standard normal Gaussian distribution (with mean zero and variance one). For ‘mpfr_grandom’, if ROP2 is a null pointer, then only one value is d2293 1 a2293 1 Gaussian distribution and then rounded in the direction RND. d2295 3 a2297 2 The ‘gmp_randstate_t’ argument should be created using the GMP ‘gmp_randinit’ function (see the GMP manual). d2299 6 a2304 6 For ‘mpfr_grandom’, the combination of the ternary values is returned like with ‘mpfr_sin_cos’. If ROP2 is a null pointer, the second ternary value is assumed to be 0 (note that the encoding of the only ternary value is not the same as the usual encoding for functions that return only one result). Otherwise the ternary value of a random number is always non-zero. a2309 10 Note: ‘mpfr_nrandom’ is much more efficient than ‘mpfr_grandom’, especially for large precision. Thus ‘mpfr_grandom’ is marked as deprecated and will be removed in a future release. -- Function: int mpfr_erandom (mpfr_t ROP1, gmp_randstate_t STATE, mpfr_rnd_t RND) Generate one random floating-point number according to an exponential distribution, with mean one. Other characteristics are identical to ‘mpfr_nrandom’. d2312 2 a2313 4 number and the significand is considered in [1/2,1). For this function, X is allowed to be outside of the current range of acceptable values. The behavior for NaN, infinity or zero is undefined. d2316 4 a2319 3 Set the exponent of X to E if X is a non-zero ordinary number and E is in the current exponent range, and return 0; otherwise, return a non-zero value (X is not changed). a2383 5 -- Function: int mpfr_buildopt_float128_p (void) Return a non-zero value if MPFR was compiled with ‘__float128’ support (that is, MPFR was built with the ‘--enable-float128’ configure option), return zero otherwise. a2393 8 -- Function: int mpfr_buildopt_sharedcache_p (void) Return a non-zero value if MPFR was compiled so that all threads share the same cache for one MPFR constant, like ‘mpfr_const_pi’ or ‘mpfr_const_log2’ (that is, MPFR was built with the ‘--enable-shared-cache’ configure option), return zero otherwise. If the return value is non-zero, MPFR applications may need to be compiled with the ‘-pthread’ option. d2419 6 a2424 19 largest exponent is not changed), and zero otherwise. For the subsequent operations, it is the user’s responsibility to check that any floating-point value used as an input is in the new exponent range (for example using ‘mpfr_check_range’). If a floating-point value outside the new exponent range is used as an input, the default behavior is undefined, in the sense of the ISO C standard; the behavior may also be explicitly documented, such as for ‘mpfr_check_range’. Note: Caches may still have values outside the current exponent range. This is not an issue as the user cannot use these caches directly via the API (MPFR extends the exponent range internally when need be). If ‘emin’ > ‘emax’ and a floating-point value needs to be produced as output, the behavior is undefined (‘mpfr_set_emin’ and ‘mpfr_set_emax’ do not check this condition as it might occur between successive calls to these two functions). d2437 1 a2437 1 This function assumes that X is the correctly rounded value of some d2461 2 a2462 3 is outside the subnormal exponent range of the emulated floating-point system, this function just propagates the *note ternary value:: T; otherwise, it rounds X to precision d2475 1 a2475 1 the current exponent range of MPFR (due to a too small ‘emax’), the a2483 5 Warning! If you change ‘emin’ (with ‘mpfr_set_emin’) just before calling ‘mpfr_subnormalize’, you need to make sure that the value is in the current exponent range of MPFR. But it is better to change ‘emin’ before any computation, if possible. d2505 3 a2507 25 Note that ‘mpfr_set_emin’ and ‘mpfr_set_emax’ are called early enough in order to make sure that all computed values are in the current exponent range. Warning! This emulates a double IEEE 754 arithmetic with correct rounding in the subnormal range, which may not be the case for your hardware. Below is another example showing how to emulate fixed-point arithmetic in a specific case. Here we compute the sine of the integers 1 to 17 with a result in a fixed-point arithmetic rounded at 2 power -42 (using the fact that the result is at most 1 in absolute value): { mpfr_t x; int i, inex; mpfr_set_emin (-41); mpfr_init2 (x, 42); for (i = 1; i <= 17; i++) { mpfr_set_ui (x, i, MPFR_RNDN); inex = mpfr_sin (x, x, MPFR_RNDZ); mpfr_subnormalize (x, inex, MPFR_RNDZ); mpfr_dump (x); } mpfr_clear (x); } d2515 2 a2516 7 Clear (lower) the underflow, overflow, divide-by-zero, invalid, inexact and _erange_ flags. -- Function: void mpfr_clear_flags (void) Clear (lower) all global flags (underflow, overflow, divide-by-zero, invalid, inexact, _erange_). Note: a group of flags can be cleared by using ‘mpfr_flags_clear’. d2524 6 a2529 2 Set (raise) the underflow, overflow, divide-by-zero, invalid, inexact and _erange_ flags. a2540 35 The ‘mpfr_flags_’ functions below that take an argument MASK can operate on any subset of the exception flags: a flag is part of this subset (or group) if and only if the corresponding bit of the argument MASK is set. The ‘MPFR_FLAGS_’ macros will normally be used to build this argument. *Note Exceptions::. -- Function: void mpfr_flags_clear (mpfr_flags_t MASK) Clear (lower) the group of flags specified by MASK. -- Function: void mpfr_flags_set (mpfr_flags_t MASK) Set (raise) the group of flags specified by MASK. -- Function: mpfr_flags_t mpfr_flags_test (mpfr_flags_t MASK) Return the flags specified by MASK. To test whether any flag from MASK is set, compare the return value to 0. You can also test individual flags by AND’ing the result with ‘MPFR_FLAGS_’ macros. Example: mpfr_flags_t t = mpfr_flags_test (MPFR_FLAGS_UNDERFLOW| MPFR_FLAGS_OVERFLOW) ... if (t) /* underflow and/or overflow (unlikely) */ { if (t & MPFR_FLAGS_UNDERFLOW) { /* handle underflow */ } if (t & MPFR_FLAGS_OVERFLOW) { /* handle overflow */ } } -- Function: mpfr_flags_t mpfr_flags_save (void) Return all the flags. It is equivalent to ‘mpfr_flags_test(MPFR_FLAGS_ALL)’. -- Function: void mpfr_flags_restore (mpfr_flags_t FLAGS, mpfr_flags_t MASK) Restore the flags specified by MASK to their state represented in FLAGS. d2552 4 a2555 3 many programs written for MPF can be compiled directly against MPFR without any changes. All operations are then performed with the default MPFR rounding mode, which can be reset with d2558 4 a2561 8 Warning! There are some differences. In particular: • The precision is different: MPFR rounds to the exact number of bits (zeroing trailing bits in the internal representation). Users may need to increase the precision of their variables. • The exponent range is also different. • The formatted output functions (‘gmp_printf’, etc.) will not work for arguments of arbitrary-precision floating-point type (‘mpf_t’, which ‘mpf2mpfr.h’ redefines as ‘mpfr_t’). d2647 13 a2659 14 • if abs(KIND) = ‘MPFR_NAN_KIND’, X is set to NaN; • if abs(KIND) = ‘MPFR_INF_KIND’, X is set to the infinity of the same sign as KIND; • if abs(KIND) = ‘MPFR_ZERO_KIND’, X is set to the zero of the same sign as KIND; • if abs(KIND) = ‘MPFR_REGULAR_KIND’, X is set to the regular number whose sign is the one of KIND, and whose exponent and significand are given by EXP and SIGNIFICAND. In all cases, SIGNIFICAND will be used directly for further computing involving X. This function does not allocate anything. A floating-point number initialized with this function cannot be resized using ‘mpfr_set_prec’ or ‘mpfr_prec_round’, or cleared using ‘mpfr_clear’! The SIGNIFICAND must have been initialized with ‘mpfr_custom_init’ using the same precision PREC. d2673 4 a2676 6 number and the significand is considered in [1/2,1). But if X is NaN, infinity or zero, contrary to ‘mpfr_get_exp’ (where the behavior is undefined), the return value is here an unspecified, valid value of the ‘mpfr_exp_t’ type. The behavior of this function for any ‘mpfr_t’ not initialized with ‘mpfr_custom_init_set’ is undefined. d2720 1 a2720 1 File: mpfr.info, Node: API Compatibility, Next: MPFR and the IEEE 754 Standard, Prev: MPFR Interface, Up: Top d2819 1 a2819 8 MPFR 3.0 (however no rounding mode ‘GMP_RNDA’ exists). Faithful rounding (‘MPFR_RNDF’) was added in MPFR 4.0, but currently, it is partially supported. The flags-related macros, whose name starts with ‘MPFR_FLAGS_’, were added in MPFR 4.0 (for the new functions ‘mpfr_flags_clear’, ‘mpfr_flags_restore’, ‘mpfr_flags_set’ and ‘mpfr_flags_test’, in particular). d2827 2 a2828 2 We give here in alphabetical order the functions (and function-like macros) that were added after MPFR 2.2, and in which MPFR version. d2836 1 a2836 7 • ‘mpfr_beta’ in MPFR 4.0 (incomplete, experimental). • ‘mpfr_buildopt_decimal_p’ in MPFR 3.0. • ‘mpfr_buildopt_float128_p’ in MPFR 4.0. • ‘mpfr_buildopt_gmpinternals_p’ in MPFR 3.1. d2838 2 a2839 5 • ‘mpfr_buildopt_sharedcache_p’ in MPFR 4.0. • ‘mpfr_buildopt_tls_p’ in MPFR 3.0. • ‘mpfr_buildopt_tune_case’ in MPFR 3.1. a2863 7 • ‘mpfr_erandom’ in MPFR 4.0. • ‘mpfr_flags_clear’, ‘mpfr_flags_restore’, ‘mpfr_flags_save’, ‘mpfr_flags_set’ and ‘mpfr_flags_test’ in MPFR 4.0. • ‘mpfr_fmma’ and ‘mpfr_fmms’ in MPFR 4.0. a2865 2 • ‘mpfr_fmodquo’ in MPFR 4.0. a2867 2 • ‘mpfr_fpif_export’ and ‘mpfr_fpif_import’ in MPFR 4.0. a2869 4 • ‘mpfr_free_cache2’ in MPFR 4.0. • ‘mpfr_free_pool’ in MPFR 4.0. a2871 5 • ‘mpfr_gamma_inc’ in MPFR 4.0. • ‘mpfr_get_float128’ in MPFR 4.0 if configured with ‘--enable-float128’. a2875 2 • ‘mpfr_get_q’ in MPFR 4.0. a2890 2 • ‘mpfr_log_ui’ in MPFR 4.0. a2894 2 • ‘mpfr_mp_memory_cleanup’ in MPFR 4.0. a2896 2 • ‘mpfr_nrandom’ in MPFR 4.0. a2904 6 • ‘mpfr_rint_roundeven’ and ‘mpfr_roundeven’ in MPFR 4.0. • ‘mpfr_round_nearest_away’ in MPFR 4.0. • ‘mpfr_rootn_ui’ in MPFR 4.0. a2906 3 • ‘mpfr_set_float128’ in MPFR 4.0 if configured with ‘--enable-float128’. a2941 5 • ‘mpfr_abs’, ‘mpfr_neg’ and ‘mpfr_set’ changed in MPFR 4.0. In previous MPFR versions, the sign bit of a NaN was unspecified; however, in practice, it was set as now specified except for ‘mpfr_neg’ with a reused argument: ‘mpfr_neg(x,x,rnd)’. a2948 4 • ‘mpfr_eint’ changed in MPFR 4.0. This function now returns the value of the E1/eint1 function for negative argument (before MPFR 4.0, it was returning NaN). a2957 5 • ‘mpfr_get_str’ changed in MPFR 4.0. This function now sets the NaN flag on NaN input (to follow the usual MPFR rules on NaN and IEEE 754-2008 recommendations on string conversions from Subclause 5.12.1) and sets the inexact flag when the conversion is inexact. a2982 10 • ‘mpfr_set_exp’ changed in MPFR 4.0. Before MPFR 4.0, the exponent was set whatever the contents of the MPFR object in argument. In practice, this could be useful as a low-level function when the MPFR number was being constructed by setting the fields of its internal structure, but the API does not provide a way to do this except by using internals. Thus, for the API, this behavior was useless and could quickly lead to undefined behavior due to the fact that the generated value could have an invalid format if the MPFR object contained a special value (NaN, infinity or zero). a3004 7 • ‘mpfr_sum’ changed in MPFR 4.0. The ‘mpfr_sum’ function has completely been rewritten for MPFR 4.0, with an update of the specification: the sign of an exact zero result is now specified, and the return value is now the usual ternary value. The old ‘mpfr_sum’ implementation could also take all the memory and crash on inputs of very different magnitude. a3013 7 • ‘mpfr_urandom’ changed in MPFR 4.0. The next random state no longer depends on the current exponent range and the rounding mode. The exceptions due to the rounding of the random number are now correctly generated, following the uniform distribution. As a consequence, the returned values can be different between MPFR 4.0 and previous MPFR versions. a3024 7 Macros ‘mpfr_add_one_ulp’ and ‘mpfr_sub_one_ulp’ have been removed in MPFR 4.0. They were no longer documented since MPFR 2.1.0 and were announced as deprecated since MPFR 3.1.0. Function ‘mpfr_grandom’ is marked as deprecated in MPFR 4.0. It will be removed in a future release. a3057 3 The way memory is allocated by MPFR should be regarded as well-specified only as of MPFR 4.0. d3059 1 a3059 51 File: mpfr.info, Node: MPFR and the IEEE 754 Standard, Next: Contributors, Prev: API Compatibility, Up: Top 7 MPFR and the IEEE 754 Standard ******************************** This section describes differences between MPFR and the IEEE 754 standard, and behaviors that are not specified yet in IEEE 754. The MPFR numbers do not include subnormals. The reason is that subnormals are less useful than in IEEE 754 as the default exponent range in MPFR is large and they would have made the implementation more complex. However, subnormals can be emulated using ‘mpfr_subnormalize’. MPFR has a single NaN. The behavior is similar either to a signaling NaN or to a quiet NaN, depending on the context. For any function returning a NaN (either produced or propagated), the NaN flag is set, while in IEEE 754, some operations are quiet (even on a signaling NaN). The ‘mpfr_rec_sqrt’ function differs from IEEE 754 on −0, where it gives +Inf (like for +0), following the usual limit rules, instead of −Inf. The ‘mpfr_root’ function predates IEEE 754-2008 and behaves differently from its rootn operation. It is deprecated and ‘mpfr_rootn_ui’ should be used instead. Operations with an unsigned zero: For functions taking an argument of integer or rational type, a zero of such a type is unsigned unlike the floating-point zero (this includes the zero of type ‘unsigned long’, which is a mathematical, exact zero, as opposed to a floating-point zero, which may come from an underflow and whose sign would correspond to the sign of the real non-zero value). Unless documented otherwise, this zero is regarded as +0, as if it were first converted to a MPFR number with ‘mpfr_set_ui’ or ‘mpfr_set_si’ (thus the result may not agree with the usual limit rules applied to a mathematical zero). This is not the case of addition and subtraction (‘mpfr_add_ui’, etc.), but for these functions, only the sign of a zero result would be affected, with +0 and −0 considered equal. Such operations are currently out of the scope of the IEEE 754 standard, and at the time of specification in MPFR, the Floating-Point Working Group in charge of the revision of IEEE 754 did not want to discuss issues with non-floating-point types in general. Note also that some obvious differences may come from the fact that in MPFR, each variable has its own precision. For instance, a subtraction of two numbers of the same sign may yield an overflow; idem for a call to ‘mpfr_set’, ‘mpfr_neg’ or ‘mpfr_abs’, if the destination variable has a smaller precision.  File: mpfr.info, Node: Contributors, Next: References, Prev: MPFR and the IEEE 754 Standard, Up: Top d3074 9 a3082 10 contributed the original version of the ‘mpfr_sum’ function (used up to MPFR 3.1). Emmanuel Jeandel, from ENS-Lyon too, contributed the generic hypergeometric code, as well as the internal function ‘mpfr_exp3’, a first implementation of the sine and cosine, and improved versions of ‘mpfr_const_log2’ and ‘mpfr_const_pi’. Ludovic Meunier helped in the design of the ‘mpfr_erf’ code. Jean-Luc Rémy contributed the ‘mpfr_zeta’ code. Fabrice Rouillier contributed the ‘mpfr_xxx_z’ and ‘mpfr_xxx_q’ functions, and helped to the Microsoft Windows porting. Damien Stehlé contributed the ‘mpfr_get_ld_2exp’ function. Charles Karney contributed the ‘mpfr_nrandom’ and ‘mpfr_erandom’ functions. d3103 1 a3103 5 2012 was partly supported by the ERC grant ANTICS of Andreas Enge. The MPFR-MPC workshop in January 2013 was partly supported by the ERC grant ANTICS, the GDR IM and the Caramel project-team, during which Mickaël Gastineau contributed the MPFRbench program, and Fredrik Johannsson a faster version of ‘mpfr_const_euler’. d3112 2 a3113 4 Cambridge University Press, Cambridge Monographs on Applied and Computational Mathematics, Number 18, 2010. Electronic version freely available at . d3122 1 a3122 1 Library", version 6.1.2, 2016, . d3137 1 a3137 1 Implementation", Birkhäuser, Boston, 3rd edition, 2016. a3613 2 * Group of flags: Nomenclature and Types. (line 39) d3623 1 a3623 1 * Integer related functions: Integer and Remainder Related Functions. a3649 2 * Remainder related functions: Integer and Remainder Related Functions. (line 3) d3651 1 a3651 1 * Rounding mode related functions: Rounding-Related Functions. d3662 1 a3662 1 * Ternary value: Rounding Modes. (line 48) d3676 3 a3678 3 (line 186) * mpfr_acos: Special Functions. (line 66) * mpfr_acosh: Special Functions. (line 130) d3691 4 a3694 4 * mpfr_agm: Special Functions. (line 250) * mpfr_ai: Special Functions. (line 269) * mpfr_asin: Special Functions. (line 67) * mpfr_asinh: Special Functions. (line 131) d3696 4 a3699 5 (line 197) * mpfr_atan: Special Functions. (line 68) * mpfr_atan2: Special Functions. (line 78) * mpfr_atanh: Special Functions. (line 132) * mpfr_beta: Special Functions. (line 192) d3701 1 a3701 3 (line 185) * mpfr_buildopt_float128_p: Miscellaneous Functions. (line 180) d3703 1 a3703 3 (line 190) * mpfr_buildopt_sharedcache_p: Miscellaneous Functions. (line 195) d3705 1 a3705 1 (line 174) d3707 2 a3708 2 (line 203) * mpfr_can_round: Rounding-Related Functions. d3711 2 a3712 2 (line 115) * mpfr_ceil: Integer and Remainder Related Functions. d3715 1 a3715 1 (line 50) d3721 1 a3721 1 (line 153) d3723 1 a3723 1 (line 156) d3725 1 a3725 1 (line 160) d3727 1 a3727 1 (line 155) d3729 1 a3729 1 (line 154) d3731 1 a3731 1 (line 152) d3733 1 a3733 1 (line 151) d3756 4 a3759 4 * mpfr_const_catalan: Special Functions. (line 280) * mpfr_const_euler: Special Functions. (line 279) * mpfr_const_log2: Special Functions. (line 277) * mpfr_const_pi: Special Functions. (line 278) d3761 10 a3770 10 (line 127) * mpfr_cos: Special Functions. (line 44) * mpfr_cosh: Special Functions. (line 110) * mpfr_cot: Special Functions. (line 62) * mpfr_coth: Special Functions. (line 126) * mpfr_csc: Special Functions. (line 61) * mpfr_csch: Special Functions. (line 125) * mpfr_custom_get_exp: Custom Interface. (line 76) * mpfr_custom_get_kind: Custom Interface. (line 66) * mpfr_custom_get_significand: Custom Interface. (line 71) d3774 1 a3774 1 * mpfr_custom_move: Custom Interface. (line 85) d3777 1 a3777 1 * mpfr_digamma: Special Functions. (line 187) d3779 1 a3779 1 (line 197) d3783 1 a3783 1 (line 176) d3785 1 a3785 1 (line 52) d3787 1 a3787 1 (line 212) d3789 1 a3789 1 (line 210) a3799 2 * mpfr_dump: Input and Output Functions. (line 68) d3804 1 a3804 1 * mpfr_eint: Special Functions. (line 140) d3806 1 a3806 1 (line 31) a3808 2 * mpfr_erandom: Miscellaneous Functions. (line 99) d3810 8 a3817 8 (line 179) * mpfr_erf: Special Functions. (line 204) * mpfr_erfc: Special Functions. (line 205) * mpfr_exp: Special Functions. (line 34) * mpfr_exp10: Special Functions. (line 36) * mpfr_exp2: Special Functions. (line 35) * mpfr_expm1: Special Functions. (line 40) * mpfr_fac_ui: Special Functions. (line 136) d3819 1 a3819 1 (line 168) d3821 1 a3821 1 (line 164) d3823 1 a3823 1 (line 162) d3825 1 a3825 1 (line 166) d3827 1 a3827 1 (line 167) d3829 1 a3829 1 (line 163) d3831 1 a3831 1 (line 161) d3833 2 a3834 14 (line 165) * mpfr_flags_clear: Exception Related Functions. (line 190) * mpfr_flags_restore: Exception Related Functions. (line 214) * mpfr_flags_save: Exception Related Functions. (line 210) * mpfr_flags_set: Exception Related Functions. (line 193) * mpfr_flags_t: Nomenclature and Types. (line 39) * mpfr_flags_test: Exception Related Functions. (line 196) * mpfr_floor: Integer and Remainder Related Functions. d3836 4 a3839 12 * mpfr_fma: Special Functions. (line 230) * mpfr_fmma: Special Functions. (line 240) * mpfr_fmms: Special Functions. (line 242) * mpfr_fmod: Integer and Remainder Related Functions. (line 106) * mpfr_fmodquo: Integer and Remainder Related Functions. (line 108) * mpfr_fms: Special Functions. (line 232) * mpfr_fpif_export: Input and Output Functions. (line 43) * mpfr_fpif_import: Input and Output Functions. (line 54) d3841 4 a3844 6 (line 161) * mpfr_frac: Integer and Remainder Related Functions. (line 89) * mpfr_free_cache: Special Functions. (line 288) * mpfr_free_cache2: Special Functions. (line 295) * mpfr_free_pool: Special Functions. (line 309) d3846 1 a3846 1 (line 156) d3848 2 a3849 3 (line 49) * mpfr_gamma: Special Functions. (line 155) * mpfr_gamma_inc: Special Functions. (line 156) d3853 1 a3853 1 (line 10) d3856 1 a3856 1 * mpfr_get_default_rounding_mode: Rounding-Related Functions. d3859 1 a3859 1 (line 36) d3863 1 a3863 1 (line 43) d3865 1 a3865 1 (line 42) d3869 1 a3869 1 (line 41) d3871 1 a3871 1 (line 40) d3873 1 a3873 1 (line 105) d3875 1 a3875 3 (line 85) * mpfr_get_float128: Conversion Functions. (line 9) d3881 1 a3881 1 (line 38) d3883 1 a3883 1 (line 165) a3885 2 * mpfr_get_q: Conversion Functions. (line 80) d3887 2 a3889 2 * mpfr_get_sj: Conversion Functions. (line 23) d3891 1 a3891 1 (line 98) d3893 2 a3895 2 * mpfr_get_uj: Conversion Functions. (line 24) d3897 1 a3897 1 (line 134) d3899 1 a3899 1 (line 71) d3901 1 a3901 1 (line 58) d3903 1 a3903 1 (line 69) d3908 1 a3908 1 * mpfr_hypot: Special Functions. (line 260) d3910 1 a3910 1 (line 178) d3941 5 a3945 5 * mpfr_integer_p: Integer and Remainder Related Functions. (line 135) * mpfr_j0: Special Functions. (line 209) * mpfr_j1: Special Functions. (line 210) * mpfr_jn: Special Functions. (line 211) d3952 7 a3958 8 * mpfr_lgamma: Special Functions. (line 177) * mpfr_li2: Special Functions. (line 150) * mpfr_lngamma: Special Functions. (line 169) * mpfr_log: Special Functions. (line 19) * mpfr_log10: Special Functions. (line 23) * mpfr_log1p: Special Functions. (line 30) * mpfr_log2: Special Functions. (line 22) * mpfr_log_ui: Special Functions. (line 20) d3963 4 a3966 5 * mpfr_min_prec: Rounding-Related Functions. (line 84) * mpfr_modf: Integer and Remainder Related Functions. (line 96) * mpfr_mp_memory_cleanup: Special Functions. (line 314) d3970 1 a3970 1 (line 50) d3972 1 a3972 1 (line 205) d3974 1 a3974 1 (line 203) d3986 1 a3986 1 (line 177) d3990 1 a3990 1 (line 185) a3996 2 * mpfr_nrandom: Miscellaneous Functions. (line 67) d4002 1 a4002 1 (line 175) d4004 1 a4004 1 (line 139) d4006 1 a4006 1 (line 143) d4008 1 a4008 1 (line 141) d4010 2 a4011 2 (line 145) * mpfr_prec_round: Rounding-Related Functions. d4016 3 a4018 3 (line 168) * mpfr_print_rnd_mode: Rounding-Related Functions. (line 88) d4020 1 a4020 1 (line 107) d4024 6 a4029 6 (line 42) * mpfr_remainder: Integer and Remainder Related Functions. (line 110) * mpfr_remquo: Integer and Remainder Related Functions. (line 112) * mpfr_rint: Integer and Remainder Related Functions. d4031 8 a4038 10 * mpfr_rint_ceil: Integer and Remainder Related Functions. (line 52) * mpfr_rint_floor: Integer and Remainder Related Functions. (line 53) * mpfr_rint_round: Integer and Remainder Related Functions. (line 55) * mpfr_rint_roundeven: Integer and Remainder Related Functions. (line 57) * mpfr_rint_trunc: Integer and Remainder Related Functions. (line 59) d4042 2 a4043 4 (line 128) * mpfr_rootn_ui: Basic Arithmetic Functions. (line 116) * mpfr_round: Integer and Remainder Related Functions. d4045 2 a4046 6 * mpfr_roundeven: Integer and Remainder Related Functions. (line 10) * mpfr_round_nearest_away: Rounding-Related Functions. (line 93) * mpfr_sec: Special Functions. (line 60) * mpfr_sech: Special Functions. (line 124) d4050 1 a4050 1 (line 121) d4054 1 a4054 1 (line 21) d4057 1 a4057 1 * mpfr_set_default_rounding_mode: Rounding-Related Functions. d4060 1 a4060 1 (line 167) d4066 1 a4066 1 (line 170) d4068 1 a4068 1 (line 112) d4070 1 a4070 3 (line 25) * mpfr_set_float128: Assignment Functions. (line 19) d4074 1 a4074 1 (line 169) d4076 1 a4076 1 (line 154) d4080 1 a4080 1 (line 153) d4082 1 a4082 1 (line 168) d4084 1 a4084 1 (line 166) d4088 1 a4088 1 (line 25) d4090 1 a4090 1 (line 24) d4094 1 a4094 1 (line 61) d4098 1 a4098 1 (line 65) d4100 1 a4100 1 (line 73) d4104 1 a4104 1 (line 59) d4108 1 a4108 1 (line 63) d4110 1 a4110 1 (line 165) d4112 1 a4112 1 (line 23) d4114 1 a4114 1 (line 155) d4116 1 a4116 1 (line 67) d4120 5 a4124 5 (line 117) * mpfr_sin: Special Functions. (line 45) * mpfr_sinh: Special Functions. (line 111) * mpfr_sinh_cosh: Special Functions. (line 116) * mpfr_sin_cos: Special Functions. (line 50) d4130 1 a4130 1 (line 184) d4132 1 a4132 1 (line 174) d4136 1 a4136 1 (line 100) d4138 1 a4138 1 (line 101) d4140 1 a4140 1 (line 91) d4144 1 a4144 1 (line 73) d4155 1 a4155 1 * mpfr_sum: Special Functions. (line 321) d4157 1 a4157 1 (line 161) d4160 4 a4163 4 * mpfr_tan: Special Functions. (line 46) * mpfr_tanh: Special Functions. (line 112) * mpfr_trunc: Integer and Remainder Related Functions. (line 11) d4167 1 a4167 1 (line 149) d4169 1 a4169 1 (line 147) d4173 1 a4173 1 (line 174) d4181 1 a4181 1 (line 198) d4183 1 a4183 1 (line 137) d4185 1 a4185 1 (line 138) d4187 2 a4189 2 * MPFR_VERSION_NUM: Miscellaneous Functions. (line 157) d4191 1 a4191 1 (line 140) d4193 1 a4193 1 (line 141) d4195 1 a4195 1 (line 162) d4197 1 a4197 1 (line 169) d4199 1 a4199 1 (line 186) d4201 4 a4204 4 (line 175) * mpfr_y0: Special Functions. (line 220) * mpfr_y1: Special Functions. (line 221) * mpfr_yn: Special Functions. (line 222) d4207 2 a4208 2 * mpfr_zeta: Special Functions. (line 198) * mpfr_zeta_ui: Special Functions. (line 199) d4216 41 a4256 43 Node: Copying2042 Node: Introduction to MPFR3805 Node: Installing MPFR6208 Node: Reporting Bugs11654 Node: MPFR Basics13684 Node: Headers and Libraries14044 Node: Nomenclature and Types17640 Node: MPFR Variable Conventions19899 Node: Rounding Modes21441 Ref: ternary value24166 Node: Floating-Point Values on Special Numbers26152 Node: Exceptions29412 Node: Memory Handling33240 Node: Getting the Best Efficiency Out of MPFR37040 Node: MPFR Interface37952 Node: Initialization Functions40238 Node: Assignment Functions47553 Node: Combined Initialization and Assignment Functions57499 Node: Conversion Functions58800 Node: Basic Arithmetic Functions69381 Node: Comparison Functions80277 Node: Special Functions83765 Node: Input and Output Functions101974 Node: Formatted Output Functions106751 Node: Integer and Remainder Related Functions116956 Node: Rounding-Related Functions124484 Node: Miscellaneous Functions131001 Node: Exception Related Functions141493 Node: Compatibility with MPF151733 Node: Custom Interface154679 Node: Internals159310 Node: API Compatibility160854 Node: Type and Macro Changes162802 Node: Added Functions165985 Node: Changed Functions170499 Node: Removed Functions177095 Node: Other Changes177825 Node: MPFR and the IEEE 754 Standard179526 Node: Contributors182143 Node: References185200 Node: GNU Free Documentation License187084 Node: Concept Index209677 Node: Function and Type Index216049 @ 1.1.1.4 log @GNU mpfr 4.1.0. main changes from 4.0: Changed __float128 to the type _Float128 specified in ISO/IEC TS 18661. __float128 is used as a fallback if _Float128 is not supported. New function mpfr_get_str_ndigits about conversion to a string of digits. New function mpfr_dot for the dot product (incomplete, experimental). New functions mpfr_get_decimal128 and mpfr_set_decimal128 (available only when MPFR has been built with decimal float support). New function mpfr_cmpabs_ui. New function mpfr_total_order_p for the IEEE 754 totalOrder predicate. The mpfr_out_str function now accepts bases from -2 to -36, in order to follow mpfr_get_str and GMP's mpf_out_str functions (these cases gave an assertion failure, as with other invalid bases). Shared caches: cleanup; really detect lock failures (abort in this case). Improved mpfr_add and mpfr_sub when all operands have a precision equal to twice the number of bits per word, e.g., 128 bits on a 64-bit platform. Optimized the tuning parameters for various architectures. @ text @d1 1 a1 1 This is mpfr.info, produced by makeinfo version 6.7 from mpfr.texi. d4 1 a4 1 Floating-Point Reliable Library, version 4.1.0. d6 1 a6 1 Copyright 1991, 1993-2020 Free Software Foundation, Inc. d26 1 a26 1 Floating-Point Reliable Library, version 4.1.0. d28 1 a28 1 Copyright 1991, 1993-2020 Free Software Foundation, Inc. d184 1 a184 1 1. To build MPFR, you first have to install GNU MP (version 5.0.0 or d220 1 a220 2 mailing-list ‘mpfr@@inria.fr’. For details, see *note Reporting Bugs::. d282 1 a282 1 . d285 2 a286 2 *Note Reporting Bugs::. Some bug fixes are available on the MPFR 4.1.0 web page . d292 1 a292 1 or . d301 4 a304 4 on the MPFR 4.1.0 web page and the FAQ : perhaps this bug is already known, in which case you may find there a workaround for it. You might also look in the archives of the MPFR mailing-list: d349 1 a349 1 * Rounding:: d438 13 a450 18 A “floating-point number”, or “float” for short, is an object representing a radix-2 floating-point number consisting of a sign, an arbitrary-precision normalized significand (also called mantissa), and an exponent (an integer in some given range); these are called “regular numbers”. Like in the IEEE 754 standard, a floating-point number can also have three kinds of special values: a signed zero, a signed infinity, and Not-a-Number (NaN). NaN can represent the default value of a floating-point object and the result of some operations for which no other results would make sense, such as 0 divided by 0 or +Infinity minus +Infinity; unless documented otherwise, the sign bit of a NaN is unspecified. Note that contrary to IEEE 754, MPFR has a single kind of NaN and does not have subnormals. Other than that, the behavior is very similar to IEEE 754, but there may be some differences. The C data type for such objects is ‘mpfr_t’, internally defined as a one-element array of a structure (so that when passed as an argument to a function, it is the pointer that is actually passed), and ‘mpfr_ptr’ is the C data type representing a pointer to this structure. d452 2 a453 2 The “precision” is the number of bits used to represent the significand of a floating-point number; the corresponding C data type is d461 9 a469 15 otherwise MPFR will abort due to an assertion failure. However, in practice, the real limitation will probably be the available memory on your platform, and in case of lack of memory, the program may abort, crash or have undefined behavior (depending on your C implementation). An “exponent” is a component of a regular floating-point number. Its C data type is ‘mpfr_exp_t’. Valid exponents are restricted to a subset of this type, and the exponent range can be changed globally as described in *note Exception Related Functions::. Special values do not have an exponent. The “rounding mode” specifies the way to round the result of a floating-point operation, in case the exact result cannot be represented exactly in the destination (*note Rounding::). The corresponding C data type is ‘mpfr_rnd_t’. d476 1 a476 1 File: mpfr.info, Node: MPFR Variable Conventions, Next: Rounding, Prev: Nomenclature and Types, Up: MPFR Basics d504 1 a504 1 File: mpfr.info, Node: Rounding, Next: Floating-Point Values on Special Numbers, Prev: MPFR Variable Conventions, Up: MPFR Basics d506 2 a507 2 4.4 Rounding ============ d510 4 a513 4 • ‘MPFR_RNDN’: round to nearest, with the even rounding rule (roundTiesToEven in IEEE 754-2008); see details below. d515 1 a515 7 IEEE 754-2008). • ‘MPFR_RNDU’: round toward plus infinity (roundTowardPositive in IEEE 754-2008). • ‘MPFR_RNDZ’: round toward zero (roundTowardZero in IEEE 754-2008). a516 1 d528 8 a535 28 Note that, in particular for a result equal to zero, the sign is preserved by the rounding operation. The ‘MPFR_RNDN’ mode works like roundTiesToEven from the IEEE 754 standard: in case the number to be rounded lies exactly in the middle between two consecutive representable numbers, it is rounded to the one with an even significand; in radix 2, this means that the least significant bit is 0. For example, the number 2.5, which is represented by (10.1) in binary, is rounded to (10.0)=2 with a precision of two bits, and not to (11.0)=3. This rule avoids the “drift” phenomenon mentioned by Knuth in volume 2 of The Art of Computer Programming (Section 4.2.2). Note: In particular for a 1-digit precision (in radix 2 or other radices, as in conversions to a string of digits), one considers the significands associated with the exponent of the number to be rounded. For instance, to round the number 95 in radix 10 with a 1-digit precision, one considers its truncated 1-digit integer significand 9 and the following integer 10 (since these are consecutive integers, exactly one of them is even). 10 is the even significand, so that 95 will be rounded to 100, not to 90. For the “directed rounding modes”, a number X is rounded to the number Y that is the closest to X such that • ‘MPFR_RNDD’: Y is less than or equal to X; • ‘MPFR_RNDU’: Y is greater than or equal to X; • ‘MPFR_RNDZ’: abs(Y) is less than or equal to abs(X); • ‘MPFR_RNDA’: abs(Y) is greater than or equal to abs(X). d561 4 a564 4 error on the result is less than or equal to 1/2 ulp (unit in the last place) of that result in the rounding to nearest mode, and less than 1 ulp of that result in the directed rounding modes (a ulp is the weight of the least significant represented bit of the result after rounding). d585 1 a585 1 File: mpfr.info, Node: Floating-Point Values on Special Numbers, Next: Exceptions, Prev: Rounding, Up: MPFR Basics d604 3 a606 3 the result is rounded as described in *note Rounding:: (but see below for the specification of the sign of an exact zero). Otherwise the general rules from this section apply unless stated otherwise in the d640 3 a642 3 ‘mpfr_hypot’ on (NaN,+Inf) gives +Inf (as specified in *note Transcendental Functions::), since for any finite or infinite input X, ‘mpfr_hypot’ on (X,+Inf) gives +Inf. d769 7 a775 6 precision, and the default rounding mode are either global (if MPFR has not been compiled as thread safe) or per-thread (thread-local storage, TLS). The initial values of TLS data after a thread is created entirely depend on the compiler and thread implementation (MPFR simply does a conventional variable initialization, the variables being declared with an implementation-defined TLS specifier). d856 1 a856 1 * Arithmetic Functions:: d858 1 a858 1 * Transcendental Functions:: a864 1 * Memory Handling Functions:: d886 2 a887 5 the precision of a variable that has already been initialized, use ‘mpfr_set_prec’ or ‘mpfr_prec_round’; note that if the precision is decreased, the unused memory will not be freed, so that it may be wise to choose a large enough initial precision in order to avoid reallocations. The precision PREC must be an integer between d1005 8 a1012 8 Set the precision of X to be *exactly* PREC bits, and set its value to NaN. The previous value stored in X is lost. It is equivalent to a call to ‘mpfr_clear(x)’ followed by a call to ‘mpfr_init2(x, prec)’, but more efficient as no allocation is done in case the current allocated space for the significand of X is enough. The precision PREC can be any integer between ‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’. In case you want to keep the previous value stored in X, use ‘mpfr_prec_round’ instead. d1041 1 a1041 1 -- Function: int mpfr_set_float128 (mpfr_t ROP, _Float128 OP, a1044 2 -- Function: int mpfr_set_decimal128 (mpfr_t ROP, _Decimal128 OP, mpfr_rnd_t RND) d1053 1 a1053 1 provides the ‘_Float128’ data type (GCC 4.3 or later supports this d1058 6 a1063 6 ‘mpfr_set_d’, ‘mpfr_set_ld’, ‘mpfr_set_decimal64’ and ‘mpfr_set_decimal128’ might not preserve the signed zeros. The ‘mpfr_set_decimal64’ and ‘mpfr_set_decimal128’ functions are built only with the configure option ‘--enable-decimal-float’, and when the compiler or system provides the ‘_Decimal64’ and ‘_Decimal128’ data type; to use those functions, one should define the macro d1065 1 a1065 1 might fail if the numerator (or the denominator) cannot be d1075 5 a1079 5 ‘mpfr_set_flt’, ‘mpfr_set_d’, ‘mpfr_set_ld’, ‘mpfr_set_decimal64’ or ‘mpfr_set_decimal128’. Otherwise the floating-point constant will be first converted into a reduced-precision (e.g., 53-bit) binary (or decimal, for ‘mpfr_set_decimal64’ and ‘mpfr_set_decimal128’) number before MPFR can work with it. d1133 20 a1152 20 digits with an optional decimal-point character, and an optional exponent consisting of an exponent prefix followed by an optional sign and a non-empty sequence of decimal digits. A significand digit is either a decimal digit or a Latin letter (62 possible characters), with ‘A’ = 10, ‘B’ = 11, ..., ‘Z’ = 35; case is ignored in bases less than or equal to 36, in bases larger than 36, ‘a’ = 36, ‘b’ = 37, ..., ‘z’ = 61. The value of a significand digit must be strictly less than the base. The decimal-point character can be either the one defined by the current locale or the period (the first one is accepted for consistency with the C standard and the practice, the second one is accepted to allow the programmer to provide MPFR numbers from strings in a way that does not depend on the current locale). The exponent prefix can be ‘e’ or ‘E’ for bases up to 10, or ‘@@’ in any base; it indicates a multiplication by a power of the base. In bases 2 and 16, the exponent prefix can also be ‘p’ or ‘P’, in which case the exponent, called _binary exponent_, indicates a multiplication by a power of 2 instead of the base (there is a difference only for base 16); in base 16 for example ‘1p2’ represents 4 whereas ‘1@@2’ represents 256. The value of an exponent is always written in base 10. d1223 1 a1223 1 File: mpfr.info, Node: Conversion Functions, Next: Arithmetic Functions, Prev: Combined Initialization and Assignment Functions, Up: MPFR Interface d1231 1 a1231 1 -- Function: _Float128 mpfr_get_float128 (mpfr_t OP, mpfr_rnd_t RND) d1233 9 a1241 12 -- Function: _Decimal128 mpfr_get_decimal128 (mpfr_t OP, mpfr_rnd_t RND) Convert OP to a ‘float’ (respectively ‘double’, ‘long double’, ‘_Decimal64’, or ‘_Decimal128’) using the rounding mode RND. If OP is NaN, some fixed NaN (either quiet or signaling) or the result of 0.0/0.0 is returned. If OP is ±Inf, an infinity of the same sign or the result of ±1.0/0.0 is returned. If OP is zero, these functions return a zero, trying to preserve its sign, if possible. The ‘mpfr_get_float128’, ‘mpfr_get_decimal64’ and ‘mpfr_get_decimal128’ functions are built only under some conditions: see the documentation of ‘mpfr_set_float128’, ‘mpfr_set_decimal64’ and ‘mpfr_set_decimal128’ respectively. d1320 8 a1327 28 -- Function: size_t mpfr_get_str_ndigits (int B, mpfr_prec_t P) Return the minimal integer m such that any number of P bits, when output with m digits in radix B with rounding to nearest, can be recovered exactly when read again, still with rounding to nearest. More precisely, we have m = 1 + ceil(P*log(2)/log(B)), with P replaced by P−1 if B is a power of 2. The argument B must be in the range 2 to 62; this is the range of bases supported by the ‘mpfr_get_str’ function. Note that contrary to the base argument of this function, negative values are not accepted. -- Function: char * mpfr_get_str (char *STR, mpfr_exp_t *EXPPTR, int BASE, size_t N, mpfr_t OP, mpfr_rnd_t RND) Convert OP to a string of digits in base abs(BASE), with rounding in the direction RND, where N is either zero (see below) or the number of significant digits output in the string. The argument BASE may vary from 2 to 62 or from −2 to −36; otherwise the function does nothing and immediately returns a null pointer. For BASE in the range 2 to 36, digits and lower-case letters are used; for −2 to −36, digits and upper-case letters are used; for 37 to 62, digits, upper-case letters, and lower-case letters, in that significance order, are used. Warning! This implies that for BASE > 10, the successor of the digit 9 depends on BASE. This choice has been done for compatibility with GMP’s ‘mpf_get_str’ function. Users who wish a more consistent behavior should write a simple wrapper. d1350 10 a1359 3 If N is zero, the number of digits of the significand is taken as ‘mpfr_get_str_ndigits(BASE,P)’ where P is the precision of OP (*note mpfr_get_str_ndigits::). d1369 2 a1370 3 replace it by ‘mpfr_get_str_ndigits(BASE,P)’ where P is the precision of OP, as mentioned above. The extra two bytes are for a possible minus sign, and for the terminating null character, and d1402 1 a1402 1 non-negative number less than or equal to ‘ULONG_MAX’. d1405 1 a1405 1 File: mpfr.info, Node: Arithmetic Functions, Next: Comparison Functions, Prev: Conversion Functions, Up: MPFR Interface d1407 2 a1408 2 5.5 Arithmetic Functions ======================== d1521 7 a1527 7 int N, mpfr_rnd_t RND) Set ROP to the Nth root (with N = 3, the cubic root, for ‘mpfr_cbrt’) of OP rounded in the direction RND. For N = 0, set ROP to NaN. For N odd (resp. even) and OP negative (including −Inf), set ROP to a negative number (resp. NaN). If OP is zero, set ROP to zero with the sign obtained by the usual limit rules, i.e., the same sign as OP if N is odd, and positive if N is even. d1530 1 a1530 3 standard and the P754/D2.41 draft of the next standard (Section 9.2). Note that it is here restricted to N >= 0. Functions allowing a negative N may be implemented in the future. d1532 1 a1532 1 -- Function: int mpfr_root (mpfr_t ROP, mpfr_t OP, unsigned long int N, d1535 1 a1535 1 and N is even: the result is −0 instead of +0 (the reason was to be d1543 46 a1621 61 -- Function: int mpfr_fac_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) Set ROP to the factorial of OP, rounded in the direction RND. -- Function: int mpfr_fma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_rnd_t RND) -- Function: int mpfr_fms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_rnd_t RND) Set ROP to (OP1 times OP2) + OP3 (resp. (OP1 times OP2) - OP3) rounded in the direction RND. Concerning special values (signed zeros, infinities, NaN), these functions behave like a multiplication followed by a separate addition or subtraction. That is, the fused operation matters only for rounding. -- Function: int mpfr_fmma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_t OP4, mpfr_rnd_t RND) -- Function: int mpfr_fmms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_t OP4, mpfr_rnd_t RND) Set ROP to (OP1 times OP2) + (OP3 times OP4) (resp. (OP1 times OP2) - (OP3 times OP4)) rounded in the direction RND. In case the computation of OP1 times OP2 overflows or underflows (or that of OP3 times OP4), the result ROP is computed as if the two intermediate products were computed with rounding toward zero. -- Function: int mpfr_hypot (mpfr_t ROP, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) Set ROP to the Euclidean norm of X and Y, i.e., the square root of the sum of the squares of X and Y, rounded in the direction RND. Special values are handled as described in the ISO C99 (Section F.9.4.3) and IEEE 754-2008 (Section 9.2.1) standards: If X or Y is an infinity, then +Inf is returned in ROP, even if the other number is NaN. -- Function: int mpfr_sum (mpfr_t ROP, const mpfr_ptr TAB[], unsigned long int N, mpfr_rnd_t RND) Set ROP to the sum of all elements of TAB, whose size is N, correctly rounded in the direction RND. Warning: for efficiency reasons, TAB is an array of pointers to ‘mpfr_t’, not an array of ‘mpfr_t’. If N = 0, then the result is +0, and if N = 1, then the function is equivalent to ‘mpfr_set’. For the special exact cases, the result is the same as the one obtained with a succession of additions (‘mpfr_add’) in infinite precision. In particular, if the result is an exact zero and N >= 1: • if all the inputs have the same sign (i.e., all +0 or all −0), then the result has the same sign as the inputs; • otherwise, either because all inputs are zeros with at least a +0 and a −0, or because some inputs are non-zero (but they globally cancel), the result is +0, except for the ‘MPFR_RNDD’ rounding mode, where it is −0. -- Function: int mpfr_dot (mpfr_t ROP, const mpfr_ptr A[], const mpfr_ptr B[], unsigned long int N, mpfr_rnd_t RND) Set ROP to the dot product of elements of A by those of B, whose common size is N, correctly rounded in the direction RND. Warning: for efficiency reasons, A and B are arrays of pointers to ‘mpfr_t’. This function is experimental, and does not yet handle intermediate overflows and underflows. For the power functions (with an integer exponent or not), see *note mpfr_pow:: in *note Transcendental Functions::. d1623 1 a1623 1 File: mpfr.info, Node: Comparison Functions, Next: Transcendental Functions, Prev: Arithmetic Functions, Up: MPFR Interface a1656 1 -- Function: int mpfr_cmpabs_ui (mpfr_t OP1, unsigned long OP2) a1694 12 -- Function: int mpfr_total_order_p (mpfr_t X, mpfr_t Y) This function implements the totalOrder predicate from IEEE 754-2008, where −NaN < −Inf < negative finite numbers < −0 < +0 < positive finite numbers < +Inf < +NaN. It returns a non-zero value (true) when X is smaller than or equal to Y for this order relation, and zero (false) otherwise. Contrary to ‘mpfr_cmp (x, y)’, which returns a ternary value, ‘mpfr_total_order_p’ returns a binary value (zero or non-zero). In particular, ‘mpfr_total_order_p (x, x)’ returns true, ‘mpfr_total_order_p (-0, +0)’ returns true and ‘mpfr_total_order_p (+0, -0)’ returns false. The sign bit of NaN also matters. d1696 1 a1696 1 File: mpfr.info, Node: Transcendental Functions, Next: Input and Output Functions, Prev: Comparison Functions, Up: MPFR Interface d1698 2 a1699 2 5.7 Transcendental Functions ============================ d1706 7 a1712 13 Important note: In some domains, computing transcendental functions (even more with correct rounding) is expensive, even in small precision, for example the trigonometric and Bessel functions with a large argument. For some functions, the algorithm complexity and memory usage does not depend only on the output precision: for instance, the memory usage of ‘mpfr_rootn_ui’ is also linear in the argument K, and the memory usage of the incomplete Gamma function also depends on the precision of the input OP. It is also theoretically possible that some functions on some particular inputs might be very hard to round (i.e. the Table Maker’s Dilemma occurs in much larger precisions than normally expected from the context), meaning that the internal precision needs to be increased even more; but it is conjectured that the needed precision has a reasonable bound. a1738 46 -- Function: int mpfr_pow (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_pow_ui (mpfr_t ROP, unsigned long int OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_pow (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to OP1 raised to OP2, rounded in the direction RND. Special values are handled as described in the ISO C99 and IEEE 754-2008 standards for the ‘pow’ function: • ‘pow(±0, Y)’ returns plus or minus infinity for Y a negative odd integer. • ‘pow(±0, Y)’ returns plus infinity for Y negative and not an odd integer. • ‘pow(±0, Y)’ returns plus or minus zero for Y a positive odd integer. • ‘pow(±0, Y)’ returns plus zero for Y positive and not an odd integer. • ‘pow(-1, ±Inf)’ returns 1. • ‘pow(+1, Y)’ returns 1 for any Y, even a NaN. • ‘pow(X, ±0)’ returns 1 for any X, even a NaN. • ‘pow(X, Y)’ returns NaN for finite negative X and finite non-integer Y. • ‘pow(X, -Inf)’ returns plus infinity for 0 < abs(x) < 1, and plus zero for abs(x) > 1. • ‘pow(X, +Inf)’ returns plus zero for 0 < abs(x) < 1, and plus infinity for abs(x) > 1. • ‘pow(-Inf, Y)’ returns minus zero for Y a negative odd integer. • ‘pow(-Inf, Y)’ returns plus zero for Y negative and not an odd integer. • ‘pow(-Inf, Y)’ returns minus infinity for Y a positive odd integer. • ‘pow(-Inf, Y)’ returns plus infinity for Y positive and not an odd integer. • ‘pow(+Inf, Y)’ returns plus zero for Y negative, and plus infinity for Y positive. Note: When 0 is of integer type, it is regarded as +0 by these functions. We do not use the usual limit rules in this case, as these rules are not used for ‘pow’. d1767 3 a1769 3 mode, this number might not be in the output range 0 <= ROP < Pi of the arc-cosine function; still, the result lies in the image of the output range by the rounding function. The same holds for d1831 4 d1925 20 d1955 9 d1983 50 d2034 1 a2034 1 File: mpfr.info, Node: Input and Output Functions, Next: Formatted Output Functions, Prev: Transcendental Functions, Up: MPFR Interface d2050 10 a2059 18 Output OP on stream STREAM as a text string in base abs(BASE), rounded in the direction RND. The base may vary from 2 to 62 or from −2 to −36 (any other value yields undefined behavior). The argument N has the same meaning as in ‘mpfr_get_str’ (*note mpfr_get_str::): Print N significant digits exactly, or if N is 0, the number ‘mpfr_get_str_ndigits(BASE,P)’ where P is the precision of OP (*note mpfr_get_str_ndigits::). If the input is NaN, +Inf, −Inf, +0, or −0, then ‘@@NaN@@’, ‘@@Inf@@’, ‘-@@Inf@@’, ‘0’, or ‘-0’ is output, respectively. For the regular numbers, the format of the output is the following: the most significant digit, then a decimal-point character (defined by the current locale), then the remaining N−1 digits (including trailing zeros), then the exponent prefix, then the exponent in decimal. The exponent prefix is ‘e’ when abs(BASE) <= 10, and ‘@@’ when abs(BASE) > 10. *Note mpfr_get_str:: for information on the digits depending on the base. d2121 1 a2121 1 is an UBF number (internal use only); ‘<’ if the exponent is less d2253 9 a2261 9 The ‘mpfr_t’ number is rounded to the given precision in the direction specified by the rounding mode (see below if the ‘precision’ field is empty). If the precision is zero with rounding to nearest mode and one of the following ‘conv’ specifiers: ‘a’, ‘A’, ‘b’, ‘e’, ‘E’, tie case is rounded to even when it lies between two consecutive values at the wanted precision which have the same exponent, otherwise, it is rounded away from zero. For instance, 85 is displayed as "8e+1" and 95 is displayed as "1e+2" with the format specification ‘"%.0RNe"’. This also applies when the ‘g’ (resp. ‘G’) conversion specifier uses the ‘e’ d2266 7 a2272 9 If the ‘precision’ field is empty with ‘conv’ specifier ‘e’ and ‘E’ (as in ‘%Re’ or ‘%.RE’), the chosen precision (i.e., the number of digits to be displayed after the initial digit and the decimal point) is ceil(P*log(2)/log(10)), where P is the precision of the input variable, matching the choice done for ‘mpfr_get_str’; thus, if rounding to nearest is used, outputting the value with an empty ‘precision’ field and reading it back will yield the original value. The chosen precision for an empty ‘precision’ field with ‘conv’ specifiers ‘f’, ‘F’, ‘g’, and ‘G’ is 6. d2488 1 a2488 1 behavior is undefined). If PREC is greater than or equal to the d2491 2 a2492 3 to precision PREC with the given direction; no memory reallocation to free the unused limbs is done. In both cases, the precision of X is changed to PREC. d2560 3 a2562 4 Return a string ("MPFR_RNDN", "MPFR_RNDZ", "MPFR_RNDU", "MPFR_RNDD", "MPFR_RNDA", "MPFR_RNDF") corresponding to the rounding mode RND, or a null pointer if RND is an invalid rounding mode. d2768 1 a2768 1 compiler-level Thread-Local Storage (that is, MPFR was built with d2773 1 a2773 1 Return a non-zero value if MPFR was compiled with ‘_Float128’ d2801 1 a2801 1 File: mpfr.info, Node: Exception Related Functions, Next: Memory Handling Functions, Prev: Miscellaneous Functions, Up: MPFR Interface d2878 1 a2878 1 ‘EXP(X)-emin+1’ according to rounding mode RND and previous ternary d2886 1 a2886 1 ‘PREC(X)’ is not modified by this function. RND and T must be the d2889 1 a2889 1 ‘emin’ to ‘emin+PREC(X)-1’. If the result cannot be represented in d3020 1 a3020 14 File: mpfr.info, Node: Memory Handling Functions, Next: Compatibility with MPF, Prev: Exception Related Functions, Up: MPFR Interface 5.14 Memory Handling Functions ============================== These are general functions concerning memory handling (*note Memory Handling::, for more information). -- Function: void mpfr_free_cache (void) Free all caches and pools used by MPFR internally (those local to the current thread and those shared by all threads). You should call this function before terminating a thread, even if you did not call ‘mpfr_const_*’ functions directly (they could have been called internally). d3022 1 a3022 30 -- Function: void mpfr_free_cache2 (mpfr_free_cache_t WAY) Free various caches and pools used by MPFR internally, as specified by WAY, which is a set of flags: • those local to the current thread if flag ‘MPFR_FREE_LOCAL_CACHE’ is set; • those shared by all threads if flag ‘MPFR_FREE_GLOBAL_CACHE’ is set. The other bits of WAY are currently ignored and are reserved for future use; they should be zero. Note: ‘mpfr_free_cache2(MPFR_FREE_LOCAL_CACHE|MPFR_FREE_GLOBAL_CACHE)’ is currently equivalent to ‘mpfr_free_cache()’. -- Function: void mpfr_free_pool (void) Free the pools used by MPFR internally. Note: This function is automatically called after the thread-local caches are freed (with ‘mpfr_free_cache’ or ‘mpfr_free_cache2’). -- Function: int mpfr_mp_memory_cleanup (void) This function should be called before calling ‘mp_set_memory_functions’. *Note Memory Handling::, for more information. Zero is returned in case of success, non-zero in case of error. Errors are currently not possible, but checking the return value is recommended for future compatibility.  