path: root/doc/mpfr.texi

\input texinfo    @c -*-texinfo-*-
@c %**start of header
@setfilename mpfr.info
@documentencoding UTF-8
@set VERSION 4.2.0-dev
@set UPDATED-MONTH August 2021
@settitle GNU MPFR @value{VERSION}
@synindex tp fn
@iftex
@afourpaper
@end iftex
@comment %**end of header

@c Note: avoid using non-ASCII characters directly when possible,
@c as the "info" utility cannot currently handle them.
@c   https://bugs.debian.org/cgi-bin/bugreport.cgi?bug=212549

@c Warning! If a macro is used with @iftex, it must also be defined for
@c info at least (e.g. with @ifnottex) in order to avoid a failure with
@c makeinfo 5.2 (there is no such problem with makeinfo 6.5).

@copying
This manual documents how to install and use the Multiple Precision
Floating-Point Reliable Library, version @value{VERSION}.

Copyright 1991, 1993-2021 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 @ref{GNU Free Documentation License}.
@end copying

@dircategory Software libraries
@direntry
* mpfr: (mpfr).                 Multiple Precision Floating-Point Reliable Library.
@end direntry

@c  html <meta name=description content="...">
@documentdescription
How to install and use GNU MPFR, a library for reliable multiple precision
floating-point arithmetic, version @value{VERSION}.
@end documentdescription

@c smallbook
@finalout
@setchapternewpage on

@ifnottex
@node Top, Copying, (dir), (dir)
@top GNU MPFR
@end ifnottex

@iftex
@titlepage
@title GNU MPFR
@subtitle The Multiple Precision Floating-Point Reliable Library
@subtitle Edition @value{VERSION}
@subtitle @value{UPDATED-MONTH}

@author The MPFR team
@email{mpfr@@inria.fr}

@c Include the Distribution inside the titlepage so
@c that headings are turned off.

@tex
\global\parindent=0pt
\global\parskip=8pt
\global\baselineskip=13pt
@end tex

@page
@vskip 0pt plus 1filll
@end iftex

@insertcopying
@ifnottex
@sp 1
@end ifnottex

@iftex
@end titlepage
@headings double
@end iftex

@c  Don't bother with contents for html, the menus seem adequate.
@ifnothtml
@contents
@end ifnothtml

@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.
* MPFR and the IEEE 754 Standard::
* Contributors::
* References::
* GNU Free Documentation License::
* Concept Index::
* Function and Type Index::
@end menu


@c  @m{T,N} is $T$ in tex or @math{N} otherwise.  This is an easy way to give
@c  different forms for math in tex and info.  Commas in N or T don't work,
@c  but @C{} can be used instead.  \, works in info but not in tex.
@iftex
@macro m {T,N}
@tex$\T\$@end tex
@end macro
@end iftex
@ifnottex
@macro m {T,N}
@math{\N\}
@end macro
@end ifnottex

@c  Usage: @GMPabs{x}
@c  Give either |x| in tex, or abs(x) in info or html.
@c  The \ensuremath is needed because the OT1 encoding is used, where
@c  the pipe character corresponds to a wide dash:
@c    https://tex.stackexchange.com/a/1775/58921
@tex
\gdef\GMPabs#1{\ensuremath{|#1|}}
@end tex
@ifnottex
@macro GMPabs {X}
@abs{}(\X\)
@end macro
@end ifnottex

@c  Usage: @GMPtimes{}
@c  Give either \times or the word "times".
@tex
\gdef\GMPtimes{\times}
@end tex
@ifnottex
@macro GMPtimes
times
@end macro
@end ifnottex

@c  New math operators.
@c  @abs{} can be used in both tex and info, or just \abs in tex.
@tex
\gdef\abs{\mathop{\rm abs}}
@end tex
@ifnottex
@macro abs
abs
@end macro
@end ifnottex

@c  @times{} made available as a "*" in info and html (already works in tex).
@ifnottex
@macro times
*
@end macro
@end ifnottex

@c  Math operators already available in tex, made available in info too.
@c  For example @log{} can be used in both tex and info.
@ifnottex
@macro le
<=
@end macro
@macro ge
>=
@end macro
@macro ne
<>
@end macro
@macro log
log
@end macro
@end ifnottex

@c  @pom{} definition
@tex
\gdef\pom{\ifmmode\pm\else$\pm$\fi}
@end tex
@ifnottex
@macro pom
±
@end macro
@end ifnottex

@c The following macro have been copied from gmp.texi
@c
@c  Usage: @MPFRpxreftop{info,title}
@c
@c  Like @pxref{}, but designed for a reference to the top of a document, not
@c  a particular section.
@c
@c  The texinfo manual recommends putting a likely section name in references
@c  like this, eg. "Introduction", but it seems better to just give the title.
@c
@iftex
@macro MPFRpxreftop{info,title}
see @cite{\title\}.
@end macro
@end iftex
@ifhtml
@macro MPFRpxreftop{info,title}
see @cite{\title\}.
@end macro
@end ifhtml
@ifnottex
@ifnothtml
@macro MPFRpxreftop{info,title}
@pxref{Top,\title\,\title\,\info\,\title\}
@end macro
@end ifnothtml
@end ifnottex

@node Copying, Introduction to MPFR, Top, Top
@comment  node-name, next, previous,  up
@unnumbered MPFR Copying Conditions
@cindex Copying conditions
@cindex Conditions for copying MPFR

The GNU MPFR library (or MPFR for short)
is @dfn{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.@refill

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.@refill

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.@refill

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.@refill

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.@refill

@node Introduction to MPFR, Installing MPFR, Copying, Top
@comment  node-name,  next,  previous,  up
@chapter 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:

@itemize @bullet

@item the MPFR code is portable, i.e., the result of any operation
does not depend on the machine word size
@code{mp_bits_per_limb} (64 on most current processors), possibly
except in faithful rounding.
It does not depend either on the machine rounding mode or rounding precision;

@item the precision in bits can be set @emph{exactly} to any valid value
for each variable (including very small precision);

@item MPFR provides the four rounding modes from the IEEE@tie{}754-1985
standard, plus away-from-zero, as well as for basic operations as for other
mathematical functions. Faithful rounding (partially supported) is provided
too, but the results may no longer be reproducible.

@end itemize

In particular, MPFR follows the specification of the IEEE@tie{}754 standard,
currently IEEE@tie{}754-2019 (which will be referred to as IEEE@tie{}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.

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.

@section How to Use This Manual

Everyone should read @ref{MPFR Basics}.  If you need to install the library
yourself, you need to read @ref{Installing MPFR}, too.
To use the library you will need to refer to @ref{MPFR Interface}.

The rest of the manual can be used for later reference, although it is
probably a good idea to glance through it.

@node Installing MPFR, Reporting Bugs, Introduction to MPFR, Top
@comment  node-name,  next,  previous,  up
@chapter Installing MPFR
@cindex Installation

The MPFR library is already installed on some GNU/Linux distributions,
but the development files necessary to the compilation such as
@file{mpfr.h} are not always present. To check that MPFR is fully
installed on your computer, you can check the presence of the file
@file{mpfr.h} in @file{/usr/include}, or try to compile a small program
having @code{#include <mpfr.h>} (since @file{mpfr.h} may be installed
somewhere else). For instance, you can try to compile:

@example
#include <stdio.h>
#include <mpfr.h>
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;
@}
@end example

@noindent
with

@example
cc -o version version.c -lmpfr -lgmp
@end example

@noindent
and if you get errors whose first line looks like

@example
version.c:2:19: error: mpfr.h: No such file or directory
@end example

@noindent
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.

@section How to Install

Here are the steps needed to install the library on Unix systems
(more details are provided in the @file{INSTALL} file):

@enumerate

@item
To build MPFR, you first have to install GNU MP (version 5.0.0 or higher)
on your computer.
You need a C compiler, preferably GCC, but any reasonable compiler should
work.  And you need the standard Unix @samp{make} command, plus some other
standard Unix utility commands.

Then, in the MPFR build directory, type the following commands.

@item
@samp{./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 @file{/usr/local}), threading support, and so on. See
the @file{INSTALL} file and/or the output of @samp{./configure --help}
for more information, in particular if you get error messages.

@item
@samp{make}

This will compile MPFR, and create a library archive file @file{libmpfr.a}.
On most platforms, a dynamic library will be produced too.

@item
@samp{make check}

This will make sure that MPFR was built correctly.
If any test fails, information about this failure can be found in the
@file{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 @samp{VERBOSE}
environment variable to 1 before running @samp{make check}, for instance
by typing:

@samp{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
@samp{mpfr@@inria.fr}. For details, see @ref{Reporting Bugs}.

@item
@samp{make install}

This will copy the files @file{mpfr.h} and @file{mpf2mpfr.h} to the directory
@file{/usr/local/include}, the library files (@file{libmpfr.a} and possibly
others) to the directory @file{/usr/local/lib}, the file @file{mpfr.info}
to the directory @file{/usr/local/share/info}, and some other documentation
files to the directory @file{/usr/local/share/doc/mpfr} (or if you passed the
@samp{--prefix} option to @file{configure}, using the prefix directory given
as argument to @samp{--prefix} instead of @file{/usr/local}).

@end enumerate

@section Other `make' Targets

There are some other useful make targets:

@itemize @bullet

@item
@samp{mpfr.info} or @samp{info}

Create or update an info version of the manual, in @file{mpfr.info}.

This file is already provided in the MPFR archives.

@item
@samp{mpfr.pdf} or @samp{pdf}

Create a PDF version of the manual, in @file{mpfr.pdf}.

@item
@samp{mpfr.dvi} or @samp{dvi}

Create a DVI version of the manual, in @file{mpfr.dvi}.

@item
@samp{mpfr.ps} or @samp{ps}

Create a Postscript version of the manual, in @file{mpfr.ps}.

@item
@samp{mpfr.html} or @samp{html}

Create a HTML version of the manual, in several pages in the directory
@file{doc/mpfr.html}; if you want only one output HTML file, then type
@samp{makeinfo --html --no-split mpfr.texi} from the @samp{doc} directory
instead.

@item
@samp{clean}

Delete all object files and archive files, but not the configuration files.

@item
@samp{distclean}

Delete all generated files not included in the distribution.

@item
@samp{uninstall}

Delete all files copied by @samp{make install}.

@end itemize


@section Build Problems

In case of problem, please read the @file{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
@url{https://www.mpfr.org/faq.html}.

@comment Warning! Do not split "MPFR ... @url{...}" across several lines
@comment as this needs to be updated with update-version.
Please report problems to the MPFR mailing-list @samp{mpfr@@inria.fr}.
@xref{Reporting Bugs}.
Some bug fixes are available on the
MPFR@tie{}4.1.0 web page @url{https://www.mpfr.org/mpfr-4.1.0/}.

@section Getting the Latest Version of MPFR

The latest version of MPFR is available from
@url{https://ftp.gnu.org/gnu/mpfr/} or @url{https://www.mpfr.org/}.

@node Reporting Bugs, MPFR Basics, Installing MPFR, Top
@comment  node-name,  next,  previous,  up
@chapter Reporting Bugs
@cindex Reporting bugs

@comment Warning! Do not split "MPFR ... @url{...}" across several lines
@comment as this needs to be updated with update-version.
If you think you have found a bug in the MPFR library, first have a look
on the MPFR@tie{}4.1.0 web page @url{https://www.mpfr.org/mpfr-4.1.0/}
and the FAQ @url{https://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:
@url{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 @samp{cc -V} on some machines, or, if you are using GCC,
@samp{gcc -v}. Also, include the output from @samp{uname -a} and the MPFR
version (the GMP version may be useful too).
If you get a failure while running @samp{make} or @samp{make check}, please
include the @file{config.log} file in your bug report, and in case of test
failure, the @file{tests/test-suite.log} file too.

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 @samp{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.

@node MPFR Basics, MPFR Interface, Reporting Bugs, Top
@comment  node-name,  next,  previous,  up
@chapter MPFR Basics

@menu
* Headers and Libraries::
* Nomenclature and Types::
* MPFR Variable Conventions::
* Rounding::
* Floating-Point Values on Special Numbers::
* Exceptions::
* Memory Handling::
* Getting the Best Efficiency Out of MPFR::
@end menu

@node Headers and Libraries, Nomenclature and Types, MPFR Basics, MPFR Basics
@comment  node-name,  next,  previous,  up
@section Headers and Libraries

@cindex @file{mpfr.h}
All declarations needed to use MPFR are collected in the include file
@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:

@example
#include <mpfr.h>
@end example

@cindex @code{stdio.h}
Note, however, that prototypes for MPFR functions with @code{FILE *} parameters
are provided only if @code{<stdio.h>} is included too (before @file{mpfr.h}):

@example
#include <stdio.h>
#include <mpfr.h>
@end example

@cindex @code{stdarg.h}
Likewise @code{<stdarg.h>} (or @code{<varargs.h>}) is required for prototypes
with @code{va_list} parameters, such as @code{mpfr_vprintf}.

@cindex @code{stdint.h}
@cindex @code{inttypes.h}
@cindex @code{intmax_t}
@cindex @code{uintmax_t}
And for any functions using @code{intmax_t}, you must include
@code{<stdint.h>} or @code{<inttypes.h>} before @file{mpfr.h}, to
allow @file{mpfr.h} to define prototypes for these functions.
Moreover, under some platforms (in particular with C++ compilers),
users may need to define
@code{MPFR_USE_INTMAX_T} (and should do it for portability) before
@file{mpfr.h} has been included; of course, it is possible to do that
on the command line, e.g., with @code{-DMPFR_USE_INTMAX_T}.