File: mpfr.info, Node: Compatibility with MPF, Next: Custom Interface, Prev: Memory Handling Functions, Up: MPFR Interface 5.15 Compatibility With MPF a3042 4 • The output of ‘mpf_out_str’ has a format slightly different from the one of ‘mpfr_out_str’ (concerning the position of the decimal-point character, trailing zeros and the output of the value 0). d3080 1 a3080 1 5.16 Custom Interface d3172 1 a3172 1 5.17 Internals d3295 3 a3297 3 The rounding modes ‘GMP_RNDx’ were renamed to ‘MPFR_RNDx’ in MPFR 3.0. However the old names ‘GMP_RNDx’ have been kept for compatibility (this might change in future versions), using: a3342 2 • ‘mpfr_cmpabs_ui’ in MPFR 4.1. a3363 2 • ‘mpfr_dot’ in MPFR 4.1 (incomplete, experimental). a3388 2 • ‘mpfr_get_decimal128’ in MPFR 4.1. a3397 2 • ‘mpfr_get_str_ndigits’ in MPFR 4.1. a3438 2 • ‘mpfr_set_decimal128’ in MPFR 4.1. a3459 2 • ‘mpfr_total_order_p’ in MPFR 4.1. a3478 9 • The formatted output functions (‘mpfr_printf’, etc.) have slightly changed in MPFR 4.1 in the case where the precision field is empty: trailing zeros were not output with the conversion specifier ‘e’ / ‘E’ (the chosen precision was not fully specified and it depended on the input value), and also on the value zero with the conversion specifiers ‘f’ / ‘F’ / ‘g’ / ‘G’ (this could partly be regarded as a bug); they are now kept in a way similar to the formatted output functions from C. d3492 2 a3493 2 value of the E1/eint1 function for negative argument (before MPFR 4.0, it was returning NaN). d3505 3 a3507 4 flag on NaN input (to follow the usual MPFR rules on NaN and IEEE 754-2008 recommendations on string conversions from Subclause 5.12.1) and sets the inexact flag when the conversion is inexact. a3533 4 • ‘mpfr_out_str’ changed in MPFR 4.1. The argument BASE can now be negative (from −2 to −36), in order to follow ‘mpfr_get_str’ and GMP’s ‘mpf_out_str’ functions. d3683 2 a3684 2 MPFR, the Floating-Point Working Group in charge of the revision of IEEE 754 did not want to discuss issues with non-floating-point types in d3742 2 a3743 3 Gastineau contributed the MPFRbench program, Fredrik Johansson a faster version of ‘mpfr_const_euler’, and Jianyang Pan a formally proven version of the ‘mpfr_add1sp1’ internal routine. d3761 1 a3761 1 2007, . d4172 1 a4172 1 . d4225 2 a4226 1 * Arithmetic functions: Arithmetic Functions. (line 3) d4228 2 d4241 8 a4248 2 * Exponent: Nomenclature and Types. (line 41) d4257 1 a4257 1 (line 52) a4282 2 * Memory handling functions: Memory Handling Functions. (line 3) d4292 1 a4292 1 (line 27) a4293 2 * Regular number: Nomenclature and Types. (line 6) a4296 2 * Rounding: Nomenclature and Types. (line 47) d4299 3 d4308 1 a4308 3 * Ternary value: Rounding. (line 79) * Transcendental functions: Transcendental Functions. (line 3) d4321 5 a4325 7 * mpfr_abs: Arithmetic Functions. (line 144) * mpfr_acos: Transcendental Functions. (line 120) * mpfr_acosh: Transcendental Functions. (line 185) * mpfr_add: Arithmetic Functions. d4327 1 a4327 1 * mpfr_add_d: Arithmetic Functions. d4329 1 a4329 1 * mpfr_add_q: Arithmetic Functions. d4331 1 a4331 1 * mpfr_add_si: Arithmetic Functions. d4333 1 a4333 1 * mpfr_add_ui: Arithmetic Functions. d4335 1 a4335 1 * mpfr_add_z: Arithmetic Functions. d4337 4 a4340 8 * mpfr_agm: Transcendental Functions. (line 281) * mpfr_ai: Transcendental Functions. (line 291) * mpfr_asin: Transcendental Functions. (line 121) * mpfr_asinh: Transcendental Functions. (line 186) d4342 5 a4346 9 (line 199) * mpfr_atan: Transcendental Functions. (line 122) * mpfr_atan2: Transcendental Functions. (line 132) * mpfr_atanh: Transcendental Functions. (line 187) * mpfr_beta: Transcendental Functions. (line 243) d4360 3 a4362 3 (line 40) * mpfr_cbrt: Arithmetic Functions. (line 116) d4368 1 a4368 1 (line 33) d4370 1 a4370 1 (line 38) a4388 2 * mpfr_cmpabs_ui: Comparison Functions. (line 35) d4407 4 a4410 8 * mpfr_const_catalan: Transcendental Functions. (line 302) * mpfr_const_euler: Transcendental Functions. (line 301) * mpfr_const_log2: Transcendental Functions. (line 299) * mpfr_const_pi: Transcendental Functions. (line 300) d4413 6 a4418 12 * mpfr_cos: Transcendental Functions. (line 98) * mpfr_cosh: Transcendental Functions. (line 165) * mpfr_cot: Transcendental Functions. (line 116) * mpfr_coth: Transcendental Functions. (line 181) * mpfr_csc: Transcendental Functions. (line 115) * mpfr_csch: Transcendental Functions. (line 180) d4427 5 a4431 6 (line 77) * mpfr_digamma: Transcendental Functions. (line 238) * mpfr_dim: Arithmetic Functions. (line 156) * mpfr_div: Arithmetic Functions. d4436 6 a4441 6 (line 56) * mpfr_div_2si: Arithmetic Functions. (line 171) * mpfr_div_2ui: Arithmetic Functions. (line 169) * mpfr_div_d: Arithmetic Functions. d4443 1 a4443 1 * mpfr_div_q: Arithmetic Functions. d4445 1 a4445 1 * mpfr_div_si: Arithmetic Functions. d4447 1 a4447 1 * mpfr_div_ui: Arithmetic Functions. d4449 1 a4449 1 * mpfr_div_z: Arithmetic Functions. a4450 2 * mpfr_dot: Arithmetic Functions. (line 228) d4452 2 a4453 2 (line 76) * mpfr_d_div: Arithmetic Functions. d4455 1 a4455 1 * mpfr_d_sub: Arithmetic Functions. d4457 1 a4457 2 * mpfr_eint: Transcendental Functions. (line 191) d4459 1 a4459 1 (line 35) d4461 1 a4461 1 (line 60) d4466 7 a4472 16 * mpfr_erf: Transcendental Functions. (line 255) * mpfr_erfc: Transcendental Functions. (line 256) * mpfr_exp: Transcendental Functions. (line 41) * mpfr_exp10: Transcendental Functions. (line 43) * mpfr_exp2: Transcendental Functions. (line 42) * mpfr_expm1: Transcendental Functions. (line 47) * mpfr_exp_t: Nomenclature and Types. (line 41) * mpfr_fac_ui: Arithmetic Functions. (line 177) d4474 1 a4474 1 (line 185) d4476 1 a4476 1 (line 181) d4478 1 a4478 1 (line 179) d4480 1 a4480 1 (line 183) d4482 1 a4482 1 (line 184) d4484 1 a4484 1 (line 180) d4486 1 a4486 1 (line 178) d4488 1 a4488 1 (line 182) d4498 1 a4498 1 (line 52) d4503 3 a4505 6 * mpfr_fma: Arithmetic Functions. (line 181) * mpfr_fmma: Arithmetic Functions. (line 191) * mpfr_fmms: Arithmetic Functions. (line 193) d4507 2 d4510 1 a4510 4 * mpfr_fmodquo: Integer and Remainder Related Functions. (line 110) * mpfr_fms: Arithmetic Functions. (line 183) d4512 1 a4512 1 (line 51) d4514 1 a4514 1 (line 62) d4516 1 a4516 1 (line 163) d4518 4 a4521 7 (line 91) * mpfr_free_cache: Memory Handling Functions. (line 9) * mpfr_free_cache2: Memory Handling Functions. (line 16) * mpfr_free_pool: Memory Handling Functions. (line 30) d4523 1 a4523 1 (line 173) d4525 3 a4527 5 (line 52) * mpfr_gamma: Transcendental Functions. (line 206) * mpfr_gamma_inc: Transcendental Functions. (line 207) a4529 2 * mpfr_get_decimal128: Conversion Functions. (line 11) d4533 1 a4533 1 (line 115) d4537 1 a4537 1 (line 39) d4553 1 a4553 1 (line 88) d4561 1 a4561 1 (line 41) d4565 1 a4565 1 (line 152) d4567 1 a4567 1 (line 83) d4569 1 a4569 1 (line 24) d4571 1 a4571 1 (line 26) d4573 1 a4573 3 (line 113) * mpfr_get_str_ndigits: Conversion Functions. (line 101) d4575 1 a4575 1 (line 25) d4577 1 a4577 1 (line 27) d4581 1 a4581 1 (line 74) d4583 1 a4583 1 (line 61) d4587 1 a4587 1 (line 57) d4589 2 a4590 3 (line 56) * mpfr_hypot: Arithmetic Functions. (line 201) d4594 1 a4594 1 (line 41) d4596 1 a4596 1 (line 56) d4600 1 a4600 1 (line 65) d4602 1 a4602 1 (line 25) d4622 1 a4622 1 (line 39) d4624 4 a4627 7 (line 137) * mpfr_j0: Transcendental Functions. (line 260) * mpfr_j1: Transcendental Functions. (line 261) * mpfr_jn: Transcendental Functions. (line 262) d4629 1 a4629 1 (line 59) d4631 1 a4631 1 (line 65) d4633 9 a4641 17 (line 58) * mpfr_lgamma: Transcendental Functions. (line 228) * mpfr_li2: Transcendental Functions. (line 201) * mpfr_lngamma: Transcendental Functions. (line 220) * mpfr_log: Transcendental Functions. (line 25) * mpfr_log10: Transcendental Functions. (line 29) * mpfr_log1p: Transcendental Functions. (line 37) * mpfr_log2: Transcendental Functions. (line 28) * mpfr_log_ui: Transcendental Functions. (line 26) d4647 1 a4647 1 (line 85) d4649 3 a4651 4 (line 98) * mpfr_mp_memory_cleanup: Memory Handling Functions. (line 35) * mpfr_mul: Arithmetic Functions. d4654 6 a4659 6 (line 54) * mpfr_mul_2si: Arithmetic Functions. (line 164) * mpfr_mul_2ui: Arithmetic Functions. (line 162) * mpfr_mul_d: Arithmetic Functions. d4661 1 a4661 1 * mpfr_mul_q: Arithmetic Functions. d4663 1 a4663 1 * mpfr_mul_si: Arithmetic Functions. d4665 1 a4665 1 * mpfr_mul_ui: Arithmetic Functions. d4667 1 a4667 1 * mpfr_mul_z: Arithmetic Functions. d4672 3 a4674 3 (line 40) * mpfr_neg: Arithmetic Functions. (line 143) d4684 1 a4684 1 (line 42) d4689 8 a4696 8 * mpfr_pow: Transcendental Functions. (line 51) * mpfr_pow_si: Transcendental Functions. (line 55) * mpfr_pow_ui: Transcendental Functions. (line 53) * mpfr_pow_z: Transcendental Functions. (line 57) d4700 1 a4700 1 (line 27) d4702 1 a4702 1 (line 170) d4704 2 a4705 4 (line 89) * mpfr_ptr: Nomenclature and Types. (line 6) * mpfr_rec_sqrt: Arithmetic Functions. d4708 1 a4708 1 (line 44) d4710 1 a4710 1 (line 46) d4712 2 a4714 2 * mpfr_remquo: Integer and Remainder Related Functions. (line 114) d4718 1 a4718 1 (line 54) d4720 2 d4723 1 a4723 1 * mpfr_rint_round: Integer and Remainder Related Functions. d4725 1 a4725 1 * mpfr_rint_roundeven: Integer and Remainder Related Functions. a4726 2 * mpfr_rint_trunc: Integer and Remainder Related Functions. (line 61) d4728 5 a4732 5 (line 47) * mpfr_root: Arithmetic Functions. (line 132) * mpfr_rootn_ui: Arithmetic Functions. (line 117) d4738 3 a4740 5 (line 95) * mpfr_sec: Transcendental Functions. (line 114) * mpfr_sech: Transcendental Functions. (line 179) a4746 2 * mpfr_set_decimal128: Assignment Functions. (line 23) d4750 1 a4750 1 (line 103) d4764 1 a4764 1 (line 27) d4772 1 a4772 1 (line 157) d4776 1 a4776 1 (line 156) d4782 1 a4782 1 (line 138) d4784 1 a4784 1 (line 29) d4786 1 a4786 1 (line 26) d4790 1 a4790 1 (line 64) d4794 1 a4794 1 (line 68) d4796 1 a4796 1 (line 76) d4800 1 a4800 1 (line 62) d4804 1 a4804 1 (line 66) d4808 1 a4808 1 (line 25) d4810 1 a4810 1 (line 158) d4812 1 a4812 1 (line 70) d4814 1 a4814 1 (line 50) d4817 5 a4821 9 * mpfr_sin: Transcendental Functions. (line 99) * mpfr_sinh: Transcendental Functions. (line 166) * mpfr_sinh_cosh: Transcendental Functions. (line 171) * mpfr_sin_cos: Transcendental Functions. (line 104) * mpfr_si_div: Arithmetic Functions. d4823 1 a4823 1 * mpfr_si_sub: Arithmetic Functions. d4826 1 a4826 1 (line 186) d4828 2 a4829 2 (line 176) * mpfr_sqr: Arithmetic Functions. d4831 1 a4831 1 * mpfr_sqrt: Arithmetic Functions. d4833 1 a4833 1 * mpfr_sqrt_ui: Arithmetic Functions. d4836 2 a4837 2 (line 94) * mpfr_sub: Arithmetic Functions. d4841 1 a4841 1 * mpfr_sub_d: Arithmetic Functions. d4843 1 a4843 1 * mpfr_sub_q: Arithmetic Functions. d4845 1 a4845 1 * mpfr_sub_si: Arithmetic Functions. d4847 1 a4847 1 * mpfr_sub_ui: Arithmetic Functions. d4849 1 a4849 1 * mpfr_sub_z: Arithmetic Functions. d4851 1 a4851 2 * mpfr_sum: Arithmetic Functions. (line 211) d4853 1 a4853 1 (line 164) d4856 2 a4857 6 * mpfr_tan: Transcendental Functions. (line 100) * mpfr_tanh: Transcendental Functions. (line 167) * mpfr_total_order_p: Comparison Functions. (line 74) d4860 1 a4860 1 * mpfr_ui_div: Arithmetic Functions. d4862 5 a4866 5 * mpfr_ui_pow: Transcendental Functions. (line 61) * mpfr_ui_pow_ui: Transcendental Functions. (line 59) * mpfr_ui_sub: Arithmetic Functions. d4871 1 a4871 1 (line 70) d4877 1 a4877 1 (line 200) d4891 1 a4891 1 (line 164) d4893 1 a4893 1 (line 171) d4895 1 a4895 1 (line 188) d4897 4 a4900 7 (line 177) * mpfr_y0: Transcendental Functions. (line 271) * mpfr_y1: Transcendental Functions. (line 272) * mpfr_yn: Transcendental Functions. (line 273) d4902 4 a4905 6 (line 43) * mpfr_zeta: Transcendental Functions. (line 249) * mpfr_zeta_ui: Transcendental Functions. (line 250) * mpfr_z_sub: Arithmetic Functions. d4915 40 a4954 44 Node: Reporting Bugs11668 Node: MPFR Basics13700 Node: Headers and Libraries14054 Node: Nomenclature and Types17650 Node: MPFR Variable Conventions20588 Node: Rounding22124 Ref: ternary value25941 Node: Floating-Point Values on Special Numbers27932 Node: Exceptions31181 Node: Memory Handling35009 Node: Getting the Best Efficiency Out of MPFR38760 Node: MPFR Interface39672 Node: Initialization Functions41989 Node: Assignment Functions49518 Node: Combined Initialization and Assignment Functions59660 Node: Conversion Functions60961 Ref: mpfr_get_str_ndigits66837 Ref: mpfr_get_str67460 Node: Arithmetic Functions72417 Node: Comparison Functions84490 Node: Transcendental Functions88781 Ref: mpfr_pow91426 Node: Input and Output Functions105400 Node: Formatted Output Functions110698 Node: Integer and Remainder Related Functions121040 Node: Rounding-Related Functions128568 Node: Miscellaneous Functions135170 Node: Exception Related Functions145661 Node: Memory Handling Functions155904 Node: Compatibility with MPF157792 Node: Custom Interface160961 Node: Internals165592 Node: API Compatibility167136 Node: Type and Macro Changes169084 Node: Added Functions172267 Node: Changed Functions177074 Node: Removed Functions184433 Node: Other Changes185163 Node: MPFR and the IEEE 754 Standard186864 Node: Contributors189481 Node: References192620 Node: GNU Free Documentation License194501 Node: Concept Index217095 Node: Function and Type Index223168 @ 1.1.1.5 log @initial import of MPFR 4.2.0. changes from 4.1.0 include: Binary compatible with MPFR 4.0.* and 4.1.*, though some minor changes in the behavior of the formatted output functions may be visible, regarded as underspecified behavior or bug fixes (see below). New functions mpfr_cosu, mpfr_sinu, mpfr_tanu, mpfr_acosu, mpfr_asinu, mpfr_atanu and mpfr_atan2u. New functions mpfr_cospi, mpfr_sinpi, mpfr_tanpi, mpfr_acospi, mpfr_asinpi, mpfr_atanpi and mpfr_atan2pi. New functions mpfr_log2p1, mpfr_log10p1, mpfr_exp2m1, mpfr_exp10m1 and mpfr_compound_si. New functions mpfr_fmod_ui, mpfr_powr, mpfr_pown, mpfr_pow_uj, mpfr_pow_sj and mpfr_rootn_si (mpfr_pown is actually a macro defined as an alias for mpfr_pow_sj). Bug fixes. - In particular, for the formatted output functions (mpfr_printf, etc.), the case where the precision consists only of a period has been fixed to be like .0 as specified in the ISO C standard, and the manual has been corrected and clarified. - The macros of the custom interface have also been fixed: they now behave like functions (except a minor limitation for mpfr_custom_init_set). @ text @d1 1 a1 1 This is mpfr.info, produced by makeinfo version 6.8 from mpfr.texi. d4 1 a4 1 Floating-Point Reliable Library, version 4.2.0. d6 1 a6 1 Copyright 1991, 1993-2023 Free Software Foundation, Inc. d26 1 a26 1 Floating-Point Reliable Library, version 4.2.0. d28 1 a28 1 Copyright 1991, 1993-2023 Free Software Foundation, Inc. d86 1 a86 1 code. See the file COPYING.LESSER.. d115 8 a122 8 In particular, MPFR follows the specification of the IEEE 754 standard, currently IEEE 754-2019 (which will be referred to as IEEE 754 in this manual), with some minor differences, such as: there is a single NaN, the default exponent range is much wider, and subnormal numbers are not implemented (but the exponent range can be reduced to any interval, and subnormals can be emulated). For instance, computations in the binary64 format (a.k.a. double precision) can be reproduced by using a precision of 53 bits. d186 3 a188 4 but any reasonable compiler should work (C++ compilers should work too, under the condition that they do not break type punning via union). And you need the standard Unix ‘make’ command, plus some other standard Unix utility commands. d194 6 a199 6 This will prepare the build and set up the options according to your system. You can give options to specify the install directories (instead of the default ‘/usr/local’), threading support, and so on. See the ‘INSTALL’ file and/or the output of ‘./configure --help’ for more information, in particular if you get error messages. d254 1 a254 1 Create a PostScript version of the manual, in ‘mpfr.ps’. d286 2 a287 2 *Note Reporting Bugs::. Some bug fixes are available on the MPFR 4.2.0 web page . d302 1 a302 1 on the MPFR 4.2.0 web page and the d308 1 a308 1 not to ask too much from you to ask you to report the bugs that you d368 1 a368 1 Note, however, that prototypes for MPFR functions with ‘FILE *’ d443 9 a451 15 numbers”. By convention, the radix point of the significand is just before the first digit (which is always 1 due to normalization), like in the C language, but unlike in IEEE 754 (thus, for a given number, the exponent values in MPFR and in IEEE 754 differ by 1). Like in the IEEE 754 standard, a floating-point number can also have three kinds of special values: a signed zero (+0 or −0), a signed infinity (+Inf or −Inf), and Not-a-Number (NaN). NaN can represent the default value of a floating-point object and the result of some operations for which no other results would make sense, such as 0 divided by 0 or +Inf minus +Inf; unless documented otherwise, the sign bit of a NaN is unspecified. Note that contrary to IEEE 754, MPFR has a single kind of NaN and does not have subnormals. Other than that, the behavior is very similar to IEEE 754, but there are some minor differences. d456 1 a456 3 is the C data type representing a pointer to this structure; ‘mpfr_srcptr’ is like ‘mpfr_ptr’, but the structure is read-only (i.e., const qualified). d524 1 a524 1 (roundTiesToEven in IEEE 754); see details below. d526 2 a527 2 • ‘MPFR_RNDD’: round toward negative infinity (roundTowardNegative in IEEE 754). d529 2 a530 2 • ‘MPFR_RNDU’: round toward positive infinity (roundTowardPositive in IEEE 754). d532 1 a532 1 • ‘MPFR_RNDZ’: round toward zero (roundTowardZero in IEEE 754). d555 2 a556 2 by (10.1) in binary, is rounded to (10.0) = 2 with a precision of two bits, and not to (11.0) = 3. This rule avoids the “drift” phenomenon a682 6 MPFR also tries to follow the specifications of the IEEE 754 standard on special values (IEEE 754 agree with the above rules in most cases). Any difference with IEEE 754 that is not explicitly mentioned, other than those due to the single NaN, is unintended and might be regarded as a bug. See also *note MPFR and the IEEE 754 Standard::. d711 1 a711 1 to the power E − 4, where E is the smallest exponent (for a d713 8 a720 9 rounding toward positive infinity. The exact result has the exponent E − 1. With the underflow before rounding, such a function call would yield an underflow, as E − 1 is outside the current exponent range. However, MPFR first considers the rounded result assuming an unbounded exponent range. The exact result cannot be represented exactly in precision 2, and here, it is rounded to 0.5 times 2 to E, which is representable in the current exponent range. As a consequence, this will not yield an underflow in MPFR. d844 1 a844 3 perform unnecessary allocations or copies. Slowdowns of up to a factor 15 have been observed on some applications with a C++ interface; d876 1 a876 1 left to the user or to a higher layer (for example, the MPFI library for d879 1 a879 1 variable with a large precision, then MPFR will still compute the result d1003 1 a1003 1 ISO C90). If PREC is not a constant expression, your compiler d1006 1 a1006 1 If you compile your program with GCC in C90 mode and with d1049 2 a1050 2 to a call to ‘mpfr_clear(X)’ followed by a call to ‘mpfr_init2(X, PREC)’, but more efficient as no allocation is done in case the d1094 2 a1095 3 ‘mpfr_set_si’, ‘mpfr_set_uj’, ‘mpfr_set_sj’, ‘mpfr_set_z’, ‘mpfr_set_q’ and ‘mpfr_set_f’, regardless of the rounding mode. The ‘mpfr_set_float128’ function is built only with the configure d1099 12 a1110 11 ‘MPFR_WANT_FLOAT128’ before including ‘mpfr.h’. If the system does not support the IEEE 754 standard, ‘mpfr_set_flt’, ‘mpfr_set_d’, ‘mpfr_set_ld’, ‘mpfr_set_decimal64’ and ‘mpfr_set_decimal128’ might not preserve the signed zeros (and in any case they don’t preserve the sign bit of NaN). The ‘mpfr_set_decimal64’ and ‘mpfr_set_decimal128’ functions are built only with the configure option ‘--enable-decimal-float’, and when the compiler or system provides the ‘_Decimal64’ and ‘_Decimal128’ data type; to use those functions, one should define the macro ‘MPFR_WANT_DECIMAL_FLOATS’ before including ‘mpfr.h’. ‘mpfr_set_q’ might fail if the numerator (or the denominator) cannot be represented as a ‘mpfr_t’. d1213 5 a1217 6 insensitive with the rules of the C locale. An ‘n-char-sequence-opt’ is a possibly empty string containing only digits, Latin letters and the underscore (0, 1, 2, ..., 9, a, b, ..., z, A, B, ..., Z, _). Note: one has an optional sign for all data, even NaN. For example, ‘-@@nAn@@(This_Is_Not_17)’ is a valid representation for NaN in base 17. d1224 2 a1225 3 positive infinity (+Inf) or positive zero (+0) iff SIGN is non-negative; in ‘mpfr_set_nan’, the sign bit of the result is unspecified. d1281 8 a1288 9 is NaN, some NaN (either quiet or signaling) or the result of 0.0/0.0 is returned (the sign bit is not preserved). If OP is ±Inf, an infinity of the same sign or the result of ±1.0/0.0 is returned. If OP is zero, these functions return a zero, trying to preserve its sign, if possible. The ‘mpfr_get_float128’, ‘mpfr_get_decimal64’ and ‘mpfr_get_decimal128’ functions are built only under some conditions: see the documentation of ‘mpfr_set_float128’, ‘mpfr_set_decimal64’ and ‘mpfr_set_decimal128’ respectively. d1290 2 a1291 2 -- Function: long int mpfr_get_si (mpfr_t OP, mpfr_rnd_t RND) -- Function: unsigned long int mpfr_get_ui (mpfr_t OP, mpfr_rnd_t RND) d1294 10 a1303 10 Convert OP to a ‘long int’, an ‘unsigned long int’, an ‘intmax_t’ or an ‘uintmax_t’ (respectively) after rounding it to an integer with respect to RND. If OP is NaN, 0 is returned and the _erange_ flag is set. If OP is too big for the return type, the function returns the maximum or the minimum of the corresponding C type, depending on the direction of the overflow; the _erange_ flag is set too. When there is no such range error, if the return value differs from OP, i.e., if OP is not an integer, the inexact flag is set. See also ‘mpfr_fits_slong_p’, ‘mpfr_fits_ulong_p’, ‘mpfr_fits_intmax_p’ and ‘mpfr_fits_uintmax_p’. d1310 4 a1313 4 that 0.5 <= abs(D) < 1 and D times 2 raised to EXP equals OP rounded to double (resp. long double) precision, using the given rounding mode. If OP is zero, then a zero of the same sign (or an unsigned zero, if the implementation does not have signed zeros) is d1321 4 a1324 4 0.5 <= abs(Y) < 1 and Y times 2 raised to EXP equals X rounded to the precision of Y, using the given rounding mode. If X is zero, then Y is set to a zero of the same sign and EXP is set to 0. If X is NaN or an infinity, then Y is set to the same value and EXP is d1332 4 a1335 4 exponent EMIN is returned. If OP is NaN or an infinity, the _erange_ flag is set, ROP is set to 0, and the minimal exponent EMIN is returned. The returned exponent may be less than the minimal exponent EMIN of MPFR numbers in the current exponent d1371 2 a1372 2 More precisely, we have m = 1 + ceil(P times log(2)/log(B)), with P replaced by P − 1 if B is a power of 2. d1390 5 a1394 5 significance order, are used. Warning! This implies that for BASE > 10, the successor of the digit 9 depends on BASE. This choice has been done for compatibility with GMP’s ‘mpf_get_str’ function. Users who wish a more consistent behavior should write a simple wrapper. d1407 1 a1407 1 number −3.1416 would be returned as ‘-31416’ in the string and 1 d1412 4 a1415 4 correspond to an even last digit: for example, with 2 digits in base 7, (14) and a half is rounded to (15), which is 12 in decimal, (16) and a half is rounded to (20), which is 14 in decimal, and (26) and a half is rounded to (26), which is 20 in decimal. d1418 1 a1418 1 ‘mpfr_get_str_ndigits (BASE, P)’, where P is the precision of OP d1428 2 a1429 2 for any value) is max(N + 2, 7) if N is not zero; if N is zero, replace it by ‘mpfr_get_str_ndigits (BASE, P)’, where P is the d1453 3 a1455 3 respectively ‘unsigned long int’, ‘long int’, ‘unsigned int’, ‘int’, ‘unsigned short’, ‘short’, ‘uintmax_t’, ‘intmax_t’, when rounded to an integer in the direction RND. For instance, with the d1485 4 a1488 4 having no signed zeros, 0 is considered unsigned (i.e., (+0) + 0 = (+0) and (−0) + 0 = (−0)). The ‘mpfr_add_d’ function assumes that the radix of the ‘double’ type is a power of 2, with a precision at most that declared by the C implementation (macro d1511 1 a1511 1 Set ROP to OP1 − OP2 rounded in the direction RND. The IEEE 754 d1513 4 a1516 4 having no signed zeros, 0 is considered unsigned (i.e., (+0) − 0 = (+0), (−0) − 0 = (−0), 0 − (+0) = (−0) and 0 − (−0) = (+0)). The same restrictions as for ‘mpfr_add_d’ apply to ‘mpfr_d_sub’ and ‘mpfr_sub_d’. d1533 2 a1534 3 positive). The same restrictions as for ‘mpfr_add_d’ apply to ‘mpfr_mul_d’. Note: when OP1 and OP2 are equal, use ‘mpfr_sqr’ instead of ‘mpfr_mul’ for better efficiency. d1557 2 a1558 2 Set ROP to OP1 / OP2 rounded in the direction RND. When a result is zero, its sign is the product of the signs of the operands. For d1562 1 a1562 1 decision on the IEEE 754 side. The same restrictions as for d1569 2 a1570 3 ROP to −0 if OP is −0, to be consistent with the IEEE 754 standard (thus this differs from ‘mpfr_rootn_ui’ and ‘mpfr_rootn_si’ with N = 2). Set ROP to NaN if OP is negative. d1577 2 a1578 3 IEEE 754 standard (Section 9.2.1), which is −Inf instead of +Inf. However, ‘mpfr_rec_sqrt’ is equivalent to ‘mpfr_rootn_si’ with N = −2. a1582 2 -- Function: int mpfr_rootn_si (mpfr_t ROP, mpfr_t OP, long int N, mpfr_rnd_t RND) d1590 4 a1593 2 These functions agree with the rootn operation of the IEEE 754 standard. d1602 3 a1604 3 This function predates IEEE 754-2008, where rootn was introduced, and behaves differently from the IEEE 754 rootn operation. It is marked as deprecated and will be removed in a future release. d1608 1 a1608 1 Set ROP to −OP and the absolute value of OP respectively, rounded d1620 1 a1620 1 Set ROP to the positive difference of OP1 and OP2, i.e., OP1 − OP2 d1647 1 a1647 1 Set ROP to (OP1 times OP2) + OP3 (resp. (OP1 times OP2) − OP3) d1657 5 a1661 5 Set ROP to (OP1 times OP2) + (OP3 times OP4) (resp. (OP1 times OP2) − (OP3 times OP4)) rounded in the direction RND. In case the computation of OP1 times OP2 overflows or underflows (or that of OP3 times OP4), the result ROP is computed as if the two intermediate products were computed with rounding toward zero. d1668 3 a1670 3 (Section F.9.4.3) and IEEE 754 (Section 9.2.1) standards: If X or Y is an infinity, then +Inf is returned in ROP, even if the other number is NaN. d1735 1 a1735 1 -- Function: int mpfr_cmpabs_ui (mpfr_t OP1, unsigned long int OP2) d1753 1 a1753 1 return zero. This is equivalent to ‘mpfr_cmp_ui (OP, 0)’, but more d1775 10 a1784 10 This function implements the totalOrder predicate from IEEE 754, where −NaN < −Inf < negative finite numbers < −0 < +0 < positive finite numbers < +Inf < +NaN. It returns a non-zero value (true) when X is smaller than or equal to Y for this order relation, and zero (false) otherwise. Contrary to ‘mpfr_cmp (X, Y)’, which returns a ternary value, ‘mpfr_total_order_p’ returns a binary value (zero or non-zero). In particular, ‘mpfr_total_order_p (X, X)’ returns true, ‘mpfr_total_order_p (-0, +0)’ returns true and ‘mpfr_total_order_p (+0, -0)’ returns false. The sign bit of NaN also matters. d1809 1 a1809 2 has a reasonable bound (and in particular, that potentially exact cases are known and can be detected efficiently). d1812 2 a1813 2 -- Function: int mpfr_log_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) d1819 2 a1820 2 IEEE 754 standards. Set ROP to −Inf if OP is ±0 (i.e., the sign of the zero has no influence on the result). d1823 2 a1824 5 -- Function: int mpfr_log2p1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_log10p1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the logarithm of one plus OP (in radix two for ‘mpfr_log2p1’, and in radix ten for ‘mpfr_log10p1’), rounded in the direction RND. Set ROP to −Inf if OP is −1. d1833 2 a1834 6 -- Function: int mpfr_exp2m1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_exp10m1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential of OP followed by a subtraction by one (resp. 2 power of OP followed by a subtraction by one, and 10 power of OP followed by a subtraction by one), rounded in the direction RND. a1837 2 -- Function: int mpfr_powr (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) a1841 6 -- Function: int mpfr_pow_uj (mpfr_t ROP, mpfr_t OP1, uintmax_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_sj (mpfr_t ROP, mpfr_t OP1, intmax_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_pown (mpfr_t ROP, mpfr_t OP1, intmax_t OP2, mpfr_rnd_t RND) d1848 10 a1857 10 Set ROP to OP1 raised to OP2, rounded in the direction RND. The ‘mpfr_powr’ function corresponds to the ‘powr’ function from IEEE 754, i.e., it computes the exponential of OP2 multiplied by the logarithm of OP1. The ‘mpfr_pown’ function is just an alias for ‘mpfr_pow_sj’ (defined with ‘#define mpfr_pown mpfr_pow_sj’), to follow the C2x function ‘pown’. Special values are handled as described in the ISO C99 and IEEE 754 standards for the ‘pow’ function: • ‘pow(±0, Y)’ returns ±Inf for Y a negative odd integer. • ‘pow(±0, Y)’ returns +Inf for Y negative and not an odd a1858 2 • ‘pow(±0, Y)’ returns ±0 for Y a positive odd integer. • ‘pow(±0, Y)’ returns +0 for Y positive and not an odd integer. d1864 5 a1868 6 • ‘pow(X, -Inf)’ returns +Inf for 0 < abs(x) < 1, and +0 for abs(x) > 1. • ‘pow(X, +Inf)’ returns +0 for 0 < abs(x) < 1, and +Inf for abs(x) > 1. • ‘pow(-Inf, Y)’ returns −0 for Y a negative odd integer. • ‘pow(-Inf, Y)’ returns +0 for Y negative and not an odd d1870 1 a1870 2 • ‘pow(-Inf, Y)’ returns −Inf for Y a positive odd integer. • ‘pow(-Inf, Y)’ returns +Inf for Y positive and not an odd d1872 6 a1877 2 • ‘pow(+Inf, Y)’ returns +0 for Y negative, and +Inf for Y positive. a1881 6 -- Function: int mpfr_compound_si (mpfr_t ROP, mpfr_t OP, long int N, mpfr_rnd_t RND) Set ROP to the power N of one plus OP, following IEEE 754 for the special cases and exceptions. When N is zero and OP is NaN or greater or equal to −1, ROP is set to 1. a1887 23 -- Function: int mpfr_cosu (mpfr_t ROP, mpfr_t OP, unsigned long int U, mpfr_rnd_t RND) -- Function: int mpfr_sinu (mpfr_t ROP, mpfr_t OP, unsigned long int U, mpfr_rnd_t RND) -- Function: int mpfr_tanu (mpfr_t ROP, mpfr_t OP, unsigned long int U, mpfr_rnd_t RND) Set ROP to the cosine (resp. sine and tangent) of OP multiplied by 2 Pi and divided by U. For example, if U equals 360, one gets the cosine (resp. sine and tangent) for OP in degrees. For ‘mpfr_cosu’, when OP multiplied by 2 and divided by U is a half-integer, the result is +0, following IEEE 754 (cosPi), so that the function is even. For ‘mpfr_sinu’, when OP multiplied by 2 and divided by U is an integer, the result is zero with the same sign as OP, following IEEE 754 (sinPi), so that the function is odd. Similarly, the function ‘mpfr_tanu’ follows IEEE 754 (tanPi). -- Function: int mpfr_cospi (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_sinpi (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_tanpi (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the cosine (resp. sine and tangent) of OP multiplied by Pi. See the description of ‘mpfr_sinu’, ‘mpfr_cosu’ and ‘mpfr_tanu’ for special values. d1893 4 a1896 4 both results are exact, more precisely it returns s + 4c where s = 0 if SOP is exact, s = 1 if SOP is larger than the sine of OP, s = 2 if SOP is smaller than the sine of OP, and similarly for c and the cosine of OP. d1913 1 a1913 1 ‘asin(-1)’, ‘asin(1)’, ‘atan(-Inf)’, ‘atan(+Inf)’ or for ‘atan(OP)’ a1915 16 -- Function: int mpfr_acosu (mpfr_t ROP, mpfr_t OP, unsigned long int U, mpfr_rnd_t RND) -- Function: int mpfr_asinu (mpfr_t ROP, mpfr_t OP, unsigned long int U, mpfr_rnd_t RND) -- Function: int mpfr_atanu (mpfr_t ROP, mpfr_t OP, unsigned long int U, mpfr_rnd_t RND) Set ROP to A multiplied by U and divided by 2 Pi, where A is the arc-cosine (resp. arc-sine and arc-tangent) of OP. For example, if U equals 360, ‘mpfr_acosu’ yields the arc-cosine in degrees. -- Function: int mpfr_acospi (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_asinpi (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_atanpi (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to ‘acos(OP)’ (resp. ‘asin(OP)’ and ‘atan(OP)’) divided by Pi. d1918 5 a1922 15 -- Function: int mpfr_atan2u (mpfr_t ROP, mpfr_t Y, mpfr_t X, unsigned long int U, mpfr_rnd_t RND) -- Function: int mpfr_atan2pi (mpfr_t ROP, mpfr_t Y, mpfr_t X, mpfr_rnd_t RND) For ‘mpfr_atan2’, set ROP to the arc-tangent2 of Y and X, rounded in the direction RND: if X > 0, then ‘atan2(Y, X)’ returns atan(Y/X); if X < 0, then ‘atan2(Y, X)’ returns the sign of Y multiplied by Pi − atan(abs(Y/X)), thus a number from −Pi to Pi. As for ‘atan’, in case the exact mathematical result is +Pi or −Pi, its rounded result might be outside the function output range. The function ‘mpfr_atan2u’ behaves similarly, except the result is multiplied by U and divided by 2 Pi; and ‘mpfr_atan2pi’ is the same as ‘mpfr_atan2u’ with U = 2. For example, if U equals 360, ‘mpfr_atan2u’ returns the arc-tangent in degrees, with values from −180 to 180. d1924 2 a1925 2 ‘atan2(Y, 0)’ does not raise any floating-point exception. Special values are handled as described in the ISO C99 and IEEE 754 d1928 1 a1928 1 • ‘atan2(-0, -0)’ returns −Pi. d1931 6 a1936 6 • ‘atan2(+0, X)’ returns +Pi for X < 0. • ‘atan2(-0, X)’ returns −Pi for X < 0. • ‘atan2(+0, X)’ returns +0 for X > 0. • ‘atan2(-0, X)’ returns −0 for X > 0. • ‘atan2(Y, 0)’ returns −Pi/2 for Y < 0. • ‘atan2(Y, 0)’ returns +Pi/2 for Y > 0. d1938 1 a1938 1 • ‘atan2(-Inf, -Inf)’ returns −3*Pi/4. d1940 7 a1946 7 • ‘atan2(-Inf, +Inf)’ returns −Pi/4. • ‘atan2(+Inf, X)’ returns +Pi/2 for finite X. • ‘atan2(-Inf, X)’ returns −Pi/2 for finite X. • ‘atan2(Y, -Inf)’ returns +Pi for finite Y > 0. • ‘atan2(Y, -Inf)’ returns −Pi for finite Y < 0. • ‘atan2(Y, +Inf)’ returns +0 for finite Y > 0. • ‘atan2(Y, +Inf)’ returns −0 for finite Y < 0. d1978 5 a1982 6 to the power k, divided by k and the factorial of k. For positive OP, it corresponds to the Ei function at OP (see formula 5.1.10 from the Handbook of Mathematical Functions from Abramowitz and Stegun), and for negative OP, to the opposite of the E1 function (sometimes called eint1) at −OP (formula 5.1.1 from the same reference). d1987 1 a1987 1 integral of −log(1−t)/t from 0 to OP. d1992 1 a1992 1 Set ROP to the value of the Gamma function on OP, resp. the d2006 1 a2006 1 all rounding modes). When OP is an infinity or a non-positive d2008 2 a2009 2 values. When −2k − 1 < OP < −2k, k being a non-negative integer, set ROP to NaN. See also ‘mpfr_lgamma’. d2017 1 a2017 1 is an infinity or a non-positive integer, set ROP to +Inf. When OP d2033 2 a2034 2 -- Function: int mpfr_zeta_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) d2045 1 a2045 1 -- Function: int mpfr_jn (mpfr_t ROP, long int N, mpfr_t OP, mpfr_rnd_t d2049 4 a2052 4 NaN, ROP is always set to NaN. When OP is positive or negative infinity, ROP is set to +0. When OP is zero, and N is not zero, ROP is set to +0 or −0 depending on the parity and sign of N, and the sign of OP. d2056 1 a2056 1 -- Function: int mpfr_yn (mpfr_t ROP, long int N, mpfr_t OP, mpfr_rnd_t d2068 5 a2072 6 limit of the sequences u_n and v_n, where u_0 = OP1, v_0 = OP2, u_(n+1) is the arithmetic mean of u_n and v_n, and v_(n+1) is the geometric mean of u_n and v_n. If any operand is negative and the other one is not zero, set ROP to NaN. If any operand is zero and the other one is finite (resp. infinite), set ROP to +0 (resp. NaN). d2115 2 a2116 2 the number ‘mpfr_get_str_ndigits (BASE, P)’, where P is the precision of OP (*note mpfr_get_str_ndigits::). d2123 1 a2123 1 by the current locale), then the remaining N − 1 digits (including d2179 2 a2180 2 remaining of the output is as follows: ‘0.’ then the p bits of the binary significand, where p is the precision of the number; if the d2215 1 a2215 3 the ‘gmp_printf’ one (itself, an extension of the ‘printf’ one). The conversion specification is of the form: a2216 1 d2218 1 a2218 1 standard ‘printf’ (in particular, notice that the precision is related d2235 4 a2238 4 and the ‘type’ specifiers defined in GMP, plus ‘R’ and ‘P’, which are specific to MPFR (the second column in the table below shows the type of the argument read in the argument list and the kind of ‘conv’ specifier to use after the ‘type’ specifier): a2261 4 Thus the ‘conv’ specifier ‘F’ is not supported (due to the use of ‘F’ as the ‘type’ specifier for ‘mpf_t’), except for the ‘type’ specifier ‘R’ (i.e., for ‘mpfr_t’ arguments). d2265 2 a2266 2 an ‘int’ or any fixed standard type. The ‘precision’ value specifies the minimum number of digits to appear. The default precision is 1. d2280 2 a2281 2 ‘U’ round toward positive infinity ‘D’ round toward negative infinity d2310 4 a2313 4 The conversion specifier ‘b’, which displays the argument in binary, is specific to ‘mpfr_t’ arguments and should not be used with other types. Other conversion specifiers have the same meaning as for a ‘double’ argument. d2322 21 a2342 26 direction specified by the rounding mode (see below if the precision is missing). Similarly to the native C types, the precision is the number of digits output after the decimal-point character, except for the ‘g’ and ‘G’ conversion specifiers, where it is the number of significant digits (but trailing zeros of the fractional part are not output by default), or 1 if the precision is zero. If the precision is zero with rounding to nearest mode and one of the following conversion specifiers: ‘a’, ‘A’, ‘b’, ‘e’, ‘E’, tie case is rounded to even when it lies between two consecutive values at the wanted precision which have the same exponent, otherwise, it is rounded away from zero. For instance, 85 is displayed as ‘8e+1’ and 95 is displayed as ‘1e+2’ with the format specification ‘"%.0RNe"’. This also applies when the ‘g’ (resp. ‘G’) conversion specifier uses the ‘e’ (resp. ‘E’) style. If the precision is set to a value greater than the maximum value for an ‘int’, it will be silently reduced down to ‘INT_MAX’. If the precision is missing, it is chosen as follows, depending on the conversion specifier. • With ‘a’, ‘A’, and ‘b’, it is chosen to have an exact representation with no trailing zeros. • With ‘e’ and ‘E’, it is ceil(p times log(2)/log(10)), where p is the precision of the input variable, matching the choice done for ‘mpfr_get_str’; thus, if rounding to nearest is used, outputting the value with a missing precision and reading it back will yield the original value. • With ‘f’, ‘F’, ‘g’, and ‘G’, it is 6. d2386 2 a2387 2 null pointer, otherwise, the first N − 1 characters are written in BUF and the N-th one is a null character. Return the number of d2425 1 a2425 1 of IEEE 754); d2445 1 a2445 2 function, you should use one of the ‘mpfr_rint_*’ functions instead. d2511 1 a2511 1 -- Function: int mpfr_fmod_ui (mpfr_t R, mpfr_t X, unsigned long int Y, a2512 2 -- Function: int mpfr_fmodquo (mpfr_t R, long int* Q, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) d2515 3 a2517 3 -- Function: int mpfr_remquo (mpfr_t R, long int* Q, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) Set R to the value of X − NY, rounded according to the direction d2519 3 a2521 3 follows: N is rounded toward zero for ‘mpfr_fmod’, ‘mpfr_fmod_ui’ and ‘mpfr_fmodquo’, and to the nearest integer (ties rounded to even) for ‘mpfr_remainder’ and ‘mpfr_remquo’. d2531 5 a2535 6 number of bits in a ‘long int’ minus one), with the sign of X divided by Y (except if those low bits are all zero, in which case zero is returned). If the result is NaN, the value of *Q is unspecified. Note that X may be so large in magnitude relative to Y that an exact representation of the quotient is not practical. The ‘mpfr_remainder’ and ‘mpfr_remquo’ functions are useful for d2568 8 a2575 8 mpfr_set_prec (t, 2 * n); mpfr_set (t, a, MPFR_RNDN); /* round a to 2n bits */ mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to 2n bits */ mpfr_ui_sub (t, 1, t, MPFR_RNDN); /* high n bits cancel with 1 */ mpfr_prec_round (t, n, MPFR_RNDN); /* t is correct to n bits */ mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to n bits */ mpfr_prec_round (x, 2 * n, MPFR_RNDN); /* exact */ mpfr_add (x, x, t, MPFR_RNDN); /* x is correct to 2n bits */ d2584 7 a2590 8 direction RND1 with error at most two to the power EXP(B) − ERR where EXP(B) is the exponent of B, return a non-zero value if one is able to round correctly X to precision PREC with the direction RND2 assuming an unbounded exponent range, and 0 otherwise (including for NaN and Inf). In other words, if the error on B is bounded by two to the power K ulps, and B has precision PREC, you should give ERR = PREC − K. This function *does not modify* its arguments. d2613 2 a2614 2 /* round the approximation b to the result r of prec bits with rounding mode rnd2 and get the ternary value inex */ d2618 1 a2618 1 PREC + 1 bits with a directed rounding: if so, one can surely round d2631 2 a2632 2 Return a string (‘"MPFR_RNDN"’, ‘"MPFR_RNDZ"’, ‘"MPFR_RNDU"’, ‘"MPFR_RNDD"’, ‘"MPFR_RNDA"’, ‘"MPFR_RNDF"’) corresponding to the d2639 1 a2639 1 ‘mpfr_t’, a ‘long int’, a ‘double’, etc.), put in ROP the d2650 6 a2655 5 the minimal exponent EMIN is the smallest possible one. This macro has been made such that the compiler is able to detect mismatch between the argument list OP and the function prototype of FOO. Multiple input arguments OP are supported only with C99 compilers. Otherwise, for C90 compilers, only one such argument is supported. a2678 5 Note: Concerning the exceptions and the sign of 0, the behavior differs from the ISO C ‘nextafter’ and ‘nexttoward’ functions. It is similar to the nextUp and nextDown operations from IEEE 754 (introduced in its 2008 revision). d2681 2 a2682 1 Equivalent to ‘mpfr_nexttoward’ where Y is +Inf (resp. −Inf). d2691 1 a2691 7 different signs, then ROP is set to −0 (resp. +0). As usual, the NaN flag is set only when the result is NaN, i.e., when both OP1 and OP2 are NaN. Note: These functions correspond to the minimumNumber and maximumNumber operations of IEEE 754-2019 for the result. But in MPFR, the NaN flag is set only when _both_ operands are NaN. d2694 6 a2699 6 Generate a uniformly distributed random float in the interval 0 <= ROP < 1. More precisely, the number can be seen as a float with a random non-normalized significand and exponent 0, which is then normalized (thus if E denotes the exponent after normalization, then the least −E significant bits of the significand are always 0). d2704 3 a2706 3 cases). The second argument is a ‘gmp_randstate_t’ structure, which should be created using the GMP ‘gmp_randinit’ function (see the GMP manual). d2719 1 a2719 1 The second argument is a ‘gmp_randstate_t’ structure, which should d2825 1 a2825 1 #if (!defined(MPFR_VERSION) || (MPFR_VERSION < MPFR_VERSION_NUM(3,0,0))) d2835 1 a2835 1 (compile-time) MPFR version are not available (however, this d2883 1 a2883 1 exponent and the largest value has the form (1 − epsilon) times 2 d2907 2 a2908 2 If EMIN > EMAX and a floating-point value needs to be produced as output, the behavior is undefined (‘mpfr_set_emin’ and d2949 17 a2965 17 ternary value:: T; otherwise, if EXP(X) denotes the exponent of X, it rounds X to precision EXP(X)−EMIN+1 according to rounding mode RND and previous ternary value T, avoiding double rounding problems. More precisely in the subnormal domain, denoting by e the value of EMIN, X is rounded in fixed-point arithmetic to an integer multiple of two to the power e − 1; as a consequence, 1.5 multiplied by two to the power e − 1 when T is zero is rounded to two to the power e with rounding to nearest. The precision PREC(X) of X is not modified by this function. RND and T must be the rounding mode and the returned ternary value used when computing X (as in ‘mpfr_check_range’). The subnormal exponent range is from EMIN to EMIN+PREC(X)−1. If the result cannot be represented in the current exponent range of MPFR (due to a too small EMAX), the behavior is undefined. Note that unlike most functions, the result is compared to the exact one, not the input value X, i.e., the ternary value is propagated. d2971 1 a2971 1 Warning! If you change EMIN (with ‘mpfr_set_emin’) just before d2974 1 a2974 1 change EMIN before any computation, if possible. d2976 2 a2977 2 This is an example of how to emulate binary64 IEEE 754 arithmetic (a.k.a. double precision) using MPFR: d3005 2 a3006 3 1 to 17 with a result in a fixed-point arithmetic rounded at two to the power −42 (using the fact that the result is at most 1 in absolute value): d3117 3 a3119 3 Note: ‘mpfr_free_cache2 (MPFR_FREE_LOCAL_CACHE | MPFR_FREE_GLOBAL_CACHE)’ is currently equivalent to ‘mpfr_free_cache()’. d3184 2 a3185 2 |OP1 − OP2| / OP1, using the precision of ROP and the rounding mode RND for all operations. d3220 1 a3220 4 macro, while ‘(mpfr_custom_init) (s, p)’ uses the function. The ‘mpfr_custom_init_set’ macro is not usable in contexts where an expression is expected, e.g., inside ‘for(...)’ or before a comma operator. d3282 4 a3285 4 collect and update its new position to ‘new_position’. However, the application has to move the significand and the ‘mpfr_t’ itself. The behavior of this function for any ‘mpfr_t’ not initialized with ‘mpfr_custom_init_set’ is undefined. d3316 1 a3316 1 different from NaN, infinity or zero) values always have the most d3414 1 a3414 1 MPFR 3.0. However, the old names ‘GMP_RNDx’ have been kept for d3421 1 a3421 1 MPFR 3.0 (however, no rounding mode ‘GMP_RNDA’ exists). Faithful a3438 2 • ‘mpfr_acospi’ and ‘mpfr_acosu’ in MPFR 4.2. a3442 2 • ‘mpfr_asinpi’ and ‘mpfr_asinu’ in MPFR 4.2. a3444 4 • ‘mpfr_atan2pi’ and ‘mpfr_atan2u’ in MPFR 4.2. • ‘mpfr_atanpi’ and ‘mpfr_atanu’ in MPFR 4.2. a3462 2 • ‘mpfr_compound_si’ in MPFR 4.2. a3467 2 • ‘mpfr_cospi’ and ‘mpfr_cosu’ in MPFR 4.2. a3487 2 • ‘mpfr_exp2m1’ and ‘mpfr_exp10m1’ in MPFR 4.2. a3496 2 • ‘mpfr_fmod_ui’ in MPFR 4.2. a3534 2 • ‘mpfr_log2p1’ and ‘mpfr_log10p1’ in MPFR 4.2. a3550 3 • ‘mpfr_powr’, ‘mpfr_pown’, ‘mpfr_pow_sj’ and ‘mpfr_pow_uj’ in MPFR 4.2. a3562 2 • ‘mpfr_rootn_si’ in MPFR 4.2. a3583 2 • ‘mpfr_sinpi’ and ‘mpfr_sinu’ in MPFR 4.2. a3587 2 • ‘mpfr_tanpi’ and ‘mpfr_tanu’ in MPFR 4.2. d3605 3 a3607 4 The following functions and function-like macros have changed after MPFR 2.2. Changes can affect the behavior of code written for some MPFR version when built and run against another MPFR version (older or newer), as described below. d3616 1 a3616 4 functions from C. Moreover, the case where the precision consists only of a period has been fixed in MPFR 4.2 to be like ‘.0’ as specified in the ISO C standard (it previously behaved as a missing precision). d3645 1 a3645 1 IEEE 754 recommendations on string conversions from d3656 1 a3656 1 bool ? mpfr_get_z(...) : mpfr_add(...); d3659 1 a3659 1 bool ? mpfr_get_z(...) : (void) mpfr_add(...); d3662 1 a3662 1 bool ? (void) mpfr_get_z(...) : (void) mpfr_add(...); a3732 4 • Up to MPFR 4.1.0, some macros of the *note Custom Interface:: had undocumented limitations. In particular, their arguments may be evaluated multiple times or none. d3761 1 a3761 1 included), ‘intmax_t’ was assumed to be defined. However, this was not d3776 1 a3776 1 The divide-by-zero exception is new in MPFR 3.1. However, it should d3810 3 a3812 3 The ‘mpfr_root’ function predates IEEE 754-2008, where rootn was introduced, and behaves differently from the IEEE 754 rootn operation. It is deprecated and ‘mpfr_rootn_ui’ should be used instead. d3916 3 a3918 7 • IEEE Standard for Floating-Point Arithmetic, IEEE Standard 754-2008, 2008. Revision of IEEE Standard 754-1985, approved June 12, 2008: IEEE-SA Standards Board, 70 pages. • IEEE Standard for Floating-Point Arithmetic, IEEE Standard 754-2019, 2019. Revision of IEEE Standard 754-2008, approved June 13, 2019: IEEE-SA Standards Board, 84 pages. d4384 1 a4384 1 (line 47) d4393 1 a4393 1 (line 58) d4430 1 a4430 1 (line 33) d4438 1 a4438 1 (line 53) d4447 1 a4447 1 * Ternary value: Rounding. (line 75) d4463 1 a4463 1 (line 145) d4465 1 a4465 1 (line 163) a4466 2 (line 253) * mpfr_acospi: Transcendental Functions. a4467 2 * mpfr_acosu: Transcendental Functions. (line 175) d4481 1 a4481 1 (line 350) d4483 1 a4483 1 (line 361) d4485 1 a4485 1 (line 164) a4486 2 (line 254) * mpfr_asinpi: Transcendental Functions. a4487 2 * mpfr_asinu: Transcendental Functions. (line 177) d4489 1 a4489 1 (line 211) d4491 1 a4491 1 (line 165) d4493 1 a4493 5 (line 191) * mpfr_atan2pi: Transcendental Functions. (line 195) * mpfr_atan2u: Transcendental Functions. (line 193) a4494 2 (line 255) * mpfr_atanpi: Transcendental Functions. a4495 2 * mpfr_atanu: Transcendental Functions. (line 179) d4497 1 a4497 1 (line 312) d4499 1 a4499 1 (line 195) d4501 2 a4503 2 * mpfr_buildopt_gmpinternals_p: Miscellaneous Functions. (line 200) d4505 1 a4505 1 (line 205) d4507 1 a4507 1 (line 184) d4509 1 a4509 1 (line 213) d4513 1 a4513 1 (line 118) d4523 1 a4523 1 (line 154) d4525 1 a4525 1 (line 157) d4527 1 a4527 1 (line 161) d4529 1 a4529 1 (line 156) d4531 1 a4531 1 (line 155) d4533 1 a4533 1 (line 153) d4535 1 a4535 1 (line 152) a4559 2 * mpfr_compound_si: Transcendental Functions. (line 112) d4561 1 a4561 1 (line 372) d4563 1 a4563 1 (line 371) d4565 1 a4565 1 (line 369) d4567 1 a4567 1 (line 370) d4569 1 a4569 1 (line 137) d4571 1 a4571 1 (line 118) d4573 1 a4573 5 (line 233) * mpfr_cospi: Transcendental Functions. (line 140) * mpfr_cosu: Transcendental Functions. (line 124) d4575 1 a4575 1 (line 159) d4577 1 a4577 1 (line 249) d4579 1 a4579 1 (line 158) d4581 8 a4588 8 (line 248) * mpfr_custom_get_exp: Custom Interface. (line 79) * mpfr_custom_get_kind: Custom Interface. (line 69) * mpfr_custom_get_significand: Custom Interface. (line 74) * mpfr_custom_get_size: Custom Interface. (line 40) * mpfr_custom_init: Custom Interface. (line 44) * mpfr_custom_init_set: Custom Interface. (line 51) * mpfr_custom_move: Custom Interface. (line 88) d4592 1 a4592 1 (line 307) d4596 1 a4596 1 (line 75) d4598 1 a4598 1 (line 177) d4606 1 a4606 1 (line 87) d4608 1 a4608 1 (line 91) d4610 1 a4610 1 (line 83) d4612 1 a4612 1 (line 79) d4614 1 a4614 1 (line 89) d4616 1 a4616 1 (line 227) d4620 1 a4620 1 (line 85) d4624 1 a4624 1 (line 259) d4630 1 a4630 1 (line 109) d4632 1 a4632 1 (line 180) d4634 1 a4634 1 (line 324) d4636 1 a4636 1 (line 325) d4638 1 a4638 1 (line 44) d4640 1 a4640 3 (line 46) * mpfr_exp10m1: Transcendental Functions. (line 52) d4642 1 a4642 3 (line 45) * mpfr_exp2m1: Transcendental Functions. (line 51) d4644 1 a4644 1 (line 50) d4646 1 a4646 1 (line 47) d4650 1 a4650 1 (line 186) d4652 1 a4652 1 (line 182) d4654 1 a4654 1 (line 180) d4656 2 a4658 2 * mpfr_fits_uintmax_p: Conversion Functions. (line 185) d4660 1 a4660 1 (line 181) d4662 1 a4662 1 (line 179) d4664 1 a4664 1 (line 183) d4666 1 a4666 1 (line 191) d4668 1 a4668 1 (line 215) d4670 1 a4670 1 (line 211) d4672 1 a4672 1 (line 194) d4674 1 a4674 1 (line 58) d4676 1 a4676 1 (line 197) d4686 1 a4686 1 (line 107) d4688 1 a4688 3 (line 111) * mpfr_fmod_ui: Integer and Remainder Related Functions. (line 109) d4696 1 a4696 1 (line 175) d4698 1 a4698 1 (line 90) d4706 1 a4706 1 (line 174) d4708 1 a4708 1 (line 53) d4710 1 a4710 1 (line 275) d4712 1 a4712 1 (line 276) d4724 1 a4724 1 (line 40) d4738 1 a4738 1 (line 115) d4740 1 a4740 1 (line 89) d4748 1 a4748 1 (line 42) d4750 1 a4750 1 (line 175) d4754 1 a4754 1 (line 84) d4756 1 a4756 1 (line 25) d4758 1 a4758 1 (line 27) d4760 1 a4760 1 (line 114) d4762 1 a4762 1 (line 102) d4764 1 a4764 1 (line 26) d4766 1 a4766 1 (line 28) d4768 1 a4768 1 (line 144) d4770 1 a4770 1 (line 75) d4772 1 a4772 1 (line 62) d4774 1 a4774 1 (line 79) d4782 1 a4782 1 (line 179) d4814 1 a4814 1 (line 139) d4816 1 a4816 1 (line 329) d4818 1 a4818 1 (line 330) d4820 1 a4820 1 (line 331) d4828 1 a4828 1 (line 297) d4830 1 a4830 1 (line 270) d4832 1 a4832 1 (line 289) d4834 1 a4834 1 (line 26) d4836 1 a4836 3 (line 30) * mpfr_log10p1: Transcendental Functions. (line 39) d4840 1 a4840 3 (line 29) * mpfr_log2p1: Transcendental Functions. (line 38) d4842 1 a4842 1 (line 27) d4844 1 a4844 1 (line 26) d4846 1 a4846 1 (line 24) d4848 1 a4848 1 (line 86) d4850 1 a4850 1 (line 97) d4872 1 a4872 1 (line 178) d4876 1 a4876 1 (line 144) d4878 1 a4878 1 (line 20) d4880 1 a4880 1 (line 21) d4884 1 a4884 1 (line 77) d4890 1 a4890 1 (line 176) d4892 1 a4892 5 (line 58) * mpfr_pown: Transcendental Functions. (line 70) * mpfr_powr: Transcendental Functions. (line 60) d4894 1 a4894 3 (line 64) * mpfr_pow_sj: Transcendental Functions. (line 68) d4896 1 a4896 3 (line 62) * mpfr_pow_uj: Transcendental Functions. (line 66) d4898 1 a4898 1 (line 72) d4902 1 a4902 1 (line 33) d4904 1 a4904 1 (line 182) d4906 1 a4906 1 (line 90) d4910 1 a4910 1 (line 109) d4916 1 a4916 1 (line 113) d4918 1 a4918 1 (line 115) d4922 1 a4922 1 (line 53) d4924 1 a4924 1 (line 54) d4926 1 a4926 1 (line 56) d4928 1 a4928 1 (line 58) d4930 1 a4930 1 (line 60) d4932 1 a4932 1 (line 53) d4934 1 a4934 3 (line 133) * mpfr_rootn_si: Arithmetic Functions. (line 121) d4936 1 a4936 1 (line 119) d4942 1 a4942 1 (line 96) d4944 1 a4944 1 (line 157) d4946 1 a4946 1 (line 247) d4950 1 a4950 1 (line 131) d4962 1 a4962 1 (line 168) d4968 1 a4968 1 (line 171) d4970 1 a4970 1 (line 122) d4978 1 a4978 1 (line 170) d4986 1 a4986 1 (line 169) d4988 1 a4988 1 (line 167) d4998 1 a4998 1 (line 63) d5002 1 a5002 1 (line 67) d5004 1 a5004 1 (line 75) d5008 1 a5008 1 (line 61) d5012 1 a5012 1 (line 65) d5014 1 a5014 1 (line 166) d5020 1 a5020 1 (line 69) d5024 1 a5024 1 (line 127) d5026 1 a5026 1 (line 119) d5028 1 a5028 1 (line 234) d5030 1 a5030 5 (line 239) * mpfr_sinpi: Transcendental Functions. (line 141) * mpfr_sinu: Transcendental Functions. (line 126) d5032 1 a5032 1 (line 147) d5034 1 a5034 1 (line 81) d5038 1 a5038 1 (line 198) d5040 1 a5040 1 (line 188) d5042 1 a5042 1 (line 72) d5044 2 a5046 4 * mpfr_sqrt_ui: Arithmetic Functions. (line 102) * mpfr_srcptr: Nomenclature and Types. (line 6) d5048 1 a5048 1 (line 93) d5064 1 a5064 1 (line 210) d5066 1 a5066 1 (line 165) d5070 1 a5070 1 (line 120) d5072 1 a5072 5 (line 235) * mpfr_tanpi: Transcendental Functions. (line 142) * mpfr_tanu: Transcendental Functions. (line 128) d5078 1 a5078 1 (line 77) d5080 1 a5080 1 (line 76) d5082 1 a5082 1 (line 74) d5086 1 a5086 1 (line 175) d5090 1 a5090 1 (line 58) d5092 1 a5092 1 (line 39) d5094 1 a5094 1 (line 212) d5096 1 a5096 1 (line 147) d5098 1 a5098 1 (line 148) d5100 1 a5100 1 (line 149) d5102 1 a5102 1 (line 167) d5104 1 a5104 1 (line 150) d5106 1 a5106 1 (line 151) d5108 1 a5108 1 (line 176) d5110 1 a5110 1 (line 183) d5112 1 a5112 1 (line 200) d5114 1 a5114 1 (line 189) d5116 1 a5116 1 (line 340) d5118 1 a5118 1 (line 341) d5120 1 a5120 1 (line 342) d5124 1 a5124 1 (line 318) d5126 1 a5126 1 (line 319) d5135 46 a5180 46 Node: Introduction to MPFR3806 Node: Installing MPFR6202 Node: Reporting Bugs11767 Node: MPFR Basics13798 Node: Headers and Libraries14152 Node: Nomenclature and Types17750 Node: MPFR Variable Conventions21068 Node: Rounding22604 Ref: ternary value26412 Node: Floating-Point Values on Special Numbers28403 Node: Exceptions31996 Node: Memory Handling35840 Node: Getting the Best Efficiency Out of MPFR39591 Node: MPFR Interface40607 Node: Initialization Functions42927 Node: Assignment Functions50456 Node: Combined Initialization and Assignment Functions60720 Node: Conversion Functions62021 Ref: mpfr_get_str_ndigits67932 Ref: mpfr_get_str68563 Node: Arithmetic Functions73535 Node: Comparison Functions85819 Node: Transcendental Functions90110 Ref: mpfr_pow93329 Node: Input and Output Functions110914 Node: Formatted Output Functions116217 Node: Integer and Remainder Related Functions127116 Node: Rounding-Related Functions134839 Node: Miscellaneous Functions141461 Node: Exception Related Functions152503 Node: Memory Handling Functions162756 Node: Compatibility with MPF164647 Node: Custom Interface167822 Node: Internals172608 Node: API Compatibility174152 Node: Type and Macro Changes176100 Node: Added Functions179285 Node: Changed Functions184857 Node: Removed Functions192624 Node: Other Changes193354 Node: MPFR and the IEEE 754 Standard195057 Node: Contributors197711 Node: References200850 Node: GNU Free Documentation License202913 Node: Concept Index225507 Node: Function and Type Index231580 @ 1.1.1.5.2.1 log @Sync with HEAD @ text @d1 1 a1 1 This is mpfr.info, produced by makeinfo version 7.0.3 from mpfr.texi. d4 1 a4 1 Floating-Point Reliable Library, version 4.2.1. d26 1 a26 1 Floating-Point Reliable Library, version 4.2.1. d287 2 a288 2 *Note Reporting Bugs::. Some bug fixes are available on the MPFR 4.2.1 web page . d303 1 a303 1 on the MPFR 4.2.1 web page and the d1161 3 a1163 4 detailed description of BASE (with its special value 0) and the valid string formats. Contrary to ‘mpfr_strtofr’, ‘mpfr_set_str’ requires the _whole_ string to represent a valid floating-point number. d1925 2 a1926 7 special cases and exceptions. In particular: • When OP < −1, ROP is set to NaN. • When N is zero and OP is NaN (like any value greater or equal to −1), ROP is set to 1. • When OP = −1, ROP is set to +Inf for N < 0, and to +0 for N > 0. The other special cases follow the usual rules. d2234 4 a2237 5 After skipping optional whitespace (as defined by ‘isspace’, which depends on the current locale), this function reads a word, defined as the longest sequence of non-whitespace characters, and parses it using ‘mpfr_set_str’. See the documentation of ‘mpfr_strtofr’ for a detailed description of the valid string formats. d2239 1 a2239 3 Return the number of bytes read (including the leading whitespace, if any), or if the string format is invalid or an error occurred, return 0. d2377 1 a2377 1 mpfr_printf ("variable x with %Pd bits", p); d2420 4 a2423 8 specified base, the exponent is always displayed in decimal. Non-real values are always displayed as ‘nan’ / ‘inf’ for the ‘a’, ‘b’, ‘e’, ‘f’, and ‘g’ specifiers, and ‘NAN’ / ‘INF’ for ‘A’, ‘E’, ‘F’, and ‘G’ specifiers, possibly preceded by a sign or a space (the minus sign when the value has a negative sign, the plus sign when the value has a positive sign and the ‘+’ flag is used, a space when the value has a positive sign and the _space_ flag is used). d4631 1 a4631 1 (line 168) d4633 1 a4633 1 (line 258) d4635 1 a4635 1 (line 190) d4637 1 a4637 1 (line 180) d4651 1 a4651 1 (line 355) d4653 1 a4653 1 (line 366) d4655 1 a4655 1 (line 169) d4657 1 a4657 1 (line 259) d4659 1 a4659 1 (line 191) d4661 1 a4661 1 (line 182) d4663 1 a4663 1 (line 215) d4665 1 a4665 1 (line 170) d4667 1 a4667 1 (line 196) d4669 1 a4669 1 (line 200) d4671 1 a4671 1 (line 198) d4673 1 a4673 1 (line 260) d4675 1 a4675 1 (line 192) d4677 1 a4677 1 (line 184) d4679 1 a4679 1 (line 317) d4745 1 a4745 1 (line 377) d4747 1 a4747 1 (line 376) d4749 1 a4749 1 (line 374) d4751 1 a4751 1 (line 375) d4755 1 a4755 1 (line 123) d4757 1 a4757 1 (line 238) d4759 1 a4759 1 (line 145) d4761 1 a4761 1 (line 129) d4763 1 a4763 1 (line 164) d4765 1 a4765 1 (line 254) d4767 1 a4767 1 (line 163) d4769 1 a4769 1 (line 253) d4780 1 a4780 1 (line 312) d4806 1 a4806 1 (line 79) d4812 1 a4812 1 (line 264) d4822 1 a4822 1 (line 329) d4824 1 a4824 1 (line 330) d4886 1 a4886 1 (line 54) d4888 1 a4888 1 (line 65) d4890 1 a4890 1 (line 179) d4904 1 a4904 1 (line 280) d4906 1 a4906 1 (line 281) d5010 1 a5010 1 (line 334) d5012 1 a5012 1 (line 335) d5014 1 a5014 1 (line 336) d5022 1 a5022 1 (line 302) d5024 1 a5024 1 (line 275) d5026 1 a5026 1 (line 294) d5110 1 a5110 1 (line 186) d5152 1 a5152 1 (line 162) d5154 1 a5154 1 (line 252) d5188 1 a5188 1 (line 158) d5192 1 a5192 1 (line 157) d5226 1 a5226 1 (line 159) d5234 1 a5234 1 (line 124) d5236 2 a5238 2 * mpfr_sinh_cosh: Transcendental Functions. (line 244) d5240 1 a5240 1 (line 146) d5242 1 a5242 1 (line 131) d5244 1 a5244 1 (line 152) d5250 1 a5250 1 (line 202) d5252 1 a5252 1 (line 192) d5262 1 a5262 1 (line 94) d5280 1 a5280 1 (line 166) d5284 1 a5284 1 (line 125) d5286 1 a5286 1 (line 240) d5288 1 a5288 1 (line 147) d5290 1 a5290 1 (line 133) d5312 1 a5312 1 (line 216) d5326 1 a5326 1 (line 180) d5328 1 a5328 1 (line 187) d5330 1 a5330 1 (line 204) d5332 1 a5332 1 (line 193) d5334 1 a5334 1 (line 345) d5336 1 a5336 1 (line 346) d5338 1 a5338 1 (line 347) d5342 1 a5342 1 (line 323) d5344 1 a5344 1 (line 324) d5351 48 a5398 48 Node: Top777 Node: Copying2044 Node: Introduction to MPFR3808 Node: Installing MPFR6204 Node: Reporting Bugs11769 Node: MPFR Basics13800 Node: Headers and Libraries14154 Node: Nomenclature and Types17752 Node: MPFR Variable Conventions21070 Node: Rounding22606 Ref: ternary value26414 Node: Floating-Point Values on Special Numbers28405 Node: Exceptions31998 Node: Memory Handling35842 Node: Getting the Best Efficiency Out of MPFR39593 Node: MPFR Interface40609 Node: Initialization Functions42929 Node: Assignment Functions50458 Node: Combined Initialization and Assignment Functions60763 Node: Conversion Functions62064 Ref: mpfr_get_str_ndigits67975 Ref: mpfr_get_str68606 Node: Arithmetic Functions73578 Node: Comparison Functions85862 Node: Transcendental Functions90153 Ref: mpfr_pow93372 Node: Input and Output Functions111192 Node: Formatted Output Functions116694 Node: Integer and Remainder Related Functions127816 Node: Rounding-Related Functions135539 Node: Miscellaneous Functions142161 Node: Exception Related Functions153203 Node: Memory Handling Functions163456 Node: Compatibility with MPF165347 Node: Custom Interface168522 Node: Internals173308 Node: API Compatibility174852 Node: Type and Macro Changes176800 Node: Added Functions179985 Node: Changed Functions185557 Node: Removed Functions193324 Node: Other Changes194054 Node: MPFR and the IEEE 754 Standard195757 Node: Contributors198411 Node: References201550 Node: GNU Free Documentation License203613 Node: Concept Index226207 Node: Function and Type Index232280 @ 1.1.1.6 log @import MPFR 4.2.1. mostly a bug-fix release, highlights include: - abort on lock failure, instead of just warn - better Inf handling - fix an unlikely stack overflow in mpfr_rec_sqrt() - fixes for mpfr_reldiff() - fix boundary error in mpfr_pow_general() - fixes to printing Nan and Inf - many manual and test updates @ text @d1 1 a1 1 This is mpfr.info, produced by makeinfo version 7.0.3 from mpfr.texi. d4 1 a4 1 Floating-Point Reliable Library, version 4.2.1. d26 1 a26 1 Floating-Point Reliable Library, version 4.2.1. d287 2 a288 2 *Note Reporting Bugs::. Some bug fixes are available on the MPFR 4.2.1 web page . d303 1 a303 1 on the MPFR 4.2.1 web page and the d1161 3 a1163 4 detailed description of BASE (with its special value 0) and the valid string formats. Contrary to ‘mpfr_strtofr’, ‘mpfr_set_str’ requires the _whole_ string to represent a valid floating-point number. d1925 2 a1926 7 special cases and exceptions. In particular: • When OP < −1, ROP is set to NaN. • When N is zero and OP is NaN (like any value greater or equal to −1), ROP is set to 1. • When OP = −1, ROP is set to +Inf for N < 0, and to +0 for N > 0. The other special cases follow the usual rules. d2234 4 a2237 5 After skipping optional whitespace (as defined by ‘isspace’, which depends on the current locale), this function reads a word, defined as the longest sequence of non-whitespace characters, and parses it using ‘mpfr_set_str’. See the documentation of ‘mpfr_strtofr’ for a detailed description of the valid string formats. d2239 1 a2239 3 Return the number of bytes read (including the leading whitespace, if any), or if the string format is invalid or an error occurred, return 0. d2377 1 a2377 1 mpfr_printf ("variable x with %Pd bits", p); d2420 4 a2423 8 specified base, the exponent is always displayed in decimal. Non-real values are always displayed as ‘nan’ / ‘inf’ for the ‘a’, ‘b’, ‘e’, ‘f’, and ‘g’ specifiers, and ‘NAN’ / ‘INF’ for ‘A’, ‘E’, ‘F’, and ‘G’ specifiers, possibly preceded by a sign or a space (the minus sign when the value has a negative sign, the plus sign when the value has a positive sign and the ‘+’ flag is used, a space when the value has a positive sign and the _space_ flag is used). d4631 1 a4631 1 (line 168) d4633 1 a4633 1 (line 258) d4635 1 a4635 1 (line 190) d4637 1 a4637 1 (line 180) d4651 1 a4651 1 (line 355) d4653 1 a4653 1 (line 366) d4655 1 a4655 1 (line 169) d4657 1 a4657 1 (line 259) d4659 1 a4659 1 (line 191) d4661 1 a4661 1 (line 182) d4663 1 a4663 1 (line 215) d4665 1 a4665 1 (line 170) d4667 1 a4667 1 (line 196) d4669 1 a4669 1 (line 200) d4671 1 a4671 1 (line 198) d4673 1 a4673 1 (line 260) d4675 1 a4675 1 (line 192) d4677 1 a4677 1 (line 184) d4679 1 a4679 1 (line 317) d4745 1 a4745 1 (line 377) d4747 1 a4747 1 (line 376) d4749 1 a4749 1 (line 374) d4751 1 a4751 1 (line 375) d4755 1 a4755 1 (line 123) d4757 1 a4757 1 (line 238) d4759 1 a4759 1 (line 145) d4761 1 a4761 1 (line 129) d4763 1 a4763 1 (line 164) d4765 1 a4765 1 (line 254) d4767 1 a4767 1 (line 163) d4769 1 a4769 1 (line 253) d4780 1 a4780 1 (line 312) d4806 1 a4806 1 (line 79) d4812 1 a4812 1 (line 264) d4822 1 a4822 1 (line 329) d4824 1 a4824 1 (line 330) d4886 1 a4886 1 (line 54) d4888 1 a4888 1 (line 65) d4890 1 a4890 1 (line 179) d4904 1 a4904 1 (line 280) d4906 1 a4906 1 (line 281) d5010 1 a5010 1 (line 334) d5012 1 a5012 1 (line 335) d5014 1 a5014 1 (line 336) d5022 1 a5022 1 (line 302) d5024 1 a5024 1 (line 275) d5026 1 a5026 1 (line 294) d5110 1 a5110 1 (line 186) d5152 1 a5152 1 (line 162) d5154 1 a5154 1 (line 252) d5188 1 a5188 1 (line 158) d5192 1 a5192 1 (line 157) d5226 1 a5226 1 (line 159) d5234 1 a5234 1 (line 124) d5236 2 a5238 2 * mpfr_sinh_cosh: Transcendental Functions. (line 244) d5240 1 a5240 1 (line 146) d5242 1 a5242 1 (line 131) d5244 1 a5244 1 (line 152) d5250 1 a5250 1 (line 202) d5252 1 a5252 1 (line 192) d5262 1 a5262 1 (line 94) d5280 1 a5280 1 (line 166) d5284 1 a5284 1 (line 125) d5286 1 a5286 1 (line 240) d5288 1 a5288 1 (line 147) d5290 1 a5290 1 (line 133) d5312 1 a5312 1 (line 216) d5326 1 a5326 1 (line 180) d5328 1 a5328 1 (line 187) d5330 1 a5330 1 (line 204) d5332 1 a5332 1 (line 193) d5334 1 a5334 1 (line 345) d5336 1 a5336 1 (line 346) d5338 1 a5338 1 (line 347) d5342 1 a5342 1 (line 323) d5344 1 a5344 1 (line 324) d5351 48 a5398 48 Node: Top777 Node: Copying2044 Node: Introduction to MPFR3808 Node: Installing MPFR6204 Node: Reporting Bugs11769 Node: MPFR Basics13800 Node: Headers and Libraries14154 Node: Nomenclature and Types17752 Node: MPFR Variable Conventions21070 Node: Rounding22606 Ref: ternary value26414 Node: Floating-Point Values on Special Numbers28405 Node: Exceptions31998 Node: Memory Handling35842 Node: Getting the Best Efficiency Out of MPFR39593 Node: MPFR Interface40609 Node: Initialization Functions42929 Node: Assignment Functions50458 Node: Combined Initialization and Assignment Functions60763 Node: Conversion Functions62064 Ref: mpfr_get_str_ndigits67975 Ref: mpfr_get_str68606 Node: Arithmetic Functions73578 Node: Comparison Functions85862 Node: Transcendental Functions90153 Ref: mpfr_pow93372 Node: Input and Output Functions111192 Node: Formatted Output Functions116694 Node: Integer and Remainder Related Functions127816 Node: Rounding-Related Functions135539 Node: Miscellaneous Functions142161 Node: Exception Related Functions153203 Node: Memory Handling Functions163456 Node: Compatibility with MPF165347 Node: Custom Interface168522 Node: Internals173308 Node: API Compatibility174852 Node: Type and Macro Changes176800 Node: Added Functions179985 Node: Changed Functions185557 Node: Removed Functions193324 Node: Other Changes194054 Node: MPFR and the IEEE 754 Standard195757 Node: Contributors198411 Node: References201550 Node: GNU Free Documentation License203613 Node: Concept Index226207 Node: Function and Type Index232280 @ 1.1.1.1.8.1 log @file mpfr.info was added on branch tls-maxphys on 2014-08-20 00:00:04 +0000 @ text @d1 4266 @ 1.1.1.1.8.2 log @Rebase to HEAD as of a few days ago. @ text @a0 4266 This is mpfr.info, produced by makeinfo version 4.13 from mpfr.texi. This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 3.1.2. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in *note GNU Free Documentation License::. INFO-DIR-SECTION Software libraries START-INFO-DIR-ENTRY * mpfr: (mpfr). Multiple Precision Floating-Point Reliable Library. END-INFO-DIR-ENTRY  File: mpfr.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir) GNU MPFR ******** This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 3.1.2. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in *note GNU Free Documentation License::. * Menu: * Copying:: MPFR Copying Conditions (LGPL). * Introduction to MPFR:: Brief introduction to GNU MPFR. * Installing MPFR:: How to configure and compile the MPFR library. * Reporting Bugs:: How to usefully report bugs. * MPFR Basics:: What every MPFR user should now. * MPFR Interface:: MPFR functions and macros. * API Compatibility:: API compatibility with previous MPFR versions. * Contributors:: * References:: * GNU Free Documentation License:: * Concept Index:: * Function and Type Index::  File: mpfr.info, Node: Copying, Next: Introduction to MPFR, Prev: Top, Up: Top MPFR Copying Conditions *********************** The GNU MPFR library (or MPFR for short) is "free"; this means that everyone is free to use it and free to redistribute it on a free basis. The library is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you. Specifically, we want to make sure that you have the right to give away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things. To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of the GNU MPFR library, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights. Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU MPFR library. If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation. The precise conditions of the license for the GNU MPFR library are found in the Lesser General Public License that accompanies the source code. See the file COPYING.LESSER.  File: mpfr.info, Node: Introduction to MPFR, Next: Installing MPFR, Prev: Copying, Up: Top 1 Introduction to MPFR ********************** MPFR is a portable library written in C for arbitrary precision arithmetic on floating-point numbers. It is based on the GNU MP library. It aims to provide a class of floating-point numbers with precise semantics. The main characteristics of MPFR, which make it differ from most arbitrary precision floating-point software tools, are: * the MPFR code is portable, i.e., the result of any operation does not depend on the machine word size `mp_bits_per_limb' (64 on most current processors); * the precision in bits can be set _exactly_ to any valid value for each variable (including very small precision); * MPFR provides the four rounding modes from the IEEE 754-1985 standard, plus away-from-zero, as well as for basic operations as for other mathematical functions. In particular, with a precision of 53 bits, MPFR is able to exactly reproduce all computations with double-precision machine floating-point numbers (e.g., `double' type in C, with a C implementation that rigorously follows Annex F of the ISO C99 standard and `FP_CONTRACT' pragma set to `OFF') on the four arithmetic operations and the square root, except the default exponent range is much wider and subnormal numbers are not implemented (but can be emulated). This version of MPFR is released under the GNU Lesser General Public License, version 3 or any later version. It is permitted to link MPFR to most non-free programs, as long as when distributing them the MPFR source code and a means to re-link with a modified MPFR library is provided. 1.1 How to Use This Manual ========================== Everyone should read *note MPFR Basics::. If you need to install the library yourself, you need to read *note Installing MPFR::, too. To use the library you will need to refer to *note MPFR Interface::. The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.  File: mpfr.info, Node: Installing MPFR, Next: Reporting Bugs, Prev: Introduction to MPFR, Up: Top 2 Installing MPFR ***************** The MPFR library is already installed on some GNU/Linux distributions, but the development files necessary to the compilation such as `mpfr.h' are not always present. To check that MPFR is fully installed on your computer, you can check the presence of the file `mpfr.h' in `/usr/include', or try to compile a small program having `#include ' (since `mpfr.h' may be installed somewhere else). For instance, you can try to compile: #include #include int main (void) { printf ("MPFR library: %-12s\nMPFR header: %s (based on %d.%d.%d)\n", mpfr_get_version (), MPFR_VERSION_STRING, MPFR_VERSION_MAJOR, MPFR_VERSION_MINOR, MPFR_VERSION_PATCHLEVEL); return 0; } with cc -o version version.c -lmpfr -lgmp and if you get errors whose first line looks like version.c:2:19: error: mpfr.h: No such file or directory then MPFR is probably not installed. Running this program will give you the MPFR version. If MPFR is not installed on your computer, or if you want to install a different version, please follow the steps below. 2.1 How to Install ================== Here are the steps needed to install the library on Unix systems (more details are provided in the `INSTALL' file): 1. To build MPFR, you first have to install GNU MP (version 4.1 or higher) on your computer. You need a C compiler, preferably GCC, but any reasonable compiler should work. And you need the standard Unix `make' command, plus some other standard Unix utility commands. Then, in the MPFR build directory, type the following commands. 2. `./configure' This will prepare the build and setup the options according to your system. You can give options to specify the install directories (instead of the default `/usr/local'), threading support, and so on. See the `INSTALL' file and/or the output of `./configure --help' for more information, in particular if you get error messages. 3. `make' This will compile MPFR, and create a library archive file `libmpfr.a'. On most platforms, a dynamic library will be produced too. 4. `make check' This will make sure MPFR was built correctly. If you get error messages, please report this to the MPFR mailing-list `mpfr@@inria.fr'. (*Note Reporting Bugs::, for information on what to include in useful bug reports.) 5. `make install' This will copy the files `mpfr.h' and `mpf2mpfr.h' to the directory `/usr/local/include', the library files (`libmpfr.a' and possibly others) to the directory `/usr/local/lib', the file `mpfr.info' to the directory `/usr/local/share/info', and some other documentation files to the directory `/usr/local/share/doc/mpfr' (or if you passed the `--prefix' option to `configure', using the prefix directory given as argument to `--prefix' instead of `/usr/local'). 2.2 Other `make' Targets ======================== There are some other useful make targets: * `mpfr.info' or `info' Create or update an info version of the manual, in `mpfr.info'. This file is already provided in the MPFR archives. * `mpfr.pdf' or `pdf' Create a PDF version of the manual, in `mpfr.pdf'. * `mpfr.dvi' or `dvi' Create a DVI version of the manual, in `mpfr.dvi'. * `mpfr.ps' or `ps' Create a Postscript version of the manual, in `mpfr.ps'. * `mpfr.html' or `html' Create a HTML version of the manual, in several pages in the directory `doc/mpfr.html'; if you want only one output HTML file, then type `makeinfo --html --no-split mpfr.texi' from the `doc' directory instead. * `clean' Delete all object files and archive files, but not the configuration files. * `distclean' Delete all generated files not included in the distribution. * `uninstall' Delete all files copied by `make install'. 2.3 Build Problems ================== In case of problem, please read the `INSTALL' file carefully before reporting a bug, in particular section "In case of problem". Some problems are due to bad configuration on the user side (not specific to MPFR). Problems are also mentioned in the FAQ `http://www.mpfr.org/faq.html'. Please report problems to the MPFR mailing-list `mpfr@@inria.fr'. *Note Reporting Bugs::. Some bug fixes are available on the MPFR 3.1.2 web page `http://www.mpfr.org/mpfr-3.1.2/'. 2.4 Getting the Latest Version of MPFR ====================================== The latest version of MPFR is available from `ftp://ftp.gnu.org/gnu/mpfr/' or `http://www.mpfr.org/'.  File: mpfr.info, Node: Reporting Bugs, Next: MPFR Basics, Prev: Installing MPFR, Up: Top 3 Reporting Bugs **************** If you think you have found a bug in the MPFR library, first have a look on the MPFR 3.1.2 web page `http://www.mpfr.org/mpfr-3.1.2/' and the FAQ `http://www.mpfr.org/faq.html': perhaps this bug is already known, in which case you may find there a workaround for it. You might also look in the archives of the MPFR mailing-list: `https://sympa.inria.fr/sympa/arc/mpfr'. Otherwise, please investigate and report it. We have made this library available to you, and it is not to ask too much from you, to ask you to report the bugs that you find. There are a few things you should think about when you put your bug report together. You have to send us a test case that makes it possible for us to reproduce the bug, i.e., a small self-content program, using no other library than MPFR. Include instructions on how to run the test case. You also have to explain what is wrong; if you get a crash, or if the results you get are incorrect and in that case, in what way. Please include compiler version information in your bug report. This can be extracted using `cc -V' on some machines, or, if you're using GCC, `gcc -v'. Also, include the output from `uname -a' and the MPFR version (the GMP version may be useful too). If you get a failure while running `make' or `make check', please include the `config.log' file in your bug report. If your bug report is good, we will do our best to help you to get a corrected version of the library; if the bug report is poor, we will not do anything about it (aside of chiding you to send better bug reports). Send your bug report to the MPFR mailing-list `mpfr@@inria.fr'. If you think something in this manual is unclear, or downright incorrect, or if the language needs to be improved, please send a note to the same address.  File: mpfr.info, Node: MPFR Basics, Next: MPFR Interface, Prev: Reporting Bugs, Up: Top 4 MPFR Basics ************* * Menu: * Headers and Libraries:: * Nomenclature and Types:: * MPFR Variable Conventions:: * Rounding Modes:: * Floating-Point Values on Special Numbers:: * Exceptions:: * Memory Handling::  File: mpfr.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: MPFR Basics, Up: MPFR Basics 4.1 Headers and Libraries ========================= All declarations needed to use MPFR are collected in the include file `mpfr.h'. It is designed to work with both C and C++ compilers. You should include that file in any program using the MPFR library: #include Note however that prototypes for MPFR functions with `FILE *' parameters are provided only if `' is included too (before `mpfr.h'): #include #include Likewise `' (or `') is required for prototypes with `va_list' parameters, such as `mpfr_vprintf'. And for any functions using `intmax_t', you must include `' or `' before `mpfr.h', to allow `mpfr.h' to define prototypes for these functions. Moreover, users of C++ compilers under some platforms may need to define `MPFR_USE_INTMAX_T' (and should do it for portability) before `mpfr.h' has been included; of course, it is possible to do that on the command line, e.g., with `-DMPFR_USE_INTMAX_T'. Note: If `mpfr.h' and/or `gmp.h' (used by `mpfr.h') are included several times (possibly from another header file), `' and/or `' (or `') should be included *before the first inclusion* of `mpfr.h' or `gmp.h'. Alternatively, you can define `MPFR_USE_FILE' (for MPFR I/O functions) and/or `MPFR_USE_VA_LIST' (for MPFR functions with `va_list' parameters) anywhere before the last inclusion of `mpfr.h'. As a consequence, if your file is a public header that includes `mpfr.h', you need to use the latter method. When calling a MPFR macro, it is not allowed to have previously defined a macro with the same name as some keywords (currently `do', `while' and `sizeof'). You can avoid the use of MPFR macros encapsulating functions by defining the `MPFR_USE_NO_MACRO' macro before `mpfr.h' is included. In general this should not be necessary, but this can be useful when debugging user code: with some macros, the compiler may emit spurious warnings with some warning options, and macros can prevent some prototype checking. All programs using MPFR must link against both `libmpfr' and `libgmp' libraries. On a typical Unix-like system this can be done with `-lmpfr -lgmp' (in that order), for example: gcc myprogram.c -lmpfr -lgmp MPFR is built using Libtool and an application can use that to link if desired, *note GNU Libtool: (libtool.info)Top. If MPFR has been installed to a non-standard location, then it may be necessary to set up environment variables such as `C_INCLUDE_PATH' and `LIBRARY_PATH', or use `-I' and `-L' compiler options, in order to point to the right directories. For a shared library, it may also be necessary to set up some sort of run-time library path (e.g., `LD_LIBRARY_PATH') on some systems. Please read the `INSTALL' file for additional information.  File: mpfr.info, Node: Nomenclature and Types, Next: MPFR Variable Conventions, Prev: Headers and Libraries, Up: MPFR Basics 4.2 Nomenclature and Types ========================== A "floating-point number", or "float" for short, is an arbitrary precision significand (also called mantissa) with a limited precision exponent. The C data type for such objects is `mpfr_t' (internally defined as a one-element array of a structure, and `mpfr_ptr' is the C data type representing a pointer to this structure). A floating-point number can have three special values: Not-a-Number (NaN) or plus or minus Infinity. NaN represents an uninitialized object, the result of an invalid operation (like 0 divided by 0), or a value that cannot be determined (like +Infinity minus +Infinity). Moreover, like in the IEEE 754 standard, zero is signed, i.e., there are both +0 and -0; the behavior is the same as in the IEEE 754 standard and it is generalized to the other functions supported by MPFR. Unless documented otherwise, the sign bit of a NaN is unspecified. The "precision" is the number of bits used to represent the significand of a floating-point number; the corresponding C data type is `mpfr_prec_t'. The precision can be any integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. In the current implementation, `MPFR_PREC_MIN' is equal to 2. Warning! MPFR needs to increase the precision internally, in order to provide accurate results (and in particular, correct rounding). Do not attempt to set the precision to any value near `MPFR_PREC_MAX', otherwise MPFR will abort due to an assertion failure. Moreover, you may reach some memory limit on your platform, in which case the program may abort, crash or have undefined behavior (depending on your C implementation). The "rounding mode" specifies the way to round the result of a floating-point operation, in case the exact result can not be represented exactly in the destination significand; the corresponding C data type is `mpfr_rnd_t'.  File: mpfr.info, Node: MPFR Variable Conventions, Next: Rounding Modes, Prev: Nomenclature and Types, Up: MPFR Basics 4.3 MPFR Variable Conventions ============================= Before you can assign to an MPFR variable, you need to initialize it by calling one of the special initialization functions. When you're done with a variable, you need to clear it out, using one of the functions for that purpose. A variable should only be initialized once, or at least cleared out between each initialization. After a variable has been initialized, it may be assigned to any number of times. For efficiency reasons, avoid to initialize and clear out a variable in loops. Instead, initialize it before entering the loop, and clear it out after the loop has exited. You do not need to be concerned about allocating additional space for MPFR variables, since any variable has a significand of fixed size. Hence unless you change its precision, or clear and reinitialize it, a floating-point variable will have the same allocated space during all its life. As a general rule, all MPFR functions expect output arguments before input arguments. This notation is based on an analogy with the assignment operator. MPFR allows you to use the same variable for both input and output in the same expression. For example, the main function for floating-point multiplication, `mpfr_mul', can be used like this: `mpfr_mul (x, x, x, rnd)'. This computes the square of X with rounding mode `rnd' and puts the result back in X.  File: mpfr.info, Node: Rounding Modes, Next: Floating-Point Values on Special Numbers, Prev: MPFR Variable Conventions, Up: MPFR Basics 4.4 Rounding Modes ================== The following five rounding modes are supported: * `MPFR_RNDN': round to nearest (roundTiesToEven in IEEE 754-2008), * `MPFR_RNDZ': round toward zero (roundTowardZero in IEEE 754-2008), * `MPFR_RNDU': round toward plus infinity (roundTowardPositive in IEEE 754-2008), * `MPFR_RNDD': round toward minus infinity (roundTowardNegative in IEEE 754-2008), * `MPFR_RNDA': round away from zero. The `round to nearest' mode works as in the IEEE 754 standard: in case the number to be rounded lies exactly in the middle of two representable numbers, it is rounded to the one with the least significant bit set to zero. For example, the number 2.5, which is represented by (10.1) in binary, is rounded to (10.0)=2 with a precision of two bits, and not to (11.0)=3. This rule avoids the "drift" phenomenon mentioned by Knuth in volume 2 of The Art of Computer Programming (Section 4.2.2). Most MPFR functions take as first argument the destination variable, as second and following arguments the input variables, as last argument a rounding mode, and have a return value of type `int', called the "ternary value". The value stored in the destination variable is correctly rounded, i.e., MPFR behaves as if it computed the result with an infinite precision, then rounded it to the precision of this variable. The input variables are regarded as exact (in particular, their precision does not affect the result). As a consequence, in case of a non-zero real rounded result, the error on the result is less or equal to 1/2 ulp (unit in the last place) of that result in the rounding to nearest mode, and less than 1 ulp of that result in the directed rounding modes (a ulp is the weight of the least significant represented bit of the result after rounding). Unless documented otherwise, functions returning an `int' return a ternary value. If the ternary value is zero, it means that the value stored in the destination variable is the exact result of the corresponding mathematical function. If the ternary value is positive (resp. negative), it means the value stored in the destination variable is greater (resp. lower) than the exact result. For example with the `MPFR_RNDU' rounding mode, the ternary value is usually positive, except when the result is exact, in which case it is zero. In the case of an infinite result, it is considered as inexact when it was obtained by overflow, and exact otherwise. A NaN result (Not-a-Number) always corresponds to an exact return value. The opposite of a returned ternary value is guaranteed to be representable in an `int'. Unless documented otherwise, functions returning as result the value `1' (or any other value specified in this manual) for special cases (like `acos(0)') yield an overflow or an underflow if that value is not representable in the current exponent range.  File: mpfr.info, Node: Floating-Point Values on Special Numbers, Next: Exceptions, Prev: Rounding Modes, Up: MPFR Basics 4.5 Floating-Point Values on Special Numbers ============================================ This section specifies the floating-point values (of type `mpfr_t') returned by MPFR functions (where by "returned" we mean here the modified value of the destination object, which should not be mixed with the ternary return value of type `int' of those functions). For functions returning several values (like `mpfr_sin_cos'), the rules apply to each result separately. Functions can have one or several input arguments. An input point is a mapping from these input arguments to the set of the MPFR numbers. When none of its components are NaN, an input point can also be seen as a tuple in the extended real numbers (the set of the real numbers with both infinities). When the input point is in the domain of the mathematical function, the result is rounded as described in Section "Rounding Modes" (but see below for the specification of the sign of an exact zero). Otherwise the general rules from this section apply unless stated otherwise in the description of the MPFR function (*note MPFR Interface::). When the input point is not in the domain of the mathematical function but is in its closure in the extended real numbers and the function can be extended by continuity, the result is the obtained limit. Examples: `mpfr_hypot' on (+Inf,0) gives +Inf. But `mpfr_pow' cannot be defined on (1,+Inf) using this rule, as one can find sequences (X_N,Y_N) such that X_N goes to 1, Y_N goes to +Inf and X_N to the Y_N goes to any positive value when N goes to the infinity. When the input point is in the closure of the domain of the mathematical function and an input argument is +0 (resp. -0), one considers the limit when the corresponding argument approaches 0 from above (resp. below). If the limit is not defined (e.g., `mpfr_log' on -0), the behavior is specified in the description of the MPFR function. When the result is equal to 0, its sign is determined by considering the limit as if the input point were not in the domain: If one approaches 0 from above (resp. below), the result is +0 (resp. -0); for example, `mpfr_sin' on +0 gives +0. In the other cases, the sign is specified in the description of the MPFR function; for example `mpfr_max' on -0 and +0 gives +0. When the input point is not in the closure of the domain of the function, the result is NaN. Example: `mpfr_sqrt' on -17 gives NaN. When an input argument is NaN, the result is NaN, possibly except when a partial function is constant on the finite floating-point numbers; such a case is always explicitly specified in *note MPFR Interface::. Example: `mpfr_hypot' on (NaN,0) gives NaN, but `mpfr_hypot' on (NaN,+Inf) gives +Inf (as specified in *note Special Functions::), since for any finite input X, `mpfr_hypot' on (X,+Inf) gives +Inf.  File: mpfr.info, Node: Exceptions, Next: Memory Handling, Prev: Floating-Point Values on Special Numbers, Up: MPFR Basics 4.6 Exceptions ============== MPFR supports 6 exception types: * Underflow: An underflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent smaller than the minimum value of the current exponent range. (In the round-to-nearest mode, the halfway case is rounded toward zero.) Note: This is not the single possible definition of the underflow. MPFR chooses to consider the underflow _after_ rounding. The underflow before rounding can also be defined. For instance, consider a function that has the exact result 7 multiplied by two to the power E-4, where E is the smallest exponent (for a significand between 1/2 and 1), with a 2-bit target precision and rounding toward plus infinity. The exact result has the exponent E-1. With the underflow before rounding, such a function call would yield an underflow, as E-1 is outside the current exponent range. However, MPFR first considers the rounded result assuming an unbounded exponent range. The exact result cannot be represented exactly in precision 2, and here, it is rounded to 0.5 times 2 to E, which is representable in the current exponent range. As a consequence, this will not yield an underflow in MPFR. * Overflow: An overflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent larger than the maximum value of the current exponent range. In the round-to-nearest mode, the result is infinite. Note: unlike the underflow case, there is only one possible definition of overflow here. * Divide-by-zero: An exact infinite result is obtained from finite inputs. * NaN: A NaN exception occurs when the result of a function is NaN. * Inexact: An inexact exception occurs when the result of a function cannot be represented exactly and must be rounded. * Range error: A range exception occurs when a function that does not return a MPFR number (such as comparisons and conversions to an integer) has an invalid result (e.g., an argument is NaN in `mpfr_cmp', or a conversion to an integer cannot be represented in the target type). MPFR has a global flag for each exception, which can be cleared, set or tested by functions described in *note Exception Related Functions::. Differences with the ISO C99 standard: * In C, only quiet NaNs are specified, and a NaN propagation does not raise an invalid exception. Unless explicitly stated otherwise, MPFR sets the NaN flag whenever a NaN is generated, even when a NaN is propagated (e.g., in NaN + NaN), as if all NaNs were signaling. * An invalid exception in C corresponds to either a NaN exception or a range error in MPFR.  File: mpfr.info, Node: Memory Handling, Prev: Exceptions, Up: MPFR Basics 4.7 Memory Handling =================== MPFR functions may create caches, e.g., when computing constants such as Pi, either because the user has called a function like `mpfr_const_pi' directly or because such a function was called internally by the MPFR library itself to compute some other function. At any time, the user can free the various caches with `mpfr_free_cache'. It is strongly advised to do that before terminating a thread, or before exiting when using tools like `valgrind' (to avoid memory leaks being reported). MPFR internal data such as flags, the exponent range, the default precision and rounding mode, and caches (i.e., data that are not accessed via parameters) are either global (if MPFR has not been compiled as thread safe) or per-thread (thread local storage, TLS). The initial values of TLS data after a thread is created entirely depend on the compiler and thread implementation (MPFR simply does a conventional variable initialization, the variables being declared with an implementation-defined TLS specifier).  File: mpfr.info, Node: MPFR Interface, Next: API Compatibility, Prev: MPFR Basics, Up: Top 5 MPFR Interface **************** The floating-point functions expect arguments of type `mpfr_t'. The MPFR floating-point functions have an interface that is similar to the GNU MP functions. The function prefix for floating-point operations is `mpfr_'. The user has to specify the precision of each variable. A computation that assigns a variable will take place with the precision of the assigned variable; the cost of that computation should not depend on the precision of variables used as input (on average). The semantics of a calculation in MPFR is specified as follows: Compute the requested operation exactly (with "infinite accuracy"), and round the result to the precision of the destination variable, with the given rounding mode. The MPFR floating-point functions are intended to be a smooth extension of the IEEE 754 arithmetic. The results obtained on a given computer are identical to those obtained on a computer with a different word size, or with a different compiler or operating system. MPFR _does not keep track_ of the accuracy of a computation. This is left to the user or to a higher layer (for example the MPFI library for interval arithmetic). As a consequence, if two variables are used to store only a few significant bits, and their product is stored in a variable with large precision, then MPFR will still compute the result with full precision. The value of the standard C macro `errno' may be set to non-zero by any MPFR function or macro, whether or not there is an error. * Menu: * Initialization Functions:: * Assignment Functions:: * Combined Initialization and Assignment Functions:: * Conversion Functions:: * Basic Arithmetic Functions:: * Comparison Functions:: * Special Functions:: * Input and Output Functions:: * Formatted Output Functions:: * Integer Related Functions:: * Rounding Related Functions:: * Miscellaneous Functions:: * Exception Related Functions:: * Compatibility with MPF:: * Custom Interface:: * Internals::  File: mpfr.info, Node: Initialization Functions, Next: Assignment Functions, Prev: MPFR Interface, Up: MPFR Interface 5.1 Initialization Functions ============================ An `mpfr_t' object must be initialized before storing the first value in it. The functions `mpfr_init' and `mpfr_init2' are used for that purpose. -- Function: void mpfr_init2 (mpfr_t X, mpfr_prec_t PREC) Initialize X, set its precision to be *exactly* PREC bits and its value to NaN. (Warning: the corresponding MPF function initializes to zero instead.) Normally, a variable should be initialized once only or at least be cleared, using `mpfr_clear', between initializations. To change the precision of a variable which has already been initialized, use `mpfr_set_prec'. The precision PREC must be an integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX' (otherwise the behavior is undefined). -- Function: void mpfr_inits2 (mpfr_prec_t PREC, mpfr_t X, ...) Initialize all the `mpfr_t' variables of the given variable argument `va_list', set their precision to be *exactly* PREC bits and their value to NaN. See `mpfr_init2' for more details. The `va_list' is assumed to be composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It begins from X, and ends when it encounters a null pointer (whose type must also be `mpfr_ptr'). -- Function: void mpfr_clear (mpfr_t X) Free the space occupied by the significand of X. Make sure to call this function for all `mpfr_t' variables when you are done with them. -- Function: void mpfr_clears (mpfr_t X, ...) Free the space occupied by all the `mpfr_t' variables of the given `va_list'. See `mpfr_clear' for more details. The `va_list' is assumed to be composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It begins from X, and ends when it encounters a null pointer (whose type must also be `mpfr_ptr'). Here is an example of how to use multiple initialization functions (since `NULL' is not necessarily defined in this context, we use `(mpfr_ptr) 0' instead, but `(mpfr_ptr) NULL' is also correct). { mpfr_t x, y, z, t; mpfr_inits2 (256, x, y, z, t, (mpfr_ptr) 0); ... mpfr_clears (x, y, z, t, (mpfr_ptr) 0); } -- Function: void mpfr_init (mpfr_t X) Initialize X, set its precision to the default precision, and set its value to NaN. The default precision can be changed by a call to `mpfr_set_default_prec'. Warning! In a given program, some other libraries might change the default precision and not restore it. Thus it is safer to use `mpfr_init2'. -- Function: void mpfr_inits (mpfr_t X, ...) Initialize all the `mpfr_t' variables of the given `va_list', set their precision to the default precision and their value to NaN. See `mpfr_init' for more details. The `va_list' is assumed to be composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It begins from X, and ends when it encounters a null pointer (whose type must also be `mpfr_ptr'). Warning! In a given program, some other libraries might change the default precision and not restore it. Thus it is safer to use `mpfr_inits2'. -- Macro: MPFR_DECL_INIT (NAME, PREC) This macro declares NAME as an automatic variable of type `mpfr_t', initializes it and sets its precision to be *exactly* PREC bits and its value to NaN. NAME must be a valid identifier. You must use this macro in the declaration section. This macro is much faster than using `mpfr_init2' but has some drawbacks: * You *must not* call `mpfr_clear' with variables created with this macro (the storage is allocated at the point of declaration and deallocated when the brace-level is exited). * You *cannot* change their precision. * You *should not* create variables with huge precision with this macro. * Your compiler must support `Non-Constant Initializers' (standard in C++ and ISO C99) and `Token Pasting' (standard in ISO C89). If PREC is not a constant expression, your compiler must support `variable-length automatic arrays' (standard in ISO C99). GCC 2.95.3 and above supports all these features. If you compile your program with GCC in C89 mode and with `-pedantic', you may want to define the `MPFR_USE_EXTENSION' macro to avoid warnings due to the `MPFR_DECL_INIT' implementation. -- Function: void mpfr_set_default_prec (mpfr_prec_t PREC) Set the default precision to be *exactly* PREC bits, where PREC can be any integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. The precision of a variable means the number of bits used to store its significand. All subsequent calls to `mpfr_init' or `mpfr_inits' will use this precision, but previously initialized variables are unaffected. The default precision is set to 53 bits initially. Note: when MPFR is built with the `--enable-thread-safe' configure option, the default precision is local to each thread. *Note Memory Handling::, for more information. -- Function: mpfr_prec_t mpfr_get_default_prec (void) Return the current default MPFR precision in bits. See the documentation of `mpfr_set_default_prec'. Here is an example on how to initialize floating-point variables: { mpfr_t x, y; mpfr_init (x); /* use default precision */ mpfr_init2 (y, 256); /* precision _exactly_ 256 bits */ ... /* When the program is about to exit, do ... */ mpfr_clear (x); mpfr_clear (y); mpfr_free_cache (); /* free the cache for constants like pi */ } The following functions are useful for changing the precision during a calculation. A typical use would be for adjusting the precision gradually in iterative algorithms like Newton-Raphson, making the computation precision closely match the actual accurate part of the numbers. -- Function: void mpfr_set_prec (mpfr_t X, mpfr_prec_t PREC) Reset the precision of X to be *exactly* PREC bits, and set its value to NaN. The previous value stored in X is lost. It is equivalent to a call to `mpfr_clear(x)' followed by a call to `mpfr_init2(x, prec)', but more efficient as no allocation is done in case the current allocated space for the significand of X is enough. The precision PREC can be any integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. In case you want to keep the previous value stored in X, use `mpfr_prec_round' instead. -- Function: mpfr_prec_t mpfr_get_prec (mpfr_t X) Return the precision of X, i.e., the number of bits used to store its significand.  