Note: If @file{mpfr.h} and/or @file{gmp.h} (used by @file{mpfr.h})
are included several times (possibly from another header file),
@code{<stdio.h>} and/or @code{<stdarg.h>} (or @code{<varargs.h>})
should be included @strong{before the first inclusion} of
@file{mpfr.h} or @file{gmp.h}.  Alternatively, you can define
@code{MPFR_USE_FILE} (for MPFR I/O functions) and/or
@code{MPFR_USE_VA_LIST} (for MPFR functions with @code{va_list}
parameters) anywhere before the last inclusion of @file{mpfr.h}.
As a consequence, if your file is a public header that includes
@file{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 @code{do},
@code{while} and @code{sizeof}).

You can avoid the use of MPFR macros encapsulating functions by defining
the @code{MPFR_USE_NO_MACRO} macro before @file{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.

@cindex Libraries
@cindex Linking
@cindex @code{libmpfr}
All programs using MPFR must link against both @file{libmpfr} and
@file{libgmp} libraries.  On a typical Unix-like system this can be
done with @samp{-lmpfr -lgmp} (in that order), for example:

@example
gcc myprogram.c -lmpfr -lgmp
@end example

@cindex Libtool
MPFR is built using Libtool and an application can use that to link if
desired, @MPFRpxreftop{libtool, GNU Libtool}
@c Note: the .info extension has been added to avoid the following bug:
@c   https://bugs.debian.org/cgi-bin/bugreport.cgi?bug=484740
@c which occurs when reading the info file from the build directory:
@c   info ./mpfr    or    info -f ./mpfr.info
@c Due to a poor design, the "info" utility will not find the correct
@c libtool info file if the .info extension is not provided, because of
@c the "libtool" script in MPFR's directory!

If MPFR has been installed to a non-standard location, then it may be
necessary to set up environment variables such as @samp{C_INCLUDE_PATH}
and @samp{LIBRARY_PATH}, or use @samp{-I} and @samp{-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.,
@samp{LD_LIBRARY_PATH}) on some systems. Please read the @file{INSTALL}
file for additional information.

Alternatively, it is possible to use @samp{pkg-config} (a file
@samp{mpfr.pc} is provided as of MPFR@tie{}4.0):

@example
cc myprogram.c $(pkg-config --cflags --libs mpfr)
@end example

Note that the @samp{MPFR_} and @samp{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.
@c Concerning system headers/libraries: those that may be used by MPFR.

@node Nomenclature and Types, MPFR Variable Conventions, Headers and Libraries, MPFR Basics
@comment  node-name,  next,  previous,  up
@section Nomenclature and Types

@cindex Floating-point number
@cindex Regular number
@tindex @code{mpfr_t}
@tindex @code{mpfr_ptr}
@tindex @code{mpfr_srcptr}
A @dfn{floating-point number}, or @dfn{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
@dfn{regular 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@tie{}754 (thus, for a given
number, the exponent values in MPFR and in IEEE@tie{}754 differ by 1).

Like in the IEEE@tie{}754 standard, a floating-point number can also
have three kinds of special values: a signed zero (+0 or @minus{}0),
a signed infinity (+Inf or @minus{}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 +Infinity minus +Infinity; unless documented
otherwise, the sign bit of a NaN is unspecified. Note that contrary to
IEEE@tie{}754, MPFR has a single kind of NaN and does not have subnormals.
Other than that, the behavior is very similar to IEEE@tie{}754, but there
are some minor differences.

The C data type for such objects is @code{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 @code{mpfr_ptr} is the C data type representing a pointer to this
structure; @code{mpfr_srcptr} is like @code{mpfr_ptr}, but the structure
is read-only (i.e., const qualified).

@cindex Precision
@tindex @code{mpfr_prec_t}
The @dfn{precision} is the number of bits used to represent the significand
of a floating-point number;
the corresponding C data type is @code{mpfr_prec_t}.
The precision can be any integer between @code{MPFR_PREC_MIN} and
@code{MPFR_PREC_MAX}. In the current implementation, @code{MPFR_PREC_MIN}
is equal to 1.

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 @code{MPFR_PREC_MAX},
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).

@cindex Exponent
@tindex @code{mpfr_exp_t}
An @dfn{exponent} is a component of a regular floating-point number.
Its C data type is @code{mpfr_exp_t}. Valid exponents are restricted
to a subset of this type, and the exponent range can be changed globally
as described in @ref{Exception Related Functions}. Special values do not
have an exponent.

@cindex Rounding
@tindex @code{mpfr_rnd_t}
The @dfn{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 (@pxref{Rounding}).
The corresponding C data type is @code{mpfr_rnd_t}.

@cindex Group of flags
@tindex @code{mpfr_flags_t}
MPFR has a global (or per-thread) flag for each supported exception and
provides operations on flags (@ref{Exceptions}). This C data type is used
to represent a group of flags (or a mask).

@node MPFR Variable Conventions, Rounding, Nomenclature and Types, MPFR Basics
@comment  node-name,  next,  previous,  up
@section MPFR Variable Conventions

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 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,
@code{mpfr_mul}, can be used like this: @code{mpfr_mul (x, x, x, rnd)}.
This computes the square of @var{x} with rounding mode @code{rnd}
and puts the result back in @var{x}.

@node Rounding, Floating-Point Values on Special Numbers, MPFR Variable Conventions, MPFR Basics
@comment  node-name,  next,  previous,  up
@section Rounding

The following rounding modes are supported:

@itemize @bullet

@item @code{MPFR_RNDN}: round to nearest, with the even rounding rule
      (roundTiesToEven in IEEE@tie{}754); see details below.

@item @code{MPFR_RNDD}: round toward negative infinity
      (roundTowardNegative in IEEE@tie{}754).

@item @code{MPFR_RNDU}: round toward positive infinity
      (roundTowardPositive in IEEE@tie{}754).

@item @code{MPFR_RNDZ}: round toward zero
      (roundTowardZero in IEEE@tie{}754).

@item @code{MPFR_RNDA}: round away from zero.

@item @code{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.

@end itemize

Note that, in particular for a result equal to zero, the sign is preserved
by the rounding operation.

@c Note: Because since 2008, IEEE 754 has 2 rounding attributes to nearest,
@c we must avoid the confusion by being explicit that one uses the even
@c rounding rule. Moreover, functions that convert a MPFR number to a
@c string of digits can output in a radix other than 2; in particular
@c because we do not always recall the rounding rule, we need to be general
@c enough here.
The @code{MPFR_RNDN} mode works like roundTiesToEven from the
IEEE@tie{}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 @dfn{drift} phenomenon mentioned by Knuth in volume 2
of The Art of Computer Programming (Section@tie{}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.

@c VL: There exist multiple equivalent definitions. I tried to give the
@c most intuitive ones, with the important requirement of being closest
@c to the input, and to make the definitions similar to each other. For
@c MPFR_RNDZ and MPFR_RNDA, the use of the absolute value allows one to
@c emphasize on the "sign symmetry" (there is no ambiguity on the sign
@c of the result due to the "closest to x" requirement, except for the
@c sign of 0, but see the note above).
For the @dfn{directed rounding modes}, a number @var{x} is rounded to
the number @var{y} that is the closest to @var{x} such that
@itemize @bullet
@item @code{MPFR_RNDD}:
      @var{y} is less than or equal to @var{x};
@item @code{MPFR_RNDU}:
      @var{y} is greater than or equal to @var{x};
@item @code{MPFR_RNDZ}:
      @GMPabs{@var{y}} is less than or equal to @GMPabs{@var{x}};
@item @code{MPFR_RNDA}:
      @GMPabs{@var{y}} is greater than or equal to @GMPabs{@var{x}}.
@end itemize

The @code{MPFR_RNDF} mode works as follows: the computed value is either
that corresponding to @code{MPFR_RNDD} or that corresponding to
@code{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 @code{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
@code{MPFR_RNDD} or @code{MPFR_RNDU}.
The results may not be reproducible.
@c Or should one guarantee reproducibility under some condition?
@c But this may be non-obvious if the caches may have an influence.

@anchor{ternary value}@cindex Ternary value
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 @code{int}, called the
@dfn{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 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).
@c Since subnormals are not supported, we must take into account the ulp of
@c the rounded result, not the one of the exact result, for full generality.

Unless documented otherwise, functions returning an @code{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 @code{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 @code{int}.

Unless documented otherwise, functions returning as result the value @code{1}
(or any other value specified in this manual)
for special cases (like @code{acos(0)}) yield an overflow or
an underflow if that value is not representable in the current exponent range.

@node Floating-Point Values on Special Numbers, Exceptions, Rounding, MPFR Basics
@comment  node-name,  next,  previous,  up
@section Floating-Point Values on Special Numbers

This section specifies the floating-point values (of type @code{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 @code{int} of those functions).
For functions returning several values (like
@code{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 @ref{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 description of the MPFR function (@ref{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: @code{mpfr_hypot} on (+Inf,0) gives +Inf. But @code{mpfr_pow}
cannot be defined on (1,+Inf) using this rule, as one can find
sequences (@m{x_n,@var{x}_@var{n}},@m{y_n,@var{y}_@var{n}}) such that
@m{x_n,@var{x}_@var{n}} goes to 1, @m{y_n,@var{y}_@var{n}} goes to +Inf
and @m{(x_n)^{y_n},@var{x}_@var{n} to the @var{y}_@var{n}} goes to any
positive value when @var{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.@: @minus{}0), one considers
the limit when the corresponding argument approaches 0 from above
(resp.@: below), if possible. If the limit is not defined (e.g.,
@code{mpfr_sqrt} and @code{mpfr_log} on @minus{}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., @code{mpfr_log} on @pom{}0
gives @minus{}Inf).

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.@: @minus{}0);
for example, @code{mpfr_sin} on @minus{}0 gives @minus{}0 and
@code{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 @code{mpfr_max} on @minus{}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: @code{mpfr_sqrt} on @minus{}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 @ref{MPFR Interface}.
@c Said otherwise, if such a case is not specified, this is a bug, thus
@c we may change the returned value after documenting it without having
@c to change the libtool interface number (this would have more drawbacks
@c that advantages in practice), like for any bug fix.
Example: @code{mpfr_hypot} on (NaN,0) gives NaN, but @code{mpfr_hypot}
on (NaN,+Inf) gives +Inf (as specified in @ref{Transcendental Functions}),
since for any finite or infinite input @var{x}, @code{mpfr_hypot} on
(@var{x},+Inf) gives +Inf.

MPFR also tries to follow the specifications of the IEEE@tie{}754 standard
on special values (IEEE@tie{}754 agree with the above rules in most cases).
Any difference with IEEE@tie{}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 @ref{MPFR and the IEEE 754 Standard}.

@node Exceptions, Memory Handling, Floating-Point Values on Special Numbers, MPFR Basics
@comment  node-name,  next,  previous,  up
@section Exceptions

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 @ref{Exception Related Functions}.

The supported exceptions are listed below. The macro associated with
each exception is in parentheses.

@itemize @bullet

@item Underflow (@code{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.)

Note: This is not the single possible definition of the underflow. MPFR chooses
to consider the underflow @emph{after} rounding. The underflow before rounding
can also be defined. For instance, consider a function that has the
exact result @m{7 \times 2^{e-4}, 7 multiplied by two to the power
@var{e}@minus{}4}, where @var{e} is the smallest exponent (for a
significand between 1/2 and 1),
with a 2-bit target precision and rounding toward positive infinity.
The exact result has the exponent @var{e}@minus{}1. With the underflow
before rounding, such a function call would yield an underflow, as
@var{e}@minus{}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 @m{0.5 \times 2^e, 0.5 times 2 to @var{e}}, which is
representable in the current exponent range. As a consequence, this will
not yield an underflow in MPFR@.

@item Overflow (@code{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.

@item Divide-by-zero (@code{MPFR_FLAGS_DIVBY0}):
An exact infinite result is obtained from finite inputs.

@item NaN (@code{MPFR_FLAGS_NAN}):
A NaN exception occurs when the result of a function is NaN@.
@c NaN is defined above. So, we don't say anything more.

@item Inexact (@code{MPFR_FLAGS_INEXACT}):
An inexact exception occurs when the result of a function cannot be
represented exactly and must be rounded.

@item Range error (@code{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 @code{mpfr_cmp}, or a
conversion to an integer cannot be represented in the target type).

@end itemize

Moreover, the group consisting of all the flags is represented by
the @code{MPFR_FLAGS_ALL} macro (if new flags are added in future
MPFR versions, they will be added to this macro too).

Differences with the ISO C99 standard:

@itemize @bullet

@item 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.

@item An invalid exception in C corresponds to either a NaN exception or
a range error in MPFR@.

@end itemize

@node Memory Handling, Getting the Best Efficiency Out of MPFR, Exceptions, MPFR Basics
@comment  node-name,  next,  previous,  up
@section Memory Handling

MPFR functions may create caches, e.g., when computing constants such
as @m{\pi,Pi}, either because the user has called a function like
@code{mpfr_const_pi} directly or because such a function was called
internally by the MPFR 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
@code{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
@code{mpfr_free_cache} and @code{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 @samp{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:
@ifinfo
@pxref{Custom Allocation,,, gmp.info,GNU MP}.
@end ifinfo
@ifnotinfo
see Section ``Custom Allocation'' in @cite{GNU MP}.
@end ifnotinfo
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
@code{mp_set_memory_functions}, it should first free all data allocated
with the current allocator: for its own data, with @code{mpfr_clear},
etc.; for the caches and pools, with @code{mpfr_mp_memory_cleanup} in
all threads where MPFR is potentially used. This function is currently
equivalent to @code{mpfr_free_cache}, but @code{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 @code{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 @code{mpfr_mp_memory_cleanup} and/or warn
their users about this issue.