File: mpfr.info, Node: Assignment Functions, Next: Combined Initialization and Assignment Functions, Prev: Initialization Functions, Up: MPFR Interface 5.2 Assignment Functions ======================== These functions assign new values to already initialized floats (*note Initialization Functions::). -- Function: int mpfr_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) -- Function: int mpfr_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t RND) -- Function: int mpfr_set_uj (mpfr_t ROP, uintmax_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_sj (mpfr_t ROP, intmax_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_flt (mpfr_t ROP, float OP, mpfr_rnd_t RND) -- Function: int mpfr_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND) -- Function: int mpfr_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t RND) -- Function: int mpfr_set_decimal64 (mpfr_t ROP, _Decimal64 OP, mpfr_rnd_t RND) -- Function: int mpfr_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND) Set the value of ROP from OP, rounded toward the given direction RND. Note that the input 0 is converted to +0 by `mpfr_set_ui', `mpfr_set_si', `mpfr_set_uj', `mpfr_set_sj', `mpfr_set_z', `mpfr_set_q' and `mpfr_set_f', regardless of the rounding mode. If the system does not support the IEEE 754 standard, `mpfr_set_flt', `mpfr_set_d', `mpfr_set_ld' and `mpfr_set_decimal64' might not preserve the signed zeros. The `mpfr_set_decimal64' function is built only with the configure option `--enable-decimal-float', which also requires `--with-gmp-build', and when the compiler or system provides the `_Decimal64' data type (recent versions of GCC support this data type); to use `mpfr_set_decimal64', one should define the macro `MPFR_WANT_DECIMAL_FLOATS' before including `mpfr.h'. `mpfr_set_q' might fail if the numerator (or the denominator) can not be represented as a `mpfr_t'. Note: If you want to store a floating-point constant to a `mpfr_t', you should use `mpfr_set_str' (or one of the MPFR constant functions, such as `mpfr_const_pi' for Pi) instead of `mpfr_set_flt', `mpfr_set_d', `mpfr_set_ld' or `mpfr_set_decimal64'. Otherwise the floating-point constant will be first converted into a reduced-precision (e.g., 53-bit) binary (or decimal, for `mpfr_set_decimal64') number before MPFR can work with it. -- Function: int mpfr_set_ui_2exp (mpfr_t ROP, unsigned long int OP, mpfr_exp_t E, mpfr_rnd_t RND) -- Function: int mpfr_set_si_2exp (mpfr_t ROP, long int OP, mpfr_exp_t E, mpfr_rnd_t RND) -- Function: int mpfr_set_uj_2exp (mpfr_t ROP, uintmax_t OP, intmax_t E, mpfr_rnd_t RND) -- Function: int mpfr_set_sj_2exp (mpfr_t ROP, intmax_t OP, intmax_t E, mpfr_rnd_t RND) -- Function: int mpfr_set_z_2exp (mpfr_t ROP, mpz_t OP, mpfr_exp_t E, mpfr_rnd_t RND) Set the value of ROP from OP multiplied by two to the power E, rounded toward the given direction RND. Note that the input 0 is converted to +0. -- Function: int mpfr_set_str (mpfr_t ROP, const char *S, int BASE, mpfr_rnd_t RND) Set ROP to the value of the string S in base BASE, rounded in the direction RND. See the documentation of `mpfr_strtofr' for a detailed description of the valid string formats. Contrary to `mpfr_strtofr', `mpfr_set_str' requires the _whole_ string to represent a valid floating-point number. The meaning of the return value differs from other MPFR functions: it is 0 if the entire string up to the final null character is a valid number in base BASE; otherwise it is -1, and ROP may have changed (users interested in the *note ternary value:: should use `mpfr_strtofr' instead). Note: it is preferable to use `mpfr_set_str' if one wants to distinguish between an infinite ROP value coming from an infinite S or from an overflow. -- Function: int mpfr_strtofr (mpfr_t ROP, const char *NPTR, char **ENDPTR, int BASE, mpfr_rnd_t RND) Read a floating-point number from a string NPTR in base BASE, rounded in the direction RND; BASE must be either 0 (to detect the base, as described below) or a number from 2 to 62 (otherwise the behavior is undefined). If NPTR starts with valid data, the result is stored in ROP and `*ENDPTR' points to the character just after the valid data (if ENDPTR is not a null pointer); otherwise ROP is set to zero (for consistency with `strtod') and the value of NPTR is stored in the location referenced by ENDPTR (if ENDPTR is not a null pointer). The usual ternary value is returned. Parsing follows the standard C `strtod' function with some extensions. After optional leading whitespace, one has a subject sequence consisting of an optional sign (`+' or `-'), and either numeric data or special data. The subject sequence is defined as the longest initial subsequence of the input string, starting with the first non-whitespace character, that is of the expected form. The form of numeric data is a non-empty sequence of significand digits with an optional decimal point, and an optional exponent consisting of an exponent prefix followed by an optional sign and a non-empty sequence of decimal digits. A significand digit is either a decimal digit or a Latin letter (62 possible characters), with `A' = 10, `B' = 11, ..., `Z' = 35; case is ignored in bases less or equal to 36, in bases larger than 36, `a' = 36, `b' = 37, ..., `z' = 61. The value of a significand digit must be strictly less than the base. The decimal point can be either the one defined by the current locale or the period (the first one is accepted for consistency with the C standard and the practice, the second one is accepted to allow the programmer to provide MPFR numbers from strings in a way that does not depend on the current locale). The exponent prefix can be `e' or `E' for bases up to 10, or `@@' in any base; it indicates a multiplication by a power of the base. In bases 2 and 16, the exponent prefix can also be `p' or `P', in which case the exponent, called _binary exponent_, indicates a multiplication by a power of 2 instead of the base (there is a difference only for base 16); in base 16 for example `1p2' represents 4 whereas `1@@2' represents 256. The value of an exponent is always written in base 10. If the argument BASE is 0, then the base is automatically detected as follows. If the significand starts with `0b' or `0B', base 2 is assumed. If the significand starts with `0x' or `0X', base 16 is assumed. Otherwise base 10 is assumed. Note: The exponent (if present) must contain at least a digit. Otherwise the possible exponent prefix and sign are not part of the number (which ends with the significand). Similarly, if `0b', `0B', `0x' or `0X' is not followed by a binary/hexadecimal digit, then the subject sequence stops at the character `0', thus 0 is read. Special data (for infinities and NaN) can be `@@inf@@' or `@@nan@@(n-char-sequence-opt)', and if BASE <= 16, it can also be `infinity', `inf', `nan' or `nan(n-char-sequence-opt)', all case insensitive. A `n-char-sequence-opt' is a possibly empty string containing only digits, Latin letters and the underscore (0, 1, 2, ..., 9, a, b, ..., z, A, B, ..., Z, _). Note: one has an optional sign for all data, even NaN. For example, `-@@nAn@@(This_Is_Not_17)' is a valid representation for NaN in base 17. -- Function: void mpfr_set_nan (mpfr_t X) -- Function: void mpfr_set_inf (mpfr_t X, int SIGN) -- Function: void mpfr_set_zero (mpfr_t X, int SIGN) Set the variable X to NaN (Not-a-Number), infinity or zero respectively. In `mpfr_set_inf' or `mpfr_set_zero', X is set to plus infinity or plus zero iff SIGN is nonnegative; in `mpfr_set_nan', the sign bit of the result is unspecified. -- Function: void mpfr_swap (mpfr_t X, mpfr_t Y) Swap the values X and Y efficiently. Warning: the precisions are exchanged too; in case the precisions are different, `mpfr_swap' is thus not equivalent to three `mpfr_set' calls using a third auxiliary variable.  File: mpfr.info, Node: Combined Initialization and Assignment Functions, Next: Conversion Functions, Prev: Assignment Functions, Up: MPFR Interface 5.3 Combined Initialization and Assignment Functions ==================================================== -- Macro: int mpfr_init_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND) Initialize ROP and set its value from OP, rounded in the direction RND. The precision of ROP will be taken from the active default precision, as set by `mpfr_set_default_prec'. -- Function: int mpfr_init_set_str (mpfr_t X, const char *S, int BASE, mpfr_rnd_t RND) Initialize X and set its value from the string S in base BASE, rounded in the direction RND. See `mpfr_set_str'.  File: mpfr.info, Node: Conversion Functions, Next: Basic Arithmetic Functions, Prev: Combined Initialization and Assignment Functions, Up: MPFR Interface 5.4 Conversion Functions ======================== -- Function: float mpfr_get_flt (mpfr_t OP, mpfr_rnd_t RND) -- Function: double mpfr_get_d (mpfr_t OP, mpfr_rnd_t RND) -- Function: long double mpfr_get_ld (mpfr_t OP, mpfr_rnd_t RND) -- Function: _Decimal64 mpfr_get_decimal64 (mpfr_t OP, mpfr_rnd_t RND) Convert OP to a `float' (respectively `double', `long double' or `_Decimal64'), using the rounding mode RND. If OP is NaN, some fixed NaN (either quiet or signaling) or the result of 0.0/0.0 is returned. If OP is ±Inf, an infinity of the same sign or the result of ±1.0/0.0 is returned. If OP is zero, these functions return a zero, trying to preserve its sign, if possible. The `mpfr_get_decimal64' function is built only under some conditions: see the documentation of `mpfr_set_decimal64'. -- Function: long mpfr_get_si (mpfr_t OP, mpfr_rnd_t RND) -- Function: unsigned long mpfr_get_ui (mpfr_t OP, mpfr_rnd_t RND) -- Function: intmax_t mpfr_get_sj (mpfr_t OP, mpfr_rnd_t RND) -- Function: uintmax_t mpfr_get_uj (mpfr_t OP, mpfr_rnd_t RND) Convert OP to a `long', an `unsigned long', an `intmax_t' or an `uintmax_t' (respectively) after rounding it with respect to RND. If OP is NaN, 0 is returned and the _erange_ flag is set. If OP is too big for the return type, the function returns the maximum or the minimum of the corresponding C type, depending on the direction of the overflow; the _erange_ flag is set too. See also `mpfr_fits_slong_p', `mpfr_fits_ulong_p', `mpfr_fits_intmax_p' and `mpfr_fits_uintmax_p'. -- Function: double mpfr_get_d_2exp (long *EXP, mpfr_t OP, mpfr_rnd_t RND) -- Function: long double mpfr_get_ld_2exp (long *EXP, mpfr_t OP, mpfr_rnd_t RND) Return D and set EXP (formally, the value pointed to by EXP) such that 0.5<=abs(D)<1 and D times 2 raised to EXP equals OP rounded to double (resp. long double) precision, using the given rounding mode. If OP is zero, then a zero of the same sign (or an unsigned zero, if the implementation does not have signed zeros) is returned, and EXP is set to 0. If OP is NaN or an infinity, then the corresponding double precision (resp. long-double precision) value is returned, and EXP is undefined. -- Function: int mpfr_frexp (mpfr_exp_t *EXP, mpfr_t Y, mpfr_t X, mpfr_rnd_t RND) Set EXP (formally, the value pointed to by EXP) and Y such that 0.5<=abs(Y)<1 and Y times 2 raised to EXP equals X rounded to the precision of Y, using the given rounding mode. If X is zero, then Y is set to a zero of the same sign and EXP is set to 0. If X is NaN or an infinity, then Y is set to the same value and EXP is undefined. -- Function: mpfr_exp_t mpfr_get_z_2exp (mpz_t ROP, mpfr_t OP) Put the scaled significand of OP (regarded as an integer, with the precision of OP) into ROP, and return the exponent EXP (which may be outside the current exponent range) such that OP exactly equals ROP times 2 raised to the power EXP. If OP is zero, the minimal exponent `emin' is returned. If OP is NaN or an infinity, the _erange_ flag is set, ROP is set to 0, and the the minimal exponent `emin' is returned. The returned exponent may be less than the minimal exponent `emin' of MPFR numbers in the current exponent range; in case the exponent is not representable in the `mpfr_exp_t' type, the _erange_ flag is set and the minimal value of the `mpfr_exp_t' type is returned. -- Function: int mpfr_get_z (mpz_t ROP, mpfr_t OP, mpfr_rnd_t RND) Convert OP to a `mpz_t', after rounding it with respect to RND. If OP is NaN or an infinity, the _erange_ flag is set, ROP is set to 0, and 0 is returned. -- Function: int mpfr_get_f (mpf_t ROP, mpfr_t OP, mpfr_rnd_t RND) Convert OP to a `mpf_t', after rounding it with respect to RND. The _erange_ flag is set if OP is NaN or an infinity, which do not exist in MPF. If OP is NaN, then ROP is undefined. If OP is an +Inf (resp. -Inf), then ROP is set to the maximum (resp. minimum) value in the precision of the MPF number; if a future MPF version supports infinities, this behavior will be considered incorrect and will change (portable programs should assume that ROP is set either to this finite number or to an infinite number). Note that since MPFR currently has the same exponent type as MPF (but not with the same radix), the range of values is much larger in MPF than in MPFR, so that an overflow or underflow is not possible. -- Function: char * mpfr_get_str (char *STR, mpfr_exp_t *EXPPTR, int B, size_t N, mpfr_t OP, mpfr_rnd_t RND) Convert OP to a string of digits in base B, with rounding in the direction RND, where N is either zero (see below) or the number of significant digits output in the string; in the latter case, N must be greater or equal to 2. The base may vary from 2 to 62. If the input number is an ordinary number, the exponent is written through the pointer EXPPTR (for input 0, the current minimal exponent is written). The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. For example, the number -3.1416 would be returned as "-31416" in the string and 1 written at EXPPTR. If RND is to nearest, and OP is exactly in the middle of two consecutive possible outputs, the one with an even significand is chosen, where both significands are considered with the exponent of OP. Note that for an odd base, this may not correspond to an even last digit: for example with 2 digits in base 7, (14) and a half is rounded to (15) which is 12 in decimal, (16) and a half is rounded to (20) which is 14 in decimal, and (26) and a half is rounded to (26) which is 20 in decimal. If N is zero, the number of digits of the significand is chosen large enough so that re-reading the printed value with the same precision, assuming both output and input use rounding to nearest, will recover the original value of OP. More precisely, in most cases, the chosen precision of STR is the minimal precision m depending only on P = PREC(OP) and B that satisfies the above property, i.e., m = 1 + ceil(P*log(2)/log(B)), with P replaced by P-1 if B is a power of 2, but in some very rare cases, it might be m+1 (the smallest case for bases up to 62 is when P equals 186564318007 for bases 7 and 49). If STR is a null pointer, space for the significand is allocated using the current allocation function, and a pointer to the string is returned. To free the returned string, you must use `mpfr_free_str'. If STR is not a null pointer, it should point to a block of storage large enough for the significand, i.e., at least `max(N + 2, 7)'. The extra two bytes are for a possible minus sign, and for the terminating null character, and the value 7 accounts for `-@@Inf@@' plus the terminating null character. A pointer to the string is returned, unless there is an error, in which case a null pointer is returned. -- Function: void mpfr_free_str (char *STR) Free a string allocated by `mpfr_get_str' using the current unallocation function. The block is assumed to be `strlen(STR)+1' bytes. For more information about how it is done: *note Custom Allocation: (gmp.info)Custom Allocation. -- Function: int mpfr_fits_ulong_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_slong_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_uint_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_sint_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_ushort_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_sshort_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_uintmax_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_intmax_p (mpfr_t OP, mpfr_rnd_t RND) Return non-zero if OP would fit in the respective C data type, respectively `unsigned long', `long', `unsigned int', `int', `unsigned short', `short', `uintmax_t', `intmax_t', when rounded to an integer in the direction RND.  File: mpfr.info, Node: Basic Arithmetic Functions, Next: Comparison Functions, Prev: Conversion Functions, Up: MPFR Interface 5.5 Basic Arithmetic Functions ============================== -- Function: int mpfr_add (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_d (mpfr_t ROP, mpfr_t OP1, double OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, mpfr_rnd_t RND) Set ROP to OP1 + OP2 rounded in the direction RND. For types having no signed zero, it is considered unsigned (i.e., (+0) + 0 = (+0) and (-0) + 0 = (-0)). The `mpfr_add_d' function assumes that the radix of the `double' type is a power of 2, with a precision at most that declared by the C implementation (macro `IEEE_DBL_MANT_DIG', and if not defined 53 bits). -- Function: int mpfr_sub (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_sub (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_si_sub (mpfr_t ROP, long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_d_sub (mpfr_t ROP, double OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_d (mpfr_t ROP, mpfr_t OP1, double OP2, mpfr_rnd_t RND) -- Function: int mpfr_z_sub (mpfr_t ROP, mpz_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, mpfr_rnd_t RND) Set ROP to OP1 - OP2 rounded in the direction RND. For types having no signed zero, it is considered unsigned (i.e., (+0) - 0 = (+0), (-0) - 0 = (-0), 0 - (+0) = (-0) and 0 - (-0) = (+0)). The same restrictions than for `mpfr_add_d' apply to `mpfr_d_sub' and `mpfr_sub_d'. -- Function: int mpfr_mul (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_d (mpfr_t ROP, mpfr_t OP1, double OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, mpfr_rnd_t RND) Set ROP to OP1 times OP2 rounded in the direction RND. When a result is zero, its sign is the product of the signs of the operands (for types having no signed zero, it is considered positive). The same restrictions than for `mpfr_add_d' apply to `mpfr_mul_d'. -- Function: int mpfr_sqr (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the square of OP rounded in the direction RND. -- Function: int mpfr_div (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_div (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_si_div (mpfr_t ROP, long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_d_div (mpfr_t ROP, double OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_d (mpfr_t ROP, mpfr_t OP1, double OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, mpfr_rnd_t RND) Set ROP to OP1/OP2 rounded in the direction RND. When a result is zero, its sign is the product of the signs of the operands (for types having no signed zero, it is considered positive). The same restrictions than for `mpfr_add_d' apply to `mpfr_d_div' and `mpfr_div_d'. -- Function: int mpfr_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_sqrt_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) Set ROP to the square root of OP rounded in the direction RND (set ROP to -0 if OP is -0, to be consistent with the IEEE 754 standard). Set ROP to NaN if OP is negative. -- Function: int mpfr_rec_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the reciprocal square root of OP rounded in the direction RND. Set ROP to +Inf if OP is ±0, +0 if OP is +Inf, and NaN if OP is negative. -- Function: int mpfr_cbrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_root (mpfr_t ROP, mpfr_t OP, unsigned long int K, mpfr_rnd_t RND) Set ROP to the cubic root (resp. the Kth root) of OP rounded in the direction RND. For K odd (resp. even) and OP negative (including -Inf), set ROP to a negative number (resp. NaN). The Kth root of -0 is defined to be -0, whatever the parity of K. -- Function: int mpfr_pow (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_pow_ui (mpfr_t ROP, unsigned long int OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_pow (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to OP1 raised to OP2, rounded in the direction RND. Special values are handled as described in the ISO C99 and IEEE 754-2008 standards for the `pow' function: * `pow(±0, Y)' returns plus or minus infinity for Y a negative odd integer. * `pow(±0, Y)' returns plus infinity for Y negative and not an odd integer. * `pow(±0, Y)' returns plus or minus zero for Y a positive odd integer. * `pow(±0, Y)' returns plus zero for Y positive and not an odd integer. * `pow(-1, ±Inf)' returns 1. * `pow(+1, Y)' returns 1 for any Y, even a NaN. * `pow(X, ±0)' returns 1 for any X, even a NaN. * `pow(X, Y)' returns NaN for finite negative X and finite non-integer Y. * `pow(X, -Inf)' returns plus infinity for 0 < abs(x) < 1, and plus zero for abs(x) > 1. * `pow(X, +Inf)' returns plus zero for 0 < abs(x) < 1, and plus infinity for abs(x) > 1. * `pow(-Inf, Y)' returns minus zero for Y a negative odd integer. * `pow(-Inf, Y)' returns plus zero for Y negative and not an odd integer. * `pow(-Inf, Y)' returns minus infinity for Y a positive odd integer. * `pow(-Inf, Y)' returns plus infinity for Y positive and not an odd integer. * `pow(+Inf, Y)' returns plus zero for Y negative, and plus infinity for Y positive. -- Function: int mpfr_neg (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_abs (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to -OP and the absolute value of OP respectively, rounded in the direction RND. Just changes or adjusts the sign if ROP and OP are the same variable, otherwise a rounding might occur if the precision of ROP is less than that of OP. -- Function: int mpfr_dim (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the positive difference of OP1 and OP2, i.e., OP1 - OP2 rounded in the direction RND if OP1 > OP2, +0 if OP1 <= OP2, and NaN if OP1 or OP2 is NaN. -- Function: int mpfr_mul_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_2si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) Set ROP to OP1 times 2 raised to OP2 rounded in the direction RND. Just increases the exponent by OP2 when ROP and OP1 are identical. -- Function: int mpfr_div_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_2si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) Set ROP to OP1 divided by 2 raised to OP2 rounded in the direction RND. Just decreases the exponent by OP2 when ROP and OP1 are identical.  File: mpfr.info, Node: Comparison Functions, Next: Special Functions, Prev: Basic Arithmetic Functions, Up: MPFR Interface 5.6 Comparison Functions ======================== -- Function: int mpfr_cmp (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_cmp_ui (mpfr_t OP1, unsigned long int OP2) -- Function: int mpfr_cmp_si (mpfr_t OP1, long int OP2) -- Function: int mpfr_cmp_d (mpfr_t OP1, double OP2) -- Function: int mpfr_cmp_ld (mpfr_t OP1, long double OP2) -- Function: int mpfr_cmp_z (mpfr_t OP1, mpz_t OP2) -- Function: int mpfr_cmp_q (mpfr_t OP1, mpq_t OP2) -- Function: int mpfr_cmp_f (mpfr_t OP1, mpf_t OP2) Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1 < OP2. Both OP1 and OP2 are considered to their full own precision, which may differ. If one of the operands is NaN, set the _erange_ flag and return zero. Note: These functions may be useful to distinguish the three possible cases. If you need to distinguish two cases only, it is recommended to use the predicate functions (e.g., `mpfr_equal_p' for the equality) described below; they behave like the IEEE 754 comparisons, in particular when one or both arguments are NaN. But only floating-point numbers can be compared (you may need to do a conversion first). -- Function: int mpfr_cmp_ui_2exp (mpfr_t OP1, unsigned long int OP2, mpfr_exp_t E) -- Function: int mpfr_cmp_si_2exp (mpfr_t OP1, long int OP2, mpfr_exp_t E) Compare OP1 and OP2 multiplied by two to the power E. Similar as above. -- Function: int mpfr_cmpabs (mpfr_t OP1, mpfr_t OP2) Compare |OP1| and |OP2|. Return a positive value if |OP1| > |OP2|, zero if |OP1| = |OP2|, and a negative value if |OP1| < |OP2|. If one of the operands is NaN, set the _erange_ flag and return zero. -- Function: int mpfr_nan_p (mpfr_t OP) -- Function: int mpfr_inf_p (mpfr_t OP) -- Function: int mpfr_number_p (mpfr_t OP) -- Function: int mpfr_zero_p (mpfr_t OP) -- Function: int mpfr_regular_p (mpfr_t OP) Return non-zero if OP is respectively NaN, an infinity, an ordinary number (i.e., neither NaN nor an infinity), zero, or a regular number (i.e., neither NaN, nor an infinity nor zero). Return zero otherwise. -- Macro: int mpfr_sgn (mpfr_t OP) Return a positive value if OP > 0, zero if OP = 0, and a negative value if OP < 0. If the operand is NaN, set the _erange_ flag and return zero. This is equivalent to `mpfr_cmp_ui (op, 0)', but more efficient. -- Function: int mpfr_greater_p (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_greaterequal_p (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_less_p (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_lessequal_p (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_equal_p (mpfr_t OP1, mpfr_t OP2) Return non-zero if OP1 > OP2, OP1 >= OP2, OP1 < OP2, OP1 <= OP2, OP1 = OP2 respectively, and zero otherwise. Those functions return zero whenever OP1 and/or OP2 is NaN. -- Function: int mpfr_lessgreater_p (mpfr_t OP1, mpfr_t OP2) Return non-zero if OP1 < OP2 or OP1 > OP2 (i.e., neither OP1, nor OP2 is NaN, and OP1 <> OP2), zero otherwise (i.e., OP1 and/or OP2 is NaN, or OP1 = OP2). -- Function: int mpfr_unordered_p (mpfr_t OP1, mpfr_t OP2) Return non-zero if OP1 or OP2 is a NaN (i.e., they cannot be compared), zero otherwise.  File: mpfr.info, Node: Special Functions, Next: Input and Output Functions, Prev: Comparison Functions, Up: MPFR Interface 5.7 Special Functions ===================== All those functions, except explicitly stated (for example `mpfr_sin_cos'), return a *note ternary value::, i.e., zero for an exact return value, a positive value for a return value larger than the exact result, and a negative value otherwise. Important note: in some domains, computing special functions (either with correct or incorrect rounding) is expensive, even for small precision, for example the trigonometric and Bessel functions for large argument. -- Function: int mpfr_log (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_log2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_log10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the natural logarithm of OP, log2(OP) or log10(OP), respectively, rounded in the direction RND. Set ROP to -Inf if OP is -0 (i.e., the sign of the zero has no influence on the result). -- Function: int mpfr_exp (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_exp2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_exp10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential of OP, to 2 power of OP or to 10 power of OP, respectively, rounded in the direction RND. -- Function: int mpfr_cos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_sin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_tan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the cosine of OP, sine of OP, tangent of OP, rounded in the direction RND. -- Function: int mpfr_sin_cos (mpfr_t SOP, mpfr_t COP, mpfr_t OP, mpfr_rnd_t RND) Set simultaneously SOP to the sine of OP and COP to the cosine of OP, rounded in the direction RND with the corresponding precisions of SOP and COP, which must be different variables. Return 0 iff both results are exact, more precisely it returns s+4c where s=0 if SOP is exact, s=1 if SOP is larger than the sine of OP, s=2 if SOP is smaller than the sine of OP, and similarly for c and the cosine of OP. -- Function: int mpfr_sec (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_csc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_cot (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the secant of OP, cosecant of OP, cotangent of OP, rounded in the direction RND. -- Function: int mpfr_acos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_asin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_atan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the arc-cosine, arc-sine or arc-tangent of OP, rounded in the direction RND. Note that since `acos(-1)' returns the floating-point number closest to Pi according to the given rounding mode, this number might not be in the output range 0 <= ROP < \pi of the arc-cosine function; still, the result lies in the image of the output range by the rounding function. The same holds for `asin(-1)', `asin(1)', `atan(-Inf)', `atan(+Inf)' or for `atan(op)' with large OP and small precision of ROP. -- Function: int mpfr_atan2 (mpfr_t ROP, mpfr_t Y, mpfr_t X, mpfr_rnd_t RND) Set ROP to the arc-tangent2 of Y and X, rounded in the direction RND: if `x > 0', `atan2(y, x) = atan (y/x)'; if `x < 0', `atan2(y, x) = sign(y)*(Pi - atan (abs(y/x)))', thus a number from -Pi to Pi. As for `atan', in case the exact mathematical result is +Pi or -Pi, its rounded result might be outside the function output range. `atan2(y, 0)' does not raise any floating-point exception. Special values are handled as described in the ISO C99 and IEEE 754-2008 standards for the `atan2' function: * `atan2(+0, -0)' returns +Pi. * `atan2(-0, -0)' returns -Pi. * `atan2(+0, +0)' returns +0. * `atan2(-0, +0)' returns -0. * `atan2(+0, x)' returns +Pi for x < 0. * `atan2(-0, x)' returns -Pi for x < 0. * `atan2(+0, x)' returns +0 for x > 0. * `atan2(-0, x)' returns -0 for x > 0. * `atan2(y, 0)' returns -Pi/2 for y < 0. * `atan2(y, 0)' returns +Pi/2 for y > 0. * `atan2(+Inf, -Inf)' returns +3*Pi/4. * `atan2(-Inf, -Inf)' returns -3*Pi/4. * `atan2(+Inf, +Inf)' returns +Pi/4. * `atan2(-Inf, +Inf)' returns -Pi/4. * `atan2(+Inf, x)' returns +Pi/2 for finite x. * `atan2(-Inf, x)' returns -Pi/2 for finite x. * `atan2(y, -Inf)' returns +Pi for finite y > 0. * `atan2(y, -Inf)' returns -Pi for finite y < 0. * `atan2(y, +Inf)' returns +0 for finite y > 0. * `atan2(y, +Inf)' returns -0 for finite y < 0. -- Function: int mpfr_cosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_sinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_tanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the hyperbolic cosine, sine or tangent of OP, rounded in the direction RND. -- Function: int mpfr_sinh_cosh (mpfr_t SOP, mpfr_t COP, mpfr_t OP, mpfr_rnd_t RND) Set simultaneously SOP to the hyperbolic sine of OP and COP to the hyperbolic cosine of OP, rounded in the direction RND with the corresponding precision of SOP and COP, which must be different variables. Return 0 iff both results are exact (see `mpfr_sin_cos' for a more detailed description of the return value). -- Function: int mpfr_sech (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_csch (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_coth (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the hyperbolic secant of OP, cosecant of OP, cotangent of OP, rounded in the direction RND. -- Function: int mpfr_acosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_asinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_atanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the inverse hyperbolic cosine, sine or tangent of OP, rounded in the direction RND. -- Function: int mpfr_fac_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) Set ROP to the factorial of OP, rounded in the direction RND. -- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the logarithm of one plus OP, rounded in the direction RND. -- Function: int mpfr_expm1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential of OP followed by a subtraction by one, rounded in the direction RND. -- Function: int mpfr_eint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential integral of OP, rounded in the direction RND. For positive OP, the exponential integral is the sum of Euler's constant, of the logarithm of OP, and of the sum for k from 1 to infinity of OP to the power k, divided by k and factorial(k). For negative OP, ROP is set to NaN (this definition for negative argument follows formula 5.1.2 from the Handbook of Mathematical Functions from Abramowitz and Stegun, a future version might use another definition). -- Function: int mpfr_li2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to real part of the dilogarithm of OP, rounded in the direction RND. MPFR defines the dilogarithm function as the integral of -log(1-t)/t from 0 to OP. -- Function: int mpfr_gamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the Gamma function on OP, rounded in the direction RND. When OP is a negative integer, ROP is set to NaN. -- Function: int mpfr_lngamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the logarithm of the Gamma function on OP, rounded in the direction RND. When -2K-1 <= OP <= -2K, K being a non-negative integer, ROP is set to NaN. See also `mpfr_lgamma'. -- Function: int mpfr_lgamma (mpfr_t ROP, int *SIGNP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the logarithm of the absolute value of the Gamma function on OP, rounded in the direction RND. The sign (1 or -1) of Gamma(OP) is returned in the object pointed to by SIGNP. When OP is an infinity or a non-positive integer, set ROP to +Inf. When OP is NaN, -Inf or a negative integer, *SIGNP is undefined, and when OP is ±0, *SIGNP is the sign of the zero. -- Function: int mpfr_digamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the Digamma (sometimes also called Psi) function on OP, rounded in the direction RND. When OP is a negative integer, set ROP to NaN. -- Function: int mpfr_zeta (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_zeta_ui (mpfr_t ROP, unsigned long OP, mpfr_rnd_t RND) Set ROP to the value of the Riemann Zeta function on OP, rounded in the direction RND. -- Function: int mpfr_erf (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_erfc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the error function on OP (resp. the complementary error function on OP) rounded in the direction RND. -- Function: int mpfr_j0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_j1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_jn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the first kind Bessel function of order 0, (resp. 1 and N) on OP, rounded in the direction RND. When OP is NaN, ROP is always set to NaN. When OP is plus or minus Infinity, ROP is set to +0. When OP is zero, and N is not zero, ROP is set to +0 or -0 depending on the parity and sign of N, and the sign of OP. -- Function: int mpfr_y0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_y1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_yn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the second kind Bessel function of order 0 (resp. 1 and N) on OP, rounded in the direction RND. When OP is NaN or negative, ROP is always set to NaN. When OP is +Inf, ROP is set to +0. When OP is zero, ROP is set to +Inf or -Inf depending on the parity and sign of N. -- Function: int mpfr_fma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_rnd_t RND) -- Function: int mpfr_fms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_rnd_t RND) Set ROP to (OP1 times OP2) + OP3 (resp. (OP1 times OP2) - OP3) rounded in the direction RND. -- Function: int mpfr_agm (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the arithmetic-geometric mean of OP1 and OP2, rounded in the direction RND. The arithmetic-geometric mean is the common limit of the sequences U_N and V_N, where U_0=OP1, V_0=OP2, U_(N+1) is the arithmetic mean of U_N and V_N, and V_(N+1) is the geometric mean of U_N and V_N. If any operand is negative, set ROP to NaN. -- Function: int mpfr_hypot (mpfr_t ROP, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) Set ROP to the Euclidean norm of X and Y, i.e., the square root of the sum of the squares of X and Y, rounded in the direction RND. Special values are handled as described in Section F.9.4.3 of the ISO C99 and IEEE 754-2008 standards: If X or Y is an infinity, then +Inf is returned in ROP, even if the other number is NaN. -- Function: int mpfr_ai (mpfr_t ROP, mpfr_t X, mpfr_rnd_t RND) Set ROP to the value of the Airy function Ai on X, rounded in the direction RND. When X is NaN, ROP is always set to NaN. When X is +Inf or -Inf, ROP is +0. The current implementation is not intended to be used with large arguments. It works with abs(X) typically smaller than 500. For larger arguments, other methods should be used and will be implemented in a future version. -- Function: int mpfr_const_log2 (mpfr_t ROP, mpfr_rnd_t RND) -- Function: int mpfr_const_pi (mpfr_t ROP, mpfr_rnd_t RND) -- Function: int mpfr_const_euler (mpfr_t ROP, mpfr_rnd_t RND) -- Function: int mpfr_const_catalan (mpfr_t ROP, mpfr_rnd_t RND) Set ROP to the logarithm of 2, the value of Pi, of Euler's constant 0.577..., of Catalan's constant 0.915..., respectively, rounded in the direction RND. These functions cache the computed values to avoid other calculations if a lower or equal precision is requested. To free these caches, use `mpfr_free_cache'. -- Function: void mpfr_free_cache (void) Free various caches used by MPFR internally, in particular the caches used by the functions computing constants (`mpfr_const_log2', `mpfr_const_pi', `mpfr_const_euler' and `mpfr_const_catalan'). You should call this function before terminating a thread, even if you did not call these functions directly (they could have been called internally). -- Function: int mpfr_sum (mpfr_t ROP, mpfr_ptr const TAB[], unsigned long int N, mpfr_rnd_t RND) Set ROP to the sum of all elements of TAB, whose size is N, rounded in the direction RND. Warning: for efficiency reasons, TAB is an array of pointers to `mpfr_t', not an array of `mpfr_t'. If the returned `int' value is zero, ROP is guaranteed to be the exact sum; otherwise ROP might be smaller than, equal to, or larger than the exact sum (in accordance to the rounding mode). However, `mpfr_sum' does guarantee the result is correctly rounded.  File: mpfr.info, Node: Input and Output Functions, Next: Formatted Output Functions, Prev: Special Functions, Up: MPFR Interface 5.8 Input and Output Functions ============================== This section describes functions that perform input from an input/output stream, and functions that output to an input/output stream. Passing a null pointer for a `stream' to any of these functions will make them read from `stdin' and write to `stdout', respectively. When using any of these functions, you must include the `' standard header before `mpfr.h', to allow `mpfr.h' to define prototypes for these functions. -- Function: size_t mpfr_out_str (FILE *STREAM, int BASE, size_t N, mpfr_t OP, mpfr_rnd_t RND) Output OP on stream STREAM, as a string of digits in base BASE, rounded in the direction RND. The base may vary from 2 to 62. Print N significant digits exactly, or if N is 0, enough digits so that OP can be read back exactly (see `mpfr_get_str'). In addition to the significant digits, a decimal point (defined by the current locale) at the right of the first digit and a trailing exponent in base 10, in the form `eNNN', are printed. If BASE is greater than 10, `@@' will be used instead of `e' as exponent delimiter. Return the number of characters written, or if an error occurred, return 0. -- Function: size_t mpfr_inp_str (mpfr_t ROP, FILE *STREAM, int BASE, mpfr_rnd_t RND) Input a string in base BASE from stream STREAM, rounded in the direction RND, and put the read float in ROP. This function reads a word (defined as a sequence of characters between whitespace) and parses it using `mpfr_set_str'. See the documentation of `mpfr_strtofr' for a detailed description of the valid string formats. Return the number of bytes read, or if an error occurred, return 0.  File: mpfr.info, Node: Formatted Output Functions, Next: Integer Related Functions, Prev: Input and Output Functions, Up: MPFR Interface 5.9 Formatted Output Functions ============================== 5.9.1 Requirements ------------------ The class of `mpfr_printf' functions provides formatted output in a similar manner as the standard C `printf'. These functions are defined only if your system supports ISO C variadic functions and the corresponding argument access macros. When using any of these functions, you must include the `' standard header before `mpfr.h', to allow `mpfr.h' to define prototypes for these functions. 5.9.2 Format String ------------------- The format specification accepted by `mpfr_printf' is an extension of the `printf' one. The conversion specification is of the form: % [flags] [width] [.[precision]] [type] [rounding] conv `flags', `width', and `precision' have the same meaning as for the standard `printf' (in particular, notice that the `precision' is related to the number of digits displayed in the base chosen by `conv' and not related to the internal precision of the `mpfr_t' variable). `mpfr_printf' accepts the same `type' specifiers as GMP (except the non-standard and deprecated `q', use `ll' instead), namely the length modifiers defined in the C standard: `h' `short' `hh' `char' `j' `intmax_t' or `uintmax_t' `l' `long' or `wchar_t' `ll' `long long' `L' `long double' `t' `ptrdiff_t' `z' `size_t' and the `type' specifiers defined in GMP plus `R' and `P' specific to MPFR (the second column in the table below shows the type of the argument read in the argument list and the kind of `conv' specifier to use after the `type' specifier): `F' `mpf_t', float conversions `Q' `mpq_t', integer conversions `M' `mp_limb_t', integer conversions `N' `mp_limb_t' array, integer conversions `Z' `mpz_t', integer conversions `P' `mpfr_prec_t', integer conversions `R' `mpfr_t', float conversions The `type' specifiers have the same restrictions as those mentioned in the GMP documentation: *note Formatted Output Strings: (gmp.info)Formatted Output Strings. In particular, the `type' specifiers (except `R' and `P') are supported only if they are supported by `gmp_printf' in your GMP build; this implies that the standard specifiers, such as `t', must _also_ be supported by your C library if you want to use them. The `rounding' field is specific to `mpfr_t' arguments and should not be used with other types. With conversion specification not involving `P' and `R' types, `mpfr_printf' behaves exactly as `gmp_printf'. The `P' type specifies that a following `o', `u', `x', or `X' conversion specifier applies to a `mpfr_prec_t' argument. It is needed because the `mpfr_prec_t' type does not necessarily correspond to an `unsigned int' or any fixed standard type. The `precision' field specifies the minimum number of digits to appear. The default `precision' is 1. For example: mpfr_t x; mpfr_prec_t p; mpfr_init (x); ... p = mpfr_get_prec (x); mpfr_printf ("variable x with %Pu bits", p); The `R' type specifies that a following `a', `A', `b', `e', `E', `f', `F', `g', `G', or `n' conversion specifier applies to a `mpfr_t' argument. The `R' type can be followed by a `rounding' specifier denoted by one of the following characters: `U' round toward plus infinity `D' round toward minus infinity `Y' round away from zero `Z' round toward zero `N' round to nearest (with ties to even) `*' rounding mode indicated by the `mpfr_rnd_t' argument just before the corresponding `mpfr_t' variable. The default rounding mode is rounding to nearest. The following three examples are equivalent: mpfr_t x; mpfr_init (x); ... mpfr_printf ("%.128Rf", x); mpfr_printf ("%.128RNf", x); mpfr_printf ("%.128R*f", MPFR_RNDN, x); Note that the rounding away from zero mode is specified with `Y' because ISO C reserves the `A' specifier for hexadecimal output (see below). The output `conv' specifiers allowed with `mpfr_t' parameter are: `a' `A' hex float, C99 style `b' binary output `e' `E' scientific format float `f' `F' fixed point float `g' `G' fixed or scientific float The conversion specifier `b' which displays the argument in binary is specific to `mpfr_t' arguments and should not be used with other types. Other conversion specifiers have the same meaning as for a `double' argument. In case of non-decimal output, only the significand is written in the specified base, the exponent is always displayed in decimal. Special values are always displayed as `nan', `-inf', and `inf' for `a', `b', `e', `f', and `g' specifiers and `NAN', `-INF', and `INF' for `A', `E', `F', and `G' specifiers. If the `precision' field is not empty, the `mpfr_t' number is rounded to the given precision in the direction specified by the rounding mode. If the precision is zero with rounding to nearest mode and one of the following `conv' specifiers: `a', `A', `b', `e', `E', tie case is rounded to even when it lies between two consecutive values at the wanted precision which have the same exponent, otherwise, it is rounded away from zero. For instance, 85 is displayed as "8e+1" and 95 is displayed as "1e+2" with the format specification `"%.0RNe"'. This also applies when the `g' (resp. `G') conversion specifier uses the `e' (resp. `E') style. If the precision is set to a value greater than the maximum value for an `int', it will be silently reduced down to `INT_MAX'. If the `precision' field is empty (as in `%Re' or `%.RE') with `conv' specifier `e' and `E', the number is displayed with enough digits so that it can be read back exactly, assuming that the input and output variables have the same precision and that the input and output rounding modes are both rounding to nearest (as for `mpfr_get_str'). The default precision for an empty `precision' field with `conv' specifiers `f', `F', `g', and `G' is 6. 5.9.3 Functions --------------- For all the following functions, if the number of characters which ought to be written appears to exceed the maximum limit for an `int', nothing is written in the stream (resp. to `stdout', to BUF, to STR), the function returns -1, sets the _erange_ flag, and (in POSIX system only) `errno' is set to `EOVERFLOW'. -- Function: int mpfr_fprintf (FILE *STREAM, const char *TEMPLATE, ...) -- Function: int mpfr_vfprintf (FILE *STREAM, const char *TEMPLATE, va_list AP) Print to the stream STREAM the optional arguments under the control of the template string TEMPLATE. Return the number of characters written or a negative value if an error occurred. -- Function: int mpfr_printf (const char *TEMPLATE, ...) -- Function: int mpfr_vprintf (const char *TEMPLATE, va_list AP) Print to `stdout' the optional arguments under the control of the template string TEMPLATE. Return the number of characters written or a negative value if an error occurred. -- Function: int mpfr_sprintf (char *BUF, const char *TEMPLATE, ...) -- Function: int mpfr_vsprintf (char *BUF, const char *TEMPLATE, va_list AP) Form a null-terminated string corresponding to the optional arguments under the control of the template string TEMPLATE, and print it in BUF. No overlap is permitted between BUF and the other arguments. Return the number of characters written in the array BUF _not counting_ the terminating null character or a negative value if an error occurred. -- Function: int mpfr_snprintf (char *BUF, size_t N, const char *TEMPLATE, ...) -- Function: int mpfr_vsnprintf (char *BUF, size_t N, const char *TEMPLATE, va_list AP) Form a null-terminated string corresponding to the optional arguments under the control of the template string TEMPLATE, and print it in BUF. If N is zero, nothing is written and BUF may be a null pointer, otherwise, the N-1 first characters are written in BUF and the N-th is a null character. Return the number of characters that would have been written had N be sufficiently large, _not counting_ the terminating null character, or a negative value if an error occurred. -- Function: int mpfr_asprintf (char **STR, const char *TEMPLATE, ...) -- Function: int mpfr_vasprintf (char **STR, const char *TEMPLATE, va_list AP) Write their output as a null terminated string in a block of memory allocated using the current allocation function. A pointer to the block is stored in STR. The block of memory must be freed using `mpfr_free_str'. The return value is the number of characters written in the string, excluding the null-terminator, or a negative value if an error occurred.  File: mpfr.info, Node: Integer Related Functions, Next: Rounding Related Functions, Prev: Formatted Output Functions, Up: MPFR Interface 5.10 Integer and Remainder Related Functions ============================================ -- Function: int mpfr_rint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_ceil (mpfr_t ROP, mpfr_t OP) -- Function: int mpfr_floor (mpfr_t ROP, mpfr_t OP) -- Function: int mpfr_round (mpfr_t ROP, mpfr_t OP) -- Function: int mpfr_trunc (mpfr_t ROP, mpfr_t OP) Set ROP to OP rounded to an integer. `mpfr_rint' rounds to the nearest representable integer in the given direction RND, `mpfr_ceil' rounds to the next higher or equal representable integer, `mpfr_floor' to the next lower or equal representable integer, `mpfr_round' to the nearest representable integer, rounding halfway cases away from zero (as in the roundTiesToAway mode of IEEE 754-2008), and `mpfr_trunc' to the next representable integer toward zero. The returned value is zero when the result is exact, positive when it is greater than the original value of OP, and negative when it is smaller. More precisely, the returned value is 0 when OP is an integer representable in ROP, 1 or -1 when OP is an integer that is not representable in ROP, 2 or -2 when OP is not an integer. Note that `mpfr_round' is different from `mpfr_rint' called with the rounding to nearest mode (where halfway cases are rounded to an even integer or significand). Note also that no double rounding is performed; for instance, 10.5 (1010.1 in binary) is rounded by `mpfr_rint' with rounding to nearest to 12 (1100 in binary) in 2-bit precision, because the two enclosing numbers representable on two bits are 8 and 12, and the closest is 12. (If one first rounded to an integer, one would round 10.5 to 10 with even rounding, and then 10 would be rounded to 8 again with even rounding.) -- Function: int mpfr_rint_ceil (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_rint_floor (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_rint_round (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_rint_trunc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to OP rounded to an integer. `mpfr_rint_ceil' rounds to the next higher or equal integer, `mpfr_rint_floor' to the next lower or equal integer, `mpfr_rint_round' to the nearest integer, rounding halfway cases away from zero, and `mpfr_rint_trunc' to the next integer toward zero. If the result is not representable, it is rounded in the direction RND. The returned value is the ternary value associated with the considered round-to-integer function (regarded in the same way as any other mathematical function). Contrary to `mpfr_rint', those functions do perform a double rounding: first OP is rounded to the nearest integer in the direction given by the function name, then this nearest integer (if not representable) is rounded in the given direction RND. For example, `mpfr_rint_round' with rounding to nearest and a precision of two bits rounds 6.5 to 7 (halfway cases away from zero), then 7 is rounded to 8 by the round-even rule, despite the fact that 6 is also representable on two bits, and is closer to 6.5 than 8. -- Function: int mpfr_frac (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the fractional part of OP, having the same sign as OP, rounded in the direction RND (unlike in `mpfr_rint', RND affects only how the exact fractional part is rounded, not how the fractional part is generated). -- Function: int mpfr_modf (mpfr_t IOP, mpfr_t FOP, mpfr_t OP, mpfr_rnd_t RND) Set simultaneously IOP to the integral part of OP and FOP to the fractional part of OP, rounded in the direction RND with the corresponding precision of IOP and FOP (equivalent to `mpfr_trunc(IOP, OP, RND)' and `mpfr_frac(FOP, OP, RND)'). The variables IOP and FOP must be different. Return 0 iff both results are exact (see `mpfr_sin_cos' for a more detailed description of the return value). -- Function: int mpfr_fmod (mpfr_t R, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) -- Function: int mpfr_remainder (mpfr_t R, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) -- Function: int mpfr_remquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) Set R to the value of X - NY, rounded according to the direction RND, where N is the integer quotient of X divided by Y, defined as follows: N is rounded toward zero for `mpfr_fmod', and to the nearest integer (ties rounded to even) for `mpfr_remainder' and `mpfr_remquo'. Special values are handled as described in Section F.9.7.1 of the ISO C99 standard: If X is infinite or Y is zero, R is NaN. If Y is infinite and X is finite, R is X rounded to the precision of R. If R is zero, it has the sign of X. The return value is the ternary value corresponding to R. Additionally, `mpfr_remquo' stores the low significant bits from the quotient N in *Q (more precisely the number of bits in a `long' minus one), with the sign of X divided by Y (except if those low bits are all zero, in which case zero is returned). Note that X may be so large in magnitude relative to Y that an exact representation of the quotient is not practical. The `mpfr_remainder' and `mpfr_remquo' functions are useful for additive argument reduction. -- Function: int mpfr_integer_p (mpfr_t OP) Return non-zero iff OP is an integer.  File: mpfr.info, Node: Rounding Related Functions, Next: Miscellaneous Functions, Prev: Integer Related Functions, Up: MPFR Interface 5.11 Rounding Related Functions =============================== -- Function: void mpfr_set_default_rounding_mode (mpfr_rnd_t RND) Set the default rounding mode to RND. The default rounding mode is to nearest initially. -- Function: mpfr_rnd_t mpfr_get_default_rounding_mode (void) Get the default rounding mode. -- Function: int mpfr_prec_round (mpfr_t X, mpfr_prec_t PREC, mpfr_rnd_t RND) Round X according to RND with precision PREC, which must be an integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX' (otherwise the behavior is undefined). If PREC is greater or equal to the precision of X, then new space is allocated for the significand, and it is filled with zeros. Otherwise, the significand is rounded to precision PREC with the given direction. In both cases, the precision of X is changed to PREC. Here is an example of how to use `mpfr_prec_round' to implement Newton's algorithm to compute the inverse of A, assuming X is already an approximation to N bits: mpfr_set_prec (t, 2 * n); mpfr_set (t, a, MPFR_RNDN); /* round a to 2n bits */ mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to 2n bits */ mpfr_ui_sub (t, 1, t, MPFR_RNDN); /* high n bits cancel with 1 */ mpfr_prec_round (t, n, MPFR_RNDN); /* t is correct to n bits */ mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to n bits */ mpfr_prec_round (x, 2 * n, MPFR_RNDN); /* exact */ mpfr_add (x, x, t, MPFR_RNDN); /* x is correct to 2n bits */ -- Function: int mpfr_can_round (mpfr_t B, mpfr_exp_t ERR, mpfr_rnd_t RND1, mpfr_rnd_t RND2, mpfr_prec_t PREC) Assuming B is an approximation of an unknown number X in the direction RND1 with error at most two to the power E(b)-ERR where E(b) is the exponent of B, return a non-zero value if one is able to round correctly X to precision PREC with the direction RND2, and 0 otherwise (including for NaN and Inf). This function *does not modify* its arguments. If RND1 is `MPFR_RNDN', then the sign of the error is unknown, but its absolute value is the same, so that the possible range is twice as large as with a directed rounding for RND1. Note: if one wants to also determine the correct *note ternary value:: when rounding B to precision PREC with rounding mode RND, a useful trick is the following: if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ, prec + (rnd == MPFR_RNDN))) ... Indeed, if RND is `MPFR_RNDN', this will check if one can round to PREC+1 bits with a directed rounding: if so, one can surely round to nearest to PREC bits, and in addition one can determine the correct ternary value, which would not be the case when B is near from a value exactly representable on PREC bits. -- Function: mpfr_prec_t mpfr_min_prec (mpfr_t X) Return the minimal number of bits required to store the significand of X, and 0 for special values, including 0. (Warning: the returned value can be less than `MPFR_PREC_MIN'.) The function name is subject to change. -- Function: const char * mpfr_print_rnd_mode (mpfr_rnd_t RND) Return a string ("MPFR_RNDD", "MPFR_RNDU", "MPFR_RNDN", "MPFR_RNDZ", "MPFR_RNDA") corresponding to the rounding mode RND, or a null pointer if RND is an invalid rounding mode.  File: mpfr.info, Node: Miscellaneous Functions, Next: Exception Related Functions, Prev: Rounding Related Functions, Up: MPFR Interface 5.12 Miscellaneous Functions ============================ -- Function: void mpfr_nexttoward (mpfr_t X, mpfr_t Y) If X or Y is NaN, set X to NaN. If X and Y are equal, X is unchanged. Otherwise, if X is different from Y, replace X by the next floating-point number (with the precision of X and the current exponent range) in the direction of Y (the infinite values are seen as the smallest and largest floating-point numbers). If the result is zero, it keeps the same sign. No underflow or overflow is generated. -- Function: void mpfr_nextabove (mpfr_t X) -- Function: void mpfr_nextbelow (mpfr_t X) Equivalent to `mpfr_nexttoward' where Y is plus infinity (resp. minus infinity). -- Function: int mpfr_min (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_max (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the minimum (resp. maximum) of OP1 and OP2. If OP1 and OP2 are both NaN, then ROP is set to NaN. If OP1 or OP2 is NaN, then ROP is set to the numeric value. If OP1 and OP2 are zeros of different signs, then ROP is set to -0 (resp. +0). -- Function: int mpfr_urandomb (mpfr_t ROP, gmp_randstate_t STATE) Generate a uniformly distributed random float in the interval 0 <= ROP < 1. More precisely, the number can be seen as a float with a random non-normalized significand and exponent 0, which is then normalized (thus if E denotes the exponent after normalization, then the least -E significant bits of the significand are always 0). Return 0, unless the exponent is not in the current exponent range, in which case ROP is set to NaN and a non-zero value is returned (this should never happen in practice, except in very specific cases). The second argument is a `gmp_randstate_t' structure which should be created using the GMP `gmp_randinit' function (see the GMP manual). Note: for a given version of MPFR, the returned value of ROP and the new value of STATE (which controls further random values) do not depend on the machine word size. -- Function: int mpfr_urandom (mpfr_t ROP, gmp_randstate_t STATE, mpfr_rnd_t RND) Generate a uniformly distributed random float. The floating-point number ROP can be seen as if a random real number is generated according to the continuous uniform distribution on the interval [0, 1] and then rounded in the direction RND. The second argument is a `gmp_randstate_t' structure which should be created using the GMP `gmp_randinit' function (see the GMP manual). Note: the note for `mpfr_urandomb' holds too. In addition, the exponent range and the rounding mode might have a side effect on the next random state. -- Function: int mpfr_grandom (mpfr_t ROP1, mpfr_t ROP2, gmp_randstate_t STATE, mpfr_rnd_t RND) Generate two random floats according to a standard normal gaussian distribution. If ROP2 is a null pointer, then only one value is generated and stored in ROP1. The floating-point number ROP1 (and ROP2) can be seen as if a random real number were generated according to the standard normal gaussian distribution and then rounded in the direction RND. The third argument is a `gmp_randstate_t' structure, which should be created using the GMP `gmp_randinit' function (see the GMP manual). The combination of the ternary values is returned like with `mpfr_sin_cos'. If ROP2 is a null pointer, the second ternary value is assumed to be 0 (note that the encoding of the only ternary value is not the same as the usual encoding for functions that return only one result). Otherwise the ternary value of a random number is always non-zero. Note: the note for `mpfr_urandomb' holds too. In addition, the exponent range and the rounding mode might have a side effect on the next random state. -- Function: mpfr_exp_t mpfr_get_exp (mpfr_t X) Return the exponent of X, assuming that X is a non-zero ordinary number and the significand is considered in [1/2,1). The behavior for NaN, infinity or zero is undefined. -- Function: int mpfr_set_exp (mpfr_t X, mpfr_exp_t E) Set the exponent of X if E is in the current exponent range, and return 0 (even if X is not a non-zero ordinary number); otherwise, return a non-zero value. The significand is assumed to be in [1/2,1). -- Function: int mpfr_signbit (mpfr_t OP) Return a non-zero value iff OP has its sign bit set (i.e., if it is negative, -0, or a NaN whose representation has its sign bit set). -- Function: int mpfr_setsign (mpfr_t ROP, mpfr_t OP, int S, mpfr_rnd_t RND) Set the value of ROP from OP, rounded toward the given direction RND, then set (resp. clear) its sign bit if S is non-zero (resp. zero), even when OP is a NaN. -- Function: int mpfr_copysign (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set the value of ROP from OP1, rounded toward the given direction RND, then set its sign bit to that of OP2 (even when OP1 or OP2 is a NaN). This function is equivalent to `mpfr_setsign (ROP, OP1, mpfr_signbit (OP2), RND)'. -- Function: const char * mpfr_get_version (void) Return the MPFR version, as a null-terminated string. -- Macro: MPFR_VERSION -- Macro: MPFR_VERSION_MAJOR -- Macro: MPFR_VERSION_MINOR -- Macro: MPFR_VERSION_PATCHLEVEL -- Macro: MPFR_VERSION_STRING `MPFR_VERSION' is the version of MPFR as a preprocessing constant. `MPFR_VERSION_MAJOR', `MPFR_VERSION_MINOR' and `MPFR_VERSION_PATCHLEVEL' are respectively the major, minor and patch level of MPFR version, as preprocessing constants. `MPFR_VERSION_STRING' is the version (with an optional suffix, used in development and pre-release versions) as a string constant, which can be compared to the result of `mpfr_get_version' to check at run time the header file and library used match: if (strcmp (mpfr_get_version (), MPFR_VERSION_STRING)) fprintf (stderr, "Warning: header and library do not match\n"); Note: Obtaining different strings is not necessarily an error, as in general, a program compiled with some old MPFR version can be dynamically linked with a newer MPFR library version (if allowed by the library versioning system). -- Macro: long MPFR_VERSION_NUM (MAJOR, MINOR, PATCHLEVEL) Create an integer in the same format as used by `MPFR_VERSION' from the given MAJOR, MINOR and PATCHLEVEL. Here is an example of how to check the MPFR version at compile time: #if (!defined(MPFR_VERSION) || (MPFR_VERSION' line, #include #include any program written for MPF can be compiled directly with MPFR without any changes (except the `gmp_printf' functions will not work for arguments of type `mpfr_t'). All operations are then performed with the default MPFR rounding mode, which can be reset with `mpfr_set_default_rounding_mode'. Warning: the `mpf_init' and `mpf_init2' functions initialize to zero, whereas the corresponding MPFR functions initialize to NaN: this is useful to detect uninitialized values, but is slightly incompatible with MPF. -- Function: void mpfr_set_prec_raw (mpfr_t X, mpfr_prec_t PREC) Reset the precision of X to be *exactly* PREC bits. The only difference with `mpfr_set_prec' is that PREC is assumed to be small enough so that the significand fits into the current allocated memory space for X. Otherwise the behavior is undefined. -- Function: int mpfr_eq (mpfr_t OP1, mpfr_t OP2, unsigned long int OP3) Return non-zero if OP1 and OP2 are both non-zero ordinary numbers with the same exponent and the same first OP3 bits, both zero, or both infinities of the same sign. Return zero otherwise. This function is defined for compatibility with MPF, we do not recommend to use it otherwise. Do not use it either if you want to know whether two numbers are close to each other; for instance, 1.011111 and 1.100000 are regarded as different for any value of OP3 larger than 1. -- Function: void mpfr_reldiff (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Compute the relative difference between OP1 and OP2 and store the result in ROP. This function does not guarantee the correct rounding on the relative difference; it just computes |OP1-OP2|/OP1, using the precision of ROP and the rounding mode RND for all operations. -- Function: int mpfr_mul_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) These functions are identical to `mpfr_mul_2ui' and `mpfr_div_2ui' respectively. These functions are only kept for compatibility with MPF, one should prefer `mpfr_mul_2ui' and `mpfr_div_2ui' otherwise.  File: mpfr.info, Node: Custom Interface, Next: Internals, Prev: Compatibility with MPF, Up: MPFR Interface 5.15 Custom Interface ===================== Some applications use a stack to handle the memory and their objects. However, the MPFR memory design is not well suited for such a thing. So that such applications are able to use MPFR, an auxiliary memory interface has been created: the Custom Interface. The following interface allows one to use MPFR in two ways: * Either directly store a floating-point number as a `mpfr_t' on the stack. * Either store its own representation on the stack and construct a new temporary `mpfr_t' each time it is needed. Nothing has to be done to destroy the floating-point numbers except garbaging the used memory: all the memory management (allocating, destroying, garbaging) is left to the application. Each function in this interface is also implemented as a macro for efficiency reasons: for example `mpfr_custom_init (s, p)' uses the macro, while `(mpfr_custom_init) (s, p)' uses the function. Note 1: MPFR functions may still initialize temporary floating-point numbers using `mpfr_init' and similar functions. See Custom Allocation (GNU MP). Note 2: MPFR functions may use the cached functions (`mpfr_const_pi' for example), even if they are not explicitly called. You have to call `mpfr_free_cache' each time you garbage the memory iff `mpfr_init', through GMP Custom Allocation, allocates its memory on the application stack. -- Function: size_t mpfr_custom_get_size (mpfr_prec_t PREC) Return the needed size in bytes to store the significand of a floating-point number of precision PREC. -- Function: void mpfr_custom_init (void *SIGNIFICAND, mpfr_prec_t PREC) Initialize a significand of precision PREC, where SIGNIFICAND must be an area of `mpfr_custom_get_size (prec)' bytes at least and be suitably aligned for an array of `mp_limb_t' (GMP type, *note Internals::). -- Function: void mpfr_custom_init_set (mpfr_t X, int KIND, mpfr_exp_t EXP, mpfr_prec_t PREC, void *SIGNIFICAND) Perform a dummy initialization of a `mpfr_t' and set it to: * if `ABS(kind) == MPFR_NAN_KIND', X is set to NaN; * if `ABS(kind) == MPFR_INF_KIND', X is set to the infinity of sign `sign(kind)'; * if `ABS(kind) == MPFR_ZERO_KIND', X is set to the zero of sign `sign(kind)'; * if `ABS(kind) == MPFR_REGULAR_KIND', X is set to a regular number: `x = sign(kind)*significand*2^exp'. In all cases, it uses SIGNIFICAND directly for further computing involving X. It will not allocate anything. A floating-point number initialized with this function cannot be resized using `mpfr_set_prec' or `mpfr_prec_round', or cleared using `mpfr_clear'! The SIGNIFICAND must have been initialized with `mpfr_custom_init' using the same precision PREC. -- Function: int mpfr_custom_get_kind (mpfr_t X) Return the current kind of a `mpfr_t' as created by `mpfr_custom_init_set'. The behavior of this function for any `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined. -- Function: void * mpfr_custom_get_significand (mpfr_t X) Return a pointer to the significand used by a `mpfr_t' initialized with `mpfr_custom_init_set'. The behavior of this function for any `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined. -- Function: mpfr_exp_t mpfr_custom_get_exp (mpfr_t X) Return the exponent of X, assuming that X is a non-zero ordinary number. The return value for NaN, Infinity or zero is unspecified but does not produce any trap. The behavior of this function for any `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined. -- Function: void mpfr_custom_move (mpfr_t X, void *NEW_POSITION) Inform MPFR that the significand of X has moved due to a garbage collect and update its new position to `new_position'. However the application has to move the significand and the `mpfr_t' itself. The behavior of this function for any `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined.  File: mpfr.info, Node: Internals, Prev: Custom Interface, Up: MPFR Interface 5.16 Internals ============== A "limb" means the part of a multi-precision number that fits in a single word. Usually a limb contains 32 or 64 bits. The C data type for a limb is `mp_limb_t'. The `mpfr_t' type is internally defined as a one-element array of a structure, and `mpfr_ptr' is the C data type representing a pointer to this structure. The `mpfr_t' type consists of four fields: * The `_mpfr_prec' field is used to store the precision of the variable (in bits); this is not less than `MPFR_PREC_MIN'. * The `_mpfr_sign' field is used to store the sign of the variable. * The `_mpfr_exp' field stores the exponent. An exponent of 0 means a radix point just above the most significant limb. Non-zero values n are a multiplier 2^n relative to that point. A NaN, an infinity and a zero are indicated by special values of the exponent field. * Finally, the `_mpfr_d' field is a pointer to the limbs, least significant limbs stored first. The number of limbs in use is controlled by `_mpfr_prec', namely ceil(`_mpfr_prec'/`mp_bits_per_limb'). Non-singular (i.e., different from NaN, Infinity or zero) values always have the most significant bit of the most significant limb set to 1. When the precision does not correspond to a whole number of limbs, the excess bits at the low end of the data are zeros.  File: mpfr.info, Node: API Compatibility, Next: Contributors, Prev: MPFR Interface, Up: Top 6 API Compatibility ******************* The goal of this section is to describe some API changes that occurred from one version of MPFR to another, and how to write code that can be compiled and run with older MPFR versions. The minimum MPFR version that is considered here is 2.2.0 (released on 20 September 2005). API changes can only occur between major or minor versions. Thus the patchlevel (the third number in the MPFR version) will be ignored in the following. If a program does not use MPFR internals, changes in the behavior between two versions differing only by the patchlevel should only result from what was regarded as a bug or unspecified behavior. As a general rule, a program written for some MPFR version should work with later versions, possibly except at a new major version, where some features (described as obsolete for some time) can be removed. In such a case, a failure should occur during compilation or linking. If a result becomes incorrect because of such a change, please look at the various changes below (they are minimal, and most software should be unaffected), at the FAQ and at the MPFR web page for your version (a bug could have been introduced and be already fixed); and if the problem is not mentioned, please send us a bug report (*note Reporting Bugs::). However, a program written for the current MPFR version (as documented by this manual) may not necessarily work with previous versions of MPFR. This section should help developers to write portable code. Note: Information given here may be incomplete. API changes are also described in the NEWS file (for each version, instead of being classified like here), together with other changes. * Menu: * Type and Macro Changes:: * Added Functions:: * Changed Functions:: * Removed Functions:: * Other Changes::  File: mpfr.info, Node: Type and Macro Changes, Next: Added Functions, Prev: API Compatibility, Up: API Compatibility 6.1 Type and Macro Changes ========================== The official type for exponent values changed from `mp_exp_t' to `mpfr_exp_t' in MPFR 3.0. The type `mp_exp_t' will remain available as it comes from GMP (with a different meaning). These types are currently the same (`mpfr_exp_t' is defined as `mp_exp_t' with `typedef'), so that programs can still use `mp_exp_t'; but this may change in the future. Alternatively, using the following code after including `mpfr.h' will work with official MPFR versions, as `mpfr_exp_t' was never defined in MPFR 2.x: #if MPFR_VERSION_MAJOR < 3 typedef mp_exp_t mpfr_exp_t; #endif The official types for precision values and for rounding modes respectively changed from `mp_prec_t' and `mp_rnd_t' to `mpfr_prec_t' and `mpfr_rnd_t' in MPFR 3.0. This change was actually done a long time ago in MPFR, at least since MPFR 2.2.0, with the following code in `mpfr.h': #ifndef mp_rnd_t # define mp_rnd_t mpfr_rnd_t #endif #ifndef mp_prec_t # define mp_prec_t mpfr_prec_t #endif This means that it is safe to use the new official types `mpfr_prec_t' and `mpfr_rnd_t' in your programs. The types `mp_prec_t' and `mp_rnd_t' (defined in MPFR only) may be removed in the future, as the prefix `mp_' is reserved by GMP. The precision type `mpfr_prec_t' (`mp_prec_t') was unsigned before MPFR 3.0; it is now signed. `MPFR_PREC_MAX' has not changed, though. Indeed the MPFR code requires that `MPFR_PREC_MAX' be representable in the exponent type, which may have the same size as `mpfr_prec_t' but has always been signed. The consequence is that valid code that does not assume anything about the signedness of `mpfr_prec_t' should work with past and new MPFR versions. This change was useful as the use of unsigned types tends to convert signed values to unsigned ones in expressions due to the usual arithmetic conversions, which can yield incorrect results if a negative value is converted in such a way. Warning! A program assuming (intentionally or not) that `mpfr_prec_t' is signed may be affected by this problem when it is built and run against MPFR 2.x. The rounding modes `GMP_RNDx' were renamed to `MPFR_RNDx' in MPFR 3.0. However the old names `GMP_RNDx' have been kept for compatibility (this might change in future versions), using: #define GMP_RNDN MPFR_RNDN #define GMP_RNDZ MPFR_RNDZ #define GMP_RNDU MPFR_RNDU #define GMP_RNDD MPFR_RNDD The rounding mode "round away from zero" (`MPFR_RNDA') was added in MPFR 3.0 (however no rounding mode `GMP_RNDA' exists).  File: mpfr.info, Node: Added Functions, Next: Changed Functions, Prev: Type and Macro Changes, Up: API Compatibility 6.2 Added Functions =================== We give here in alphabetical order the functions that were added after MPFR 2.2, and in which MPFR version. * `mpfr_add_d' in MPFR 2.4. * `mpfr_ai' in MPFR 3.0 (incomplete, experimental). * `mpfr_asprintf' in MPFR 2.4. * `mpfr_buildopt_decimal_p' and `mpfr_buildopt_tls_p' in MPFR 3.0. * `mpfr_buildopt_gmpinternals_p' and `mpfr_buildopt_tune_case' in MPFR 3.1. * `mpfr_clear_divby0' in MPFR 3.1 (new divide-by-zero exception). * `mpfr_copysign' in MPFR 2.3. Note: MPFR 2.2 had a `mpfr_copysign' function that was available, but not documented, and with a slight difference in the semantics (when the second input operand is a NaN). * `mpfr_custom_get_significand' in MPFR 3.0. This function was named `mpfr_custom_get_mantissa' in previous versions; `mpfr_custom_get_mantissa' is still available via a macro in `mpfr.h': #define mpfr_custom_get_mantissa mpfr_custom_get_significand Thus code that needs to work with both MPFR 2.x and MPFR 3.x should use `mpfr_custom_get_mantissa'. * `mpfr_d_div' and `mpfr_d_sub' in MPFR 2.4. * `mpfr_digamma' in MPFR 3.0. * `mpfr_divby0_p' in MPFR 3.1 (new divide-by-zero exception). * `mpfr_div_d' in MPFR 2.4. * `mpfr_fmod' in MPFR 2.4. * `mpfr_fms' in MPFR 2.3. * `mpfr_fprintf' in MPFR 2.4. * `mpfr_frexp' in MPFR 3.1. * `mpfr_get_flt' in MPFR 3.0. * `mpfr_get_patches' in MPFR 2.3. * `mpfr_get_z_2exp' in MPFR 3.0. This function was named `mpfr_get_z_exp' in previous versions; `mpfr_get_z_exp' is still available via a macro in `mpfr.h': #define mpfr_get_z_exp mpfr_get_z_2exp Thus code that needs to work with both MPFR 2.x and MPFR 3.x should use `mpfr_get_z_exp'. * `mpfr_grandom' in MPFR 3.1. * `mpfr_j0', `mpfr_j1' and `mpfr_jn' in MPFR 2.3. * `mpfr_lgamma' in MPFR 2.3. * `mpfr_li2' in MPFR 2.4. * `mpfr_min_prec' in MPFR 3.0. * `mpfr_modf' in MPFR 2.4. * `mpfr_mul_d' in MPFR 2.4. * `mpfr_printf' in MPFR 2.4. * `mpfr_rec_sqrt' in MPFR 2.4. * `mpfr_regular_p' in MPFR 3.0. * `mpfr_remainder' and `mpfr_remquo' in MPFR 2.3. * `mpfr_set_divby0' in MPFR 3.1 (new divide-by-zero exception). * `mpfr_set_flt' in MPFR 3.0. * `mpfr_set_z_2exp' in MPFR 3.0. * `mpfr_set_zero' in MPFR 3.0. * `mpfr_setsign' in MPFR 2.3. * `mpfr_signbit' in MPFR 2.3. * `mpfr_sinh_cosh' in MPFR 2.4. * `mpfr_snprintf' and `mpfr_sprintf' in MPFR 2.4. * `mpfr_sub_d' in MPFR 2.4. * `mpfr_urandom' in MPFR 3.0. * `mpfr_vasprintf', `mpfr_vfprintf', `mpfr_vprintf', `mpfr_vsprintf' and `mpfr_vsnprintf' in MPFR 2.4. * `mpfr_y0', `mpfr_y1' and `mpfr_yn' in MPFR 2.3. * `mpfr_z_sub' in MPFR 3.1.  File: mpfr.info, Node: Changed Functions, Next: Removed Functions, Prev: Added Functions, Up: API Compatibility 6.3 Changed Functions ===================== The following functions have changed after MPFR 2.2. Changes can affect the behavior of code written for some MPFR version when built and run against another MPFR version (older or newer), as described below. * `mpfr_check_range' changed in MPFR 2.3.2 and MPFR 2.4. If the value is an inexact infinity, the overflow flag is now set (in case it was lost), while it was previously left unchanged. This is really what is expected in practice (and what the MPFR code was expecting), so that the previous behavior was regarded as a bug. Hence the change in MPFR 2.3.2. * `mpfr_get_f' changed in MPFR 3.0. This function was returning zero, except for NaN and Inf, which do not exist in MPF. The _erange_ flag is now set in these cases, and `mpfr_get_f' now returns the usual ternary value. * `mpfr_get_si', `mpfr_get_sj', `mpfr_get_ui' and `mpfr_get_uj' changed in MPFR 3.0. In previous MPFR versions, the cases where the _erange_ flag is set were unspecified. * `mpfr_get_z' changed in MPFR 3.0. The return type was `void'; it is now `int', and the usual ternary value is returned. Thus programs that need to work with both MPFR 2.x and 3.x must not use the return value. Even in this case, C code using `mpfr_get_z' as the second or third term of a conditional operator may also be affected. For instance, the following is correct with MPFR 3.0, but not with MPFR 2.x: bool ? mpfr_get_z(...) : mpfr_add(...); On the other hand, the following is correct with MPFR 2.x, but not with MPFR 3.0: bool ? mpfr_get_z(...) : (void) mpfr_add(...); Portable code should cast `mpfr_get_z(...)' to `void' to use the type `void' for both terms of the conditional operator, as in: bool ? (void) mpfr_get_z(...) : (void) mpfr_add(...); Alternatively, `if ... else' can be used instead of the conditional operator. Moreover the cases where the _erange_ flag is set were unspecified in MPFR 2.x. * `mpfr_get_z_exp' changed in MPFR 3.0. In previous MPFR versions, the cases where the _erange_ flag is set were unspecified. Note: this function has been renamed to `mpfr_get_z_2exp' in MPFR 3.0, but `mpfr_get_z_exp' is still available for compatibility reasons. * `mpfr_strtofr' changed in MPFR 2.3.1 and MPFR 2.4. This was actually a bug fix since the code and the documentation did not match. But both were changed in order to have a more consistent and useful behavior. The main changes in the code are as follows. The binary exponent is now accepted even without the `0b' or `0x' prefix. Data corresponding to NaN can now have an optional sign (such data were previously invalid). * `mpfr_strtofr' changed in MPFR 3.0. This function now accepts bases from 37 to 62 (no changes for the other bases). Note: if an unsupported base is provided to this function, the behavior is undefined; more precisely, in MPFR 2.3.1 and later, providing an unsupported base yields an assertion failure (this behavior may change in the future). * `mpfr_subnormalize' changed in MPFR 3.1. This was actually regarded as a bug fix. The `mpfr_subnormalize' implementation up to MPFR 3.0.0 did not change the flags. In particular, it did not follow the generic rule concerning the inexact flag (and no special behavior was specified). The case of the underflow flag was more a lack of specification. * `mpfr_urandom' and `mpfr_urandomb' changed in MPFR 3.1. Their behavior no longer depends on the platform (assuming this is also true for GMP's random generator, which is not the case between GMP 4.1 and 4.2 if `gmp_randinit_default' is used). As a consequence, the returned values can be different between MPFR 3.1 and previous MPFR versions. Note: as the reproducibility of these functions was not specified before MPFR 3.1, the MPFR 3.1 behavior is _not_ regarded as backward incompatible with previous versions.  File: mpfr.info, Node: Removed Functions, Next: Other Changes, Prev: Changed Functions, Up: API Compatibility 6.4 Removed Functions ===================== Functions `mpfr_random' and `mpfr_random2' have been removed in MPFR 3.0 (this only affects old code built against MPFR 3.0 or later). (The function `mpfr_random' had been deprecated since at least MPFR 2.2.0, and `mpfr_random2' since MPFR 2.4.0.)  File: mpfr.info, Node: Other Changes, Prev: Removed Functions, Up: API Compatibility 6.5 Other Changes ================= For users of a C++ compiler, the way how the availability of `intmax_t' is detected has changed in MPFR 3.0. In MPFR 2.x, if a macro `INTMAX_C' or `UINTMAX_C' was defined (e.g. when the `__STDC_CONSTANT_MACROS' macro had been defined before `' or `' has been included), `intmax_t' was assumed to be defined. However this was not always the case (more precisely, `intmax_t' can be defined only in the namespace `std', as with Boost), so that compilations could fail. Thus the check for `INTMAX_C' or `UINTMAX_C' is now disabled for C++ compilers, with the following consequences: * Programs written for MPFR 2.x that need `intmax_t' may no longer be compiled against MPFR 3.0: a `#define MPFR_USE_INTMAX_T' may be necessary before `mpfr.h' is included. * The compilation of programs that work with MPFR 3.0 may fail with MPFR 2.x due to the problem described above. Workarounds are possible, such as defining `intmax_t' and `uintmax_t' in the global namespace, though this is not clean. The divide-by-zero exception is new in MPFR 3.1. However it should not introduce incompatible changes for programs that strictly follow the MPFR API since the exception can only be seen via new functions. As of MPFR 3.1, the `mpfr.h' header can be included several times, while still supporting optional functions (*note Headers and Libraries::).  File: mpfr.info, Node: Contributors, Next: References, Prev: API Compatibility, Up: Top Contributors ************ The main developers of MPFR are Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, Philippe Théveny and Paul Zimmermann. Sylvie Boldo from ENS-Lyon, France, contributed the functions `mpfr_agm' and `mpfr_log'. Sylvain Chevillard contributed the `mpfr_ai' function. David Daney contributed the hyperbolic and inverse hyperbolic functions, the base-2 exponential, and the factorial function. Alain Delplanque contributed the new version of the `mpfr_get_str' function. Mathieu Dutour contributed the functions `mpfr_acos', `mpfr_asin' and `mpfr_atan', and a previous version of `mpfr_gamma'. Laurent Fousse contributed the `mpfr_sum' function. Emmanuel Jeandel, from ENS-Lyon too, contributed the generic hypergeometric code, as well as the internal function `mpfr_exp3', a first implementation of the sine and cosine, and improved versions of `mpfr_const_log2' and `mpfr_const_pi'. Ludovic Meunier helped in the design of the `mpfr_erf' code. Jean-Luc Rémy contributed the `mpfr_zeta' code. Fabrice Rouillier contributed the `mpfr_xxx_z' and `mpfr_xxx_q' functions, and helped to the Microsoft Windows porting. Damien Stehlé contributed the `mpfr_get_ld_2exp' function. We would like to thank Jean-Michel Muller and Joris van der Hoeven for very fruitful discussions at the beginning of that project, Torbjörn Granlund and Kevin Ryde for their help about design issues, and Nathalie Revol for her careful reading of a previous version of this documentation. In particular Kevin Ryde did a tremendous job for the portability of MPFR in 2002-2004. The development of the MPFR library would not have been possible without the continuous support of INRIA, and of the LORIA (Nancy, France) and LIP (Lyon, France) laboratories. In particular the main authors were or are members of the PolKA, Spaces, Cacao and Caramel project-teams at LORIA and of the Arénaire and AriC project-teams at LIP. This project was started during the Fiable (reliable in French) action supported by INRIA, and continued during the AOC action. The development of MPFR was also supported by a grant (202F0659 00 MPN 121) from the Conseil Régional de Lorraine in 2002, from INRIA by an "associate engineer" grant (2003-2005), an "opération de développement logiciel" grant (2007-2009), and the post-doctoral grant of Sylvain Chevillard in 2009-2010. The MPFR-MPC workshop in June 2012 was partly supported by the ERC grant ANTICS of Andreas Enge.  File: mpfr.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top References ********** * Richard Brent and Paul Zimmermann, "Modern Computer Arithmetic", Cambridge University Press (to appear), also available from the authors' web pages. * Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier and Paul Zimmermann, "MPFR: A Multiple-Precision Binary Floating-Point Library With Correct Rounding", ACM Transactions on Mathematical Software, volume 33, issue 2, article 13, 15 pages, 2007, `http://doi.acm.org/10.1145/1236463.1236468'. * Torbjörn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic Library", version 5.0.1, 2010, `http://gmplib.org'. * IEEE standard for binary floating-point arithmetic, Technical Report ANSI-IEEE Standard 754-1985, New York, 1985. Approved March 21, 1985: IEEE Standards Board; approved July 26, 1985: American National Standards Institute, 18 pages. * IEEE Standard for Floating-Point Arithmetic, ANSI-IEEE Standard 754-2008, 2008. Revision of ANSI-IEEE Standard 754-1985, approved June 12, 2008: IEEE Standards Board, 70 pages. * Donald E. Knuth, "The Art of Computer Programming", vol 2, "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981. * Jean-Michel Muller, "Elementary Functions, Algorithms and Implementation", Birkhäuser, Boston, 2nd edition, 2006. * Jean-Michel Muller, Nicolas Brisebarre, Florent de Dinechin, Claude-Pierre Jeannerod, Vincent Lefèvre, Guillaume Melquiond, Nathalie Revol, Damien Stehlé and Serge Torrès, "Handbook of Floating-Point Arithmetic", Birkhäuser, Boston, 2009.  File: mpfr.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top Appendix A GNU Free Documentation License ***************************************** Version 1.2, November 2002 Copyright (C) 2000,2001,2002 Free Software Foundation, Inc. 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 0. PREAMBLE The purpose of this License is to make a manual, textbook, or other functional and useful document "free" in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. 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Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled ``GNU Free Documentation License''. If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the "with...Texts." line with this: with the Invariant Sections being LIST THEIR TITLES, with the Front-Cover Texts being LIST, and with the Back-Cover Texts being LIST. If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation. If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.  File: mpfr.info, Node: Concept Index, Next: Function and Type Index, Prev: GNU Free Documentation License, Up: Top Concept Index ************* [index] * Menu: * Accuracy: MPFR Interface. (line 25) * Arithmetic functions: Basic Arithmetic Functions. (line 3) * Assignment functions: Assignment Functions. (line 3) * Basic arithmetic functions: Basic Arithmetic Functions. (line 3) * Combined initialization and assignment functions: Combined Initialization and Assignment Functions. (line 3) * Comparison functions: Comparison Functions. (line 3) * Compatibility with MPF: Compatibility with MPF. (line 3) * Conditions for copying MPFR: Copying. (line 6) * Conversion functions: Conversion Functions. (line 3) * Copying conditions: Copying. (line 6) * Custom interface: Custom Interface. (line 3) * Exception related functions: Exception Related Functions. (line 3) * Float arithmetic functions: Basic Arithmetic Functions. (line 3) * Float comparisons functions: Comparison Functions. (line 3) * Float functions: MPFR Interface. (line 6) * Float input and output functions: Input and Output Functions. (line 3) * Float output functions: Formatted Output Functions. (line 3) * Floating-point functions: MPFR Interface. (line 6) * Floating-point number: Nomenclature and Types. (line 6) * GNU Free Documentation License: GNU Free Documentation License. (line 6) * I/O functions <1>: Formatted Output Functions. (line 3) * I/O functions: Input and Output Functions. (line 3) * Initialization functions: Initialization Functions. (line 3) * Input functions: Input and Output Functions. (line 3) * Installation: Installing MPFR. (line 6) * Integer related functions: Integer Related Functions. (line 3) * Internals: Internals. (line 3) * intmax_t: Headers and Libraries. (line 22) * inttypes.h: Headers and Libraries. (line 22) * libmpfr: Headers and Libraries. (line 50) * Libraries: Headers and Libraries. (line 50) * Libtool: Headers and Libraries. (line 56) * Limb: Internals. (line 6) * Linking: Headers and Libraries. (line 50) * Miscellaneous float functions: Miscellaneous Functions. (line 3) * mpfr.h: Headers and Libraries. (line 6) * Output functions <1>: Formatted Output Functions. (line 3) * Output functions: Input and Output Functions. (line 3) * Precision <1>: MPFR Interface. (line 17) * Precision: Nomenclature and Types. (line 20) * Reporting bugs: Reporting Bugs. (line 6) * Rounding mode related functions: Rounding Related Functions. (line 3) * Rounding Modes: Nomenclature and Types. (line 34) * Special functions: Special Functions. (line 3) * stdarg.h: Headers and Libraries. (line 19) * stdint.h: Headers and Libraries. (line 22) * stdio.h: Headers and Libraries. (line 12) * Ternary value: Rounding Modes. (line 29) * uintmax_t: Headers and Libraries. (line 22)  File: mpfr.info, Node: Function and Type Index, Prev: Concept Index, Up: Top Function and Type Index *********************** [index] * Menu: * mpfr_abs: Basic Arithmetic Functions. (line 175) * mpfr_acos: Special Functions. (line 52) * mpfr_acosh: Special Functions. (line 136) * mpfr_add: Basic Arithmetic Functions. (line 8) * mpfr_add_d: Basic Arithmetic Functions. (line 14) * mpfr_add_q: Basic Arithmetic Functions. (line 18) * mpfr_add_si: Basic Arithmetic Functions. (line 12) * mpfr_add_ui: Basic Arithmetic Functions. (line 10) * mpfr_add_z: Basic Arithmetic Functions. (line 16) * mpfr_agm: Special Functions. (line 232) * mpfr_ai: Special Functions. (line 248) * mpfr_asin: Special Functions. (line 53) * mpfr_asinh: Special Functions. (line 137) * mpfr_asprintf: Formatted Output Functions. (line 194) * mpfr_atan: Special Functions. (line 54) * mpfr_atan2: Special Functions. (line 65) * mpfr_atanh: Special Functions. (line 138) * mpfr_buildopt_decimal_p: Miscellaneous Functions. (line 163) * mpfr_buildopt_gmpinternals_p: Miscellaneous Functions. (line 168) * mpfr_buildopt_tls_p: Miscellaneous Functions. (line 157) * mpfr_buildopt_tune_case: Miscellaneous Functions. (line 173) * mpfr_can_round: Rounding Related Functions. (line 37) * mpfr_cbrt: Basic Arithmetic Functions. (line 109) * mpfr_ceil: Integer Related Functions. (line 8) * mpfr_check_range: Exception Related Functions. (line 38) * mpfr_clear: Initialization Functions. (line 31) * mpfr_clear_divby0: Exception Related Functions. (line 113) * mpfr_clear_erangeflag: Exception Related Functions. (line 116) * mpfr_clear_flags: Exception Related Functions. (line 129) * mpfr_clear_inexflag: Exception Related Functions. (line 115) * mpfr_clear_nanflag: Exception Related Functions. (line 114) * mpfr_clear_overflow: Exception Related Functions. (line 112) * mpfr_clear_underflow: Exception Related Functions. (line 111) * mpfr_clears: Initialization Functions. (line 36) * mpfr_cmp: Comparison Functions. (line 7) * mpfr_cmp_d: Comparison Functions. (line 10) * mpfr_cmp_f: Comparison Functions. (line 14) * mpfr_cmp_ld: Comparison Functions. (line 11) * mpfr_cmp_q: Comparison Functions. (line 13) * mpfr_cmp_si: Comparison Functions. (line 9) * mpfr_cmp_si_2exp: Comparison Functions. (line 31) * mpfr_cmp_ui: Comparison Functions. (line 8) * mpfr_cmp_ui_2exp: Comparison Functions. (line 29) * mpfr_cmp_z: Comparison Functions. (line 12) * mpfr_cmpabs: Comparison Functions. (line 35) * mpfr_const_catalan: Special Functions. (line 259) * mpfr_const_euler: Special Functions. (line 258) * mpfr_const_log2: Special Functions. (line 256) * mpfr_const_pi: Special Functions. (line 257) * mpfr_copysign: Miscellaneous Functions. (line 111) * mpfr_cos: Special Functions. (line 30) * mpfr_cosh: Special Functions. (line 115) * mpfr_cot: Special Functions. (line 48) * mpfr_coth: Special Functions. (line 132) * mpfr_csc: Special Functions. (line 47) * mpfr_csch: Special Functions. (line 131) * mpfr_custom_get_exp: Custom Interface. (line 78) * mpfr_custom_get_kind: Custom Interface. (line 67) * mpfr_custom_get_significand: Custom Interface. (line 72) * mpfr_custom_get_size: Custom Interface. (line 36) * mpfr_custom_init: Custom Interface. (line 41) * mpfr_custom_init_set: Custom Interface. (line 48) * mpfr_custom_move: Custom Interface. (line 85) * mpfr_d_div: Basic Arithmetic Functions. (line 84) * mpfr_d_sub: Basic Arithmetic Functions. (line 37) * MPFR_DECL_INIT: Initialization Functions. (line 75) * mpfr_digamma: Special Functions. (line 187) * mpfr_dim: Basic Arithmetic Functions. (line 182) * mpfr_div: Basic Arithmetic Functions. (line 74) * mpfr_div_2exp: Compatibility with MPF. (line 51) * mpfr_div_2si: Basic Arithmetic Functions. (line 197) * mpfr_div_2ui: Basic Arithmetic Functions. (line 195) * mpfr_div_d: Basic Arithmetic Functions. (line 86) * mpfr_div_q: Basic Arithmetic Functions. (line 90) * mpfr_div_si: Basic Arithmetic Functions. (line 82) * mpfr_div_ui: Basic Arithmetic Functions. (line 78) * mpfr_div_z: Basic Arithmetic Functions. (line 88) * mpfr_divby0_p: Exception Related Functions. (line 135) * mpfr_eint: Special Functions. (line 154) * mpfr_eq: Compatibility with MPF. (line 30) * mpfr_equal_p: Comparison Functions. (line 61) * mpfr_erangeflag_p: Exception Related Functions. (line 138) * mpfr_erf: Special Functions. (line 198) * mpfr_erfc: Special Functions. (line 199) * mpfr_exp: Special Functions. (line 24) * mpfr_exp10: Special Functions. (line 26) * mpfr_exp2: Special Functions. (line 25) * mpfr_expm1: Special Functions. (line 150) * mpfr_fac_ui: Special Functions. (line 143) * mpfr_fits_intmax_p: Conversion Functions. (line 146) * mpfr_fits_sint_p: Conversion Functions. (line 142) * mpfr_fits_slong_p: Conversion Functions. (line 140) * mpfr_fits_sshort_p: Conversion Functions. (line 144) * mpfr_fits_uint_p: Conversion Functions. (line 141) * mpfr_fits_uintmax_p: Conversion Functions. (line 145) * mpfr_fits_ulong_p: Conversion Functions. (line 139) * mpfr_fits_ushort_p: Conversion Functions. (line 143) * mpfr_floor: Integer Related Functions. (line 9) * mpfr_fma: Special Functions. (line 225) * mpfr_fmod: Integer Related Functions. (line 79) * mpfr_fms: Special Functions. (line 227) * mpfr_fprintf: Formatted Output Functions. (line 158) * mpfr_frac: Integer Related Functions. (line 62) * mpfr_free_cache: Special Functions. (line 266) * mpfr_free_str: Conversion Functions. (line 133) * mpfr_frexp: Conversion Functions. (line 47) * mpfr_gamma: Special Functions. (line 169) * mpfr_get_d: Conversion Functions. (line 8) * mpfr_get_d_2exp: Conversion Functions. (line 34) * mpfr_get_decimal64: Conversion Functions. (line 10) * mpfr_get_default_prec: Initialization Functions. (line 114) * mpfr_get_default_rounding_mode: Rounding Related Functions. (line 11) * mpfr_get_emax: Exception Related Functions. (line 8) * mpfr_get_emax_max: Exception Related Functions. (line 31) * mpfr_get_emax_min: Exception Related Functions. (line 30) * mpfr_get_emin: Exception Related Functions. (line 7) * mpfr_get_emin_max: Exception Related Functions. (line 29) * mpfr_get_emin_min: Exception Related Functions. (line 28) * mpfr_get_exp: Miscellaneous Functions. (line 89) * mpfr_get_f: Conversion Functions. (line 73) * mpfr_get_flt: Conversion Functions. (line 7) * mpfr_get_ld: Conversion Functions. (line 9) * mpfr_get_ld_2exp: Conversion Functions. (line 36) * mpfr_get_patches: Miscellaneous Functions. (line 148) * mpfr_get_prec: Initialization Functions. (line 147) * mpfr_get_si: Conversion Functions. (line 20) * mpfr_get_sj: Conversion Functions. (line 22) * mpfr_get_str: Conversion Functions. (line 87) * mpfr_get_ui: Conversion Functions. (line 21) * mpfr_get_uj: Conversion Functions. (line 23) * mpfr_get_version: Miscellaneous Functions. (line 117) * mpfr_get_z: Conversion Functions. (line 68) * mpfr_get_z_2exp: Conversion Functions. (line 55) * mpfr_grandom: Miscellaneous Functions. (line 65) * mpfr_greater_p: Comparison Functions. (line 57) * mpfr_greaterequal_p: Comparison Functions. (line 58) * mpfr_hypot: Special Functions. (line 241) * mpfr_inexflag_p: Exception Related Functions. (line 137) * mpfr_inf_p: Comparison Functions. (line 42) * mpfr_init: Initialization Functions. (line 54) * mpfr_init2: Initialization Functions. (line 11) * mpfr_init_set: Combined Initialization and Assignment Functions. (line 7) * mpfr_init_set_d: Combined Initialization and Assignment Functions. (line 12) * mpfr_init_set_f: Combined Initialization and Assignment Functions. (line 17) * mpfr_init_set_ld: Combined Initialization and Assignment Functions. (line 14) * mpfr_init_set_q: Combined Initialization and Assignment Functions. (line 16) * mpfr_init_set_si: Combined Initialization and Assignment Functions. (line 11) * mpfr_init_set_str: Combined Initialization and Assignment Functions. (line 23) * mpfr_init_set_ui: Combined Initialization and Assignment Functions. (line 9) * mpfr_init_set_z: Combined Initialization and Assignment Functions. (line 15) * mpfr_inits: Initialization Functions. (line 63) * mpfr_inits2: Initialization Functions. (line 23) * mpfr_inp_str: Input and Output Functions. (line 33) * mpfr_integer_p: Integer Related Functions. (line 105) * mpfr_j0: Special Functions. (line 203) * mpfr_j1: Special Functions. (line 204) * mpfr_jn: Special Functions. (line 206) * mpfr_less_p: Comparison Functions. (line 59) * mpfr_lessequal_p: Comparison Functions. (line 60) * mpfr_lessgreater_p: Comparison Functions. (line 66) * mpfr_lgamma: Special Functions. (line 179) * mpfr_li2: Special Functions. (line 164) * mpfr_lngamma: Special Functions. (line 173) * mpfr_log: Special Functions. (line 17) * mpfr_log10: Special Functions. (line 19) * mpfr_log1p: Special Functions. (line 146) * mpfr_log2: Special Functions. (line 18) * mpfr_max: Miscellaneous Functions. (line 24) * mpfr_min: Miscellaneous Functions. (line 22) * mpfr_min_prec: Rounding Related Functions. (line 59) * mpfr_modf: Integer Related Functions. (line 69) * mpfr_mul: Basic Arithmetic Functions. (line 53) * mpfr_mul_2exp: Compatibility with MPF. (line 49) * mpfr_mul_2si: Basic Arithmetic Functions. (line 190) * mpfr_mul_2ui: Basic Arithmetic Functions. (line 188) * mpfr_mul_d: Basic Arithmetic Functions. (line 59) * mpfr_mul_q: Basic Arithmetic Functions. (line 63) * mpfr_mul_si: Basic Arithmetic Functions. (line 57) * mpfr_mul_ui: Basic Arithmetic Functions. (line 55) * mpfr_mul_z: Basic Arithmetic Functions. (line 61) * mpfr_nan_p: Comparison Functions. (line 41) * mpfr_nanflag_p: Exception Related Functions. (line 136) * mpfr_neg: Basic Arithmetic Functions. (line 174) * mpfr_nextabove: Miscellaneous Functions. (line 16) * mpfr_nextbelow: Miscellaneous Functions. (line 17) * mpfr_nexttoward: Miscellaneous Functions. (line 7) * mpfr_number_p: Comparison Functions. (line 43) * mpfr_out_str: Input and Output Functions. (line 17) * mpfr_overflow_p: Exception Related Functions. (line 134) * mpfr_pow: Basic Arithmetic Functions. (line 118) * mpfr_pow_si: Basic Arithmetic Functions. (line 122) * mpfr_pow_ui: Basic Arithmetic Functions. (line 120) * mpfr_pow_z: Basic Arithmetic Functions. (line 124) * mpfr_prec_round: Rounding Related Functions. (line 15) * mpfr_prec_t: Nomenclature and Types. (line 20) * mpfr_print_rnd_mode: Rounding Related Functions. (line 66) * mpfr_printf: Formatted Output Functions. (line 165) * mpfr_rec_sqrt: Basic Arithmetic Functions. (line 104) * mpfr_regular_p: Comparison Functions. (line 45) * mpfr_reldiff: Compatibility with MPF. (line 41) * mpfr_remainder: Integer Related Functions. (line 81) * mpfr_remquo: Integer Related Functions. (line 83) * mpfr_rint: Integer Related Functions. (line 7) * mpfr_rint_ceil: Integer Related Functions. (line 38) * mpfr_rint_floor: Integer Related Functions. (line 40) * mpfr_rint_round: Integer Related Functions. (line 42) * mpfr_rint_trunc: Integer Related Functions. (line 44) * mpfr_rnd_t: Nomenclature and Types. (line 34) * mpfr_root: Basic Arithmetic Functions. (line 111) * mpfr_round: Integer Related Functions. (line 10) * mpfr_sec: Special Functions. (line 46) * mpfr_sech: Special Functions. (line 130) * mpfr_set: Assignment Functions. (line 10) * mpfr_set_d: Assignment Functions. (line 17) * mpfr_set_decimal64: Assignment Functions. (line 21) * mpfr_set_default_prec: Initialization Functions. (line 101) * mpfr_set_default_rounding_mode: Rounding Related Functions. (line 7) * mpfr_set_divby0: Exception Related Functions. (line 122) * mpfr_set_emax: Exception Related Functions. (line 17) * mpfr_set_emin: Exception Related Functions. (line 16) * mpfr_set_erangeflag: Exception Related Functions. (line 125) * mpfr_set_exp: Miscellaneous Functions. (line 94) * mpfr_set_f: Assignment Functions. (line 24) * mpfr_set_flt: Assignment Functions. (line 16) * mpfr_set_inexflag: Exception Related Functions. (line 124) * mpfr_set_inf: Assignment Functions. (line 147) * mpfr_set_ld: Assignment Functions. (line 19) * mpfr_set_nan: Assignment Functions. (line 146) * mpfr_set_nanflag: Exception Related Functions. (line 123) * mpfr_set_overflow: Exception Related Functions. (line 121) * mpfr_set_prec: Initialization Functions. (line 137) * mpfr_set_prec_raw: Compatibility with MPF. (line 23) * mpfr_set_q: Assignment Functions. (line 23) * mpfr_set_si: Assignment Functions. (line 13) * mpfr_set_si_2exp: Assignment Functions. (line 53) * mpfr_set_sj: Assignment Functions. (line 15) * mpfr_set_sj_2exp: Assignment Functions. (line 57) * mpfr_set_str: Assignment Functions. (line 65) * mpfr_set_ui: Assignment Functions. (line 12) * mpfr_set_ui_2exp: Assignment Functions. (line 51) * mpfr_set_uj: Assignment Functions. (line 14) * mpfr_set_uj_2exp: Assignment Functions. (line 55) * mpfr_set_underflow: Exception Related Functions. (line 120) * mpfr_set_z: Assignment Functions. (line 22) * mpfr_set_z_2exp: Assignment Functions. (line 59) * mpfr_set_zero: Assignment Functions. (line 148) * mpfr_setsign: Miscellaneous Functions. (line 105) * mpfr_sgn: Comparison Functions. (line 51) * mpfr_si_div: Basic Arithmetic Functions. (line 80) * mpfr_si_sub: Basic Arithmetic Functions. (line 33) * mpfr_signbit: Miscellaneous Functions. (line 100) * mpfr_sin: Special Functions. (line 31) * mpfr_sin_cos: Special Functions. (line 37) * mpfr_sinh: Special Functions. (line 116) * mpfr_sinh_cosh: Special Functions. (line 122) * mpfr_snprintf: Formatted Output Functions. (line 182) * mpfr_sprintf: Formatted Output Functions. (line 171) * mpfr_sqr: Basic Arithmetic Functions. (line 70) * mpfr_sqrt: Basic Arithmetic Functions. (line 97) * mpfr_sqrt_ui: Basic Arithmetic Functions. (line 99) * mpfr_strtofr: Assignment Functions. (line 83) * mpfr_sub: Basic Arithmetic Functions. (line 27) * mpfr_sub_d: Basic Arithmetic Functions. (line 39) * mpfr_sub_q: Basic Arithmetic Functions. (line 45) * mpfr_sub_si: Basic Arithmetic Functions. (line 35) * mpfr_sub_ui: Basic Arithmetic Functions. (line 31) * mpfr_sub_z: Basic Arithmetic Functions. (line 43) * mpfr_subnormalize: Exception Related Functions. (line 61) * mpfr_sum: Special Functions. (line 275) * mpfr_swap: Assignment Functions. (line 154) * mpfr_t: Nomenclature and Types. (line 6) * mpfr_tan: Special Functions. (line 32) * mpfr_tanh: Special Functions. (line 117) * mpfr_trunc: Integer Related Functions. (line 11) * mpfr_ui_div: Basic Arithmetic Functions. (line 76) * mpfr_ui_pow: Basic Arithmetic Functions. (line 128) * mpfr_ui_pow_ui: Basic Arithmetic Functions. (line 126) * mpfr_ui_sub: Basic Arithmetic Functions. (line 29) * mpfr_underflow_p: Exception Related Functions. (line 133) * mpfr_unordered_p: Comparison Functions. (line 71) * mpfr_urandom: Miscellaneous Functions. (line 50) * mpfr_urandomb: Miscellaneous Functions. (line 30) * mpfr_vasprintf: Formatted Output Functions. (line 196) * MPFR_VERSION: Miscellaneous Functions. (line 120) * MPFR_VERSION_MAJOR: Miscellaneous Functions. (line 121) * MPFR_VERSION_MINOR: Miscellaneous Functions. (line 122) * MPFR_VERSION_NUM: Miscellaneous Functions. (line 140) * MPFR_VERSION_PATCHLEVEL: Miscellaneous Functions. (line 123) * MPFR_VERSION_STRING: Miscellaneous Functions. (line 124) * mpfr_vfprintf: Formatted Output Functions. (line 160) * mpfr_vprintf: Formatted Output Functions. (line 166) * mpfr_vsnprintf: Formatted Output Functions. (line 184) * mpfr_vsprintf: Formatted Output Functions. (line 173) * mpfr_y0: Special Functions. (line 214) * mpfr_y1: Special Functions. (line 215) * mpfr_yn: Special Functions. (line 217) * mpfr_z_sub: Basic Arithmetic Functions. (line 41) * mpfr_zero_p: Comparison Functions. (line 44) * mpfr_zeta: Special Functions. (line 192) * mpfr_zeta_ui: Special Functions. (line 194)  Tag Table: Node: Top892 Node: Copying2243 Node: Introduction to MPFR4003 Node: Installing MPFR6092 Node: Reporting Bugs10914 Node: MPFR Basics12843 Node: Headers and Libraries13159 Node: Nomenclature and Types16143 Node: MPFR Variable Conventions18147 Node: Rounding Modes19677 Ref: ternary value20774 Node: Floating-Point Values on Special Numbers22727 Node: Exceptions25703 Node: Memory Handling28855 Node: MPFR Interface29987 Node: Initialization Functions32083 Node: Assignment Functions38997 Node: Combined Initialization and Assignment Functions47651 Node: Conversion Functions48944 Node: Basic Arithmetic Functions57496 Node: Comparison Functions66504 Node: Special Functions69986 Node: Input and Output Functions83739 Node: Formatted Output Functions85662 Node: Integer Related Functions94781 Node: Rounding Related Functions100543 Node: Miscellaneous Functions104157 Node: Exception Related Functions112724 Node: Compatibility with MPF119478 Node: Custom Interface122166 Node: Internals126411 Node: API Compatibility127895 Node: Type and Macro Changes129825 Node: Added Functions132546 Node: Changed Functions135489 Node: Removed Functions139770 Node: Other Changes140182 Node: Contributors141711 Node: References144285 Node: GNU Free Documentation License146026 Node: Concept Index168469 Node: Function and Type Index174388  End Tag Table  Local Variables: coding: utf-8 End: @ 1.1.1.1.4.1 log @file mpfr.info was added on branch yamt-pagecache on 2014-05-22 14:09:15 +0000 @ text @d1 4266 @ 1.1.1.1.4.2 log @sync with head. for a reference, the tree before this commit was tagged as yamt-pagecache-tag8. this commit was splitted into small chunks to avoid a limitation of cvs. ("Protocol error: too many arguments") @ text @a0 4266 This is mpfr.info, produced by makeinfo version 4.13 from mpfr.texi. This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 3.1.2. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in *note GNU Free Documentation License::. INFO-DIR-SECTION Software libraries START-INFO-DIR-ENTRY * mpfr: (mpfr). Multiple Precision Floating-Point Reliable Library. END-INFO-DIR-ENTRY  File: mpfr.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir) GNU MPFR ******** This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 3.1.2. Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 2013 Free Software Foundation, Inc. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in *note GNU Free Documentation License::. * Menu: * Copying:: MPFR Copying Conditions (LGPL). * Introduction to MPFR:: Brief introduction to GNU MPFR. * Installing MPFR:: How to configure and compile the MPFR library. * Reporting Bugs:: How to usefully report bugs. * MPFR Basics:: What every MPFR user should now. * MPFR Interface:: MPFR functions and macros. * API Compatibility:: API compatibility with previous MPFR versions. * Contributors:: * References:: * GNU Free Documentation License:: * Concept Index:: * Function and Type Index::  File: mpfr.info, Node: Copying, Next: Introduction to MPFR, Prev: Top, Up: Top MPFR Copying Conditions *********************** The GNU MPFR library (or MPFR for short) is "free"; this means that everyone is free to use it and free to redistribute it on a free basis. The library is not in the public domain; it is copyrighted and there are restrictions on its distribution, but these restrictions are designed to permit everything that a good cooperating citizen would want to do. What is not allowed is to try to prevent others from further sharing any version of this library that they might get from you. Specifically, we want to make sure that you have the right to give away copies of the library, that you receive source code or else can get it if you want it, that you can change this library or use pieces of it in new free programs, and that you know you can do these things. To make sure that everyone has such rights, we have to forbid you to deprive anyone else of these rights. For example, if you distribute copies of the GNU MPFR library, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights. Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the GNU MPFR library. If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation. The precise conditions of the license for the GNU MPFR library are found in the Lesser General Public License that accompanies the source code. See the file COPYING.LESSER.  File: mpfr.info, Node: Introduction to MPFR, Next: Installing MPFR, Prev: Copying, Up: Top 1 Introduction to MPFR ********************** MPFR is a portable library written in C for arbitrary precision arithmetic on floating-point numbers. It is based on the GNU MP library. It aims to provide a class of floating-point numbers with precise semantics. The main characteristics of MPFR, which make it differ from most arbitrary precision floating-point software tools, are: * the MPFR code is portable, i.e., the result of any operation does not depend on the machine word size `mp_bits_per_limb' (64 on most current processors); * the precision in bits can be set _exactly_ to any valid value for each variable (including very small precision); * MPFR provides the four rounding modes from the IEEE 754-1985 standard, plus away-from-zero, as well as for basic operations as for other mathematical functions. In particular, with a precision of 53 bits, MPFR is able to exactly reproduce all computations with double-precision machine floating-point numbers (e.g., `double' type in C, with a C implementation that rigorously follows Annex F of the ISO C99 standard and `FP_CONTRACT' pragma set to `OFF') on the four arithmetic operations and the square root, except the default exponent range is much wider and subnormal numbers are not implemented (but can be emulated). This version of MPFR is released under the GNU Lesser General Public License, version 3 or any later version. It is permitted to link MPFR to most non-free programs, as long as when distributing them the MPFR source code and a means to re-link with a modified MPFR library is provided. 1.1 How to Use This Manual ========================== Everyone should read *note MPFR Basics::. If you need to install the library yourself, you need to read *note Installing MPFR::, too. To use the library you will need to refer to *note MPFR Interface::. The rest of the manual can be used for later reference, although it is probably a good idea to glance through it.  File: mpfr.info, Node: Installing MPFR, Next: Reporting Bugs, Prev: Introduction to MPFR, Up: Top 2 Installing MPFR ***************** The MPFR library is already installed on some GNU/Linux distributions, but the development files necessary to the compilation such as `mpfr.h' are not always present. To check that MPFR is fully installed on your computer, you can check the presence of the file `mpfr.h' in `/usr/include', or try to compile a small program having `#include ' (since `mpfr.h' may be installed somewhere else). For instance, you can try to compile: #include #include int main (void) { printf ("MPFR library: %-12s\nMPFR header: %s (based on %d.%d.%d)\n", mpfr_get_version (), MPFR_VERSION_STRING, MPFR_VERSION_MAJOR, MPFR_VERSION_MINOR, MPFR_VERSION_PATCHLEVEL); return 0; } with cc -o version version.c -lmpfr -lgmp and if you get errors whose first line looks like version.c:2:19: error: mpfr.h: No such file or directory then MPFR is probably not installed. Running this program will give you the MPFR version. If MPFR is not installed on your computer, or if you want to install a different version, please follow the steps below. 2.1 How to Install ================== Here are the steps needed to install the library on Unix systems (more details are provided in the `INSTALL' file): 1. To build MPFR, you first have to install GNU MP (version 4.1 or higher) on your computer. You need a C compiler, preferably GCC, but any reasonable compiler should work. And you need the standard Unix `make' command, plus some other standard Unix utility commands. Then, in the MPFR build directory, type the following commands. 2. `./configure' This will prepare the build and setup the options according to your system. You can give options to specify the install directories (instead of the default `/usr/local'), threading support, and so on. See the `INSTALL' file and/or the output of `./configure --help' for more information, in particular if you get error messages. 3. `make' This will compile MPFR, and create a library archive file `libmpfr.a'. On most platforms, a dynamic library will be produced too. 4. `make check' This will make sure MPFR was built correctly. If you get error messages, please report this to the MPFR mailing-list `mpfr@@inria.fr'. (*Note Reporting Bugs::, for information on what to include in useful bug reports.) 5. `make install' This will copy the files `mpfr.h' and `mpf2mpfr.h' to the directory `/usr/local/include', the library files (`libmpfr.a' and possibly others) to the directory `/usr/local/lib', the file `mpfr.info' to the directory `/usr/local/share/info', and some other documentation files to the directory `/usr/local/share/doc/mpfr' (or if you passed the `--prefix' option to `configure', using the prefix directory given as argument to `--prefix' instead of `/usr/local'). 2.2 Other `make' Targets ======================== There are some other useful make targets: * `mpfr.info' or `info' Create or update an info version of the manual, in `mpfr.info'. This file is already provided in the MPFR archives. * `mpfr.pdf' or `pdf' Create a PDF version of the manual, in `mpfr.pdf'. * `mpfr.dvi' or `dvi' Create a DVI version of the manual, in `mpfr.dvi'. * `mpfr.ps' or `ps' Create a Postscript version of the manual, in `mpfr.ps'. * `mpfr.html' or `html' Create a HTML version of the manual, in several pages in the directory `doc/mpfr.html'; if you want only one output HTML file, then type `makeinfo --html --no-split mpfr.texi' from the `doc' directory instead. * `clean' Delete all object files and archive files, but not the configuration files. * `distclean' Delete all generated files not included in the distribution. * `uninstall' Delete all files copied by `make install'. 2.3 Build Problems ================== In case of problem, please read the `INSTALL' file carefully before reporting a bug, in particular section "In case of problem". Some problems are due to bad configuration on the user side (not specific to MPFR). Problems are also mentioned in the FAQ `http://www.mpfr.org/faq.html'. Please report problems to the MPFR mailing-list `mpfr@@inria.fr'. *Note Reporting Bugs::. Some bug fixes are available on the MPFR 3.1.2 web page `http://www.mpfr.org/mpfr-3.1.2/'. 2.4 Getting the Latest Version of MPFR ====================================== The latest version of MPFR is available from `ftp://ftp.gnu.org/gnu/mpfr/' or `http://www.mpfr.org/'.  File: mpfr.info, Node: Reporting Bugs, Next: MPFR Basics, Prev: Installing MPFR, Up: Top 3 Reporting Bugs **************** If you think you have found a bug in the MPFR library, first have a look on the MPFR 3.1.2 web page `http://www.mpfr.org/mpfr-3.1.2/' and the FAQ `http://www.mpfr.org/faq.html': perhaps this bug is already known, in which case you may find there a workaround for it. You might also look in the archives of the MPFR mailing-list: `https://sympa.inria.fr/sympa/arc/mpfr'. Otherwise, please investigate and report it. We have made this library available to you, and it is not to ask too much from you, to ask you to report the bugs that you find. There are a few things you should think about when you put your bug report together. You have to send us a test case that makes it possible for us to reproduce the bug, i.e., a small self-content program, using no other library than MPFR. Include instructions on how to run the test case. You also have to explain what is wrong; if you get a crash, or if the results you get are incorrect and in that case, in what way. Please include compiler version information in your bug report. This can be extracted using `cc -V' on some machines, or, if you're using GCC, `gcc -v'. Also, include the output from `uname -a' and the MPFR version (the GMP version may be useful too). If you get a failure while running `make' or `make check', please include the `config.log' file in your bug report. If your bug report is good, we will do our best to help you to get a corrected version of the library; if the bug report is poor, we will not do anything about it (aside of chiding you to send better bug reports). Send your bug report to the MPFR mailing-list `mpfr@@inria.fr'. If you think something in this manual is unclear, or downright incorrect, or if the language needs to be improved, please send a note to the same address.  File: mpfr.info, Node: MPFR Basics, Next: MPFR Interface, Prev: Reporting Bugs, Up: Top 4 MPFR Basics ************* * Menu: * Headers and Libraries:: * Nomenclature and Types:: * MPFR Variable Conventions:: * Rounding Modes:: * Floating-Point Values on Special Numbers:: * Exceptions:: * Memory Handling::  File: mpfr.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: MPFR Basics, Up: MPFR Basics 4.1 Headers and Libraries ========================= All declarations needed to use MPFR are collected in the include file `mpfr.h'. It is designed to work with both C and C++ compilers. You should include that file in any program using the MPFR library: #include Note however that prototypes for MPFR functions with `FILE *' parameters are provided only if `' is included too (before `mpfr.h'): #include #include Likewise `' (or `') is required for prototypes with `va_list' parameters, such as `mpfr_vprintf'. And for any functions using `intmax_t', you must include `' or `' before `mpfr.h', to allow `mpfr.h' to define prototypes for these functions. Moreover, users of C++ compilers under some platforms may need to define `MPFR_USE_INTMAX_T' (and should do it for portability) before `mpfr.h' has been included; of course, it is possible to do that on the command line, e.g., with `-DMPFR_USE_INTMAX_T'. Note: If `mpfr.h' and/or `gmp.h' (used by `mpfr.h') are included several times (possibly from another header file), `' and/or `' (or `') should be included *before the first inclusion* of `mpfr.h' or `gmp.h'. Alternatively, you can define `MPFR_USE_FILE' (for MPFR I/O functions) and/or `MPFR_USE_VA_LIST' (for MPFR functions with `va_list' parameters) anywhere before the last inclusion of `mpfr.h'. As a consequence, if your file is a public header that includes `mpfr.h', you need to use the latter method. When calling a MPFR macro, it is not allowed to have previously defined a macro with the same name as some keywords (currently `do', `while' and `sizeof'). You can avoid the use of MPFR macros encapsulating functions by defining the `MPFR_USE_NO_MACRO' macro before `mpfr.h' is included. In general this should not be necessary, but this can be useful when debugging user code: with some macros, the compiler may emit spurious warnings with some warning options, and macros can prevent some prototype checking. All programs using MPFR must link against both `libmpfr' and `libgmp' libraries. On a typical Unix-like system this can be done with `-lmpfr -lgmp' (in that order), for example: gcc myprogram.c -lmpfr -lgmp MPFR is built using Libtool and an application can use that to link if desired, *note GNU Libtool: (libtool.info)Top. If MPFR has been installed to a non-standard location, then it may be necessary to set up environment variables such as `C_INCLUDE_PATH' and `LIBRARY_PATH', or use `-I' and `-L' compiler options, in order to point to the right directories. For a shared library, it may also be necessary to set up some sort of run-time library path (e.g., `LD_LIBRARY_PATH') on some systems. Please read the `INSTALL' file for additional information.  File: mpfr.info, Node: Nomenclature and Types, Next: MPFR Variable Conventions, Prev: Headers and Libraries, Up: MPFR Basics 4.2 Nomenclature and Types ========================== A "floating-point number", or "float" for short, is an arbitrary precision significand (also called mantissa) with a limited precision exponent. The C data type for such objects is `mpfr_t' (internally defined as a one-element array of a structure, and `mpfr_ptr' is the C data type representing a pointer to this structure). A floating-point number can have three special values: Not-a-Number (NaN) or plus or minus Infinity. NaN represents an uninitialized object, the result of an invalid operation (like 0 divided by 0), or a value that cannot be determined (like +Infinity minus +Infinity). Moreover, like in the IEEE 754 standard, zero is signed, i.e., there are both +0 and -0; the behavior is the same as in the IEEE 754 standard and it is generalized to the other functions supported by MPFR. Unless documented otherwise, the sign bit of a NaN is unspecified. The "precision" is the number of bits used to represent the significand of a floating-point number; the corresponding C data type is `mpfr_prec_t'. The precision can be any integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. In the current implementation, `MPFR_PREC_MIN' is equal to 2. Warning! MPFR needs to increase the precision internally, in order to provide accurate results (and in particular, correct rounding). Do not attempt to set the precision to any value near `MPFR_PREC_MAX', otherwise MPFR will abort due to an assertion failure. Moreover, you may reach some memory limit on your platform, in which case the program may abort, crash or have undefined behavior (depending on your C implementation). The "rounding mode" specifies the way to round the result of a floating-point operation, in case the exact result can not be represented exactly in the destination significand; the corresponding C data type is `mpfr_rnd_t'.  File: mpfr.info, Node: MPFR Variable Conventions, Next: Rounding Modes, Prev: Nomenclature and Types, Up: MPFR Basics 4.3 MPFR Variable Conventions ============================= Before you can assign to an MPFR variable, you need to initialize it by calling one of the special initialization functions. When you're done with a variable, you need to clear it out, using one of the functions for that purpose. A variable should only be initialized once, or at least cleared out between each initialization. After a variable has been initialized, it may be assigned to any number of times. For efficiency reasons, avoid to initialize and clear out a variable in loops. Instead, initialize it before entering the loop, and clear it out after the loop has exited. You do not need to be concerned about allocating additional space for MPFR variables, since any variable has a significand of fixed size. Hence unless you change its precision, or clear and reinitialize it, a floating-point variable will have the same allocated space during all its life. As a general rule, all MPFR functions expect output arguments before input arguments. This notation is based on an analogy with the assignment operator. MPFR allows you to use the same variable for both input and output in the same expression. For example, the main function for floating-point multiplication, `mpfr_mul', can be used like this: `mpfr_mul (x, x, x, rnd)'. This computes the square of X with rounding mode `rnd' and puts the result back in X.  File: mpfr.info, Node: Rounding Modes, Next: Floating-Point Values on Special Numbers, Prev: MPFR Variable Conventions, Up: MPFR Basics 4.4 Rounding Modes ================== The following five rounding modes are supported: * `MPFR_RNDN': round to nearest (roundTiesToEven in IEEE 754-2008), * `MPFR_RNDZ': round toward zero (roundTowardZero in IEEE 754-2008), * `MPFR_RNDU': round toward plus infinity (roundTowardPositive in IEEE 754-2008), * `MPFR_RNDD': round toward minus infinity (roundTowardNegative in IEEE 754-2008), * `MPFR_RNDA': round away from zero. The `round to nearest' mode works as in the IEEE 754 standard: in case the number to be rounded lies exactly in the middle of two representable numbers, it is rounded to the one with the least significant bit set to zero. For example, the number 2.5, which is represented by (10.1) in binary, is rounded to (10.0)=2 with a precision of two bits, and not to (11.0)=3. This rule avoids the "drift" phenomenon mentioned by Knuth in volume 2 of The Art of Computer Programming (Section 4.2.2). Most MPFR functions take as first argument the destination variable, as second and following arguments the input variables, as last argument a rounding mode, and have a return value of type `int', called the "ternary value". The value stored in the destination variable is correctly rounded, i.e., MPFR behaves as if it computed the result with an infinite precision, then rounded it to the precision of this variable. The input variables are regarded as exact (in particular, their precision does not affect the result). As a consequence, in case of a non-zero real rounded result, the error on the result is less or equal to 1/2 ulp (unit in the last place) of that result in the rounding to nearest mode, and less than 1 ulp of that result in the directed rounding modes (a ulp is the weight of the least significant represented bit of the result after rounding). Unless documented otherwise, functions returning an `int' return a ternary value. If the ternary value is zero, it means that the value stored in the destination variable is the exact result of the corresponding mathematical function. If the ternary value is positive (resp. negative), it means the value stored in the destination variable is greater (resp. lower) than the exact result. For example with the `MPFR_RNDU' rounding mode, the ternary value is usually positive, except when the result is exact, in which case it is zero. In the case of an infinite result, it is considered as inexact when it was obtained by overflow, and exact otherwise. A NaN result (Not-a-Number) always corresponds to an exact return value. The opposite of a returned ternary value is guaranteed to be representable in an `int'. Unless documented otherwise, functions returning as result the value `1' (or any other value specified in this manual) for special cases (like `acos(0)') yield an overflow or an underflow if that value is not representable in the current exponent range.  File: mpfr.info, Node: Floating-Point Values on Special Numbers, Next: Exceptions, Prev: Rounding Modes, Up: MPFR Basics 4.5 Floating-Point Values on Special Numbers ============================================ This section specifies the floating-point values (of type `mpfr_t') returned by MPFR functions (where by "returned" we mean here the modified value of the destination object, which should not be mixed with the ternary return value of type `int' of those functions). For functions returning several values (like `mpfr_sin_cos'), the rules apply to each result separately. Functions can have one or several input arguments. An input point is a mapping from these input arguments to the set of the MPFR numbers. When none of its components are NaN, an input point can also be seen as a tuple in the extended real numbers (the set of the real numbers with both infinities). When the input point is in the domain of the mathematical function, the result is rounded as described in Section "Rounding Modes" (but see below for the specification of the sign of an exact zero). Otherwise the general rules from this section apply unless stated otherwise in the description of the MPFR function (*note MPFR Interface::). When the input point is not in the domain of the mathematical function but is in its closure in the extended real numbers and the function can be extended by continuity, the result is the obtained limit. Examples: `mpfr_hypot' on (+Inf,0) gives +Inf. But `mpfr_pow' cannot be defined on (1,+Inf) using this rule, as one can find sequences (X_N,Y_N) such that X_N goes to 1, Y_N goes to +Inf and X_N to the Y_N goes to any positive value when N goes to the infinity. When the input point is in the closure of the domain of the mathematical function and an input argument is +0 (resp. -0), one considers the limit when the corresponding argument approaches 0 from above (resp. below). If the limit is not defined (e.g., `mpfr_log' on -0), the behavior is specified in the description of the MPFR function. When the result is equal to 0, its sign is determined by considering the limit as if the input point were not in the domain: If one approaches 0 from above (resp. below), the result is +0 (resp. -0); for example, `mpfr_sin' on +0 gives +0. In the other cases, the sign is specified in the description of the MPFR function; for example `mpfr_max' on -0 and +0 gives +0. When the input point is not in the closure of the domain of the function, the result is NaN. Example: `mpfr_sqrt' on -17 gives NaN. When an input argument is NaN, the result is NaN, possibly except when a partial function is constant on the finite floating-point numbers; such a case is always explicitly specified in *note MPFR Interface::. Example: `mpfr_hypot' on (NaN,0) gives NaN, but `mpfr_hypot' on (NaN,+Inf) gives +Inf (as specified in *note Special Functions::), since for any finite input X, `mpfr_hypot' on (X,+Inf) gives +Inf.  File: mpfr.info, Node: Exceptions, Next: Memory Handling, Prev: Floating-Point Values on Special Numbers, Up: MPFR Basics 4.6 Exceptions ============== MPFR supports 6 exception types: * Underflow: An underflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent smaller than the minimum value of the current exponent range. (In the round-to-nearest mode, the halfway case is rounded toward zero.) Note: This is not the single possible definition of the underflow. MPFR chooses to consider the underflow _after_ rounding. The underflow before rounding can also be defined. For instance, consider a function that has the exact result 7 multiplied by two to the power E-4, where E is the smallest exponent (for a significand between 1/2 and 1), with a 2-bit target precision and rounding toward plus infinity. The exact result has the exponent E-1. With the underflow before rounding, such a function call would yield an underflow, as E-1 is outside the current exponent range. However, MPFR first considers the rounded result assuming an unbounded exponent range. The exact result cannot be represented exactly in precision 2, and here, it is rounded to 0.5 times 2 to E, which is representable in the current exponent range. As a consequence, this will not yield an underflow in MPFR. * Overflow: An overflow occurs when the exact result of a function is a non-zero real number and the result obtained after the rounding, assuming an unbounded exponent range (for the rounding), has an exponent larger than the maximum value of the current exponent range. In the round-to-nearest mode, the result is infinite. Note: unlike the underflow case, there is only one possible definition of overflow here. * Divide-by-zero: An exact infinite result is obtained from finite inputs. * NaN: A NaN exception occurs when the result of a function is NaN. * Inexact: An inexact exception occurs when the result of a function cannot be represented exactly and must be rounded. * Range error: A range exception occurs when a function that does not return a MPFR number (such as comparisons and conversions to an integer) has an invalid result (e.g., an argument is NaN in `mpfr_cmp', or a conversion to an integer cannot be represented in the target type). MPFR has a global flag for each exception, which can be cleared, set or tested by functions described in *note Exception Related Functions::. Differences with the ISO C99 standard: * In C, only quiet NaNs are specified, and a NaN propagation does not raise an invalid exception. Unless explicitly stated otherwise, MPFR sets the NaN flag whenever a NaN is generated, even when a NaN is propagated (e.g., in NaN + NaN), as if all NaNs were signaling. * An invalid exception in C corresponds to either a NaN exception or a range error in MPFR.  File: mpfr.info, Node: Memory Handling, Prev: Exceptions, Up: MPFR Basics 4.7 Memory Handling =================== MPFR functions may create caches, e.g., when computing constants such as Pi, either because the user has called a function like `mpfr_const_pi' directly or because such a function was called internally by the MPFR library itself to compute some other function. At any time, the user can free the various caches with `mpfr_free_cache'. It is strongly advised to do that before terminating a thread, or before exiting when using tools like `valgrind' (to avoid memory leaks being reported). MPFR internal data such as flags, the exponent range, the default precision and rounding mode, and caches (i.e., data that are not accessed via parameters) are either global (if MPFR has not been compiled as thread safe) or per-thread (thread local storage, TLS). The initial values of TLS data after a thread is created entirely depend on the compiler and thread implementation (MPFR simply does a conventional variable initialization, the variables being declared with an implementation-defined TLS specifier).  File: mpfr.info, Node: MPFR Interface, Next: API Compatibility, Prev: MPFR Basics, Up: Top 5 MPFR Interface **************** The floating-point functions expect arguments of type `mpfr_t'. The MPFR floating-point functions have an interface that is similar to the GNU MP functions. The function prefix for floating-point operations is `mpfr_'. The user has to specify the precision of each variable. A computation that assigns a variable will take place with the precision of the assigned variable; the cost of that computation should not depend on the precision of variables used as input (on average). The semantics of a calculation in MPFR is specified as follows: Compute the requested operation exactly (with "infinite accuracy"), and round the result to the precision of the destination variable, with the given rounding mode. The MPFR floating-point functions are intended to be a smooth extension of the IEEE 754 arithmetic. The results obtained on a given computer are identical to those obtained on a computer with a different word size, or with a different compiler or operating system. MPFR _does not keep track_ of the accuracy of a computation. This is left to the user or to a higher layer (for example the MPFI library for interval arithmetic). As a consequence, if two variables are used to store only a few significant bits, and their product is stored in a variable with large precision, then MPFR will still compute the result with full precision. The value of the standard C macro `errno' may be set to non-zero by any MPFR function or macro, whether or not there is an error. * Menu: * Initialization Functions:: * Assignment Functions:: * Combined Initialization and Assignment Functions:: * Conversion Functions:: * Basic Arithmetic Functions:: * Comparison Functions:: * Special Functions:: * Input and Output Functions:: * Formatted Output Functions:: * Integer Related Functions:: * Rounding Related Functions:: * Miscellaneous Functions:: * Exception Related Functions:: * Compatibility with MPF:: * Custom Interface:: * Internals::  File: mpfr.info, Node: Initialization Functions, Next: Assignment Functions, Prev: MPFR Interface, Up: MPFR Interface 5.1 Initialization Functions ============================ An `mpfr_t' object must be initialized before storing the first value in it. The functions `mpfr_init' and `mpfr_init2' are used for that purpose. -- Function: void mpfr_init2 (mpfr_t X, mpfr_prec_t PREC) Initialize X, set its precision to be *exactly* PREC bits and its value to NaN. (Warning: the corresponding MPF function initializes to zero instead.) Normally, a variable should be initialized once only or at least be cleared, using `mpfr_clear', between initializations. To change the precision of a variable which has already been initialized, use `mpfr_set_prec'. The precision PREC must be an integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX' (otherwise the behavior is undefined). -- Function: void mpfr_inits2 (mpfr_prec_t PREC, mpfr_t X, ...) Initialize all the `mpfr_t' variables of the given variable argument `va_list', set their precision to be *exactly* PREC bits and their value to NaN. See `mpfr_init2' for more details. The `va_list' is assumed to be composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It begins from X, and ends when it encounters a null pointer (whose type must also be `mpfr_ptr'). -- Function: void mpfr_clear (mpfr_t X) Free the space occupied by the significand of X. Make sure to call this function for all `mpfr_t' variables when you are done with them. -- Function: void mpfr_clears (mpfr_t X, ...) Free the space occupied by all the `mpfr_t' variables of the given `va_list'. See `mpfr_clear' for more details. The `va_list' is assumed to be composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It begins from X, and ends when it encounters a null pointer (whose type must also be `mpfr_ptr'). Here is an example of how to use multiple initialization functions (since `NULL' is not necessarily defined in this context, we use `(mpfr_ptr) 0' instead, but `(mpfr_ptr) NULL' is also correct). { mpfr_t x, y, z, t; mpfr_inits2 (256, x, y, z, t, (mpfr_ptr) 0); ... mpfr_clears (x, y, z, t, (mpfr_ptr) 0); } -- Function: void mpfr_init (mpfr_t X) Initialize X, set its precision to the default precision, and set its value to NaN. The default precision can be changed by a call to `mpfr_set_default_prec'. Warning! In a given program, some other libraries might change the default precision and not restore it. Thus it is safer to use `mpfr_init2'. -- Function: void mpfr_inits (mpfr_t X, ...) Initialize all the `mpfr_t' variables of the given `va_list', set their precision to the default precision and their value to NaN. See `mpfr_init' for more details. The `va_list' is assumed to be composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It begins from X, and ends when it encounters a null pointer (whose type must also be `mpfr_ptr'). Warning! In a given program, some other libraries might change the default precision and not restore it. Thus it is safer to use `mpfr_inits2'. -- Macro: MPFR_DECL_INIT (NAME, PREC) This macro declares NAME as an automatic variable of type `mpfr_t', initializes it and sets its precision to be *exactly* PREC bits and its value to NaN. NAME must be a valid identifier. You must use this macro in the declaration section. This macro is much faster than using `mpfr_init2' but has some drawbacks: * You *must not* call `mpfr_clear' with variables created with this macro (the storage is allocated at the point of declaration and deallocated when the brace-level is exited). * You *cannot* change their precision. * You *should not* create variables with huge precision with this macro. * Your compiler must support `Non-Constant Initializers' (standard in C++ and ISO C99) and `Token Pasting' (standard in ISO C89). If PREC is not a constant expression, your compiler must support `variable-length automatic arrays' (standard in ISO C99). GCC 2.95.3 and above supports all these features. If you compile your program with GCC in C89 mode and with `-pedantic', you may want to define the `MPFR_USE_EXTENSION' macro to avoid warnings due to the `MPFR_DECL_INIT' implementation. -- Function: void mpfr_set_default_prec (mpfr_prec_t PREC) Set the default precision to be *exactly* PREC bits, where PREC can be any integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. The precision of a variable means the number of bits used to store its significand. All subsequent calls to `mpfr_init' or `mpfr_inits' will use this precision, but previously initialized variables are unaffected. The default precision is set to 53 bits initially. Note: when MPFR is built with the `--enable-thread-safe' configure option, the default precision is local to each thread. *Note Memory Handling::, for more information. -- Function: mpfr_prec_t mpfr_get_default_prec (void) Return the current default MPFR precision in bits. See the documentation of `mpfr_set_default_prec'. Here is an example on how to initialize floating-point variables: { mpfr_t x, y; mpfr_init (x); /* use default precision */ mpfr_init2 (y, 256); /* precision _exactly_ 256 bits */ ... /* When the program is about to exit, do ... */ mpfr_clear (x); mpfr_clear (y); mpfr_free_cache (); /* free the cache for constants like pi */ } The following functions are useful for changing the precision during a calculation. A typical use would be for adjusting the precision gradually in iterative algorithms like Newton-Raphson, making the computation precision closely match the actual accurate part of the numbers. -- Function: void mpfr_set_prec (mpfr_t X, mpfr_prec_t PREC) Reset the precision of X to be *exactly* PREC bits, and set its value to NaN. The previous value stored in X is lost. It is equivalent to a call to `mpfr_clear(x)' followed by a call to `mpfr_init2(x, prec)', but more efficient as no allocation is done in case the current allocated space for the significand of X is enough. The precision PREC can be any integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. In case you want to keep the previous value stored in X, use `mpfr_prec_round' instead. -- Function: mpfr_prec_t mpfr_get_prec (mpfr_t X) Return the precision of X, i.e., the number of bits used to store its significand.  File: mpfr.info, Node: Assignment Functions, Next: Combined Initialization and Assignment Functions, Prev: Initialization Functions, Up: MPFR Interface 5.2 Assignment Functions ======================== These functions assign new values to already initialized floats (*note Initialization Functions::). -- Function: int mpfr_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) -- Function: int mpfr_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t RND) -- Function: int mpfr_set_uj (mpfr_t ROP, uintmax_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_sj (mpfr_t ROP, intmax_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_flt (mpfr_t ROP, float OP, mpfr_rnd_t RND) -- Function: int mpfr_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND) -- Function: int mpfr_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t RND) -- Function: int mpfr_set_decimal64 (mpfr_t ROP, _Decimal64 OP, mpfr_rnd_t RND) -- Function: int mpfr_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND) -- Function: int mpfr_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND) Set the value of ROP from OP, rounded toward the given direction RND. Note that the input 0 is converted to +0 by `mpfr_set_ui', `mpfr_set_si', `mpfr_set_uj', `mpfr_set_sj', `mpfr_set_z', `mpfr_set_q' and `mpfr_set_f', regardless of the rounding mode. If the system does not support the IEEE 754 standard, `mpfr_set_flt', `mpfr_set_d', `mpfr_set_ld' and `mpfr_set_decimal64' might not preserve the signed zeros. The `mpfr_set_decimal64' function is built only with the configure option `--enable-decimal-float', which also requires `--with-gmp-build', and when the compiler or system provides the `_Decimal64' data type (recent versions of GCC support this data type); to use `mpfr_set_decimal64', one should define the macro `MPFR_WANT_DECIMAL_FLOATS' before including `mpfr.h'. `mpfr_set_q' might fail if the numerator (or the denominator) can not be represented as a `mpfr_t'. Note: If you want to store a floating-point constant to a `mpfr_t', you should use `mpfr_set_str' (or one of the MPFR constant functions, such as `mpfr_const_pi' for Pi) instead of `mpfr_set_flt', `mpfr_set_d', `mpfr_set_ld' or `mpfr_set_decimal64'. Otherwise the floating-point constant will be first converted into a reduced-precision (e.g., 53-bit) binary (or decimal, for `mpfr_set_decimal64') number before MPFR can work with it. -- Function: int mpfr_set_ui_2exp (mpfr_t ROP, unsigned long int OP, mpfr_exp_t E, mpfr_rnd_t RND) -- Function: int mpfr_set_si_2exp (mpfr_t ROP, long int OP, mpfr_exp_t E, mpfr_rnd_t RND) -- Function: int mpfr_set_uj_2exp (mpfr_t ROP, uintmax_t OP, intmax_t E, mpfr_rnd_t RND) -- Function: int mpfr_set_sj_2exp (mpfr_t ROP, intmax_t OP, intmax_t E, mpfr_rnd_t RND) -- Function: int mpfr_set_z_2exp (mpfr_t ROP, mpz_t OP, mpfr_exp_t E, mpfr_rnd_t RND) Set the value of ROP from OP multiplied by two to the power E, rounded toward the given direction RND. Note that the input 0 is converted to +0. -- Function: int mpfr_set_str (mpfr_t ROP, const char *S, int BASE, mpfr_rnd_t RND) Set ROP to the value of the string S in base BASE, rounded in the direction RND. See the documentation of `mpfr_strtofr' for a detailed description of the valid string formats. Contrary to `mpfr_strtofr', `mpfr_set_str' requires the _whole_ string to represent a valid floating-point number. The meaning of the return value differs from other MPFR functions: it is 0 if the entire string up to the final null character is a valid number in base BASE; otherwise it is -1, and ROP may have changed (users interested in the *note ternary value:: should use `mpfr_strtofr' instead). Note: it is preferable to use `mpfr_set_str' if one wants to distinguish between an infinite ROP value coming from an infinite S or from an overflow. -- Function: int mpfr_strtofr (mpfr_t ROP, const char *NPTR, char **ENDPTR, int BASE, mpfr_rnd_t RND) Read a floating-point number from a string NPTR in base BASE, rounded in the direction RND; BASE must be either 0 (to detect the base, as described below) or a number from 2 to 62 (otherwise the behavior is undefined). If NPTR starts with valid data, the result is stored in ROP and `*ENDPTR' points to the character just after the valid data (if ENDPTR is not a null pointer); otherwise ROP is set to zero (for consistency with `strtod') and the value of NPTR is stored in the location referenced by ENDPTR (if ENDPTR is not a null pointer). The usual ternary value is returned. Parsing follows the standard C `strtod' function with some extensions. After optional leading whitespace, one has a subject sequence consisting of an optional sign (`+' or `-'), and either numeric data or special data. The subject sequence is defined as the longest initial subsequence of the input string, starting with the first non-whitespace character, that is of the expected form. The form of numeric data is a non-empty sequence of significand digits with an optional decimal point, and an optional exponent consisting of an exponent prefix followed by an optional sign and a non-empty sequence of decimal digits. A significand digit is either a decimal digit or a Latin letter (62 possible characters), with `A' = 10, `B' = 11, ..., `Z' = 35; case is ignored in bases less or equal to 36, in bases larger than 36, `a' = 36, `b' = 37, ..., `z' = 61. The value of a significand digit must be strictly less than the base. The decimal point can be either the one defined by the current locale or the period (the first one is accepted for consistency with the C standard and the practice, the second one is accepted to allow the programmer to provide MPFR numbers from strings in a way that does not depend on the current locale). The exponent prefix can be `e' or `E' for bases up to 10, or `@@' in any base; it indicates a multiplication by a power of the base. In bases 2 and 16, the exponent prefix can also be `p' or `P', in which case the exponent, called _binary exponent_, indicates a multiplication by a power of 2 instead of the base (there is a difference only for base 16); in base 16 for example `1p2' represents 4 whereas `1@@2' represents 256. The value of an exponent is always written in base 10. If the argument BASE is 0, then the base is automatically detected as follows. If the significand starts with `0b' or `0B', base 2 is assumed. If the significand starts with `0x' or `0X', base 16 is assumed. Otherwise base 10 is assumed. Note: The exponent (if present) must contain at least a digit. Otherwise the possible exponent prefix and sign are not part of the number (which ends with the significand). Similarly, if `0b', `0B', `0x' or `0X' is not followed by a binary/hexadecimal digit, then the subject sequence stops at the character `0', thus 0 is read. Special data (for infinities and NaN) can be `@@inf@@' or `@@nan@@(n-char-sequence-opt)', and if BASE <= 16, it can also be `infinity', `inf', `nan' or `nan(n-char-sequence-opt)', all case insensitive. A `n-char-sequence-opt' is a possibly empty string containing only digits, Latin letters and the underscore (0, 1, 2, ..., 9, a, b, ..., z, A, B, ..., Z, _). Note: one has an optional sign for all data, even NaN. For example, `-@@nAn@@(This_Is_Not_17)' is a valid representation for NaN in base 17. -- Function: void mpfr_set_nan (mpfr_t X) -- Function: void mpfr_set_inf (mpfr_t X, int SIGN) -- Function: void mpfr_set_zero (mpfr_t X, int SIGN) Set the variable X to NaN (Not-a-Number), infinity or zero respectively. In `mpfr_set_inf' or `mpfr_set_zero', X is set to plus infinity or plus zero iff SIGN is nonnegative; in `mpfr_set_nan', the sign bit of the result is unspecified. -- Function: void mpfr_swap (mpfr_t X, mpfr_t Y) Swap the values X and Y efficiently. Warning: the precisions are exchanged too; in case the precisions are different, `mpfr_swap' is thus not equivalent to three `mpfr_set' calls using a third auxiliary variable.  