@c This is important for shared caches.
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.

MPFR internal data such as flags, the exponent range, the default 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).
@c References to TLS specification or documentation can be given here.
@c Concerning some thread implementations under Unix, POSIX specifies
@c the thread interface only; TLS variables (with the __thread specifier)
@c is just a GCC extension. There is currently no clear documentation
@c about TLS variable initialization.

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.

@node Getting the Best Efficiency Out of MPFR, , Memory Handling, MPFR Basics
@comment  node-name,  next,  previous,  up
@section Getting the Best Efficiency Out of MPFR

Here are a few hints to get the best efficiency out of MPFR:

@itemize @bullet

@item 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;

@item use @code{mpfr_swap} instead of @code{mpfr_set} whenever possible.
      This will avoid copying the significands;

@item avoid using MPFR from C++, or make sure your C++ interface does not
      perform unnecessary allocations or copies. Slowdowns of up to a
      factor 15 have been observed on some applications with a C++ interface;

@item MPFR functions work in-place: to compute @code{a = a + b} you don't
      need an auxiliary variable, you can directly write
      @code{mpfr_add (a, a, b, ...)}.

@end itemize

@node MPFR Interface, API Compatibility, MPFR Basics, Top
@comment  node-name,  next,  previous,  up
@chapter MPFR Interface
@cindex Floating-point functions

The floating-point functions expect arguments of type @code{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 @code{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).

@cindex Precision
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@tie{}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.

@cindex Accuracy
MPFR @emph{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 a
large precision, then MPFR will still compute the result with full precision.

The value of the standard C macro @code{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 @code{errno}, but functions called by the
MPFR code (libc functions, memory allocator, etc.) may do so.

@menu
* Initialization Functions::
* Assignment Functions::
* Combined Initialization and Assignment Functions::
* Conversion Functions::
* Arithmetic Functions::
* Comparison Functions::
* Transcendental Functions::
* Input and Output Functions::
* Formatted Output Functions::
* Integer and Remainder Related Functions::
* Rounding-Related Functions::
* Miscellaneous Functions::
* Exception Related Functions::
* Memory Handling Functions::
* Compatibility with MPF::
* Custom Interface::
* Internals::
@end menu

@node Initialization Functions, Assignment Functions, MPFR Interface, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Initialization functions
@section Initialization Functions

An @code{mpfr_t} object must be initialized before storing the first value in
it.  The functions @code{mpfr_init} and @code{mpfr_init2} are used for that
purpose.

@deftypefun void mpfr_init2 (mpfr_t @var{x}, mpfr_prec_t @var{prec})
Initialize @var{x}, set its precision to be @strong{exactly}
@var{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 @code{mpfr_clear}, between initializations.
To change the precision of a variable that has already been initialized,
use @code{mpfr_set_prec} or @code{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 @var{prec} must be an integer between @code{MPFR_PREC_MIN} and
@code{MPFR_PREC_MAX} (otherwise the behavior is undefined).
@end deftypefun

@deftypefun void mpfr_inits2 (mpfr_prec_t @var{prec}, mpfr_t @var{x}, ...)
Initialize all the @code{mpfr_t} variables of the given variable
argument @code{va_list}, set their precision to be @strong{exactly}
@var{prec} bits and their value to NaN@.
See @code{mpfr_init2} for more details.
The @code{va_list} is assumed to be composed only of type @code{mpfr_t}
(or equivalently @code{mpfr_ptr}).
It begins from @var{x}, and ends when it encounters a null pointer (whose
type must also be @code{mpfr_ptr}).
@end deftypefun

@deftypefun void mpfr_clear (mpfr_t @var{x})
Free the space occupied by the significand of
@var{x}.  Make sure to call this function for all
@code{mpfr_t} variables when you are done with them.
@end deftypefun

@deftypefun void mpfr_clears (mpfr_t @var{x}, ...)
Free the space occupied by all the @code{mpfr_t} variables of the given
@code{va_list}. See @code{mpfr_clear} for more details.
The @code{va_list} is assumed to be composed only of type @code{mpfr_t}
(or equivalently @code{mpfr_ptr}).
It begins from @var{x}, and ends when it encounters a null pointer (whose
type must also be @code{mpfr_ptr}).
@end deftypefun

Here is an example of how to use multiple initialization functions
(since @code{NULL} is not necessarily defined in this context, we use
@code{(mpfr_ptr) 0} instead, but @code{(mpfr_ptr) NULL} is also correct).

@example
@{
  mpfr_t x, y, z, t;
  mpfr_inits2 (256, x, y, z, t, (mpfr_ptr) 0);
  @dots{}
  mpfr_clears (x, y, z, t, (mpfr_ptr) 0);
@}
@end example

@deftypefun void mpfr_init (mpfr_t @var{x})
Initialize @var{x}, set its precision to the default precision,
and set its value to NaN@.
The default precision can be changed by a call to @code{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 @code{mpfr_init2}.
@end deftypefun

@deftypefun void mpfr_inits (mpfr_t @var{x}, ...)
Initialize all the @code{mpfr_t} variables of the given @code{va_list},
set their precision to the default precision and their value to NaN@.
See @code{mpfr_init} for more details.
The @code{va_list} is assumed to be composed only of type @code{mpfr_t}
(or equivalently @code{mpfr_ptr}).
It begins from @var{x}, and ends when it encounters a null pointer (whose
type must also be @code{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 @code{mpfr_inits2}.
@end deftypefun

@defmac MPFR_DECL_INIT (@var{name}, @var{prec})
This macro declares @var{name} as an automatic variable of type @code{mpfr_t},
initializes it and sets its precision to be @strong{exactly} @var{prec} bits
and its value to NaN@. @var{name} must be a valid identifier.
You must use this macro in the declaration section.
This macro is much faster than using @code{mpfr_init2} but has some
drawbacks:

@itemize @bullet

@item You @strong{must not} call @code{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).

@item You @strong{cannot} change their precision.

@item You @strong{should not} create variables with huge precision with this
macro.

@item Your compiler must support @samp{Non-Constant Initializers} (standard
in C++ and ISO C99) and @samp{Token Pasting}
(standard in ISO C90). If @var{prec} is not a constant expression, your
compiler must support @samp{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 C90 mode and with @samp{-pedantic},
you may want to define the @code{MPFR_USE_EXTENSION} macro to avoid warnings
due to the @code{MPFR_DECL_INIT} implementation.

@end itemize
@end defmac

@deftypefun void mpfr_set_default_prec (mpfr_prec_t @var{prec})
Set the default precision to be @strong{exactly} @var{prec} bits, where
@var{prec} can be any integer between @code{MPFR_PREC_MIN} and
@code{MPFR_PREC_MAX}.
The
precision of a variable means the number of bits used to store its significand.
All
subsequent calls to @code{mpfr_init} or @code{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 @samp{--enable-thread-safe} configure option,
the default precision is local to each thread. @xref{Memory Handling}, for
more information.
@end deftypefun

@deftypefun mpfr_prec_t mpfr_get_default_prec (void)
Return the current default MPFR precision in bits.
See the documentation of @code{mpfr_set_default_prec}.
@end deftypefun

@need 2000
Here is an example on how to initialize floating-point variables:

@example
@{
  mpfr_t x, y;
  mpfr_init (x);                /* use default precision */
  mpfr_init2 (y, 256);          /* precision @emph{exactly} 256 bits */
  @dots{}
  /* When the program is about to exit, do ... */
  mpfr_clear (x);
  mpfr_clear (y);
  mpfr_free_cache ();           /* free the cache for constants like pi */
@}
@end example

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.

@deftypefun void mpfr_set_prec (mpfr_t @var{x}, mpfr_prec_t @var{prec})
Set the precision of @var{x} to be @strong{exactly} @var{prec} bits,
and set its value to NaN@.
The previous value stored in @var{x} is lost. It is equivalent to
a call to @code{mpfr_clear(x)} followed by a call to
@code{mpfr_init2(x, prec)}, but more efficient as no allocation is done in
case the current allocated space for the significand of @var{x} is enough.
The precision @var{prec} can be any integer between @code{MPFR_PREC_MIN} and
@code{MPFR_PREC_MAX}.
In case you want to keep the previous value stored in @var{x},
use @code{mpfr_prec_round} instead.

Warning! You must not use this function if @var{x} was initialized
with @code{MPFR_DECL_INIT} or with @code{mpfr_custom_init_set}
(@pxref{Custom Interface}).
@end deftypefun

@deftypefun mpfr_prec_t mpfr_get_prec (mpfr_t @var{x})
Return the precision of @var{x}, i.e., the
number of bits used to store its significand.
@end deftypefun

@node Assignment Functions, Combined Initialization and Assignment Functions, Initialization Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Assignment functions
@section Assignment Functions

These functions assign new values to already initialized floats
(@pxref{Initialization Functions}).

@deftypefun int mpfr_set (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_ui (mpfr_t @var{rop}, unsigned long int @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_si (mpfr_t @var{rop}, long int @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_uj (mpfr_t @var{rop}, uintmax_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_sj (mpfr_t @var{rop}, intmax_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_flt (mpfr_t @var{rop}, float @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_d (mpfr_t @var{rop}, double @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_ld (mpfr_t @var{rop}, long double @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_float128 (mpfr_t @var{rop}, _Float128 @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_decimal64 (mpfr_t @var{rop}, _Decimal64 @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_decimal128 (mpfr_t @var{rop}, _Decimal128 @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_z (mpfr_t @var{rop}, mpz_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_q (mpfr_t @var{rop}, mpq_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_f (mpfr_t @var{rop}, mpf_t @var{op}, mpfr_rnd_t @var{rnd})
Set the value of @var{rop} from @var{op}, rounded
toward the given direction @var{rnd}.
Note that the input 0 is converted to +0 by @code{mpfr_set_ui},
@code{mpfr_set_si}, @code{mpfr_set_uj}, @code{mpfr_set_sj},
@code{mpfr_set_z}, @code{mpfr_set_q} and
@code{mpfr_set_f}, regardless of the rounding mode.
The @code{mpfr_set_float128} function is built only with the configure
option @samp{--enable-float128}, which requires the compiler or
system provides the @samp{_Float128} data type
(GCC 4.3 or later supports this data type);
to use @code{mpfr_set_float128}, one should define the macro
@code{MPFR_WANT_FLOAT128} before including @file{mpfr.h}.
If the system does not support the IEEE@tie{}754 standard,
@code{mpfr_set_flt}, @code{mpfr_set_d}, @code{mpfr_set_ld},
@code{mpfr_set_decimal64} and @code{mpfr_set_decimal128}
might not preserve the signed zeros
(and in any case they don't preserve the sign bit of NaN)@.
The @code{mpfr_set_decimal64} and @code{mpfr_set_decimal128}
functions are built only with the configure
option @samp{--enable-decimal-float}, and when the compiler or
system provides the @samp{_Decimal64} and @samp{_Decimal128} data type;
to use those functions, one should define the macro
@code{MPFR_WANT_DECIMAL_FLOATS} before including @file{mpfr.h}.
@code{mpfr_set_q} might fail if the numerator (or the
denominator) cannot be represented as a @code{mpfr_t}.

For @code{mpfr_set}, the sign of a NaN is propagated in order to mimic the
IEEE@tie{}754 @code{copy} operation. But contrary to IEEE@tie{}754, the
NaN flag is set as usual.

Note: If you want to store a floating-point constant to a @code{mpfr_t},
you should use @code{mpfr_set_str} (or one of the MPFR constant functions,
such as @code{mpfr_const_pi} for @m{\pi,Pi}) instead of
@code{mpfr_set_flt}, @code{mpfr_set_d},
@code{mpfr_set_ld}, @code{mpfr_set_decimal64} or
@code{mpfr_set_decimal128}.
Otherwise the floating-point constant will be first
converted into a reduced-precision (e.g., 53-bit) binary
(or decimal, for @code{mpfr_set_decimal64} and @code{mpfr_set_decimal128})
number before MPFR can work with it.
@end deftypefun

@deftypefun int mpfr_set_ui_2exp (mpfr_t @var{rop}, unsigned long int @var{op}, mpfr_exp_t @var{e}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_si_2exp (mpfr_t @var{rop}, long int @var{op}, mpfr_exp_t @var{e}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_uj_2exp (mpfr_t @var{rop}, uintmax_t @var{op}, intmax_t @var{e}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_sj_2exp (mpfr_t @var{rop}, intmax_t @var{op}, intmax_t @var{e}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_set_z_2exp (mpfr_t @var{rop}, mpz_t @var{op}, mpfr_exp_t @var{e}, mpfr_rnd_t @var{rnd})
Set the value of @var{rop} from @m{@var{op} \times 2^e, @var{op} multiplied by
two to the power @var{e}}, rounded toward the given direction @var{rnd}.
Note that the input 0 is converted to +0.
@end deftypefun

@deftypefun int mpfr_set_str (mpfr_t @var{rop}, const char *@var{s}, int @var{base}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the value of the string @var{s} in base @var{base},
rounded in the direction @var{rnd}.
See the documentation of @code{mpfr_strtofr} for a detailed description
of the valid string formats.
Contrary to @code{mpfr_strtofr}, @code{mpfr_set_str} requires the
@emph{whole} string to represent a valid floating-point number.
@c Additionally, special values
@c @samp{@@NaN@@}, @samp{@@Inf@@}, @samp{+@@Inf@@} and @samp{-@@Inf@@},
@c all case insensitive, without leading whitespace and possibly followed by
@c other characters, are accepted too (it may change).

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 @var{base}; otherwise it is @minus{}1, and
@var{rop} may have changed (users interested in the @ref{ternary value}
should use @code{mpfr_strtofr} instead).

Note: it is preferable to use @code{mpfr_strtofr} if one wants to distinguish
between an infinite @var{rop} value coming from an infinite @var{s} or from
an overflow.
@end deftypefun

@deftypefun int mpfr_strtofr (mpfr_t @var{rop}, const char *@var{nptr}, char **@var{endptr}, int @var{base}, mpfr_rnd_t @var{rnd})
Read a floating-point number from a string @var{nptr} in base @var{base},
rounded in the direction @var{rnd}; @var{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 @var{nptr} starts with valid data, the
result is stored in @var{rop} and @code{*@var{endptr}} points to the
character just after the valid data (if @var{endptr} is not a null pointer);
otherwise @var{rop} is set to zero (for consistency with @code{strtod})
and the value of @var{nptr} is stored
in the location referenced by @var{endptr} (if @var{endptr} is not a null
pointer). The usual ternary value is returned.

Parsing follows the standard C @code{strtod} function with some extensions.
After optional leading whitespace, one has a subject sequence consisting of an
optional sign (@samp{+} or @samp{-}), 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 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 @samp{A} = 10, @samp{B} = 11, @dots{},
@samp{Z} = 35; case is ignored in bases less than or equal to 36, in bases
larger than 36, @samp{a} = 36, @samp{b} = 37, @dots{}, @samp{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 @samp{e} or @samp{E} for bases up to 10, or
@samp{@@} in any base; it indicates a multiplication by a power of the
base. In bases 2 and 16, the exponent prefix can also be @samp{p} or @samp{P},
in which case the exponent, called @emph{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 @samp{1p2} represents 4 whereas
@samp{1@@2} represents 256. The value of an exponent is always written in
base 10.

If the argument @var{base} is 0, then the base is automatically detected
as follows. If the significand starts with @samp{0b} or @samp{0B}, base 2
is assumed. If the significand starts with @samp{0x} or @samp{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 @samp{0b}, @samp{0B}, @samp{0x} or @samp{0X}
is not followed by a binary/hexadecimal digit, then the subject sequence
stops at the character @samp{0}, thus 0 is read.

Special data (for infinities and NaN) can be @samp{@@inf@@} or
@samp{@@nan@@(n-char-sequence-opt)}, and if @math{@var{base} @le{} 16},
it can also be @samp{infinity}, @samp{inf}, @samp{nan} or
@samp{nan(n-char-sequence-opt)}, all case insensitive with the rules of
the C locale.
A @samp{n-char-sequence-opt} is a possibly empty string containing only digits,
Latin letters and the underscore (0, 1, 2, @dots{}, 9, a, b, @dots{}, z,
A, B, @dots{}, Z, _). Note: one has an optional sign for all data, even
NaN@.
For example, @samp{-@@nAn@@(This_Is_Not_17)} is a valid representation for NaN
in base 17.

@c Note about the "case insensitive with the rules of the C locale":
@c The reason is that in Turkish locales on some systems, the uppercase
@c version of "i" is an "I" with a dot above, and the lowercase version
@c of "I" is a dotless "i". We do not follow these rules here.
@c See README.dev for additional information.

@end deftypefun

@deftypefun void mpfr_set_nan (mpfr_t @var{x})
@deftypefunx void mpfr_set_inf (mpfr_t @var{x}, int @var{sign})
@deftypefunx void mpfr_set_zero (mpfr_t @var{x}, int @var{sign})
Set the variable @var{x} to NaN (Not-a-Number), infinity or zero respectively.
In @code{mpfr_set_inf} or @code{mpfr_set_zero}, @var{x} is set to plus
infinity or plus zero iff @var{sign} is nonnegative;
in @code{mpfr_set_nan}, the sign bit of the result is unspecified.
@end deftypefun

@deftypefun void mpfr_swap (mpfr_t @var{x}, mpfr_t @var{y})
Swap the structures pointed to by @var{x} and @var{y}. In particular,
the values are exchanged without rounding (this may be different from
three @code{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 @var{x}
and/or @var{y} does not permit it. This is the case when @var{x} and/or
@var{y} were declared and initialized with @code{MPFR_DECL_INIT}, and
possibly with @code{mpfr_custom_init_set} (@pxref{Custom Interface}).
@end deftypefun

@node Combined Initialization and Assignment Functions, Conversion Functions, Assignment Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Combined initialization and assignment functions
@section Combined Initialization and Assignment Functions

@deftypefn Macro int mpfr_init_set (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_ui (mpfr_t @var{rop}, unsigned long int @var{op}, mpfr_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_si (mpfr_t @var{rop}, long int @var{op}, mpfr_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_d (mpfr_t @var{rop}, double @var{op}, mpfr_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_ld (mpfr_t @var{rop}, long double @var{op}, mpfr_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_z (mpfr_t @var{rop}, mpz_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_q (mpfr_t @var{rop}, mpq_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_f (mpfr_t @var{rop}, mpf_t @var{op}, mpfr_rnd_t @var{rnd})
Initialize @var{rop} and set its value from @var{op}, rounded in the direction
@var{rnd}.
The precision of @var{rop} will be taken from the active default precision,
as set by @code{mpfr_set_default_prec}.
@end deftypefn

@deftypefun int mpfr_init_set_str (mpfr_t @var{x}, const char *@var{s}, int @var{base}, mpfr_rnd_t @var{rnd})
Initialize @var{x} and set its value from
the string @var{s} in base @var{base},
rounded in the direction @var{rnd}.
See @code{mpfr_set_str}.
@end deftypefun

@node Conversion Functions, Arithmetic Functions, Combined Initialization and Assignment Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Conversion functions
@section Conversion Functions

@deftypefun float mpfr_get_flt (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx double mpfr_get_d (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx {long double} mpfr_get_ld (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx {_Float128} mpfr_get_float128 (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx _Decimal64 mpfr_get_decimal64 (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx _Decimal128 mpfr_get_decimal128 (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Convert @var{op} to a @code{float} (respectively @code{double},
@code{long double}, @code{_Decimal64}, or @code{_Decimal128})
using the rounding mode @var{rnd}.
If @var{op} 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 @var{op} is @pom{}Inf, an infinity of the same
sign or the result of @pom{}1.0/0.0 is returned. If @var{op} is zero, these
functions return a zero, trying to preserve its sign, if possible.
The @code{mpfr_get_float128}, @code{mpfr_get_decimal64} and
@code{mpfr_get_decimal128} functions are built
only under some conditions: see the documentation of @code{mpfr_set_float128},
@code{mpfr_set_decimal64} and @code{mpfr_set_decimal128} respectively.
@end deftypefun

@deftypefun {long int} mpfr_get_si (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx {unsigned long int} mpfr_get_ui (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx intmax_t mpfr_get_sj (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx uintmax_t mpfr_get_uj (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Convert @var{op} to a @code{long int}, an @code{unsigned long int},
an @code{intmax_t} or an @code{uintmax_t} (respectively) after rounding
it to an integer with respect to @var{rnd}.
If @var{op} is NaN, 0 is returned and the @emph{erange} flag is set.
If @var{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 @emph{erange} flag is set too.
When there is no such range error, if the return value differs from
@var{op}, i.e., if @var{op} is not an integer, the inexact flag is set.
@c For the flag specification, we simply followed the historical behavior.
@c See <https://sympa.inria.fr/sympa/arc/mpfr/2015-07/msg00010.html>.
@c In summary, this was a consequence of the use of mpfr_rint in case of
@c no range error. IEEE 754 specifies two kinds of operations: with
@c inexact flag either affected or not. Here this is the former kind of
@c operations. The easiest way to get the latter kind of operations is to
@c save the status of the inexact flag just before the call and restore it
@c just after (but in user code, there may be other possibilities); this
@c can be done with mpfr_inexflag_p and mpfr_set_inexflag (knowing that
@c the inexact flag can only be set, never cleared). A more readable way
@c with MPFR 4.0+ is to use mpfr_flags_test or mpfr_flags_save to save
@c the status of the inexact flag, and mpfr_flags_restore to restore it.
@c The mpfr_get_z function follows the same specification.
See also @code{mpfr_fits_slong_p}, @code{mpfr_fits_ulong_p},
@code{mpfr_fits_intmax_p} and @code{mpfr_fits_uintmax_p}.
@end deftypefun

@deftypefun double mpfr_get_d_2exp (long *@var{exp}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx {long double} mpfr_get_ld_2exp (long *@var{exp}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Return @var{d} and set @var{exp}
(formally, the value pointed to by @var{exp})
such that @math{0.5@le{}@GMPabs{@var{d}}<1}
and @m{@var{d}\times 2^{@var{exp}}, @var{d} times 2 raised to @var{exp}} equals
@var{op} rounded to double (resp.@: long double)
precision, using the given rounding mode.
@comment See ISO C standard, frexp function.
If @var{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
@var{exp} is set to 0.
If @var{op} is NaN or an infinity, then the corresponding double precision
(resp.@: long-double precision)
value is returned, and @var{exp} is undefined.
@end deftypefun

@deftypefun int mpfr_frexp (mpfr_exp_t *@var{exp}, mpfr_t @var{y}, mpfr_t @var{x}, mpfr_rnd_t @var{rnd})
Set @var{exp}
(formally, the value pointed to by @var{exp}) and @var{y}
such that @math{0.5@le{}@GMPabs{@var{y}}<1}
and @m{@var{y}\times 2^{@var{exp}}, @var{y} times 2 raised to @var{exp}} equals
@var{x} rounded to the precision of @var{y}, using the given rounding mode.
@comment See ISO C standard, frexp function.
If @var{x} is zero, then @var{y} is set to a zero of the same sign and
@var{exp} is set to 0.
If @var{x} is NaN or an infinity, then @var{y} is set to the same value
and @var{exp} is undefined.
@end deftypefun

@deftypefun mpfr_exp_t mpfr_get_z_2exp (mpz_t @var{rop}, mpfr_t @var{op})
Put the scaled significand of @var{op} (regarded as an integer, with the
precision of @var{op}) into @var{rop}, and return the exponent @var{exp}
(which may be outside the current exponent range) such that @var{op}
exactly equals @m{@var{rop} \times 2^{@var{exp}},@var{rop} times 2 raised
to the power @var{exp}}.
If @var{op} is zero, the minimal exponent @code{emin} is returned.
If @var{op} is NaN or an infinity, the @emph{erange} flag is set, @var{rop}
is set to 0, and the the minimal exponent @code{emin} is returned.
The returned exponent may be less than the minimal exponent @code{emin}
of MPFR numbers in the current exponent range; in case the exponent is
not representable in the @code{mpfr_exp_t} type, the @emph{erange} flag
is set and the minimal value of the @code{mpfr_exp_t} type is returned.
@end deftypefun

@deftypefun int mpfr_get_z (mpz_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Convert @var{op} to a @code{mpz_t}, after rounding it with respect to
@var{rnd}. If @var{op} is NaN or an infinity, the @emph{erange} flag is
set, @var{rop} is set to 0, and 0 is returned. Otherwise the return
value is zero when @var{rop} is equal to @var{op} (i.e., when @var{op}
is an integer), positive when it is greater than @var{op}, and negative
when it is smaller than @var{op}; moreover, if @var{rop} differs from
@var{op}, i.e., if @var{op} is not an integer, the inexact flag is set.
@end deftypefun

@deftypefun void mpfr_get_q (mpq_t @var{rop}, mpfr_t @var{op})
Convert @var{op} to a @code{mpq_t}.
If @var{op} is NaN or an infinity, the @emph{erange} flag is
set and @var{rop} is set to 0. Otherwise the conversion is always exact.
@end deftypefun

@deftypefun int mpfr_get_f (mpf_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Convert @var{op} to a @code{mpf_t}, after rounding it with respect to
@var{rnd}.
The @emph{erange} flag is set if @var{op} is NaN or an infinity, which
do not exist in MPF@.  If @var{op} is NaN, then @var{rop} is undefined.
If @var{op} is +Inf (resp.@: @minus{}Inf), then @var{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 @var{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.
@end deftypefun

@anchor{mpfr_get_str_ndigits}
@deftypefun {size_t} mpfr_get_str_ndigits (int @var{b}, mpfr_prec_t @var{p})
Return the minimal integer @math{m} such that any number
of @var{p} bits, when output with @math{m} digits in radix @var{b} with
rounding to nearest, can be recovered exactly when read again,
still with rounding to nearest.
More precisely, we have
@ifnottex
m = 1 + ceil(@var{p}*log(2)/log(@var{b})),
@end ifnottex
@tex
$m = 1 + \left\lceil @var{p} {\log 2 \over \log @var{b}} \right\rceil$,
@end tex
with @var{p} replaced by @var{p}@minus{}1 if @var{b} is a power of 2.

The argument @var{b} must be in the range 2 to 62; this is the range of bases
supported by the @code{mpfr_get_str} function. Note that contrary to the base
argument of this function, negative values are not accepted.
@end deftypefun

@anchor{mpfr_get_str}
@deftypefun {char *} mpfr_get_str (char *@var{str}, mpfr_exp_t *@var{expptr}, int @var{base}, size_t @var{n}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Convert @var{op} to a string of digits in base @GMPabs{@var{base}},
with rounding in the direction @var{rnd}, where @var{n} is either zero
(see below) or the number of significant digits output in the string.
The argument @var{base} may vary from 2 to 62 or from @minus{}2 to @minus{}36;
otherwise the function does nothing and immediately returns a null pointer.