File: mpfr.info, Node: Combined Initialization and Assignment Functions, Next: Conversion Functions, Prev: Assignment Functions, Up: MPFR Interface 5.3 Combined Initialization and Assignment Functions ==================================================== -- Macro: int mpfr_init_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND) -- Macro: int mpfr_init_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND) Initialize ROP and set its value from OP, rounded in the direction RND. The precision of ROP will be taken from the active default precision, as set by `mpfr_set_default_prec'. -- Function: int mpfr_init_set_str (mpfr_t X, const char *S, int BASE, mpfr_rnd_t RND) Initialize X and set its value from the string S in base BASE, rounded in the direction RND. See `mpfr_set_str'.  File: mpfr.info, Node: Conversion Functions, Next: Basic Arithmetic Functions, Prev: Combined Initialization and Assignment Functions, Up: MPFR Interface 5.4 Conversion Functions ======================== -- Function: float mpfr_get_flt (mpfr_t OP, mpfr_rnd_t RND) -- Function: double mpfr_get_d (mpfr_t OP, mpfr_rnd_t RND) -- Function: long double mpfr_get_ld (mpfr_t OP, mpfr_rnd_t RND) -- Function: _Decimal64 mpfr_get_decimal64 (mpfr_t OP, mpfr_rnd_t RND) Convert OP to a `float' (respectively `double', `long double' or `_Decimal64'), using the rounding mode RND. If OP is NaN, some fixed NaN (either quiet or signaling) or the result of 0.0/0.0 is returned. If OP is ±Inf, an infinity of the same sign or the result of ±1.0/0.0 is returned. If OP is zero, these functions return a zero, trying to preserve its sign, if possible. The `mpfr_get_decimal64' function is built only under some conditions: see the documentation of `mpfr_set_decimal64'. -- Function: long mpfr_get_si (mpfr_t OP, mpfr_rnd_t RND) -- Function: unsigned long mpfr_get_ui (mpfr_t OP, mpfr_rnd_t RND) -- Function: intmax_t mpfr_get_sj (mpfr_t OP, mpfr_rnd_t RND) -- Function: uintmax_t mpfr_get_uj (mpfr_t OP, mpfr_rnd_t RND) Convert OP to a `long', an `unsigned long', an `intmax_t' or an `uintmax_t' (respectively) after rounding it with respect to RND. If OP is NaN, 0 is returned and the _erange_ flag is set. If OP is too big for the return type, the function returns the maximum or the minimum of the corresponding C type, depending on the direction of the overflow; the _erange_ flag is set too. See also `mpfr_fits_slong_p', `mpfr_fits_ulong_p', `mpfr_fits_intmax_p' and `mpfr_fits_uintmax_p'. -- Function: double mpfr_get_d_2exp (long *EXP, mpfr_t OP, mpfr_rnd_t RND) -- Function: long double mpfr_get_ld_2exp (long *EXP, mpfr_t OP, mpfr_rnd_t RND) Return D and set EXP (formally, the value pointed to by EXP) such that 0.5<=abs(D)<1 and D times 2 raised to EXP equals OP rounded to double (resp. long double) precision, using the given rounding mode. If OP is zero, then a zero of the same sign (or an unsigned zero, if the implementation does not have signed zeros) is returned, and EXP is set to 0. If OP is NaN or an infinity, then the corresponding double precision (resp. long-double precision) value is returned, and EXP is undefined. -- Function: int mpfr_frexp (mpfr_exp_t *EXP, mpfr_t Y, mpfr_t X, mpfr_rnd_t RND) Set EXP (formally, the value pointed to by EXP) and Y such that 0.5<=abs(Y)<1 and Y times 2 raised to EXP equals X rounded to the precision of Y, using the given rounding mode. If X is zero, then Y is set to a zero of the same sign and EXP is set to 0. If X is NaN or an infinity, then Y is set to the same value and EXP is undefined. -- Function: mpfr_exp_t mpfr_get_z_2exp (mpz_t ROP, mpfr_t OP) Put the scaled significand of OP (regarded as an integer, with the precision of OP) into ROP, and return the exponent EXP (which may be outside the current exponent range) such that OP exactly equals ROP times 2 raised to the power EXP. If OP is zero, the minimal exponent `emin' is returned. If OP is NaN or an infinity, the _erange_ flag is set, ROP is set to 0, and the the minimal exponent `emin' is returned. The returned exponent may be less than the minimal exponent `emin' of MPFR numbers in the current exponent range; in case the exponent is not representable in the `mpfr_exp_t' type, the _erange_ flag is set and the minimal value of the `mpfr_exp_t' type is returned. -- Function: int mpfr_get_z (mpz_t ROP, mpfr_t OP, mpfr_rnd_t RND) Convert OP to a `mpz_t', after rounding it with respect to RND. If OP is NaN or an infinity, the _erange_ flag is set, ROP is set to 0, and 0 is returned. -- Function: int mpfr_get_f (mpf_t ROP, mpfr_t OP, mpfr_rnd_t RND) Convert OP to a `mpf_t', after rounding it with respect to RND. The _erange_ flag is set if OP is NaN or an infinity, which do not exist in MPF. If OP is NaN, then ROP is undefined. If OP is an +Inf (resp. -Inf), then ROP is set to the maximum (resp. minimum) value in the precision of the MPF number; if a future MPF version supports infinities, this behavior will be considered incorrect and will change (portable programs should assume that ROP is set either to this finite number or to an infinite number). Note that since MPFR currently has the same exponent type as MPF (but not with the same radix), the range of values is much larger in MPF than in MPFR, so that an overflow or underflow is not possible. -- Function: char * mpfr_get_str (char *STR, mpfr_exp_t *EXPPTR, int B, size_t N, mpfr_t OP, mpfr_rnd_t RND) Convert OP to a string of digits in base B, with rounding in the direction RND, where N is either zero (see below) or the number of significant digits output in the string; in the latter case, N must be greater or equal to 2. The base may vary from 2 to 62. If the input number is an ordinary number, the exponent is written through the pointer EXPPTR (for input 0, the current minimal exponent is written). The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. For example, the number -3.1416 would be returned as "-31416" in the string and 1 written at EXPPTR. If RND is to nearest, and OP is exactly in the middle of two consecutive possible outputs, the one with an even significand is chosen, where both significands are considered with the exponent of OP. Note that for an odd base, this may not correspond to an even last digit: for example with 2 digits in base 7, (14) and a half is rounded to (15) which is 12 in decimal, (16) and a half is rounded to (20) which is 14 in decimal, and (26) and a half is rounded to (26) which is 20 in decimal. If N is zero, the number of digits of the significand is chosen large enough so that re-reading the printed value with the same precision, assuming both output and input use rounding to nearest, will recover the original value of OP. More precisely, in most cases, the chosen precision of STR is the minimal precision m depending only on P = PREC(OP) and B that satisfies the above property, i.e., m = 1 + ceil(P*log(2)/log(B)), with P replaced by P-1 if B is a power of 2, but in some very rare cases, it might be m+1 (the smallest case for bases up to 62 is when P equals 186564318007 for bases 7 and 49). If STR is a null pointer, space for the significand is allocated using the current allocation function, and a pointer to the string is returned. To free the returned string, you must use `mpfr_free_str'. If STR is not a null pointer, it should point to a block of storage large enough for the significand, i.e., at least `max(N + 2, 7)'. The extra two bytes are for a possible minus sign, and for the terminating null character, and the value 7 accounts for `-@@Inf@@' plus the terminating null character. A pointer to the string is returned, unless there is an error, in which case a null pointer is returned. -- Function: void mpfr_free_str (char *STR) Free a string allocated by `mpfr_get_str' using the current unallocation function. The block is assumed to be `strlen(STR)+1' bytes. For more information about how it is done: *note Custom Allocation: (gmp.info)Custom Allocation. -- Function: int mpfr_fits_ulong_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_slong_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_uint_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_sint_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_ushort_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_sshort_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_uintmax_p (mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_fits_intmax_p (mpfr_t OP, mpfr_rnd_t RND) Return non-zero if OP would fit in the respective C data type, respectively `unsigned long', `long', `unsigned int', `int', `unsigned short', `short', `uintmax_t', `intmax_t', when rounded to an integer in the direction RND.  File: mpfr.info, Node: Basic Arithmetic Functions, Next: Comparison Functions, Prev: Conversion Functions, Up: MPFR Interface 5.5 Basic Arithmetic Functions ============================== -- Function: int mpfr_add (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_d (mpfr_t ROP, mpfr_t OP1, double OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_add_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, mpfr_rnd_t RND) Set ROP to OP1 + OP2 rounded in the direction RND. For types having no signed zero, it is considered unsigned (i.e., (+0) + 0 = (+0) and (-0) + 0 = (-0)). The `mpfr_add_d' function assumes that the radix of the `double' type is a power of 2, with a precision at most that declared by the C implementation (macro `IEEE_DBL_MANT_DIG', and if not defined 53 bits). -- Function: int mpfr_sub (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_sub (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_si_sub (mpfr_t ROP, long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_d_sub (mpfr_t ROP, double OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_d (mpfr_t ROP, mpfr_t OP1, double OP2, mpfr_rnd_t RND) -- Function: int mpfr_z_sub (mpfr_t ROP, mpz_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_sub_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, mpfr_rnd_t RND) Set ROP to OP1 - OP2 rounded in the direction RND. For types having no signed zero, it is considered unsigned (i.e., (+0) - 0 = (+0), (-0) - 0 = (-0), 0 - (+0) = (-0) and 0 - (-0) = (+0)). The same restrictions than for `mpfr_add_d' apply to `mpfr_d_sub' and `mpfr_sub_d'. -- Function: int mpfr_mul (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_d (mpfr_t ROP, mpfr_t OP1, double OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, mpfr_rnd_t RND) Set ROP to OP1 times OP2 rounded in the direction RND. When a result is zero, its sign is the product of the signs of the operands (for types having no signed zero, it is considered positive). The same restrictions than for `mpfr_add_d' apply to `mpfr_mul_d'. -- Function: int mpfr_sqr (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the square of OP rounded in the direction RND. -- Function: int mpfr_div (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_div (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_si_div (mpfr_t ROP, long int OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_d_div (mpfr_t ROP, double OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_d (mpfr_t ROP, mpfr_t OP1, double OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, mpfr_rnd_t RND) Set ROP to OP1/OP2 rounded in the direction RND. When a result is zero, its sign is the product of the signs of the operands (for types having no signed zero, it is considered positive). The same restrictions than for `mpfr_add_d' apply to `mpfr_d_div' and `mpfr_div_d'. -- Function: int mpfr_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_sqrt_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) Set ROP to the square root of OP rounded in the direction RND (set ROP to -0 if OP is -0, to be consistent with the IEEE 754 standard). Set ROP to NaN if OP is negative. -- Function: int mpfr_rec_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the reciprocal square root of OP rounded in the direction RND. Set ROP to +Inf if OP is ±0, +0 if OP is +Inf, and NaN if OP is negative. -- Function: int mpfr_cbrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_root (mpfr_t ROP, mpfr_t OP, unsigned long int K, mpfr_rnd_t RND) Set ROP to the cubic root (resp. the Kth root) of OP rounded in the direction RND. For K odd (resp. even) and OP negative (including -Inf), set ROP to a negative number (resp. NaN). The Kth root of -0 is defined to be -0, whatever the parity of K. -- Function: int mpfr_pow (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_pow_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_pow_ui (mpfr_t ROP, unsigned long int OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_ui_pow (mpfr_t ROP, unsigned long int OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to OP1 raised to OP2, rounded in the direction RND. Special values are handled as described in the ISO C99 and IEEE 754-2008 standards for the `pow' function: * `pow(±0, Y)' returns plus or minus infinity for Y a negative odd integer. * `pow(±0, Y)' returns plus infinity for Y negative and not an odd integer. * `pow(±0, Y)' returns plus or minus zero for Y a positive odd integer. * `pow(±0, Y)' returns plus zero for Y positive and not an odd integer. * `pow(-1, ±Inf)' returns 1. * `pow(+1, Y)' returns 1 for any Y, even a NaN. * `pow(X, ±0)' returns 1 for any X, even a NaN. * `pow(X, Y)' returns NaN for finite negative X and finite non-integer Y. * `pow(X, -Inf)' returns plus infinity for 0 < abs(x) < 1, and plus zero for abs(x) > 1. * `pow(X, +Inf)' returns plus zero for 0 < abs(x) < 1, and plus infinity for abs(x) > 1. * `pow(-Inf, Y)' returns minus zero for Y a negative odd integer. * `pow(-Inf, Y)' returns plus zero for Y negative and not an odd integer. * `pow(-Inf, Y)' returns minus infinity for Y a positive odd integer. * `pow(-Inf, Y)' returns plus infinity for Y positive and not an odd integer. * `pow(+Inf, Y)' returns plus zero for Y negative, and plus infinity for Y positive. -- Function: int mpfr_neg (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_abs (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to -OP and the absolute value of OP respectively, rounded in the direction RND. Just changes or adjusts the sign if ROP and OP are the same variable, otherwise a rounding might occur if the precision of ROP is less than that of OP. -- Function: int mpfr_dim (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the positive difference of OP1 and OP2, i.e., OP1 - OP2 rounded in the direction RND if OP1 > OP2, +0 if OP1 <= OP2, and NaN if OP1 or OP2 is NaN. -- Function: int mpfr_mul_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_mul_2si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) Set ROP to OP1 times 2 raised to OP2 rounded in the direction RND. Just increases the exponent by OP2 when ROP and OP1 are identical. -- Function: int mpfr_div_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_2si (mpfr_t ROP, mpfr_t OP1, long int OP2, mpfr_rnd_t RND) Set ROP to OP1 divided by 2 raised to OP2 rounded in the direction RND. Just decreases the exponent by OP2 when ROP and OP1 are identical.  File: mpfr.info, Node: Comparison Functions, Next: Special Functions, Prev: Basic Arithmetic Functions, Up: MPFR Interface 5.6 Comparison Functions ======================== -- Function: int mpfr_cmp (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_cmp_ui (mpfr_t OP1, unsigned long int OP2) -- Function: int mpfr_cmp_si (mpfr_t OP1, long int OP2) -- Function: int mpfr_cmp_d (mpfr_t OP1, double OP2) -- Function: int mpfr_cmp_ld (mpfr_t OP1, long double OP2) -- Function: int mpfr_cmp_z (mpfr_t OP1, mpz_t OP2) -- Function: int mpfr_cmp_q (mpfr_t OP1, mpq_t OP2) -- Function: int mpfr_cmp_f (mpfr_t OP1, mpf_t OP2) Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if OP1 = OP2, and a negative value if OP1 < OP2. Both OP1 and OP2 are considered to their full own precision, which may differ. If one of the operands is NaN, set the _erange_ flag and return zero. Note: These functions may be useful to distinguish the three possible cases. If you need to distinguish two cases only, it is recommended to use the predicate functions (e.g., `mpfr_equal_p' for the equality) described below; they behave like the IEEE 754 comparisons, in particular when one or both arguments are NaN. But only floating-point numbers can be compared (you may need to do a conversion first). -- Function: int mpfr_cmp_ui_2exp (mpfr_t OP1, unsigned long int OP2, mpfr_exp_t E) -- Function: int mpfr_cmp_si_2exp (mpfr_t OP1, long int OP2, mpfr_exp_t E) Compare OP1 and OP2 multiplied by two to the power E. Similar as above. -- Function: int mpfr_cmpabs (mpfr_t OP1, mpfr_t OP2) Compare |OP1| and |OP2|. Return a positive value if |OP1| > |OP2|, zero if |OP1| = |OP2|, and a negative value if |OP1| < |OP2|. If one of the operands is NaN, set the _erange_ flag and return zero. -- Function: int mpfr_nan_p (mpfr_t OP) -- Function: int mpfr_inf_p (mpfr_t OP) -- Function: int mpfr_number_p (mpfr_t OP) -- Function: int mpfr_zero_p (mpfr_t OP) -- Function: int mpfr_regular_p (mpfr_t OP) Return non-zero if OP is respectively NaN, an infinity, an ordinary number (i.e., neither NaN nor an infinity), zero, or a regular number (i.e., neither NaN, nor an infinity nor zero). Return zero otherwise. -- Macro: int mpfr_sgn (mpfr_t OP) Return a positive value if OP > 0, zero if OP = 0, and a negative value if OP < 0. If the operand is NaN, set the _erange_ flag and return zero. This is equivalent to `mpfr_cmp_ui (op, 0)', but more efficient. -- Function: int mpfr_greater_p (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_greaterequal_p (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_less_p (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_lessequal_p (mpfr_t OP1, mpfr_t OP2) -- Function: int mpfr_equal_p (mpfr_t OP1, mpfr_t OP2) Return non-zero if OP1 > OP2, OP1 >= OP2, OP1 < OP2, OP1 <= OP2, OP1 = OP2 respectively, and zero otherwise. Those functions return zero whenever OP1 and/or OP2 is NaN. -- Function: int mpfr_lessgreater_p (mpfr_t OP1, mpfr_t OP2) Return non-zero if OP1 < OP2 or OP1 > OP2 (i.e., neither OP1, nor OP2 is NaN, and OP1 <> OP2), zero otherwise (i.e., OP1 and/or OP2 is NaN, or OP1 = OP2). -- Function: int mpfr_unordered_p (mpfr_t OP1, mpfr_t OP2) Return non-zero if OP1 or OP2 is a NaN (i.e., they cannot be compared), zero otherwise.  File: mpfr.info, Node: Special Functions, Next: Input and Output Functions, Prev: Comparison Functions, Up: MPFR Interface 5.7 Special Functions ===================== All those functions, except explicitly stated (for example `mpfr_sin_cos'), return a *note ternary value::, i.e., zero for an exact return value, a positive value for a return value larger than the exact result, and a negative value otherwise. Important note: in some domains, computing special functions (either with correct or incorrect rounding) is expensive, even for small precision, for example the trigonometric and Bessel functions for large argument. -- Function: int mpfr_log (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_log2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_log10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the natural logarithm of OP, log2(OP) or log10(OP), respectively, rounded in the direction RND. Set ROP to -Inf if OP is -0 (i.e., the sign of the zero has no influence on the result). -- Function: int mpfr_exp (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_exp2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_exp10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential of OP, to 2 power of OP or to 10 power of OP, respectively, rounded in the direction RND. -- Function: int mpfr_cos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_sin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_tan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the cosine of OP, sine of OP, tangent of OP, rounded in the direction RND. -- Function: int mpfr_sin_cos (mpfr_t SOP, mpfr_t COP, mpfr_t OP, mpfr_rnd_t RND) Set simultaneously SOP to the sine of OP and COP to the cosine of OP, rounded in the direction RND with the corresponding precisions of SOP and COP, which must be different variables. Return 0 iff both results are exact, more precisely it returns s+4c where s=0 if SOP is exact, s=1 if SOP is larger than the sine of OP, s=2 if SOP is smaller than the sine of OP, and similarly for c and the cosine of OP. -- Function: int mpfr_sec (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_csc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_cot (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the secant of OP, cosecant of OP, cotangent of OP, rounded in the direction RND. -- Function: int mpfr_acos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_asin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_atan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the arc-cosine, arc-sine or arc-tangent of OP, rounded in the direction RND. Note that since `acos(-1)' returns the floating-point number closest to Pi according to the given rounding mode, this number might not be in the output range 0 <= ROP < \pi of the arc-cosine function; still, the result lies in the image of the output range by the rounding function. The same holds for `asin(-1)', `asin(1)', `atan(-Inf)', `atan(+Inf)' or for `atan(op)' with large OP and small precision of ROP. -- Function: int mpfr_atan2 (mpfr_t ROP, mpfr_t Y, mpfr_t X, mpfr_rnd_t RND) Set ROP to the arc-tangent2 of Y and X, rounded in the direction RND: if `x > 0', `atan2(y, x) = atan (y/x)'; if `x < 0', `atan2(y, x) = sign(y)*(Pi - atan (abs(y/x)))', thus a number from -Pi to Pi. As for `atan', in case the exact mathematical result is +Pi or -Pi, its rounded result might be outside the function output range. `atan2(y, 0)' does not raise any floating-point exception. Special values are handled as described in the ISO C99 and IEEE 754-2008 standards for the `atan2' function: * `atan2(+0, -0)' returns +Pi. * `atan2(-0, -0)' returns -Pi. * `atan2(+0, +0)' returns +0. * `atan2(-0, +0)' returns -0. * `atan2(+0, x)' returns +Pi for x < 0. * `atan2(-0, x)' returns -Pi for x < 0. * `atan2(+0, x)' returns +0 for x > 0. * `atan2(-0, x)' returns -0 for x > 0. * `atan2(y, 0)' returns -Pi/2 for y < 0. * `atan2(y, 0)' returns +Pi/2 for y > 0. * `atan2(+Inf, -Inf)' returns +3*Pi/4. * `atan2(-Inf, -Inf)' returns -3*Pi/4. * `atan2(+Inf, +Inf)' returns +Pi/4. * `atan2(-Inf, +Inf)' returns -Pi/4. * `atan2(+Inf, x)' returns +Pi/2 for finite x. * `atan2(-Inf, x)' returns -Pi/2 for finite x. * `atan2(y, -Inf)' returns +Pi for finite y > 0. * `atan2(y, -Inf)' returns -Pi for finite y < 0. * `atan2(y, +Inf)' returns +0 for finite y > 0. * `atan2(y, +Inf)' returns -0 for finite y < 0. -- Function: int mpfr_cosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_sinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_tanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the hyperbolic cosine, sine or tangent of OP, rounded in the direction RND. -- Function: int mpfr_sinh_cosh (mpfr_t SOP, mpfr_t COP, mpfr_t OP, mpfr_rnd_t RND) Set simultaneously SOP to the hyperbolic sine of OP and COP to the hyperbolic cosine of OP, rounded in the direction RND with the corresponding precision of SOP and COP, which must be different variables. Return 0 iff both results are exact (see `mpfr_sin_cos' for a more detailed description of the return value). -- Function: int mpfr_sech (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_csch (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_coth (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the hyperbolic secant of OP, cosecant of OP, cotangent of OP, rounded in the direction RND. -- Function: int mpfr_acosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_asinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_atanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the inverse hyperbolic cosine, sine or tangent of OP, rounded in the direction RND. -- Function: int mpfr_fac_ui (mpfr_t ROP, unsigned long int OP, mpfr_rnd_t RND) Set ROP to the factorial of OP, rounded in the direction RND. -- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the logarithm of one plus OP, rounded in the direction RND. -- Function: int mpfr_expm1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential of OP followed by a subtraction by one, rounded in the direction RND. -- Function: int mpfr_eint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the exponential integral of OP, rounded in the direction RND. For positive OP, the exponential integral is the sum of Euler's constant, of the logarithm of OP, and of the sum for k from 1 to infinity of OP to the power k, divided by k and factorial(k). For negative OP, ROP is set to NaN (this definition for negative argument follows formula 5.1.2 from the Handbook of Mathematical Functions from Abramowitz and Stegun, a future version might use another definition). -- Function: int mpfr_li2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to real part of the dilogarithm of OP, rounded in the direction RND. MPFR defines the dilogarithm function as the integral of -log(1-t)/t from 0 to OP. -- Function: int mpfr_gamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the Gamma function on OP, rounded in the direction RND. When OP is a negative integer, ROP is set to NaN. -- Function: int mpfr_lngamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the logarithm of the Gamma function on OP, rounded in the direction RND. When -2K-1 <= OP <= -2K, K being a non-negative integer, ROP is set to NaN. See also `mpfr_lgamma'. -- Function: int mpfr_lgamma (mpfr_t ROP, int *SIGNP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the logarithm of the absolute value of the Gamma function on OP, rounded in the direction RND. The sign (1 or -1) of Gamma(OP) is returned in the object pointed to by SIGNP. When OP is an infinity or a non-positive integer, set ROP to +Inf. When OP is NaN, -Inf or a negative integer, *SIGNP is undefined, and when OP is ±0, *SIGNP is the sign of the zero. -- Function: int mpfr_digamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the Digamma (sometimes also called Psi) function on OP, rounded in the direction RND. When OP is a negative integer, set ROP to NaN. -- Function: int mpfr_zeta (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_zeta_ui (mpfr_t ROP, unsigned long OP, mpfr_rnd_t RND) Set ROP to the value of the Riemann Zeta function on OP, rounded in the direction RND. -- Function: int mpfr_erf (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_erfc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the error function on OP (resp. the complementary error function on OP) rounded in the direction RND. -- Function: int mpfr_j0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_j1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_jn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the first kind Bessel function of order 0, (resp. 1 and N) on OP, rounded in the direction RND. When OP is NaN, ROP is always set to NaN. When OP is plus or minus Infinity, ROP is set to +0. When OP is zero, and N is not zero, ROP is set to +0 or -0 depending on the parity and sign of N, and the sign of OP. -- Function: int mpfr_y0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_y1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_yn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the value of the second kind Bessel function of order 0 (resp. 1 and N) on OP, rounded in the direction RND. When OP is NaN or negative, ROP is always set to NaN. When OP is +Inf, ROP is set to +0. When OP is zero, ROP is set to +Inf or -Inf depending on the parity and sign of N. -- Function: int mpfr_fma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_rnd_t RND) -- Function: int mpfr_fms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t OP3, mpfr_rnd_t RND) Set ROP to (OP1 times OP2) + OP3 (resp. (OP1 times OP2) - OP3) rounded in the direction RND. -- Function: int mpfr_agm (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the arithmetic-geometric mean of OP1 and OP2, rounded in the direction RND. The arithmetic-geometric mean is the common limit of the sequences U_N and V_N, where U_0=OP1, V_0=OP2, U_(N+1) is the arithmetic mean of U_N and V_N, and V_(N+1) is the geometric mean of U_N and V_N. If any operand is negative, set ROP to NaN. -- Function: int mpfr_hypot (mpfr_t ROP, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) Set ROP to the Euclidean norm of X and Y, i.e., the square root of the sum of the squares of X and Y, rounded in the direction RND. Special values are handled as described in Section F.9.4.3 of the ISO C99 and IEEE 754-2008 standards: If X or Y is an infinity, then +Inf is returned in ROP, even if the other number is NaN. -- Function: int mpfr_ai (mpfr_t ROP, mpfr_t X, mpfr_rnd_t RND) Set ROP to the value of the Airy function Ai on X, rounded in the direction RND. When X is NaN, ROP is always set to NaN. When X is +Inf or -Inf, ROP is +0. The current implementation is not intended to be used with large arguments. It works with abs(X) typically smaller than 500. For larger arguments, other methods should be used and will be implemented in a future version. -- Function: int mpfr_const_log2 (mpfr_t ROP, mpfr_rnd_t RND) -- Function: int mpfr_const_pi (mpfr_t ROP, mpfr_rnd_t RND) -- Function: int mpfr_const_euler (mpfr_t ROP, mpfr_rnd_t RND) -- Function: int mpfr_const_catalan (mpfr_t ROP, mpfr_rnd_t RND) Set ROP to the logarithm of 2, the value of Pi, of Euler's constant 0.577..., of Catalan's constant 0.915..., respectively, rounded in the direction RND. These functions cache the computed values to avoid other calculations if a lower or equal precision is requested. To free these caches, use `mpfr_free_cache'. -- Function: void mpfr_free_cache (void) Free various caches used by MPFR internally, in particular the caches used by the functions computing constants (`mpfr_const_log2', `mpfr_const_pi', `mpfr_const_euler' and `mpfr_const_catalan'). You should call this function before terminating a thread, even if you did not call these functions directly (they could have been called internally). -- Function: int mpfr_sum (mpfr_t ROP, mpfr_ptr const TAB[], unsigned long int N, mpfr_rnd_t RND) Set ROP to the sum of all elements of TAB, whose size is N, rounded in the direction RND. Warning: for efficiency reasons, TAB is an array of pointers to `mpfr_t', not an array of `mpfr_t'. If the returned `int' value is zero, ROP is guaranteed to be the exact sum; otherwise ROP might be smaller than, equal to, or larger than the exact sum (in accordance to the rounding mode). However, `mpfr_sum' does guarantee the result is correctly rounded.  File: mpfr.info, Node: Input and Output Functions, Next: Formatted Output Functions, Prev: Special Functions, Up: MPFR Interface 5.8 Input and Output Functions ============================== This section describes functions that perform input from an input/output stream, and functions that output to an input/output stream. Passing a null pointer for a `stream' to any of these functions will make them read from `stdin' and write to `stdout', respectively. When using any of these functions, you must include the `' standard header before `mpfr.h', to allow `mpfr.h' to define prototypes for these functions. -- Function: size_t mpfr_out_str (FILE *STREAM, int BASE, size_t N, mpfr_t OP, mpfr_rnd_t RND) Output OP on stream STREAM, as a string of digits in base BASE, rounded in the direction RND. The base may vary from 2 to 62. Print N significant digits exactly, or if N is 0, enough digits so that OP can be read back exactly (see `mpfr_get_str'). In addition to the significant digits, a decimal point (defined by the current locale) at the right of the first digit and a trailing exponent in base 10, in the form `eNNN', are printed. If BASE is greater than 10, `@@' will be used instead of `e' as exponent delimiter. Return the number of characters written, or if an error occurred, return 0. -- Function: size_t mpfr_inp_str (mpfr_t ROP, FILE *STREAM, int BASE, mpfr_rnd_t RND) Input a string in base BASE from stream STREAM, rounded in the direction RND, and put the read float in ROP. This function reads a word (defined as a sequence of characters between whitespace) and parses it using `mpfr_set_str'. See the documentation of `mpfr_strtofr' for a detailed description of the valid string formats. Return the number of bytes read, or if an error occurred, return 0.  File: mpfr.info, Node: Formatted Output Functions, Next: Integer Related Functions, Prev: Input and Output Functions, Up: MPFR Interface 5.9 Formatted Output Functions ============================== 5.9.1 Requirements ------------------ The class of `mpfr_printf' functions provides formatted output in a similar manner as the standard C `printf'. These functions are defined only if your system supports ISO C variadic functions and the corresponding argument access macros. When using any of these functions, you must include the `' standard header before `mpfr.h', to allow `mpfr.h' to define prototypes for these functions. 5.9.2 Format String ------------------- The format specification accepted by `mpfr_printf' is an extension of the `printf' one. The conversion specification is of the form: % [flags] [width] [.[precision]] [type] [rounding] conv `flags', `width', and `precision' have the same meaning as for the standard `printf' (in particular, notice that the `precision' is related to the number of digits displayed in the base chosen by `conv' and not related to the internal precision of the `mpfr_t' variable). `mpfr_printf' accepts the same `type' specifiers as GMP (except the non-standard and deprecated `q', use `ll' instead), namely the length modifiers defined in the C standard: `h' `short' `hh' `char' `j' `intmax_t' or `uintmax_t' `l' `long' or `wchar_t' `ll' `long long' `L' `long double' `t' `ptrdiff_t' `z' `size_t' and the `type' specifiers defined in GMP plus `R' and `P' specific to MPFR (the second column in the table below shows the type of the argument read in the argument list and the kind of `conv' specifier to use after the `type' specifier): `F' `mpf_t', float conversions `Q' `mpq_t', integer conversions `M' `mp_limb_t', integer conversions `N' `mp_limb_t' array, integer conversions `Z' `mpz_t', integer conversions `P' `mpfr_prec_t', integer conversions `R' `mpfr_t', float conversions The `type' specifiers have the same restrictions as those mentioned in the GMP documentation: *note Formatted Output Strings: (gmp.info)Formatted Output Strings. In particular, the `type' specifiers (except `R' and `P') are supported only if they are supported by `gmp_printf' in your GMP build; this implies that the standard specifiers, such as `t', must _also_ be supported by your C library if you want to use them. The `rounding' field is specific to `mpfr_t' arguments and should not be used with other types. With conversion specification not involving `P' and `R' types, `mpfr_printf' behaves exactly as `gmp_printf'. The `P' type specifies that a following `o', `u', `x', or `X' conversion specifier applies to a `mpfr_prec_t' argument. It is needed because the `mpfr_prec_t' type does not necessarily correspond to an `unsigned int' or any fixed standard type. The `precision' field specifies the minimum number of digits to appear. The default `precision' is 1. For example: mpfr_t x; mpfr_prec_t p; mpfr_init (x); ... p = mpfr_get_prec (x); mpfr_printf ("variable x with %Pu bits", p); The `R' type specifies that a following `a', `A', `b', `e', `E', `f', `F', `g', `G', or `n' conversion specifier applies to a `mpfr_t' argument. The `R' type can be followed by a `rounding' specifier denoted by one of the following characters: `U' round toward plus infinity `D' round toward minus infinity `Y' round away from zero `Z' round toward zero `N' round to nearest (with ties to even) `*' rounding mode indicated by the `mpfr_rnd_t' argument just before the corresponding `mpfr_t' variable. The default rounding mode is rounding to nearest. The following three examples are equivalent: mpfr_t x; mpfr_init (x); ... mpfr_printf ("%.128Rf", x); mpfr_printf ("%.128RNf", x); mpfr_printf ("%.128R*f", MPFR_RNDN, x); Note that the rounding away from zero mode is specified with `Y' because ISO C reserves the `A' specifier for hexadecimal output (see below). The output `conv' specifiers allowed with `mpfr_t' parameter are: `a' `A' hex float, C99 style `b' binary output `e' `E' scientific format float `f' `F' fixed point float `g' `G' fixed or scientific float The conversion specifier `b' which displays the argument in binary is specific to `mpfr_t' arguments and should not be used with other types. Other conversion specifiers have the same meaning as for a `double' argument. In case of non-decimal output, only the significand is written in the specified base, the exponent is always displayed in decimal. Special values are always displayed as `nan', `-inf', and `inf' for `a', `b', `e', `f', and `g' specifiers and `NAN', `-INF', and `INF' for `A', `E', `F', and `G' specifiers. If the `precision' field is not empty, the `mpfr_t' number is rounded to the given precision in the direction specified by the rounding mode. If the precision is zero with rounding to nearest mode and one of the following `conv' specifiers: `a', `A', `b', `e', `E', tie case is rounded to even when it lies between two consecutive values at the wanted precision which have the same exponent, otherwise, it is rounded away from zero. For instance, 85 is displayed as "8e+1" and 95 is displayed as "1e+2" with the format specification `"%.0RNe"'. This also applies when the `g' (resp. `G') conversion specifier uses the `e' (resp. `E') style. If the precision is set to a value greater than the maximum value for an `int', it will be silently reduced down to `INT_MAX'. If the `precision' field is empty (as in `%Re' or `%.RE') with `conv' specifier `e' and `E', the number is displayed with enough digits so that it can be read back exactly, assuming that the input and output variables have the same precision and that the input and output rounding modes are both rounding to nearest (as for `mpfr_get_str'). The default precision for an empty `precision' field with `conv' specifiers `f', `F', `g', and `G' is 6. 5.9.3 Functions --------------- For all the following functions, if the number of characters which ought to be written appears to exceed the maximum limit for an `int', nothing is written in the stream (resp. to `stdout', to BUF, to STR), the function returns -1, sets the _erange_ flag, and (in POSIX system only) `errno' is set to `EOVERFLOW'. -- Function: int mpfr_fprintf (FILE *STREAM, const char *TEMPLATE, ...) -- Function: int mpfr_vfprintf (FILE *STREAM, const char *TEMPLATE, va_list AP) Print to the stream STREAM the optional arguments under the control of the template string TEMPLATE. Return the number of characters written or a negative value if an error occurred. -- Function: int mpfr_printf (const char *TEMPLATE, ...) -- Function: int mpfr_vprintf (const char *TEMPLATE, va_list AP) Print to `stdout' the optional arguments under the control of the template string TEMPLATE. Return the number of characters written or a negative value if an error occurred. -- Function: int mpfr_sprintf (char *BUF, const char *TEMPLATE, ...) -- Function: int mpfr_vsprintf (char *BUF, const char *TEMPLATE, va_list AP) Form a null-terminated string corresponding to the optional arguments under the control of the template string TEMPLATE, and print it in BUF. No overlap is permitted between BUF and the other arguments. Return the number of characters written in the array BUF _not counting_ the terminating null character or a negative value if an error occurred. -- Function: int mpfr_snprintf (char *BUF, size_t N, const char *TEMPLATE, ...) -- Function: int mpfr_vsnprintf (char *BUF, size_t N, const char *TEMPLATE, va_list AP) Form a null-terminated string corresponding to the optional arguments under the control of the template string TEMPLATE, and print it in BUF. If N is zero, nothing is written and BUF may be a null pointer, otherwise, the N-1 first characters are written in BUF and the N-th is a null character. Return the number of characters that would have been written had N be sufficiently large, _not counting_ the terminating null character, or a negative value if an error occurred. -- Function: int mpfr_asprintf (char **STR, const char *TEMPLATE, ...) -- Function: int mpfr_vasprintf (char **STR, const char *TEMPLATE, va_list AP) Write their output as a null terminated string in a block of memory allocated using the current allocation function. A pointer to the block is stored in STR. The block of memory must be freed using `mpfr_free_str'. The return value is the number of characters written in the string, excluding the null-terminator, or a negative value if an error occurred.  File: mpfr.info, Node: Integer Related Functions, Next: Rounding Related Functions, Prev: Formatted Output Functions, Up: MPFR Interface 5.10 Integer and Remainder Related Functions ============================================ -- Function: int mpfr_rint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_ceil (mpfr_t ROP, mpfr_t OP) -- Function: int mpfr_floor (mpfr_t ROP, mpfr_t OP) -- Function: int mpfr_round (mpfr_t ROP, mpfr_t OP) -- Function: int mpfr_trunc (mpfr_t ROP, mpfr_t OP) Set ROP to OP rounded to an integer. `mpfr_rint' rounds to the nearest representable integer in the given direction RND, `mpfr_ceil' rounds to the next higher or equal representable integer, `mpfr_floor' to the next lower or equal representable integer, `mpfr_round' to the nearest representable integer, rounding halfway cases away from zero (as in the roundTiesToAway mode of IEEE 754-2008), and `mpfr_trunc' to the next representable integer toward zero. The returned value is zero when the result is exact, positive when it is greater than the original value of OP, and negative when it is smaller. More precisely, the returned value is 0 when OP is an integer representable in ROP, 1 or -1 when OP is an integer that is not representable in ROP, 2 or -2 when OP is not an integer. Note that `mpfr_round' is different from `mpfr_rint' called with the rounding to nearest mode (where halfway cases are rounded to an even integer or significand). Note also that no double rounding is performed; for instance, 10.5 (1010.1 in binary) is rounded by `mpfr_rint' with rounding to nearest to 12 (1100 in binary) in 2-bit precision, because the two enclosing numbers representable on two bits are 8 and 12, and the closest is 12. (If one first rounded to an integer, one would round 10.5 to 10 with even rounding, and then 10 would be rounded to 8 again with even rounding.) -- Function: int mpfr_rint_ceil (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_rint_floor (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_rint_round (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) -- Function: int mpfr_rint_trunc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to OP rounded to an integer. `mpfr_rint_ceil' rounds to the next higher or equal integer, `mpfr_rint_floor' to the next lower or equal integer, `mpfr_rint_round' to the nearest integer, rounding halfway cases away from zero, and `mpfr_rint_trunc' to the next integer toward zero. If the result is not representable, it is rounded in the direction RND. The returned value is the ternary value associated with the considered round-to-integer function (regarded in the same way as any other mathematical function). Contrary to `mpfr_rint', those functions do perform a double rounding: first OP is rounded to the nearest integer in the direction given by the function name, then this nearest integer (if not representable) is rounded in the given direction RND. For example, `mpfr_rint_round' with rounding to nearest and a precision of two bits rounds 6.5 to 7 (halfway cases away from zero), then 7 is rounded to 8 by the round-even rule, despite the fact that 6 is also representable on two bits, and is closer to 6.5 than 8. -- Function: int mpfr_frac (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) Set ROP to the fractional part of OP, having the same sign as OP, rounded in the direction RND (unlike in `mpfr_rint', RND affects only how the exact fractional part is rounded, not how the fractional part is generated). -- Function: int mpfr_modf (mpfr_t IOP, mpfr_t FOP, mpfr_t OP, mpfr_rnd_t RND) Set simultaneously IOP to the integral part of OP and FOP to the fractional part of OP, rounded in the direction RND with the corresponding precision of IOP and FOP (equivalent to `mpfr_trunc(IOP, OP, RND)' and `mpfr_frac(FOP, OP, RND)'). The variables IOP and FOP must be different. Return 0 iff both results are exact (see `mpfr_sin_cos' for a more detailed description of the return value). -- Function: int mpfr_fmod (mpfr_t R, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) -- Function: int mpfr_remainder (mpfr_t R, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) -- Function: int mpfr_remquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y, mpfr_rnd_t RND) Set R to the value of X - NY, rounded according to the direction RND, where N is the integer quotient of X divided by Y, defined as follows: N is rounded toward zero for `mpfr_fmod', and to the nearest integer (ties rounded to even) for `mpfr_remainder' and `mpfr_remquo'. Special values are handled as described in Section F.9.7.1 of the ISO C99 standard: If X is infinite or Y is zero, R is NaN. If Y is infinite and X is finite, R is X rounded to the precision of R. If R is zero, it has the sign of X. The return value is the ternary value corresponding to R. Additionally, `mpfr_remquo' stores the low significant bits from the quotient N in *Q (more precisely the number of bits in a `long' minus one), with the sign of X divided by Y (except if those low bits are all zero, in which case zero is returned). Note that X may be so large in magnitude relative to Y that an exact representation of the quotient is not practical. The `mpfr_remainder' and `mpfr_remquo' functions are useful for additive argument reduction. -- Function: int mpfr_integer_p (mpfr_t OP) Return non-zero iff OP is an integer.  File: mpfr.info, Node: Rounding Related Functions, Next: Miscellaneous Functions, Prev: Integer Related Functions, Up: MPFR Interface 5.11 Rounding Related Functions =============================== -- Function: void mpfr_set_default_rounding_mode (mpfr_rnd_t RND) Set the default rounding mode to RND. The default rounding mode is to nearest initially. -- Function: mpfr_rnd_t mpfr_get_default_rounding_mode (void) Get the default rounding mode. -- Function: int mpfr_prec_round (mpfr_t X, mpfr_prec_t PREC, mpfr_rnd_t RND) Round X according to RND with precision PREC, which must be an integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX' (otherwise the behavior is undefined). If PREC is greater or equal to the precision of X, then new space is allocated for the significand, and it is filled with zeros. Otherwise, the significand is rounded to precision PREC with the given direction. In both cases, the precision of X is changed to PREC. Here is an example of how to use `mpfr_prec_round' to implement Newton's algorithm to compute the inverse of A, assuming X is already an approximation to N bits: mpfr_set_prec (t, 2 * n); mpfr_set (t, a, MPFR_RNDN); /* round a to 2n bits */ mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to 2n bits */ mpfr_ui_sub (t, 1, t, MPFR_RNDN); /* high n bits cancel with 1 */ mpfr_prec_round (t, n, MPFR_RNDN); /* t is correct to n bits */ mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to n bits */ mpfr_prec_round (x, 2 * n, MPFR_RNDN); /* exact */ mpfr_add (x, x, t, MPFR_RNDN); /* x is correct to 2n bits */ -- Function: int mpfr_can_round (mpfr_t B, mpfr_exp_t ERR, mpfr_rnd_t RND1, mpfr_rnd_t RND2, mpfr_prec_t PREC) Assuming B is an approximation of an unknown number X in the direction RND1 with error at most two to the power E(b)-ERR where E(b) is the exponent of B, return a non-zero value if one is able to round correctly X to precision PREC with the direction RND2, and 0 otherwise (including for NaN and Inf). This function *does not modify* its arguments. If RND1 is `MPFR_RNDN', then the sign of the error is unknown, but its absolute value is the same, so that the possible range is twice as large as with a directed rounding for RND1. Note: if one wants to also determine the correct *note ternary value:: when rounding B to precision PREC with rounding mode RND, a useful trick is the following: if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ, prec + (rnd == MPFR_RNDN))) ... Indeed, if RND is `MPFR_RNDN', this will check if one can round to PREC+1 bits with a directed rounding: if so, one can surely round to nearest to PREC bits, and in addition one can determine the correct ternary value, which would not be the case when B is near from a value exactly representable on PREC bits. -- Function: mpfr_prec_t mpfr_min_prec (mpfr_t X) Return the minimal number of bits required to store the significand of X, and 0 for special values, including 0. (Warning: the returned value can be less than `MPFR_PREC_MIN'.) The function name is subject to change. -- Function: const char * mpfr_print_rnd_mode (mpfr_rnd_t RND) Return a string ("MPFR_RNDD", "MPFR_RNDU", "MPFR_RNDN", "MPFR_RNDZ", "MPFR_RNDA") corresponding to the rounding mode RND, or a null pointer if RND is an invalid rounding mode.  