For @var{base} in the range 2 to 36, digits and lower-case letters are used;
for @minus{}2 to @minus{}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
@math{@var{base} > 10}, the successor of the digit 9 depends on @var{base}.
This choice has been done for compatibility with GMP's @code{mpf_get_str}
function. Users who wish a more consistent behavior should write a simple
wrapper.

If the input is NaN, then the returned string is @samp{@@NaN@@} and the
NaN flag is set. If the input is +Inf (resp.@: @minus{}Inf), then the
returned string is @samp{@@Inf@@} (resp.@: @samp{-@@Inf@@}).

If the input number is a finite number, the exponent is written through
the pointer @var{expptr} (for input 0, the current minimal exponent is
written); the type @code{mpfr_exp_t} is large enough to hold the exponent
in all cases.

The generated string is a fraction, with an implicit radix point immediately
to the left of the first digit.  For example, the number @minus{}3.1416 would
be returned as "@minus{}31416" in the string and 1 written at @var{expptr}.
If @var{rnd} is to nearest, and @var{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 @var{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,
@c The following example duplicates (16) and a half
@c (36) and a half is rounded to (40) which is 28 in decimal,
and (26) and a half is rounded to (26), which is 20 in decimal.

If @var{n} is zero, the number of digits of the significand is taken as
@code{mpfr_get_str_ndigits(@var{base},@var{p})} where @var{p} is the precision
of @var{op} (@pxref{mpfr_get_str_ndigits}).

If @var{str} is a null pointer, space for the significand is allocated using
the allocation function (@pxref{Memory Handling}) and a pointer to the string
is returned (unless the base is invalid).
To free the returned string, you must use @code{mpfr_free_str}.

If @var{str} is not a null pointer, it should point to a block of storage
large enough for the significand. A safe block size (sufficient for any
value) is @code{max(@var{n} + 2, 7)} if @var{n} is not zero; if @var{n} is
zero, replace it by @code{mpfr_get_str_ndigits(@var{base},@var{p})} where
@var{p} is the precision of @var{op}, as mentioned above.
The extra two bytes are
for a possible minus sign, and for the terminating null character, and the
value 7 accounts for @samp{-@@Inf@@} plus the terminating null character.
The pointer to the string @var{str} is returned (unless the base is invalid).

Like in usual functions, the inexact flag is set iff the result is inexact.
@end deftypefun

@deftypefun void mpfr_free_str (char *@var{str})
Free a string allocated by @code{mpfr_get_str} using the unallocation
function (@pxref{Memory Handling}).
The block is assumed to be @code{strlen(@var{str})+1} bytes.
@end deftypefun

@deftypefun int mpfr_fits_ulong_p (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_slong_p (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_uint_p (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_sint_p (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_ushort_p (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_sshort_p (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_uintmax_p (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_intmax_p (mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Return non-zero if @var{op} would fit in the respective C data type,
respectively @code{unsigned long int}, @code{long int}, @code{unsigned int},
@code{int}, @code{unsigned short}, @code{short}, @code{uintmax_t},
@code{intmax_t}, when rounded to an integer in the direction @var{rnd}.
For instance, with the @code{MPFR_RNDU} rounding mode on @minus{}0.5,
the result will be non-zero for all these functions.
For @code{MPFR_RNDF}, those functions return non-zero when it is guaranteed
that the corresponding conversion function (for example @code{mpfr_get_ui}
for @code{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 @code{MPFR_RNDF}, @code{mpfr_fits_ulong_p} will return
non-zero for a non-negative number less than or equal to @code{ULONG_MAX}.
@end deftypefun

@node Arithmetic Functions, Comparison Functions, Conversion Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Arithmetic functions
@section Arithmetic Functions

@deftypefun int mpfr_add (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_add_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_add_si (mpfr_t @var{rop}, mpfr_t @var{op1}, long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_add_d (mpfr_t @var{rop}, mpfr_t @var{op1}, double @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_add_z (mpfr_t @var{rop}, mpfr_t @var{op1}, mpz_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_add_q (mpfr_t @var{rop}, mpfr_t @var{op1}, mpq_t @var{op2}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @math{@var{op1} + @var{op2}} rounded in the direction
@var{rnd}.  The IEEE@tie{}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 (@minus{}0) + 0 = (@minus{}0)).
The @code{mpfr_add_d} function assumes that the radix of the @code{double} type
is a power of 2, with a precision at most that declared by the C implementation
(macro @code{IEEE_DBL_MANT_DIG}, and if not defined 53 bits).
@end deftypefun

@deftypefun int mpfr_sub (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_ui_sub (mpfr_t @var{rop}, unsigned long int @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_sub_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_si_sub (mpfr_t @var{rop}, long int @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_sub_si (mpfr_t @var{rop}, mpfr_t @var{op1}, long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_d_sub (mpfr_t @var{rop}, double @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_sub_d (mpfr_t @var{rop}, mpfr_t @var{op1}, double @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_z_sub (mpfr_t @var{rop}, mpz_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_sub_z (mpfr_t @var{rop}, mpfr_t @var{op1}, mpz_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_sub_q (mpfr_t @var{rop}, mpfr_t @var{op1}, mpq_t @var{op2}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @math{@var{op1} - @var{op2}} rounded in the direction
@var{rnd}.  The IEEE@tie{}754 rules are used, in particular for signed zeros.
But for types having no signed zeros, 0 is considered unsigned
(i.e., (+0) @minus{} 0 = (+0), (@minus{}0) @minus{} 0 = (@minus{}0),
0 @minus{} (+0) = (@minus{}0) and 0 @minus{} (@minus{}0) = (+0)).
The same restrictions than for @code{mpfr_add_d} apply to @code{mpfr_d_sub}
and @code{mpfr_sub_d}.
@end deftypefun

@deftypefun int mpfr_mul (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_mul_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_mul_si (mpfr_t @var{rop}, mpfr_t @var{op1}, long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_mul_d (mpfr_t @var{rop}, mpfr_t @var{op1}, double @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_mul_z (mpfr_t @var{rop}, mpfr_t @var{op1}, mpz_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_mul_q (mpfr_t @var{rop}, mpfr_t @var{op1}, mpq_t @var{op2}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}} rounded in the
direction @var{rnd}.
When a result is zero, its sign is the product of the signs of the operands
(for types having no signed zeros, 0 is considered positive).
The same restrictions than for @code{mpfr_add_d} apply to @code{mpfr_mul_d}.
Note: when @var{op1} and @var{op2} are equal, use @code{mpfr_sqr} instead of
@code{mpfr_mul} for better efficiency.
@end deftypefun

@deftypefun int mpfr_sqr (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @m{@var{op}^{2}, the square of @var{op}}
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_div (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_ui_div (mpfr_t @var{rop}, unsigned long int @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_div_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_si_div (mpfr_t @var{rop}, long int @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_div_si (mpfr_t @var{rop}, mpfr_t @var{op1}, long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_d_div (mpfr_t @var{rop}, double @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_div_d (mpfr_t @var{rop}, mpfr_t @var{op1}, double @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_div_z (mpfr_t @var{rop}, mpfr_t @var{op1}, mpz_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_div_q (mpfr_t @var{rop}, mpfr_t @var{op1}, mpq_t @var{op2}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @math{@var{op1}/@var{op2}} rounded in the direction @var{rnd}.
When a result is 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
@var{op1} is non-zero and @var{op2} is zero, the result might change from
@pom{}Inf to NaN in future MPFR versions if there is an opposite decision
on the IEEE@tie{}754 side.
The same restrictions than for @code{mpfr_add_d} apply to @code{mpfr_d_div}
and @code{mpfr_div_d}.
@end deftypefun

@deftypefun int mpfr_sqrt (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_sqrt_ui (mpfr_t @var{rop}, unsigned long int @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @m{\sqrt{@var{op}}, the square root of @var{op}}
rounded in the direction @var{rnd}.  Set @var{rop} to @minus{}0 if
@var{op} is @minus{}0, to be consistent with the IEEE@tie{}754 standard.
Set @var{rop} to NaN if @var{op} is negative.
@end deftypefun

@deftypefun int mpfr_rec_sqrt (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @m{1/\sqrt{@var{op}}, the reciprocal square root of @var{op}}
rounded in the direction @var{rnd}.  Set @var{rop} to +Inf if @var{op} is
@pom{}0, +0 if @var{op} is +Inf, and NaN if @var{op} is negative.  Warning!
Therefore the result on @minus{}0 is different from the one of the rSqrt
function recommended by the IEEE@tie{}754 standard (Section@tie{}9.2.1),
which is @minus{}Inf instead of +Inf.
@end deftypefun

@deftypefun int mpfr_cbrt (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_rootn_ui (mpfr_t @var{rop}, mpfr_t @var{op}, unsigned long int @var{n}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the @var{n}th root (with @math{@var{n} = 3}, the cubic root,
for @code{mpfr_cbrt}) of @var{op} rounded in the direction @var{rnd}.
For @math{@var{n} = 0}, set @var{rop} to NaN@.
For @var{n} odd (resp.@: even) and @var{op} negative (including @minus{}Inf),
set @var{rop} to a negative number (resp.@: NaN)@.
If @var{op} is zero, set @var{rop} to zero with the sign obtained by the
usual limit rules, i.e., the same sign as @var{op} if @var{n} is odd, and
positive if @var{n} is even.

These functions agree with the rootn operation of the IEEE@tie{}754 standard.
Note that it is here restricted to @math{@var{n} @ge{} 0}. Functions
allowing a negative @var{n} may be implemented in the future.
@end deftypefun

@deftypefun int mpfr_root (mpfr_t @var{rop}, mpfr_t @var{op}, unsigned long int @var{n}, mpfr_rnd_t @var{rnd})
This function is the same as @code{mpfr_rootn_ui} except when @var{op}
is @minus{}0 and @var{n} is even: the result is @minus{}0 instead of +0
(the reason was to be consistent with @code{mpfr_sqrt}). Said otherwise,
if @var{op} is zero, set @var{rop} to @var{op}.

This function predates IEEE@tie{}754-2008, where rootn was introduced, and
behaves differently from the IEEE@tie{}754 rootn operation. It is marked as
deprecated and will be removed in a future release.
@end deftypefun

@deftypefun int mpfr_neg (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_abs (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @math{-@var{op}} and the absolute value of @var{op}
respectively, rounded in the direction @var{rnd}.
Just changes or adjusts
the sign if @var{rop} and @var{op} are the same variable,
otherwise a rounding might occur if the precision of @var{rop} is less than
that of @var{op}.

The sign rule also applies to NaN in order to mimic the IEEE@tie{}754
@code{negate} and @code{abs} operations, i.e., for @code{mpfr_neg}, the
sign is reversed, and for @code{mpfr_abs}, the sign is set to positive.
But contrary to IEEE@tie{}754, the NaN flag is set as usual.
@end deftypefun

@deftypefun int mpfr_dim (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the positive difference of @var{op1} and @var{op2}, i.e.,
@math{@var{op1} - @var{op2}} rounded in the direction @var{rnd}
if @math{@var{op1} > @var{op2}}, +0 if @math{@var{op1} @le{} @var{op2}},
and NaN if @var{op1} or @var{op2} is NaN@.
@end deftypefun

@deftypefun int mpfr_mul_2ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_mul_2si (mpfr_t @var{rop}, mpfr_t @var{op1}, long int @var{op2}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @m{@var{op1} \times 2^{@var{op2}}, @var{op1} times 2 raised
to @var{op2}}
rounded in the direction @var{rnd}. Just increases the exponent by @var{op2}
when @var{rop} and @var{op1} are identical.
@end deftypefun

@deftypefun int mpfr_div_2ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_div_2si (mpfr_t @var{rop}, mpfr_t @var{op1}, long int @var{op2}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @m{@var{op1}/2^{@var{op2}}, @var{op1} divided by 2 raised
to @var{op2}}
rounded in the direction @var{rnd}. Just decreases the exponent by @var{op2}
when @var{rop} and @var{op1} are identical.
@end deftypefun

@deftypefun int mpfr_fac_ui (mpfr_t @var{rop}, unsigned long int @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the factorial of @var{op}, rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_fma (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_t @var{op3}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_fms (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_t @var{op3}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @math{(@var{op1} @GMPtimes{} @var{op2}) + @var{op3}}
(resp.@: @math{(@var{op1} @GMPtimes{} @var{op2}) - @var{op3}})
rounded in the direction @var{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.
@end deftypefun

@deftypefun int mpfr_fmma (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_t @var{op3}, mpfr_t @var{op4}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_fmms (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_t @var{op3}, mpfr_t @var{op4}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @math{(@var{op1} @GMPtimes{} @var{op2}) + (@var{op3} @GMPtimes{} @var{op4})}
(resp.@: @math{(@var{op1} @GMPtimes{} @var{op2}) - (@var{op3} @GMPtimes{} @var{op4})})
rounded in the direction @var{rnd}.
In case the computation of @math{@var{op1} @GMPtimes{} @var{op2}} overflows or
underflows (or that of @math{@var{op3} @GMPtimes{} @var{op4}}), the result
@var{rop} is computed as if the two intermediate products were computed with
rounding toward zero.
@end deftypefun

@deftypefun int mpfr_hypot (mpfr_t @var{rop}, mpfr_t @var{x}, mpfr_t @var{y}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the Euclidean norm of @var{x} and @var{y},
@ifnottex
i.e., the square root of the sum of the squares of @var{x} and @var{y},
@end ifnottex
@tex
i.e., $\sqrt{x^2+y^2}$,
@end tex
rounded in the direction @var{rnd}.
Special values are handled as described in the ISO C99 (Section@tie{}F.9.4.3)
and IEEE@tie{}754 (Section@tie{}9.2.1) standards:
If @var{x} or @var{y} is an infinity, then +Inf is returned in @var{rop},
even if the other number is NaN@.
@end deftypefun

@deftypefun int mpfr_sum (mpfr_t @var{rop}, const mpfr_ptr @var{tab}[], unsigned long int @var{n}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the sum of all elements of @var{tab}, whose size is @var{n},
correctly rounded in the direction @var{rnd}. Warning: for efficiency reasons,
@var{tab} is an array of pointers
to @code{mpfr_t}, not an array of @code{mpfr_t}.
If @var{n} = 0, then the result is +0, and if @var{n} = 1, then the function
is equivalent to @code{mpfr_set}.
For the special exact cases, the result is the same as the one obtained
with a succession of additions (@code{mpfr_add}) in infinite precision.
In particular, if the result is an exact zero and @math{@var{n} @ge{} 1}:
@itemize @bullet
@item if all the inputs have the same sign (i.e., all +0 or all @minus{}0),
then the result has the same sign as the inputs;
@item otherwise, either because all inputs are zeros with at least a +0 and
a @minus{}0, or because some inputs are non-zero (but they globally cancel),
the result is +0, except for the @code{MPFR_RNDD} rounding mode, where it is
@minus{}0.
@end itemize
@end deftypefun

@deftypefun int mpfr_dot (mpfr_t @var{rop}, const mpfr_ptr @var{a}[], const mpfr_ptr @var{b}[], unsigned long int @var{n}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the dot product of elements of @var{a} by those of @var{b},
whose common size is @var{n},
correctly rounded in the direction @var{rnd}. Warning: for efficiency reasons,
@var{a} and @var{b} are arrays of pointers to @code{mpfr_t}.
This function is experimental, and does not yet handle intermediate overflows
and underflows.
@end deftypefun

For the power functions (with an integer exponent or not), see @ref{mpfr_pow}
in @ref{Transcendental Functions}.

@node Comparison Functions, Transcendental Functions, Arithmetic Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Comparison functions
@section Comparison Functions

@deftypefun int mpfr_cmp (mpfr_t @var{op1}, mpfr_t @var{op2})
@deftypefunx int mpfr_cmp_ui (mpfr_t @var{op1}, unsigned long int @var{op2})
@deftypefunx int mpfr_cmp_si (mpfr_t @var{op1}, long int @var{op2})
@deftypefunx int mpfr_cmp_d (mpfr_t @var{op1}, double @var{op2})
@deftypefunx int mpfr_cmp_ld (mpfr_t @var{op1}, long double @var{op2})
@deftypefunx int mpfr_cmp_z (mpfr_t @var{op1}, mpz_t @var{op2})
@deftypefunx int mpfr_cmp_q (mpfr_t @var{op1}, mpq_t @var{op2})
@deftypefunx int mpfr_cmp_f (mpfr_t @var{op1}, mpf_t @var{op2})
Compare @var{op1} and @var{op2}.  Return a positive value if @math{@var{op1} >
@var{op2}}, zero if @math{@var{op1} = @var{op2}}, and a negative value if
@math{@var{op1} < @var{op2}}.
Both @var{op1} and @var{op2} are considered to their full own precision,
which may differ.
If one of the operands is NaN, set the @emph{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., @code{mpfr_equal_p} for the equality) described
below; they behave like the IEEE@tie{}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).
@end deftypefun

@deftypefun int mpfr_cmp_ui_2exp (mpfr_t @var{op1}, unsigned long int @var{op2}, mpfr_exp_t @var{e})
@deftypefunx int mpfr_cmp_si_2exp (mpfr_t @var{op1}, long int @var{op2}, mpfr_exp_t @var{e})
Compare @var{op1} and @m{@var{op2} \times 2^{@var{e}}, @var{op2} multiplied by two to
the power @var{e}}. Similar as above.
@end deftypefun

@deftypefun int mpfr_cmpabs (mpfr_t @var{op1}, mpfr_t @var{op2})
@deftypefunx int mpfr_cmpabs_ui (mpfr_t @var{op1}, unsigned long int @var{op2})
Compare @math{|@var{op1}|} and @math{|@var{op2}|}.  Return a positive value if
@math{|@var{op1}| > |@var{op2}|}, zero if @math{|@var{op1}| = |@var{op2}|}, and
a negative value if @math{|@var{op1}| < |@var{op2}|}.
If one of the operands is NaN, set the @emph{erange} flag and return zero.
@end deftypefun

@deftypefun int mpfr_nan_p (mpfr_t @var{op})
@deftypefunx int mpfr_inf_p (mpfr_t @var{op})
@deftypefunx int mpfr_number_p (mpfr_t @var{op})
@deftypefunx int mpfr_zero_p (mpfr_t @var{op})
@deftypefunx int mpfr_regular_p (mpfr_t @var{op})
Return non-zero if @var{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.
@end deftypefun

@deftypefn Macro int mpfr_sgn (mpfr_t @var{op})
Return a positive value if @math{@var{op} > 0}, zero if @math{@var{op} = 0},
and a negative value if @math{@var{op} < 0}.
If the operand is NaN, set the @emph{erange} flag and return zero.
This is equivalent to @code{mpfr_cmp_ui (op, 0)}, but more efficient.
@end deftypefn

@deftypefun int mpfr_greater_p (mpfr_t @var{op1}, mpfr_t @var{op2})
@deftypefunx int mpfr_greaterequal_p (mpfr_t @var{op1}, mpfr_t @var{op2})
@deftypefunx int mpfr_less_p (mpfr_t @var{op1}, mpfr_t @var{op2})
@deftypefunx int mpfr_lessequal_p (mpfr_t @var{op1}, mpfr_t @var{op2})
@deftypefunx int mpfr_equal_p (mpfr_t @var{op1}, mpfr_t @var{op2})
Return non-zero if
@math{@var{op1} > @var{op2}},
@math{@var{op1} @ge{} @var{op2}},
@math{@var{op1} < @var{op2}},
@math{@var{op1} @le{} @var{op2}},
@math{@var{op1} = @var{op2}} respectively,
and zero otherwise.
Those functions return zero whenever @var{op1} and/or @var{op2} is NaN@.
@end deftypefun

@deftypefun int mpfr_lessgreater_p (mpfr_t @var{op1}, mpfr_t @var{op2})
Return non-zero if @math{@var{op1} < @var{op2}} or
@math{@var{op1} > @var{op2}} (i.e., neither @var{op1}, nor @var{op2} is
NaN, and @math{@var{op1} @ne{} @var{op2}}), zero otherwise (i.e., @var{op1}
and/or @var{op2} is NaN, or @math{@var{op1} = @var{op2}}).
@end deftypefun

@deftypefun int mpfr_unordered_p (mpfr_t @var{op1}, mpfr_t @var{op2})
Return non-zero if @var{op1} or @var{op2} is a NaN (i.e., they cannot be
compared), zero otherwise.
@end deftypefun

@deftypefun int mpfr_total_order_p (mpfr_t @var{x}, mpfr_t @var{y})
This function implements the totalOrder predicate from IEEE@tie{}754,
where @minus{}NaN < @minus{}Inf < negative finite numbers < @minus{}0 <
+0 < positive finite numbers < +Inf < +NaN@.
It returns a non-zero value (true) when @var{x} is smaller than or equal
to @var{y} for this order relation, and zero (false) otherwise.
Contrary to @code{mpfr_cmp (x, y)}, which returns a ternary value,
@code{mpfr_total_order_p} returns a binary value (zero or non-zero).
In particular, @code{mpfr_total_order_p (x, x)} returns true,
@code{mpfr_total_order_p (-0, +0)} returns true and
@code{mpfr_total_order_p (+0, -0)} returns false.
The sign bit of NaN also matters.
@end deftypefun

@node Transcendental Functions, Input and Output Functions, Comparison Functions, MPFR Interface
@cindex Transcendental functions
@section Transcendental Functions

All those functions, except explicitly stated (for example
@code{mpfr_sin_cos}), return a @ref{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 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
@code{mpfr_rootn_ui} is also linear in the argument @var{k}, and the
memory usage of the incomplete Gamma function also depends on the
precision of the input @var{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 (and in particular, that potentially
exact cases are known and can be detected efficiently).
@c Let's not give too many details, but by context, it is implied that
@c the input precision is involved if the cases have been built so that
@c they are hard to round, like in function bad_cases in the test suite
@c (tests/tests.c).

@deftypefun int mpfr_log (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_log_ui (mpfr_t @var{rop}, unsigned long int @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_log2 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_log10 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the natural logarithm of @var{op},
@m{\log_2 @var{op}, log2(@var{op})} or
@m{\log_{10} @var{op}, log10(@var{op})}, respectively,
rounded in the direction @var{rnd}.
Set @var{rop} to +0 if @var{op} is 1 (in all rounding modes),
for consistency with the ISO C99 and IEEE@tie{}754 standards.
Set @var{rop} to @minus{}Inf if @var{op} is @pom{}0
(i.e., the sign of the zero has no influence on the result).
@end deftypefun

@deftypefun int mpfr_log1p (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_log2p1 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_log10p1 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the logarithm of one plus @var{op} (in radix two for
@code{mpfr_log2p1}, and in radix ten for @code{mpfr_log10p1}), rounded in the
direction @var{rnd}.
Set @var{rop} to @minus{}Inf if @var{op} is @minus{}1.
@end deftypefun

@deftypefun int mpfr_exp (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_exp2 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_exp10 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the exponential of @var{op},
 to @m{2^{@var{op}}, 2 power of @var{op}}
or to @m{10^{@var{op}}, 10 power of @var{op}}, respectively,
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_expm1 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_exp2m1 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_exp10m1 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @m{e^{@var{op}}-1,the exponential of @var{op} followed by a
subtraction by one}
(resp.@: @m{2^{@var{op}}-1,2 power of @var{op} followed by a
subtraction by one},
and @m{10^{@var{op}}-1,10 power of @var{op} followed by a subtraction by one}),
rounded in the direction @var{rnd}.
@end deftypefun

@anchor{mpfr_pow}
@deftypefun int mpfr_pow (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_powr (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_pow_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_pow_si (mpfr_t @var{rop}, mpfr_t @var{op1}, long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_pow_uj (mpfr_t @var{rop}, mpfr_t @var{op1}, uintmax_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_pow_sj (mpfr_t @var{rop}, mpfr_t @var{op1}, intmax_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_pown (mpfr_t @var{rop}, mpfr_t @var{op1}, intmax_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_pow_z (mpfr_t @var{rop}, mpfr_t @var{op1}, mpz_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_ui_pow_ui (mpfr_t @var{rop}, unsigned long int @var{op1}, unsigned long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_ui_pow (mpfr_t @var{rop}, unsigned long int @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @m{@var{op1}^{@var{op2}}, @var{op1} raised to @var{op2}},
rounded in the direction @var{rnd}.
The @code{mpfr_powr} function corresponds to the @code{powr} function
from IEEE@tie{}754, i.e., it computes the exponential of
@var{op2} multiplied by the logarithm of @var{op1}.
The @code{mpfr_pown} function is just an alias (currently implemented as
a macro) for @code{mpfr_pow_sj}, to follow the C2x function @code{pown}.
Special values are handled as described in the ISO C99 and IEEE@tie{}754
standards for the @code{pow} function:
@itemize @bullet
@item @code{pow(@pom{}0, @var{y})} returns plus or negative infinity for @var{y} a negative odd integer.
@item @code{pow(@pom{}0, @var{y})} returns positive infinity for @var{y} negative and not an odd integer.
@item @code{pow(@pom{}0, @var{y})} returns plus or minus zero for @var{y} a positive odd integer.
@item @code{pow(@pom{}0, @var{y})} returns plus zero for @var{y} positive and not an odd integer.
@item @code{pow(-1, @pom{}Inf)} returns 1.
@item @code{pow(+1, @var{y})} returns 1 for any @var{y}, even a NaN@.
@item @code{pow(@var{x}, @pom{}0)} returns 1 for any @var{x}, even a NaN@.
@item @code{pow(@var{x}, @var{y})} returns NaN for finite negative @var{x} and finite non-integer @var{y}.
@item @code{pow(@var{x}, -Inf)} returns positive infinity for @math{0 < @GMPabs{x} < 1}, and plus zero for @math{@GMPabs{x} > 1}.
@item @code{pow(@var{x}, +Inf)} returns plus zero for @math{0 < @GMPabs{x} < 1}, and positive infinity for @math{@GMPabs{x} > 1}.
@item @code{pow(-Inf, @var{y})} returns minus zero for @var{y} a negative odd integer.
@item @code{pow(-Inf, @var{y})} returns plus zero for @var{y} negative and not an odd integer.
@item @code{pow(-Inf, @var{y})} returns negative infinity for @var{y} a positive odd integer.
@item @code{pow(-Inf, @var{y})} returns positive infinity for @var{y} positive and not an odd integer.
@item @code{pow(+Inf, @var{y})} returns plus zero for @var{y} negative, and positive infinity for @var{y} positive.
@end itemize
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 @code{pow}.
@end deftypefun