File: mpfr.info, Node: Miscellaneous Functions, Next: Exception Related Functions, Prev: Rounding Related Functions, Up: MPFR Interface 5.12 Miscellaneous Functions ============================ -- Function: void mpfr_nexttoward (mpfr_t X, mpfr_t Y) If X or Y is NaN, set X to NaN. If X and Y are equal, X is unchanged. Otherwise, if X is different from Y, replace X by the next floating-point number (with the precision of X and the current exponent range) in the direction of Y (the infinite values are seen as the smallest and largest floating-point numbers). If the result is zero, it keeps the same sign. No underflow or overflow is generated. -- Function: void mpfr_nextabove (mpfr_t X) -- Function: void mpfr_nextbelow (mpfr_t X) Equivalent to `mpfr_nexttoward' where Y is plus infinity (resp. minus infinity). -- Function: int mpfr_min (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) -- Function: int mpfr_max (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set ROP to the minimum (resp. maximum) of OP1 and OP2. If OP1 and OP2 are both NaN, then ROP is set to NaN. If OP1 or OP2 is NaN, then ROP is set to the numeric value. If OP1 and OP2 are zeros of different signs, then ROP is set to -0 (resp. +0). -- Function: int mpfr_urandomb (mpfr_t ROP, gmp_randstate_t STATE) Generate a uniformly distributed random float in the interval 0 <= ROP < 1. More precisely, the number can be seen as a float with a random non-normalized significand and exponent 0, which is then normalized (thus if E denotes the exponent after normalization, then the least -E significant bits of the significand are always 0). Return 0, unless the exponent is not in the current exponent range, in which case ROP is set to NaN and a non-zero value is returned (this should never happen in practice, except in very specific cases). The second argument is a `gmp_randstate_t' structure which should be created using the GMP `gmp_randinit' function (see the GMP manual). Note: for a given version of MPFR, the returned value of ROP and the new value of STATE (which controls further random values) do not depend on the machine word size. -- Function: int mpfr_urandom (mpfr_t ROP, gmp_randstate_t STATE, mpfr_rnd_t RND) Generate a uniformly distributed random float. The floating-point number ROP can be seen as if a random real number is generated according to the continuous uniform distribution on the interval [0, 1] and then rounded in the direction RND. The second argument is a `gmp_randstate_t' structure which should be created using the GMP `gmp_randinit' function (see the GMP manual). Note: the note for `mpfr_urandomb' holds too. In addition, the exponent range and the rounding mode might have a side effect on the next random state. -- Function: int mpfr_grandom (mpfr_t ROP1, mpfr_t ROP2, gmp_randstate_t STATE, mpfr_rnd_t RND) Generate two random floats according to a standard normal gaussian distribution. If ROP2 is a null pointer, then only one value is generated and stored in ROP1. The floating-point number ROP1 (and ROP2) can be seen as if a random real number were generated according to the standard normal gaussian distribution and then rounded in the direction RND. The third argument is a `gmp_randstate_t' structure, which should be created using the GMP `gmp_randinit' function (see the GMP manual). The combination of the ternary values is returned like with `mpfr_sin_cos'. If ROP2 is a null pointer, the second ternary value is assumed to be 0 (note that the encoding of the only ternary value is not the same as the usual encoding for functions that return only one result). Otherwise the ternary value of a random number is always non-zero. Note: the note for `mpfr_urandomb' holds too. In addition, the exponent range and the rounding mode might have a side effect on the next random state. -- Function: mpfr_exp_t mpfr_get_exp (mpfr_t X) Return the exponent of X, assuming that X is a non-zero ordinary number and the significand is considered in [1/2,1). The behavior for NaN, infinity or zero is undefined. -- Function: int mpfr_set_exp (mpfr_t X, mpfr_exp_t E) Set the exponent of X if E is in the current exponent range, and return 0 (even if X is not a non-zero ordinary number); otherwise, return a non-zero value. The significand is assumed to be in [1/2,1). -- Function: int mpfr_signbit (mpfr_t OP) Return a non-zero value iff OP has its sign bit set (i.e., if it is negative, -0, or a NaN whose representation has its sign bit set). -- Function: int mpfr_setsign (mpfr_t ROP, mpfr_t OP, int S, mpfr_rnd_t RND) Set the value of ROP from OP, rounded toward the given direction RND, then set (resp. clear) its sign bit if S is non-zero (resp. zero), even when OP is a NaN. -- Function: int mpfr_copysign (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Set the value of ROP from OP1, rounded toward the given direction RND, then set its sign bit to that of OP2 (even when OP1 or OP2 is a NaN). This function is equivalent to `mpfr_setsign (ROP, OP1, mpfr_signbit (OP2), RND)'. -- Function: const char * mpfr_get_version (void) Return the MPFR version, as a null-terminated string. -- Macro: MPFR_VERSION -- Macro: MPFR_VERSION_MAJOR -- Macro: MPFR_VERSION_MINOR -- Macro: MPFR_VERSION_PATCHLEVEL -- Macro: MPFR_VERSION_STRING `MPFR_VERSION' is the version of MPFR as a preprocessing constant. `MPFR_VERSION_MAJOR', `MPFR_VERSION_MINOR' and `MPFR_VERSION_PATCHLEVEL' are respectively the major, minor and patch level of MPFR version, as preprocessing constants. `MPFR_VERSION_STRING' is the version (with an optional suffix, used in development and pre-release versions) as a string constant, which can be compared to the result of `mpfr_get_version' to check at run time the header file and library used match: if (strcmp (mpfr_get_version (), MPFR_VERSION_STRING)) fprintf (stderr, "Warning: header and library do not match\n"); Note: Obtaining different strings is not necessarily an error, as in general, a program compiled with some old MPFR version can be dynamically linked with a newer MPFR library version (if allowed by the library versioning system). -- Macro: long MPFR_VERSION_NUM (MAJOR, MINOR, PATCHLEVEL) Create an integer in the same format as used by `MPFR_VERSION' from the given MAJOR, MINOR and PATCHLEVEL. Here is an example of how to check the MPFR version at compile time: #if (!defined(MPFR_VERSION) || (MPFR_VERSION' line, #include #include any program written for MPF can be compiled directly with MPFR without any changes (except the `gmp_printf' functions will not work for arguments of type `mpfr_t'). All operations are then performed with the default MPFR rounding mode, which can be reset with `mpfr_set_default_rounding_mode'. Warning: the `mpf_init' and `mpf_init2' functions initialize to zero, whereas the corresponding MPFR functions initialize to NaN: this is useful to detect uninitialized values, but is slightly incompatible with MPF. -- Function: void mpfr_set_prec_raw (mpfr_t X, mpfr_prec_t PREC) Reset the precision of X to be *exactly* PREC bits. The only difference with `mpfr_set_prec' is that PREC is assumed to be small enough so that the significand fits into the current allocated memory space for X. Otherwise the behavior is undefined. -- Function: int mpfr_eq (mpfr_t OP1, mpfr_t OP2, unsigned long int OP3) Return non-zero if OP1 and OP2 are both non-zero ordinary numbers with the same exponent and the same first OP3 bits, both zero, or both infinities of the same sign. Return zero otherwise. This function is defined for compatibility with MPF, we do not recommend to use it otherwise. Do not use it either if you want to know whether two numbers are close to each other; for instance, 1.011111 and 1.100000 are regarded as different for any value of OP3 larger than 1. -- Function: void mpfr_reldiff (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_rnd_t RND) Compute the relative difference between OP1 and OP2 and store the result in ROP. This function does not guarantee the correct rounding on the relative difference; it just computes |OP1-OP2|/OP1, using the precision of ROP and the rounding mode RND for all operations. -- Function: int mpfr_mul_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) -- Function: int mpfr_div_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long int OP2, mpfr_rnd_t RND) These functions are identical to `mpfr_mul_2ui' and `mpfr_div_2ui' respectively. These functions are only kept for compatibility with MPF, one should prefer `mpfr_mul_2ui' and `mpfr_div_2ui' otherwise.  File: mpfr.info, Node: Custom Interface, Next: Internals, Prev: Compatibility with MPF, Up: MPFR Interface 5.15 Custom Interface ===================== Some applications use a stack to handle the memory and their objects. However, the MPFR memory design is not well suited for such a thing. So that such applications are able to use MPFR, an auxiliary memory interface has been created: the Custom Interface. The following interface allows one to use MPFR in two ways: * Either directly store a floating-point number as a `mpfr_t' on the stack. * Either store its own representation on the stack and construct a new temporary `mpfr_t' each time it is needed. Nothing has to be done to destroy the floating-point numbers except garbaging the used memory: all the memory management (allocating, destroying, garbaging) is left to the application. Each function in this interface is also implemented as a macro for efficiency reasons: for example `mpfr_custom_init (s, p)' uses the macro, while `(mpfr_custom_init) (s, p)' uses the function. Note 1: MPFR functions may still initialize temporary floating-point numbers using `mpfr_init' and similar functions. See Custom Allocation (GNU MP). Note 2: MPFR functions may use the cached functions (`mpfr_const_pi' for example), even if they are not explicitly called. You have to call `mpfr_free_cache' each time you garbage the memory iff `mpfr_init', through GMP Custom Allocation, allocates its memory on the application stack. -- Function: size_t mpfr_custom_get_size (mpfr_prec_t PREC) Return the needed size in bytes to store the significand of a floating-point number of precision PREC. -- Function: void mpfr_custom_init (void *SIGNIFICAND, mpfr_prec_t PREC) Initialize a significand of precision PREC, where SIGNIFICAND must be an area of `mpfr_custom_get_size (prec)' bytes at least and be suitably aligned for an array of `mp_limb_t' (GMP type, *note Internals::). -- Function: void mpfr_custom_init_set (mpfr_t X, int KIND, mpfr_exp_t EXP, mpfr_prec_t PREC, void *SIGNIFICAND) Perform a dummy initialization of a `mpfr_t' and set it to: * if `ABS(kind) == MPFR_NAN_KIND', X is set to NaN; * if `ABS(kind) == MPFR_INF_KIND', X is set to the infinity of sign `sign(kind)'; * if `ABS(kind) == MPFR_ZERO_KIND', X is set to the zero of sign `sign(kind)'; * if `ABS(kind) == MPFR_REGULAR_KIND', X is set to a regular number: `x = sign(kind)*significand*2^exp'. In all cases, it uses SIGNIFICAND directly for further computing involving X. It will not allocate anything. A floating-point number initialized with this function cannot be resized using `mpfr_set_prec' or `mpfr_prec_round', or cleared using `mpfr_clear'! The SIGNIFICAND must have been initialized with `mpfr_custom_init' using the same precision PREC. -- Function: int mpfr_custom_get_kind (mpfr_t X) Return the current kind of a `mpfr_t' as created by `mpfr_custom_init_set'. The behavior of this function for any `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined. -- Function: void * mpfr_custom_get_significand (mpfr_t X) Return a pointer to the significand used by a `mpfr_t' initialized with `mpfr_custom_init_set'. The behavior of this function for any `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined. -- Function: mpfr_exp_t mpfr_custom_get_exp (mpfr_t X) Return the exponent of X, assuming that X is a non-zero ordinary number. The return value for NaN, Infinity or zero is unspecified but does not produce any trap. The behavior of this function for any `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined. -- Function: void mpfr_custom_move (mpfr_t X, void *NEW_POSITION) Inform MPFR that the significand of X has moved due to a garbage collect and update its new position to `new_position'. However the application has to move the significand and the `mpfr_t' itself. The behavior of this function for any `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined.  File: mpfr.info, Node: Internals, Prev: Custom Interface, Up: MPFR Interface 5.16 Internals ============== A "limb" means the part of a multi-precision number that fits in a single word. Usually a limb contains 32 or 64 bits. The C data type for a limb is `mp_limb_t'. The `mpfr_t' type is internally defined as a one-element array of a structure, and `mpfr_ptr' is the C data type representing a pointer to this structure. The `mpfr_t' type consists of four fields: * The `_mpfr_prec' field is used to store the precision of the variable (in bits); this is not less than `MPFR_PREC_MIN'. * The `_mpfr_sign' field is used to store the sign of the variable. * The `_mpfr_exp' field stores the exponent. An exponent of 0 means a radix point just above the most significant limb. Non-zero values n are a multiplier 2^n relative to that point. A NaN, an infinity and a zero are indicated by special values of the exponent field. * Finally, the `_mpfr_d' field is a pointer to the limbs, least significant limbs stored first. The number of limbs in use is controlled by `_mpfr_prec', namely ceil(`_mpfr_prec'/`mp_bits_per_limb'). Non-singular (i.e., different from NaN, Infinity or zero) values always have the most significant bit of the most significant limb set to 1. When the precision does not correspond to a whole number of limbs, the excess bits at the low end of the data are zeros.  File: mpfr.info, Node: API Compatibility, Next: Contributors, Prev: MPFR Interface, Up: Top 6 API Compatibility ******************* The goal of this section is to describe some API changes that occurred from one version of MPFR to another, and how to write code that can be compiled and run with older MPFR versions. The minimum MPFR version that is considered here is 2.2.0 (released on 20 September 2005). API changes can only occur between major or minor versions. Thus the patchlevel (the third number in the MPFR version) will be ignored in the following. If a program does not use MPFR internals, changes in the behavior between two versions differing only by the patchlevel should only result from what was regarded as a bug or unspecified behavior. As a general rule, a program written for some MPFR version should work with later versions, possibly except at a new major version, where some features (described as obsolete for some time) can be removed. In such a case, a failure should occur during compilation or linking. If a result becomes incorrect because of such a change, please look at the various changes below (they are minimal, and most software should be unaffected), at the FAQ and at the MPFR web page for your version (a bug could have been introduced and be already fixed); and if the problem is not mentioned, please send us a bug report (*note Reporting Bugs::). However, a program written for the current MPFR version (as documented by this manual) may not necessarily work with previous versions of MPFR. This section should help developers to write portable code. Note: Information given here may be incomplete. API changes are also described in the NEWS file (for each version, instead of being classified like here), together with other changes. * Menu: * Type and Macro Changes:: * Added Functions:: * Changed Functions:: * Removed Functions:: * Other Changes::  File: mpfr.info, Node: Type and Macro Changes, Next: Added Functions, Prev: API Compatibility, Up: API Compatibility 6.1 Type and Macro Changes ========================== The official type for exponent values changed from `mp_exp_t' to `mpfr_exp_t' in MPFR 3.0. The type `mp_exp_t' will remain available as it comes from GMP (with a different meaning). These types are currently the same (`mpfr_exp_t' is defined as `mp_exp_t' with `typedef'), so that programs can still use `mp_exp_t'; but this may change in the future. Alternatively, using the following code after including `mpfr.h' will work with official MPFR versions, as `mpfr_exp_t' was never defined in MPFR 2.x: #if MPFR_VERSION_MAJOR < 3 typedef mp_exp_t mpfr_exp_t; #endif The official types for precision values and for rounding modes respectively changed from `mp_prec_t' and `mp_rnd_t' to `mpfr_prec_t' and `mpfr_rnd_t' in MPFR 3.0. This change was actually done a long time ago in MPFR, at least since MPFR 2.2.0, with the following code in `mpfr.h': #ifndef mp_rnd_t # define mp_rnd_t mpfr_rnd_t #endif #ifndef mp_prec_t # define mp_prec_t mpfr_prec_t #endif This means that it is safe to use the new official types `mpfr_prec_t' and `mpfr_rnd_t' in your programs. The types `mp_prec_t' and `mp_rnd_t' (defined in MPFR only) may be removed in the future, as the prefix `mp_' is reserved by GMP. The precision type `mpfr_prec_t' (`mp_prec_t') was unsigned before MPFR 3.0; it is now signed. `MPFR_PREC_MAX' has not changed, though. Indeed the MPFR code requires that `MPFR_PREC_MAX' be representable in the exponent type, which may have the same size as `mpfr_prec_t' but has always been signed. The consequence is that valid code that does not assume anything about the signedness of `mpfr_prec_t' should work with past and new MPFR versions. This change was useful as the use of unsigned types tends to convert signed values to unsigned ones in expressions due to the usual arithmetic conversions, which can yield incorrect results if a negative value is converted in such a way. Warning! A program assuming (intentionally or not) that `mpfr_prec_t' is signed may be affected by this problem when it is built and run against MPFR 2.x. The rounding modes `GMP_RNDx' were renamed to `MPFR_RNDx' in MPFR 3.0. However the old names `GMP_RNDx' have been kept for compatibility (this might change in future versions), using: #define GMP_RNDN MPFR_RNDN #define GMP_RNDZ MPFR_RNDZ #define GMP_RNDU MPFR_RNDU #define GMP_RNDD MPFR_RNDD The rounding mode "round away from zero" (`MPFR_RNDA') was added in MPFR 3.0 (however no rounding mode `GMP_RNDA' exists).  File: mpfr.info, Node: Added Functions, Next: Changed Functions, Prev: Type and Macro Changes, Up: API Compatibility 6.2 Added Functions =================== We give here in alphabetical order the functions that were added after MPFR 2.2, and in which MPFR version. * `mpfr_add_d' in MPFR 2.4. * `mpfr_ai' in MPFR 3.0 (incomplete, experimental). * `mpfr_asprintf' in MPFR 2.4. * `mpfr_buildopt_decimal_p' and `mpfr_buildopt_tls_p' in MPFR 3.0. * `mpfr_buildopt_gmpinternals_p' and `mpfr_buildopt_tune_case' in MPFR 3.1. * `mpfr_clear_divby0' in MPFR 3.1 (new divide-by-zero exception). * `mpfr_copysign' in MPFR 2.3. Note: MPFR 2.2 had a `mpfr_copysign' function that was available, but not documented, and with a slight difference in the semantics (when the second input operand is a NaN). * `mpfr_custom_get_significand' in MPFR 3.0. This function was named `mpfr_custom_get_mantissa' in previous versions; `mpfr_custom_get_mantissa' is still available via a macro in `mpfr.h': #define mpfr_custom_get_mantissa mpfr_custom_get_significand Thus code that needs to work with both MPFR 2.x and MPFR 3.x should use `mpfr_custom_get_mantissa'. * `mpfr_d_div' and `mpfr_d_sub' in MPFR 2.4. * `mpfr_digamma' in MPFR 3.0. * `mpfr_divby0_p' in MPFR 3.1 (new divide-by-zero exception). * `mpfr_div_d' in MPFR 2.4. * `mpfr_fmod' in MPFR 2.4. * `mpfr_fms' in MPFR 2.3. * `mpfr_fprintf' in MPFR 2.4. * `mpfr_frexp' in MPFR 3.1. * `mpfr_get_flt' in MPFR 3.0. * `mpfr_get_patches' in MPFR 2.3. * `mpfr_get_z_2exp' in MPFR 3.0. This function was named `mpfr_get_z_exp' in previous versions; `mpfr_get_z_exp' is still available via a macro in `mpfr.h': #define mpfr_get_z_exp mpfr_get_z_2exp Thus code that needs to work with both MPFR 2.x and MPFR 3.x should use `mpfr_get_z_exp'. * `mpfr_grandom' in MPFR 3.1. * `mpfr_j0', `mpfr_j1' and `mpfr_jn' in MPFR 2.3. * `mpfr_lgamma' in MPFR 2.3. * `mpfr_li2' in MPFR 2.4. * `mpfr_min_prec' in MPFR 3.0. * `mpfr_modf' in MPFR 2.4. * `mpfr_mul_d' in MPFR 2.4. * `mpfr_printf' in MPFR 2.4. * `mpfr_rec_sqrt' in MPFR 2.4. * `mpfr_regular_p' in MPFR 3.0. * `mpfr_remainder' and `mpfr_remquo' in MPFR 2.3. * `mpfr_set_divby0' in MPFR 3.1 (new divide-by-zero exception). * `mpfr_set_flt' in MPFR 3.0. * `mpfr_set_z_2exp' in MPFR 3.0. * `mpfr_set_zero' in MPFR 3.0. * `mpfr_setsign' in MPFR 2.3. * `mpfr_signbit' in MPFR 2.3. * `mpfr_sinh_cosh' in MPFR 2.4. * `mpfr_snprintf' and `mpfr_sprintf' in MPFR 2.4. * `mpfr_sub_d' in MPFR 2.4. * `mpfr_urandom' in MPFR 3.0. * `mpfr_vasprintf', `mpfr_vfprintf', `mpfr_vprintf', `mpfr_vsprintf' and `mpfr_vsnprintf' in MPFR 2.4. * `mpfr_y0', `mpfr_y1' and `mpfr_yn' in MPFR 2.3. * `mpfr_z_sub' in MPFR 3.1.  File: mpfr.info, Node: Changed Functions, Next: Removed Functions, Prev: Added Functions, Up: API Compatibility 6.3 Changed Functions ===================== The following functions have changed after MPFR 2.2. Changes can affect the behavior of code written for some MPFR version when built and run against another MPFR version (older or newer), as described below. * `mpfr_check_range' changed in MPFR 2.3.2 and MPFR 2.4. If the value is an inexact infinity, the overflow flag is now set (in case it was lost), while it was previously left unchanged. This is really what is expected in practice (and what the MPFR code was expecting), so that the previous behavior was regarded as a bug. Hence the change in MPFR 2.3.2. * `mpfr_get_f' changed in MPFR 3.0. This function was returning zero, except for NaN and Inf, which do not exist in MPF. The _erange_ flag is now set in these cases, and `mpfr_get_f' now returns the usual ternary value. * `mpfr_get_si', `mpfr_get_sj', `mpfr_get_ui' and `mpfr_get_uj' changed in MPFR 3.0. In previous MPFR versions, the cases where the _erange_ flag is set were unspecified. * `mpfr_get_z' changed in MPFR 3.0. The return type was `void'; it is now `int', and the usual ternary value is returned. Thus programs that need to work with both MPFR 2.x and 3.x must not use the return value. Even in this case, C code using `mpfr_get_z' as the second or third term of a conditional operator may also be affected. For instance, the following is correct with MPFR 3.0, but not with MPFR 2.x: bool ? mpfr_get_z(...) : mpfr_add(...); On the other hand, the following is correct with MPFR 2.x, but not with MPFR 3.0: bool ? mpfr_get_z(...) : (void) mpfr_add(...); Portable code should cast `mpfr_get_z(...)' to `void' to use the type `void' for both terms of the conditional operator, as in: bool ? (void) mpfr_get_z(...) : (void) mpfr_add(...); Alternatively, `if ... else' can be used instead of the conditional operator. Moreover the cases where the _erange_ flag is set were unspecified in MPFR 2.x. * `mpfr_get_z_exp' changed in MPFR 3.0. In previous MPFR versions, the cases where the _erange_ flag is set were unspecified. Note: this function has been renamed to `mpfr_get_z_2exp' in MPFR 3.0, but `mpfr_get_z_exp' is still available for compatibility reasons. * `mpfr_strtofr' changed in MPFR 2.3.1 and MPFR 2.4. This was actually a bug fix since the code and the documentation did not match. But both were changed in order to have a more consistent and useful behavior. The main changes in the code are as follows. The binary exponent is now accepted even without the `0b' or `0x' prefix. Data corresponding to NaN can now have an optional sign (such data were previously invalid). * `mpfr_strtofr' changed in MPFR 3.0. This function now accepts bases from 37 to 62 (no changes for the other bases). Note: if an unsupported base is provided to this function, the behavior is undefined; more precisely, in MPFR 2.3.1 and later, providing an unsupported base yields an assertion failure (this behavior may change in the future). * `mpfr_subnormalize' changed in MPFR 3.1. This was actually regarded as a bug fix. The `mpfr_subnormalize' implementation up to MPFR 3.0.0 did not change the flags. In particular, it did not follow the generic rule concerning the inexact flag (and no special behavior was specified). The case of the underflow flag was more a lack of specification. * `mpfr_urandom' and `mpfr_urandomb' changed in MPFR 3.1. Their behavior no longer depends on the platform (assuming this is also true for GMP's random generator, which is not the case between GMP 4.1 and 4.2 if `gmp_randinit_default' is used). As a consequence, the returned values can be different between MPFR 3.1 and previous MPFR versions. Note: as the reproducibility of these functions was not specified before MPFR 3.1, the MPFR 3.1 behavior is _not_ regarded as backward incompatible with previous versions.  File: mpfr.info, Node: Removed Functions, Next: Other Changes, Prev: Changed Functions, Up: API Compatibility 6.4 Removed Functions ===================== Functions `mpfr_random' and `mpfr_random2' have been removed in MPFR 3.0 (this only affects old code built against MPFR 3.0 or later). (The function `mpfr_random' had been deprecated since at least MPFR 2.2.0, and `mpfr_random2' since MPFR 2.4.0.)  File: mpfr.info, Node: Other Changes, Prev: Removed Functions, Up: API Compatibility 6.5 Other Changes ================= For users of a C++ compiler, the way how the availability of `intmax_t' is detected has changed in MPFR 3.0. In MPFR 2.x, if a macro `INTMAX_C' or `UINTMAX_C' was defined (e.g. when the `__STDC_CONSTANT_MACROS' macro had been defined before `' or `' has been included), `intmax_t' was assumed to be defined. However this was not always the case (more precisely, `intmax_t' can be defined only in the namespace `std', as with Boost), so that compilations could fail. Thus the check for `INTMAX_C' or `UINTMAX_C' is now disabled for C++ compilers, with the following consequences: * Programs written for MPFR 2.x that need `intmax_t' may no longer be compiled against MPFR 3.0: a `#define MPFR_USE_INTMAX_T' may be necessary before `mpfr.h' is included. * The compilation of programs that work with MPFR 3.0 may fail with MPFR 2.x due to the problem described above. Workarounds are possible, such as defining `intmax_t' and `uintmax_t' in the global namespace, though this is not clean. The divide-by-zero exception is new in MPFR 3.1. However it should not introduce incompatible changes for programs that strictly follow the MPFR API since the exception can only be seen via new functions. As of MPFR 3.1, the `mpfr.h' header can be included several times, while still supporting optional functions (*note Headers and Libraries::).  File: mpfr.info, Node: Contributors, Next: References, Prev: API Compatibility, Up: Top Contributors ************ The main developers of MPFR are Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, Philippe Théveny and Paul Zimmermann. Sylvie Boldo from ENS-Lyon, France, contributed the functions `mpfr_agm' and `mpfr_log'. Sylvain Chevillard contributed the `mpfr_ai' function. David Daney contributed the hyperbolic and inverse hyperbolic functions, the base-2 exponential, and the factorial function. Alain Delplanque contributed the new version of the `mpfr_get_str' function. Mathieu Dutour contributed the functions `mpfr_acos', `mpfr_asin' and `mpfr_atan', and a previous version of `mpfr_gamma'. Laurent Fousse contributed the `mpfr_sum' function. Emmanuel Jeandel, from ENS-Lyon too, contributed the generic hypergeometric code, as well as the internal function `mpfr_exp3', a first implementation of the sine and cosine, and improved versions of `mpfr_const_log2' and `mpfr_const_pi'. Ludovic Meunier helped in the design of the `mpfr_erf' code. Jean-Luc Rémy contributed the `mpfr_zeta' code. Fabrice Rouillier contributed the `mpfr_xxx_z' and `mpfr_xxx_q' functions, and helped to the Microsoft Windows porting. Damien Stehlé contributed the `mpfr_get_ld_2exp' function. We would like to thank Jean-Michel Muller and Joris van der Hoeven for very fruitful discussions at the beginning of that project, Torbjörn Granlund and Kevin Ryde for their help about design issues, and Nathalie Revol for her careful reading of a previous version of this documentation. In particular Kevin Ryde did a tremendous job for the portability of MPFR in 2002-2004. The development of the MPFR library would not have been possible without the continuous support of INRIA, and of the LORIA (Nancy, France) and LIP (Lyon, France) laboratories. In particular the main authors were or are members of the PolKA, Spaces, Cacao and Caramel project-teams at LORIA and of the Arénaire and AriC project-teams at LIP. This project was started during the Fiable (reliable in French) action supported by INRIA, and continued during the AOC action. The development of MPFR was also supported by a grant (202F0659 00 MPN 121) from the Conseil Régional de Lorraine in 2002, from INRIA by an "associate engineer" grant (2003-2005), an "opération de développement logiciel" grant (2007-2009), and the post-doctoral grant of Sylvain Chevillard in 2009-2010. The MPFR-MPC workshop in June 2012 was partly supported by the ERC grant ANTICS of Andreas Enge.  File: mpfr.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top References ********** * Richard Brent and Paul Zimmermann, "Modern Computer Arithmetic", Cambridge University Press (to appear), also available from the authors' web pages. * Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier and Paul Zimmermann, "MPFR: A Multiple-Precision Binary Floating-Point Library With Correct Rounding", ACM Transactions on Mathematical Software, volume 33, issue 2, article 13, 15 pages, 2007, `http://doi.acm.org/10.1145/1236463.1236468'. * Torbjörn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic Library", version 5.0.1, 2010, `http://gmplib.org'. * IEEE standard for binary floating-point arithmetic, Technical Report ANSI-IEEE Standard 754-1985, New York, 1985. Approved March 21, 1985: IEEE Standards Board; approved July 26, 1985: American National Standards Institute, 18 pages. * IEEE Standard for Floating-Point Arithmetic, ANSI-IEEE Standard 754-2008, 2008. Revision of ANSI-IEEE Standard 754-1985, approved June 12, 2008: IEEE Standards Board, 70 pages. * Donald E. Knuth, "The Art of Computer Programming", vol 2, "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981. * Jean-Michel Muller, "Elementary Functions, Algorithms and Implementation", Birkhäuser, Boston, 2nd edition, 2006. * Jean-Michel Muller, Nicolas Brisebarre, Florent de Dinechin, Claude-Pierre Jeannerod, Vincent Lefèvre, Guillaume Melquiond, Nathalie Revol, Damien Stehlé and Serge Torrès, "Handbook of Floating-Point Arithmetic", Birkhäuser, Boston, 2009.  File: mpfr.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top Appendix A GNU Free Documentation License ***************************************** Version 1.2, November 2002 Copyright (C) 2000,2001,2002 Free Software Foundation, Inc. 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 0. PREAMBLE The purpose of this License is to make a manual, textbook, or other functional and useful document "free" in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others. 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File: mpfr.info, Node: Concept Index, Next: Function and Type Index, Prev: GNU Free Documentation License, Up: Top Concept Index ************* [index] * Menu: * Accuracy: MPFR Interface. (line 25) * Arithmetic functions: Basic Arithmetic Functions. (line 3) * Assignment functions: Assignment Functions. (line 3) * Basic arithmetic functions: Basic Arithmetic Functions. (line 3) * Combined initialization and assignment functions: Combined Initialization and Assignment Functions. (line 3) * Comparison functions: Comparison Functions. (line 3) * Compatibility with MPF: Compatibility with MPF. (line 3) * Conditions for copying MPFR: Copying. (line 6) * Conversion functions: Conversion Functions. (line 3) * Copying conditions: Copying. (line 6) * Custom interface: Custom Interface. (line 3) * Exception related functions: Exception Related Functions. (line 3) * Float arithmetic functions: Basic Arithmetic Functions. (line 3) * Float comparisons functions: Comparison Functions. (line 3) * Float functions: MPFR Interface. (line 6) * Float input and output functions: Input and Output Functions. (line 3) * Float output functions: Formatted Output Functions. (line 3) * Floating-point functions: MPFR Interface. (line 6) * Floating-point number: Nomenclature and Types. (line 6) * GNU Free Documentation License: GNU Free Documentation License. (line 6) * I/O functions <1>: Formatted Output Functions. (line 3) * I/O functions: Input and Output Functions. (line 3) * Initialization functions: Initialization Functions. (line 3) * Input functions: Input and Output Functions. (line 3) * Installation: Installing MPFR. (line 6) * Integer related functions: Integer Related Functions. (line 3) * Internals: Internals. (line 3) * intmax_t: Headers and Libraries. (line 22) * inttypes.h: Headers and Libraries. (line 22) * libmpfr: Headers and Libraries. (line 50) * Libraries: Headers and Libraries. (line 50) * Libtool: Headers and Libraries. (line 56) * Limb: Internals. (line 6) * Linking: Headers and Libraries. (line 50) * Miscellaneous float functions: Miscellaneous Functions. (line 3) * mpfr.h: Headers and Libraries. (line 6) * Output functions <1>: Formatted Output Functions. (line 3) * Output functions: Input and Output Functions. (line 3) * Precision <1>: MPFR Interface. (line 17) * Precision: Nomenclature and Types. (line 20) * Reporting bugs: Reporting Bugs. (line 6) * Rounding mode related functions: Rounding Related Functions. (line 3) * Rounding Modes: Nomenclature and Types. (line 34) * Special functions: Special Functions. (line 3) * stdarg.h: Headers and Libraries. (line 19) * stdint.h: Headers and Libraries. (line 22) * stdio.h: Headers and Libraries. (line 12) * Ternary value: Rounding Modes. (line 29) * uintmax_t: Headers and Libraries. (line 22)  File: mpfr.info, Node: Function and Type Index, Prev: Concept Index, Up: Top Function and Type Index *********************** [index] * Menu: * mpfr_abs: Basic Arithmetic Functions. (line 175) * mpfr_acos: Special Functions. (line 52) * mpfr_acosh: Special Functions. (line 136) * mpfr_add: Basic Arithmetic Functions. (line 8) * mpfr_add_d: Basic Arithmetic Functions. (line 14) * mpfr_add_q: Basic Arithmetic Functions. (line 18) * mpfr_add_si: Basic Arithmetic Functions. (line 12) * mpfr_add_ui: Basic Arithmetic Functions. (line 10) * mpfr_add_z: Basic Arithmetic Functions. (line 16) * mpfr_agm: Special Functions. (line 232) * mpfr_ai: Special Functions. (line 248) * mpfr_asin: Special Functions. (line 53) * mpfr_asinh: Special Functions. (line 137) * mpfr_asprintf: Formatted Output Functions. (line 194) * mpfr_atan: Special Functions. (line 54) * mpfr_atan2: Special Functions. (line 65) * mpfr_atanh: Special Functions. (line 138) * mpfr_buildopt_decimal_p: Miscellaneous Functions. (line 163) * mpfr_buildopt_gmpinternals_p: Miscellaneous Functions. (line 168) * mpfr_buildopt_tls_p: Miscellaneous Functions. (line 157) * mpfr_buildopt_tune_case: Miscellaneous Functions. (line 173) * mpfr_can_round: Rounding Related Functions. (line 37) * mpfr_cbrt: Basic Arithmetic Functions. (line 109) * mpfr_ceil: Integer Related Functions. (line 8) * mpfr_check_range: Exception Related Functions. (line 38) * mpfr_clear: Initialization Functions. (line 31) * mpfr_clear_divby0: Exception Related Functions. (line 113) * mpfr_clear_erangeflag: Exception Related Functions. (line 116) * mpfr_clear_flags: Exception Related Functions. (line 129) * mpfr_clear_inexflag: Exception Related Functions. (line 115) * mpfr_clear_nanflag: Exception Related Functions. (line 114) * mpfr_clear_overflow: Exception Related Functions. (line 112) * mpfr_clear_underflow: Exception Related Functions. (line 111) * mpfr_clears: Initialization Functions. (line 36) * mpfr_cmp: Comparison Functions. (line 7) * mpfr_cmp_d: Comparison Functions. (line 10) * mpfr_cmp_f: Comparison Functions. (line 14) * mpfr_cmp_ld: Comparison Functions. (line 11) * mpfr_cmp_q: Comparison Functions. (line 13) * mpfr_cmp_si: Comparison Functions. (line 9) * mpfr_cmp_si_2exp: Comparison Functions. (line 31) * mpfr_cmp_ui: Comparison Functions. (line 8) * mpfr_cmp_ui_2exp: Comparison Functions. (line 29) * mpfr_cmp_z: Comparison Functions. (line 12) * mpfr_cmpabs: Comparison Functions. (line 35) * mpfr_const_catalan: Special Functions. (line 259) * mpfr_const_euler: Special Functions. (line 258) * mpfr_const_log2: Special Functions. (line 256) * mpfr_const_pi: Special Functions. (line 257) * mpfr_copysign: Miscellaneous Functions. (line 111) * mpfr_cos: Special Functions. (line 30) * mpfr_cosh: Special Functions. (line 115) * mpfr_cot: Special Functions. (line 48) * mpfr_coth: Special Functions. (line 132) * mpfr_csc: Special Functions. (line 47) * mpfr_csch: Special Functions. (line 131) * mpfr_custom_get_exp: Custom Interface. (line 78) * mpfr_custom_get_kind: Custom Interface. (line 67) * mpfr_custom_get_significand: Custom Interface. (line 72) * mpfr_custom_get_size: Custom Interface. (line 36) * mpfr_custom_init: Custom Interface. (line 41) * mpfr_custom_init_set: Custom Interface. (line 48) * mpfr_custom_move: Custom Interface. (line 85) * mpfr_d_div: Basic Arithmetic Functions. (line 84) * mpfr_d_sub: Basic Arithmetic Functions. (line 37) * MPFR_DECL_INIT: Initialization Functions. (line 75) * mpfr_digamma: Special Functions. (line 187) * mpfr_dim: Basic Arithmetic Functions. (line 182) * mpfr_div: Basic Arithmetic Functions. (line 74) * mpfr_div_2exp: Compatibility with MPF. (line 51) * mpfr_div_2si: Basic Arithmetic Functions. (line 197) * mpfr_div_2ui: Basic Arithmetic Functions. (line 195) * mpfr_div_d: Basic Arithmetic Functions. (line 86) * mpfr_div_q: Basic Arithmetic Functions. (line 90) * mpfr_div_si: Basic Arithmetic Functions. (line 82) * mpfr_div_ui: Basic Arithmetic Functions. (line 78) * mpfr_div_z: Basic Arithmetic Functions. (line 88) * mpfr_divby0_p: Exception Related Functions. (line 135) * mpfr_eint: Special Functions. (line 154) * mpfr_eq: Compatibility with MPF. (line 30) * mpfr_equal_p: Comparison Functions. (line 61) * mpfr_erangeflag_p: Exception Related Functions. (line 138) * mpfr_erf: Special Functions. (line 198) * mpfr_erfc: Special Functions. (line 199) * mpfr_exp: Special Functions. (line 24) * mpfr_exp10: Special Functions. (line 26) * mpfr_exp2: Special Functions. (line 25) * mpfr_expm1: Special Functions. (line 150) * mpfr_fac_ui: Special Functions. (line 143) * mpfr_fits_intmax_p: Conversion Functions. (line 146) * mpfr_fits_sint_p: Conversion Functions. (line 142) * mpfr_fits_slong_p: Conversion Functions. (line 140) * mpfr_fits_sshort_p: Conversion Functions. (line 144) * mpfr_fits_uint_p: Conversion Functions. (line 141) * mpfr_fits_uintmax_p: Conversion Functions. (line 145) * mpfr_fits_ulong_p: Conversion Functions. (line 139) * mpfr_fits_ushort_p: Conversion Functions. (line 143) * mpfr_floor: Integer Related Functions. (line 9) * mpfr_fma: Special Functions. (line 225) * mpfr_fmod: Integer Related Functions. (line 79) * mpfr_fms: Special Functions. (line 227) * mpfr_fprintf: Formatted Output Functions. (line 158) * mpfr_frac: Integer Related Functions. (line 62) * mpfr_free_cache: Special Functions. (line 266) * mpfr_free_str: Conversion Functions. (line 133) * mpfr_frexp: Conversion Functions. (line 47) * mpfr_gamma: Special Functions. (line 169) * mpfr_get_d: Conversion Functions. (line 8) * mpfr_get_d_2exp: Conversion Functions. (line 34) * mpfr_get_decimal64: Conversion Functions. (line 10) * mpfr_get_default_prec: Initialization Functions. (line 114) * mpfr_get_default_rounding_mode: Rounding Related Functions. (line 11) * mpfr_get_emax: Exception Related Functions. (line 8) * mpfr_get_emax_max: Exception Related Functions. (line 31) * mpfr_get_emax_min: Exception Related Functions. (line 30) * mpfr_get_emin: Exception Related Functions. (line 7) * mpfr_get_emin_max: Exception Related Functions. (line 29) * mpfr_get_emin_min: Exception Related Functions. (line 28) * mpfr_get_exp: Miscellaneous Functions. (line 89) * mpfr_get_f: Conversion Functions. (line 73) * mpfr_get_flt: Conversion Functions. (line 7) * mpfr_get_ld: Conversion Functions. (line 9) * mpfr_get_ld_2exp: Conversion Functions. (line 36) * mpfr_get_patches: Miscellaneous Functions. (line 148) * mpfr_get_prec: Initialization Functions. (line 147) * mpfr_get_si: Conversion Functions. (line 20) * mpfr_get_sj: Conversion Functions. (line 22) * mpfr_get_str: Conversion Functions. (line 87) * mpfr_get_ui: Conversion Functions. (line 21) * mpfr_get_uj: Conversion Functions. (line 23) * mpfr_get_version: Miscellaneous Functions. (line 117) * mpfr_get_z: Conversion Functions. (line 68) * mpfr_get_z_2exp: Conversion Functions. (line 55) * mpfr_grandom: Miscellaneous Functions. (line 65) * mpfr_greater_p: Comparison Functions. (line 57) * mpfr_greaterequal_p: Comparison Functions. (line 58) * mpfr_hypot: Special Functions. (line 241) * mpfr_inexflag_p: Exception Related Functions. (line 137) * mpfr_inf_p: Comparison Functions. (line 42) * mpfr_init: Initialization Functions. (line 54) * mpfr_init2: Initialization Functions. (line 11) * mpfr_init_set: Combined Initialization and Assignment Functions. (line 7) * mpfr_init_set_d: Combined Initialization and Assignment Functions. (line 12) * mpfr_init_set_f: Combined Initialization and Assignment Functions. (line 17) * mpfr_init_set_ld: Combined Initialization and Assignment Functions. (line 14) * mpfr_init_set_q: Combined Initialization and Assignment Functions. (line 16) * mpfr_init_set_si: Combined Initialization and Assignment Functions. (line 11) * mpfr_init_set_str: Combined Initialization and Assignment Functions. (line 23) * mpfr_init_set_ui: Combined Initialization and Assignment Functions. (line 9) * mpfr_init_set_z: Combined Initialization and Assignment Functions. (line 15) * mpfr_inits: Initialization Functions. (line 63) * mpfr_inits2: Initialization Functions. (line 23) * mpfr_inp_str: Input and Output Functions. (line 33) * mpfr_integer_p: Integer Related Functions. (line 105) * mpfr_j0: Special Functions. (line 203) * mpfr_j1: Special Functions. (line 204) * mpfr_jn: Special Functions. (line 206) * mpfr_less_p: Comparison Functions. (line 59) * mpfr_lessequal_p: Comparison Functions. (line 60) * mpfr_lessgreater_p: Comparison Functions. (line 66) * mpfr_lgamma: Special Functions. (line 179) * mpfr_li2: Special Functions. (line 164) * mpfr_lngamma: Special Functions. (line 173) * mpfr_log: Special Functions. (line 17) * mpfr_log10: Special Functions. (line 19) * mpfr_log1p: Special Functions. (line 146) * mpfr_log2: Special Functions. (line 18) * mpfr_max: Miscellaneous Functions. (line 24) * mpfr_min: Miscellaneous Functions. (line 22) * mpfr_min_prec: Rounding Related Functions. (line 59) * mpfr_modf: Integer Related Functions. (line 69) * mpfr_mul: Basic Arithmetic Functions. (line 53) * mpfr_mul_2exp: Compatibility with MPF. (line 49) * mpfr_mul_2si: Basic Arithmetic Functions. (line 190) * mpfr_mul_2ui: Basic Arithmetic Functions. (line 188) * mpfr_mul_d: Basic Arithmetic Functions. (line 59) * mpfr_mul_q: Basic Arithmetic Functions. (line 63) * mpfr_mul_si: Basic Arithmetic Functions. (line 57) * mpfr_mul_ui: Basic Arithmetic Functions. (line 55) * mpfr_mul_z: Basic Arithmetic Functions. (line 61) * mpfr_nan_p: Comparison Functions. (line 41) * mpfr_nanflag_p: Exception Related Functions. (line 136) * mpfr_neg: Basic Arithmetic Functions. (line 174) * mpfr_nextabove: Miscellaneous Functions. (line 16) * mpfr_nextbelow: Miscellaneous Functions. (line 17) * mpfr_nexttoward: Miscellaneous Functions. (line 7) * mpfr_number_p: Comparison Functions. (line 43) * mpfr_out_str: Input and Output Functions. (line 17) * mpfr_overflow_p: Exception Related Functions. (line 134) * mpfr_pow: Basic Arithmetic Functions. (line 118) * mpfr_pow_si: Basic Arithmetic Functions. (line 122) * mpfr_pow_ui: Basic Arithmetic Functions. (line 120) * mpfr_pow_z: Basic Arithmetic Functions. (line 124) * mpfr_prec_round: Rounding Related Functions. (line 15) * mpfr_prec_t: Nomenclature and Types. (line 20) * mpfr_print_rnd_mode: Rounding Related Functions. (line 66) * mpfr_printf: Formatted Output Functions. (line 165) * mpfr_rec_sqrt: Basic Arithmetic Functions. (line 104) * mpfr_regular_p: Comparison Functions. (line 45) * mpfr_reldiff: Compatibility with MPF. (line 41) * mpfr_remainder: Integer Related Functions. (line 81) * mpfr_remquo: Integer Related Functions. (line 83) * mpfr_rint: Integer Related Functions. (line 7) * mpfr_rint_ceil: Integer Related Functions. (line 38) * mpfr_rint_floor: Integer Related Functions. (line 40) * mpfr_rint_round: Integer Related Functions. (line 42) * mpfr_rint_trunc: Integer Related Functions. (line 44) * mpfr_rnd_t: Nomenclature and Types. (line 34) * mpfr_root: Basic Arithmetic Functions. (line 111) * mpfr_round: Integer Related Functions. (line 10) * mpfr_sec: Special Functions. (line 46) * mpfr_sech: Special Functions. (line 130) * mpfr_set: Assignment Functions. (line 10) * mpfr_set_d: Assignment Functions. (line 17) * mpfr_set_decimal64: Assignment Functions. (line 21) * mpfr_set_default_prec: Initialization Functions. (line 101) * mpfr_set_default_rounding_mode: Rounding Related Functions. (line 7) * mpfr_set_divby0: Exception Related Functions. (line 122) * mpfr_set_emax: Exception Related Functions. (line 17) * mpfr_set_emin: Exception Related Functions. (line 16) * mpfr_set_erangeflag: Exception Related Functions. (line 125) * mpfr_set_exp: Miscellaneous Functions. (line 94) * mpfr_set_f: Assignment Functions. (line 24) * mpfr_set_flt: Assignment Functions. (line 16) * mpfr_set_inexflag: Exception Related Functions. (line 124) * mpfr_set_inf: Assignment Functions. (line 147) * mpfr_set_ld: Assignment Functions. (line 19) * mpfr_set_nan: Assignment Functions. (line 146) * mpfr_set_nanflag: Exception Related Functions. (line 123) * mpfr_set_overflow: Exception Related Functions. (line 121) * mpfr_set_prec: Initialization Functions. (line 137) * mpfr_set_prec_raw: Compatibility with MPF. (line 23) * mpfr_set_q: Assignment Functions. (line 23) * mpfr_set_si: Assignment Functions. (line 13) * mpfr_set_si_2exp: Assignment Functions. (line 53) * mpfr_set_sj: Assignment Functions. (line 15) * mpfr_set_sj_2exp: Assignment Functions. (line 57) * mpfr_set_str: Assignment Functions. (line 65) * mpfr_set_ui: Assignment Functions. (line 12) * mpfr_set_ui_2exp: Assignment Functions. (line 51) * mpfr_set_uj: Assignment Functions. (line 14) * mpfr_set_uj_2exp: Assignment Functions. (line 55) * mpfr_set_underflow: Exception Related Functions. (line 120) * mpfr_set_z: Assignment Functions. (line 22) * mpfr_set_z_2exp: Assignment Functions. (line 59) * mpfr_set_zero: Assignment Functions. (line 148) * mpfr_setsign: Miscellaneous Functions. (line 105) * mpfr_sgn: Comparison Functions. (line 51) * mpfr_si_div: Basic Arithmetic Functions. (line 80) * mpfr_si_sub: Basic Arithmetic Functions. (line 33) * mpfr_signbit: Miscellaneous Functions. (line 100) * mpfr_sin: Special Functions. (line 31) * mpfr_sin_cos: Special Functions. (line 37) * mpfr_sinh: Special Functions. (line 116) * mpfr_sinh_cosh: Special Functions. (line 122) * mpfr_snprintf: Formatted Output Functions. (line 182) * mpfr_sprintf: Formatted Output Functions. (line 171) * mpfr_sqr: Basic Arithmetic Functions. (line 70) * mpfr_sqrt: Basic Arithmetic Functions. (line 97) * mpfr_sqrt_ui: Basic Arithmetic Functions. (line 99) * mpfr_strtofr: Assignment Functions. (line 83) * mpfr_sub: Basic Arithmetic Functions. (line 27) * mpfr_sub_d: Basic Arithmetic Functions. (line 39) * mpfr_sub_q: Basic Arithmetic Functions. (line 45) * mpfr_sub_si: Basic Arithmetic Functions. (line 35) * mpfr_sub_ui: Basic Arithmetic Functions. (line 31) * mpfr_sub_z: Basic Arithmetic Functions. (line 43) * mpfr_subnormalize: Exception Related Functions. (line 61) * mpfr_sum: Special Functions. (line 275) * mpfr_swap: Assignment Functions. (line 154) * mpfr_t: Nomenclature and Types. (line 6) * mpfr_tan: Special Functions. (line 32) * mpfr_tanh: Special Functions. (line 117) * mpfr_trunc: Integer Related Functions. (line 11) * mpfr_ui_div: Basic Arithmetic Functions. (line 76) * mpfr_ui_pow: Basic Arithmetic Functions. (line 128) * mpfr_ui_pow_ui: Basic Arithmetic Functions. (line 126) * mpfr_ui_sub: Basic Arithmetic Functions. (line 29) * mpfr_underflow_p: Exception Related Functions. (line 133) * mpfr_unordered_p: Comparison Functions. (line 71) * mpfr_urandom: Miscellaneous Functions. (line 50) * mpfr_urandomb: Miscellaneous Functions. (line 30) * mpfr_vasprintf: Formatted Output Functions. (line 196) * MPFR_VERSION: Miscellaneous Functions. (line 120) * MPFR_VERSION_MAJOR: Miscellaneous Functions. (line 121) * MPFR_VERSION_MINOR: Miscellaneous Functions. (line 122) * MPFR_VERSION_NUM: Miscellaneous Functions. (line 140) * MPFR_VERSION_PATCHLEVEL: Miscellaneous Functions. (line 123) * MPFR_VERSION_STRING: Miscellaneous Functions. (line 124) * mpfr_vfprintf: Formatted Output Functions. (line 160) * mpfr_vprintf: Formatted Output Functions. (line 166) * mpfr_vsnprintf: Formatted Output Functions. (line 184) * mpfr_vsprintf: Formatted Output Functions. (line 173) * mpfr_y0: Special Functions. (line 214) * mpfr_y1: Special Functions. (line 215) * mpfr_yn: Special Functions. (line 217) * mpfr_z_sub: Basic Arithmetic Functions. (line 41) * mpfr_zero_p: Comparison Functions. (line 44) * mpfr_zeta: Special Functions. (line 192) * mpfr_zeta_ui: Special Functions. (line 194)  Tag Table: Node: Top892 Node: Copying2243 Node: Introduction to MPFR4003 Node: Installing MPFR6092 Node: Reporting Bugs10914 Node: MPFR Basics12843 Node: Headers and Libraries13159 Node: Nomenclature and Types16143 Node: MPFR Variable Conventions18147 Node: Rounding Modes19677 Ref: ternary value20774 Node: Floating-Point Values on Special Numbers22727 Node: Exceptions25703 Node: Memory Handling28855 Node: MPFR Interface29987 Node: Initialization Functions32083 Node: Assignment Functions38997 Node: Combined Initialization and Assignment Functions47651 Node: Conversion Functions48944 Node: Basic Arithmetic Functions57496 Node: Comparison Functions66504 Node: Special Functions69986 Node: Input and Output Functions83739 Node: Formatted Output Functions85662 Node: Integer Related Functions94781 Node: Rounding Related Functions100543 Node: Miscellaneous Functions104157 Node: Exception Related Functions112724 Node: Compatibility with MPF119478 Node: Custom Interface122166 Node: Internals126411 Node: API Compatibility127895 Node: Type and Macro Changes129825 Node: Added Functions132546 Node: Changed Functions135489 Node: Removed Functions139770 Node: Other Changes140182 Node: Contributors141711 Node: References144285 Node: GNU Free Documentation License146026 Node: Concept Index168469 Node: Function and Type Index174388  End Tag Table  Local Variables: coding: utf-8 End: @