@deftypefun int mpfr_compound_si (mpfr_t @var{rop}, mpfr_t @var{op}, long int @var{n}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the power @var{n} of one plus @var{op},
following IEEE@tie{}754 for the special cases and exceptions.
When @var{n} is zero and @var{op} is NaN or greater or equal to @minus{}1,
@var{rop} is set to 1.
@end deftypefun

@deftypefun int mpfr_cos (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_sin (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_tan (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the cosine of @var{op}, sine of @var{op},
tangent of @var{op}, rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_cosu (mpfr_t @var{rop}, mpfr_t @var{op}, unsigned long int @var{u}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_sinu (mpfr_t @var{rop}, mpfr_t @var{op}, unsigned long int @var{u}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_tanu (mpfr_t @var{rop}, mpfr_t @var{op}, unsigned long int @var{u}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the cosine (resp.@: sine and tangent) of
@m{@var{op} \times 2\pi/u,@var{op} multiplied by 2@tie{}Pi and divided
by @var{u}}. For example, if @var{u} equals 360, one gets the cosine
(resp.@: sine and tangent) for @var{op} in degrees. For @code{mpfr_cosu},
when @m{@var{op} \times 2/u,@var{op} multiplied by 2 and divided by @var{u}}
is a half-integer, the result is +0, following IEEE@tie{}754 (cosPi),
so that the function is even. For @code{mpfr_sinu},
when @m{@var{op} \times 2/u,@var{op} multiplied by 2 and divided by @var{u}}
is an integer, the result is zero with the same sign as @var{op}, following
IEEE@tie{}754 (sinPi), so that the function is odd.
Similarly, the function @code{mpfr_tanu} follows IEEE@tie{}754 (tanPi).
@end deftypefun

@deftypefun int mpfr_cospi (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_sinpi (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_tanpi (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the cosine (resp.@: sine and tangent) of
@m{@var{op} \times \pi,@var{op} multiplied by Pi}. See the description of
@code{mpfr_sinu}, @code{mpfr_cosu} and @code{mpfr_tanu} for special values.
@end deftypefun

@deftypefun int mpfr_sin_cos (mpfr_t @var{sop}, mpfr_t @var{cop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set simultaneously @var{sop} to the sine of @var{op} and @var{cop} to the
cosine of @var{op}, rounded in the direction @var{rnd} with the corresponding
precisions of @var{sop} and @var{cop}, which must be different variables.
Return 0 iff both results are exact, more precisely it returns @math{s+4c}
where @math{s=0} if @var{sop} is exact, @math{s=1} if @var{sop} is larger
than the sine of @var{op}, @math{s=2} if @var{sop} is smaller than the sine
of @var{op}, and similarly for @math{c} and the cosine of @var{op}.
@end deftypefun

@deftypefun int mpfr_sec (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_csc (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_cot (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the secant of @var{op}, cosecant of @var{op},
cotangent of @var{op}, rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_acos (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_asin (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_atan (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the arc-cosine, arc-sine or arc-tangent of @var{op},
rounded in the direction @var{rnd}.
Note that since @code{acos(-1)} returns the floating-point number closest to
@m{\pi,Pi} according to the given rounding mode, this number might not be
in the output range @m{0 @le{} @var{rop} < \pi,0 @le{} @var{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 @code{asin(-1)}, @code{asin(1)}, @code{atan(-Inf)},
@code{atan(+Inf)} or for @code{atan(op)} with large @var{op} and
small precision of @var{rop}.
@c PZ: check the above is correct
@end deftypefun

@deftypefun int mpfr_acosu (mpfr_t @var{rop}, mpfr_t @var{op}, unsigned long int @var{u}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_asinu (mpfr_t @var{rop}, mpfr_t @var{op}, unsigned long int @var{u}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_atanu (mpfr_t @var{rop}, mpfr_t @var{op}, unsigned long int @var{u}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @var{a} multiplied by @var{u} and divided by
@m{2\pi,2@tie{}Pi}, where @var{a} is the arc-cosine (resp.@: arc-sine and
arc-tangent) of @var{op}.
For example, if @var{u} equals 360, @code{mpfr_acosu} yields the arc-cosine in
degrees.
@end deftypefun

@deftypefun int mpfr_acospi (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_asinpi (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_atanpi (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @code{acos(op)} (resp.@: @code{asin(op)} and @code{atan(op)})
divided by @m{\pi,Pi}.
@end deftypefun

@deftypefun int mpfr_atan2 (mpfr_t @var{rop}, mpfr_t @var{y}, mpfr_t @var{x}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_atan2u (mpfr_t @var{rop}, mpfr_t @var{y}, mpfr_t @var{x}, unsigned long int @var{u}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_atan2pi (mpfr_t @var{rop}, mpfr_t @var{y}, mpfr_t @var{x}, mpfr_rnd_t @var{rnd})
For @code{mpfr_atan2}, set @var{rop} to the arc-tangent2 of @var{y} and
@var{x}, rounded in the direction @var{rnd}:
if @code{x > 0}, @code{atan2(y, x) = atan(y/x)};
if @code{x < 0}, @code{atan2(y, x) = sign(y)*(Pi - atan(@GMPabs{y/x}))},
thus a number from @m{-\pi,-Pi} to @m{\pi,Pi}.
As for @code{atan}, in case the exact mathematical result is @m{+\pi,+Pi} or
@m{-\pi,-Pi},
its rounded result might be outside the function output range.
The function @code{mpfr_atan2u} behaves similarly, except the result is
multiplied by @var{u} and divided by 2@tie{}Pi; and @code{mpfr_atan2pi}
is the same as @code{mpfr_atan2u} with @code{u=2}.
For example, if @var{u} equals 360, @code{mpfr_atan2u} returns the
arc-tangent in degrees, with values from @minus{}180 to 180.

@code{atan2(y, 0)} does not raise any floating-point exception.
Special values are handled as described in the ISO C99 and IEEE@tie{}754
standards for the @code{atan2} function:
@itemize @bullet
@item @code{atan2(+0, -0)} returns @m{+\pi,+Pi}.
@item @code{atan2(-0, -0)} returns @m{-\pi,-Pi}.
@item @code{atan2(+0, +0)} returns +0.
@item @code{atan2(-0, +0)} returns @minus{}0.
@item @code{atan2(+0, x)} returns @m{+\pi,+Pi} for @math{x < 0}.
@item @code{atan2(-0, x)} returns @m{-\pi,-Pi} for @math{x < 0}.
@item @code{atan2(+0, x)} returns +0 for @math{x > 0}.
@item @code{atan2(-0, x)} returns @minus{}0 for @math{x > 0}.
@item @code{atan2(y, 0)} returns @m{-\pi/2,-Pi/2} for @math{y < 0}.
@item @code{atan2(y, 0)} returns @m{+\pi/2,+Pi/2} for @math{y > 0}.
@item @code{atan2(+Inf, -Inf)} returns @m{+3\pi/4,+3*Pi/4}.
@item @code{atan2(-Inf, -Inf)} returns @m{-3\pi/4,-3*Pi/4}.
@item @code{atan2(+Inf, +Inf)} returns @m{+\pi/4,+Pi/4}.
@item @code{atan2(-Inf, +Inf)} returns @m{-\pi/4,-Pi/4}.
@item @code{atan2(+Inf, x)} returns @m{+\pi/2,+Pi/2} for finite @math{x}.
@item @code{atan2(-Inf, x)} returns @m{-\pi/2,-Pi/2} for finite @math{x}.
@item @code{atan2(y, -Inf)} returns @m{+\pi,+Pi} for finite @math{y > 0}.
@item @code{atan2(y, -Inf)} returns @m{-\pi,-Pi} for finite @math{y < 0}.
@item @code{atan2(y, +Inf)} returns +0 for finite @math{y > 0}.
@item @code{atan2(y, +Inf)} returns @minus{}0 for finite @math{y < 0}.
@end itemize
@end deftypefun

@deftypefun int mpfr_cosh (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_sinh (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_tanh (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the hyperbolic cosine, sine or tangent of @var{op},
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_sinh_cosh (mpfr_t @var{sop}, mpfr_t @var{cop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set simultaneously @var{sop} to the hyperbolic sine of @var{op} and
@var{cop} to the hyperbolic cosine of @var{op},
rounded in the direction @var{rnd} with the corresponding precision of
@var{sop} and @var{cop}, which must be different variables.
Return 0 iff both results are exact (see @code{mpfr_sin_cos} for a more
detailed description of the return value).
@end deftypefun

@deftypefun int mpfr_sech (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_csch (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_coth (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the hyperbolic secant of @var{op}, cosecant of @var{op},
cotangent of @var{op}, rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_acosh (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_asinh (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_atanh (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the inverse hyperbolic cosine, sine or tangent of @var{op},
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_eint (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the exponential integral of @var{op},
rounded in the direction @var{rnd}.
This is the sum of Euler's constant, of the logarithm
of the absolute value of @var{op}, and of the sum for k from 1 to infinity of
@ifnottex
@var{op} to the power k, divided by k and factorial(k).
@end ifnottex
@tex
$@var{op}^k/(k \cdot k!)$.
@end tex
For positive @var{op}, it corresponds to the Ei function at @var{op}
(see formula 5.1.10 from the Handbook of Mathematical Functions from
Abramowitz and Stegun),
and for negative @var{op}, to the opposite of the
E1 function (sometimes called eint1)
at @minus{}@var{op} (formula 5.1.1 from the same reference).
@end deftypefun

@deftypefun int mpfr_li2 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to real part of the dilogarithm of @var{op}, rounded in the
direction @var{rnd}. MPFR defines the dilogarithm function as
@m{-\int_{t=0}^{@var{op}} \log(1-t)/t\ dt,the integral of -log(1-t)/t from 0
to @var{op}}.
@end deftypefun

@deftypefun int mpfr_gamma (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_gamma_inc (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the value of the Gamma function on @var{op}, resp.@: the
incomplete Gamma function on @var{op} and @var{op2},
rounded in the direction @var{rnd}.
(In the literature, @code{mpfr_gamma_inc} is called upper
incomplete Gamma function,
or sometimes complementary incomplete Gamma function.)
For @code{mpfr_gamma} (and @code{mpfr_gamma_inc} when @var{op2} is zero),
when @var{op} is a negative integer, @var{rop} is set to NaN@.

Note: the current implementation of @code{mpfr_gamma_inc} is slow for
large values of @var{rop} or @var{op}, in which case some internal overflow
might also occur.
@end deftypefun

@deftypefun int mpfr_lngamma (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the value of the logarithm of the Gamma function on @var{op},
rounded in the direction @var{rnd}.
When @var{op} is 1 or 2, set @var{rop} to +0 (in all rounding modes).
When @var{op} is an infinity or a nonpositive integer, set @var{rop} to +Inf,
following the general rules on special values.
When @math{@minus{}2@var{k}@minus{}1 < @var{op} < @minus{}2@var{k}},
@var{k} being a nonnegative integer, set @var{rop} to NaN@.
See also @code{mpfr_lgamma}.
@end deftypefun

@deftypefun int mpfr_lgamma (mpfr_t @var{rop}, int *@var{signp}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the value of the logarithm of the absolute value of the
Gamma function on @var{op}, rounded in the direction @var{rnd}. The sign
(1 or @minus{}1) of Gamma(@var{op}) is returned in the object pointed to
by @var{signp}.
When @var{op} is 1 or 2, set @var{rop} to +0 (in all rounding modes).
When @var{op} is an infinity or a nonpositive integer, set @var{rop} to +Inf.
When @var{op} is NaN, @minus{}Inf or a negative integer, *@var{signp} is
undefined, and when @var{op} is @pom{}0, *@var{signp} is the sign of the zero.
@end deftypefun

@deftypefun int mpfr_digamma (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the value of the Digamma (sometimes also called Psi)
function on @var{op}, rounded in the direction @var{rnd}.
When @var{op} is a negative integer, set @var{rop} to NaN@.
@end deftypefun

@deftypefun int mpfr_beta (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the value of the Beta function at arguments @var{op1} and
@var{op2}.
Note: the current code does not try to avoid internal overflow or underflow,
and might use a huge internal precision in some cases.
@end deftypefun

@deftypefun int mpfr_zeta (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_zeta_ui (mpfr_t @var{rop}, unsigned long int @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the value of the Riemann Zeta function on @var{op},
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_erf (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_erfc (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the value of the error function on @var{op}
(resp.@: the complementary error function on @var{op})
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_j0 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_j1 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_jn (mpfr_t @var{rop}, long int @var{n}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the value of the first kind Bessel function of order 0,
(resp.@: 1 and @var{n})
on @var{op}, rounded in the direction @var{rnd}. When @var{op} is
NaN, @var{rop} is always set to NaN@. When @var{op} is plus or negative infinity,
@var{rop} is set to +0. When @var{op} is zero, and @var{n} is not zero,
@var{rop} is set to +0 or @minus{}0 depending on the parity and sign of @var{n},
and the sign of @var{op}.
@end deftypefun

@deftypefun int mpfr_y0 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_y1 (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_yn (mpfr_t @var{rop}, long int @var{n}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the value of the second kind Bessel function of order 0
(resp.@: 1 and @var{n})
on @var{op}, rounded in the direction @var{rnd}. When @var{op} is
NaN or negative, @var{rop} is always set to NaN@. When @var{op} is +Inf,
@var{rop} is set to +0. When @var{op} is zero, @var{rop} is set to +Inf
or @minus{}Inf depending on the parity and sign of @var{n}.
@end deftypefun

@deftypefun int mpfr_agm (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the arithmetic-geometric mean of @var{op1} and @var{op2},
rounded in the direction @var{rnd}.
The arithmetic-geometric mean is the common limit of the sequences
@m{u_n,@var{u}_@var{n}} and @m{v_n,@var{v}_@var{n}},
where @m{u_0,@var{u}_@var{0}}=@var{op1}, @m{v_0,@var{v}_@var{0}}=@var{op2},
@m{u_{n+1},@var{u}_(@var{n}+1)} is the
arithmetic mean of @m{u_n,@var{u}_@var{n}} and @m{v_n,@var{v}_@var{n}},
and @m{v_{n+1},@var{v}_(@var{n}+1)} is the geometric mean of
@m{u_n,@var{u}_@var{n}} and @m{v_n,@var{v}_@var{n}}.
If any operand is negative and the other one is not zero,
set @var{rop} to NaN@.
If any operand is zero and the other one is finite (resp.@: infinite),
set @var{rop} to +0 (resp.@: NaN)@.
@end deftypefun

@deftypefun int mpfr_ai (mpfr_t @var{rop}, mpfr_t @var{x}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the value of the Airy function Ai
 on @var{x}, rounded in the direction @var{rnd}.
When @var{x} is
NaN,
@var{rop} is always set to NaN@. When @var{x} is +Inf or @minus{}Inf,
@var{rop} is +0.
The current implementation is not intended to be used with large arguments.
It works with @GMPabs{@var{x}} typically smaller than 500. For larger arguments,
other methods should be used and will be implemented in a future version.
@end deftypefun

@deftypefun int mpfr_const_log2 (mpfr_t @var{rop}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_const_pi (mpfr_t @var{rop}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_const_euler (mpfr_t @var{rop}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_const_catalan (mpfr_t @var{rop}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the logarithm of 2, the value of @m{\pi,Pi},
of Euler's constant 0.577@dots{}, of Catalan's constant 0.915@dots{},
respectively, rounded in the direction
@var{rnd}. These functions cache the computed values to avoid other
calculations if a lower or equal precision is requested. To free these caches,
use @code{mpfr_free_cache} or @code{mpfr_free_cache2}.
@end deftypefun

@node Input and Output Functions, Formatted Output Functions, Transcendental Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Input functions
@cindex Output functions
@cindex I/O functions
@section 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 @code{stream} to any of these functions will make
them read from @code{stdin} and write to @code{stdout}, respectively.

When using a function that takes a @code{FILE *} argument, you must
include the @code{<stdio.h>} standard header before @file{mpfr.h},
to allow @file{mpfr.h} to define prototypes for these functions.

@deftypefun size_t mpfr_out_str (FILE *@var{stream}, int @var{base}, size_t @var{n}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Output @var{op} on stream @var{stream} as a text string in
base @GMPabs{@var{base}}, rounded in the direction @var{rnd}.
The base may vary from 2 to 62 or from @minus{}2 to @minus{}36
(any other value yields undefined behavior). The argument @var{n} has
the same meaning as in @code{mpfr_get_str} (@pxref{mpfr_get_str}):
Print @var{n} significant digits exactly, or if @var{n} is 0, the number
@code{mpfr_get_str_ndigits(@var{base},@var{p})} where @var{p} is the
precision of @var{op} (@pxref{mpfr_get_str_ndigits}).

If the input is NaN, +Inf, @minus{}Inf, +0, or @minus{}0, then
@samp{@@NaN@@}, @samp{@@Inf@@}, @samp{-@@Inf@@}, @samp{0}, or
@samp{-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 @var{n}@minus{}1 digits (including
trailing zeros), then the exponent prefix, then the exponent in decimal.
The exponent prefix is @samp{e} when @math{@GMPabs{@var{base}} @le{} 10},
and @samp{@@} when @math{@GMPabs{@var{base}} > 10}. @xref{mpfr_get_str} for
information on the digits depending on the base.
@c The term "exponent prefix" is used in the mpfr_strtofr description.

Return the number of characters written, or if an error occurred, return 0.
@end deftypefun

@deftypefun size_t mpfr_inp_str (mpfr_t @var{rop}, FILE *@var{stream}, int @var{base}, mpfr_rnd_t @var{rnd})
Input a string in base @var{base} from stream @var{stream},
rounded in the direction @var{rnd}, and put the
read float in @var{rop}.
@c The argument @var{base} must be in the range 2 to 62.

@c The string is of the form @samp{M@@N} or, if the
@c base is 10 or less, alternatively @samp{MeN} or @samp{MEN}, or, if the base
@c is 16, alternatively @samp{MpB} or @samp{MPB}.
@c @samp{M} is the significand in the specified base, @samp{N} is the exponent
@c written in decimal for the specified base, and in base 16, @samp{B} is the
@c binary exponent written in decimal (i.e., it indicates the power of 2 by
@c which the significand is to be scaled).
This function reads a word (defined as a sequence of characters between
whitespace) and parses it using @code{mpfr_set_str}.
See the documentation of @code{mpfr_strtofr} for a detailed description
of the valid string formats.
@c Special values can be read as follows (the case does not matter):
@c @samp{@@NaN@@}, @samp{@@Inf@@}, @samp{+@@Inf@@} and @samp{-@@Inf@@},
@c possibly followed by other characters; if the base is smaller than
@c or equal to 16, the following strings are accepted too: @samp{NaN},
@c @samp{Inf}, @samp{+Inf} and @samp{-Inf}.

Return the number of bytes read, or if an error occurred, return 0.
@end deftypefun

@c @deftypefun void mpfr_inp_raw (mpfr_t @var{float}, FILE *@var{stream})
@c Input from stdio stream @var{stream} in the format written by
@c @code{mpfr_out_raw}, and put the result in @var{float}.
@c @end deftypefun

@deftypefun int mpfr_fpif_export (FILE *@var{stream}, mpfr_t @var{op})
Export the number @var{op} to the stream @var{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 @var{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.
@end deftypefun

@deftypefun int mpfr_fpif_import (mpfr_t @var{op}, FILE *@var{stream})
Import the number @var{op} from the stream @var{stream} in a floating-point
interchange format (see @code{mpfr_fpif_export}).
Note that the precision of @var{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 @code{MPFR_PREC_MAX}, the
function fails (it returns non-zero) and @var{op} is unchanged. If the
function fails for another reason, @var{op} is set to NaN and it is
unspecified whether the precision of @var{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.
@end deftypefun

@deftypefun void mpfr_dump (mpfr_t @var{op})
Output @var{op} on @code{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); @samp{@@NaN@@}, @samp{@@Inf@@} or @samp{0} if the
argument is NaN, an infinity or zero, respectively; otherwise the remaining
of the output is as follows: @samp{0.} then the @var{p} bits of the binary
significand, where @var{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 @samp{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 (@samp{!!!}),
followed by flags, followed by three exclamation marks (@samp{!!!}) again.
These flags are: @samp{N} if the most significant bit of the significand
is 0 (i.e., the number is not normalized); @samp{T} if there are non-zero
trailing bits; @samp{U} if this is an UBF number (internal use only);
@samp{<} if the exponent is less than the current minimum exponent;
@samp{>} if the exponent is greater than the current maximum exponent.
@end deftypefun

@node Formatted Output Functions, Integer and Remainder Related Functions, Input and Output Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Output functions
@cindex I/O functions
@section Formatted Output Functions

@subsection Requirements
The class of @code{mpfr_printf} functions provides formatted output in a
similar manner as the standard C @code{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 @code{<stdio.h>}
standard header before @file{mpfr.h}, to allow @file{mpfr.h} to define
prototypes for these functions.

@subsection Format String
The format specification accepted by @code{mpfr_printf} is an extension of the
@code{printf} one. The conversion specification is of the form:

@example
% [flags] [width] [.[precision]] [type] [rounding] conv
@end example

@samp{flags}, @samp{width}, and @samp{precision} have the same meaning as for
the standard @code{printf} (in particular, notice that the precision is
related to the number of digits displayed in the base chosen by @samp{conv}
and not related to the internal precision of the @code{mpfr_t} variable), but
note that for @samp{Re}, the default precision is not the same as the one for
@samp{e}.
@code{mpfr_printf} accepts the same @samp{type} specifiers as GMP (except the
non-standard and deprecated @samp{q}, use @samp{ll} instead), namely the
length modifiers defined in the C standard:

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @samp{h}  @tab @code{short}
@item @samp{hh} @tab @code{char}
@item @samp{j}  @tab @code{intmax_t} or @code{uintmax_t}
@item @samp{l}  @tab @code{long} or @code{wchar_t}
@item @samp{ll} @tab @code{long long}
@item @samp{L}  @tab @code{long double}
@item @samp{t}  @tab @code{ptrdiff_t}
@item @samp{z}  @tab @code{size_t}
@end multitable
@end quotation

@noindent
and the @samp{type} specifiers defined in GMP, plus @samp{R} and @samp{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 @samp{conv}
specifier to use after the @samp{type} specifier):

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @samp{F}  @tab @code{mpf_t}, float conversions
@item @samp{Q}  @tab @code{mpq_t}, integer conversions
@item @samp{M}  @tab @code{mp_limb_t}, integer conversions
@item @samp{N}  @tab @code{mp_limb_t} array, integer conversions
@item @samp{Z}  @tab @code{mpz_t}, integer conversions

@item @samp{P}  @tab @code{mpfr_prec_t}, integer conversions
@item @samp{R}  @tab @code{mpfr_t}, float conversions
@end multitable
@end quotation

The @samp{type} specifiers have the same restrictions as those
mentioned in the GMP documentation:
@ifinfo
@pxref{Formatted Output Strings,,, gmp.info,GNU MP}.
@end ifinfo
@ifnotinfo
see Section ``Formatted Output Strings'' in @cite{GNU MP}.
@end ifnotinfo
In particular, the @samp{type} specifiers (except @samp{R} and @samp{P}) are
supported only if they are supported by @code{gmp_printf} in your GMP build;
this implies that the standard specifiers, such as @samp{t}, must @emph{also}
be supported by your C library if you want to use them.

The @samp{rounding} field is specific to @code{mpfr_t} arguments and should
not be used with other types.

With conversion specification not involving @samp{P} and @samp{R} types,
@code{mpfr_printf} behaves exactly as @code{gmp_printf}.

The @samp{P} type specifies that a following @samp{d}, @samp{i},
@samp{o}, @samp{u}, @samp{x}, or @samp{X} conversion specifier applies
to a @code{mpfr_prec_t} argument.
It is needed because the @code{mpfr_prec_t} type does not necessarily
correspond to an @code{int} or any fixed standard type.
The @samp{precision} value specifies the minimum number of digits to
appear. The default precision is 1.
For example:
@example
mpfr_t x;
mpfr_prec_t p;
mpfr_init (x);
@dots{}
p = mpfr_get_prec (x);
mpfr_printf ("variable x with %Pu bits", p);
@end example

The @samp{R} type specifies that a following @samp{a}, @samp{A}, @samp{b},
@samp{e}, @samp{E}, @samp{f}, @samp{F}, @samp{g}, @samp{G}, or @samp{n}
conversion specifier applies to a @code{mpfr_t} argument.
The @samp{R} type can be followed by a @samp{rounding} specifier denoted by
one of the following characters:

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @samp{U}  @tab round toward positive infinity
@item @samp{D}  @tab round toward negative infinity
@item @samp{Y}  @tab round away from zero
@item @samp{Z}  @tab round toward zero
@item @samp{N}  @tab round to nearest (with ties to even)
@item @samp{*}  @tab rounding mode indicated by the @code{mpfr_rnd_t} argument
just before the corresponding @code{mpfr_t} variable.
@end multitable
@end quotation

The default rounding mode is rounding to nearest.
The following three examples are equivalent:
@example
mpfr_t x;
mpfr_init (x);
@dots{}
mpfr_printf ("%.128Rf", x);
mpfr_printf ("%.128RNf", x);
mpfr_printf ("%.128R*f", MPFR_RNDN, x);
@end example

Note that the rounding away from zero mode is specified with @samp{Y}
because ISO C reserves the @samp{A} specifier for hexadecimal output (see
below).

The output @samp{conv} specifiers allowed with @code{mpfr_t} parameter are:

@quotation
@multitable {(space)} {MMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMMM}
@item @samp{a} @samp{A} @tab hex float, C99 style
@item @samp{b}          @tab binary output
@item @samp{e} @samp{E} @tab scientific-format float
@item @samp{f} @samp{F} @tab fixed-point float
@item @samp{g} @samp{G} @tab fixed-point or scientific float
@end multitable
@end quotation

The conversion specifier @samp{b}, which displays the argument in binary, is
specific to @code{mpfr_t} arguments and should not be used with other types.
Other conversion specifiers have the same meaning as for a @code{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 @code{nan}, @code{-inf}, and @code{inf}
for @samp{a}, @samp{b}, @samp{e}, @samp{f}, and @samp{g} specifiers and
@code{NAN}, @code{-INF}, and @code{INF} for @samp{A}, @samp{E}, @samp{F}, and
@samp{G} specifiers.

The @code{mpfr_t} number is rounded to the given precision in the 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, except for the @samp{g} and @samp{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: @samp{a}, @samp{A}, @samp{b}, @samp{e},
@samp{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 @code{"%.0RNe"}.
This also applies when the @samp{g} (resp.@: @samp{G}) conversion specifier
uses the @samp{e} (resp.@: @samp{E}) style.
If the precision is set to a value greater than the maximum value for an
@code{int}, it will be silently reduced down to @code{INT_MAX}.

If the precision is missing, it is chosen as follows, depending on the
conversion specifier.
@itemize @bullet
@item With @samp{a}, @samp{A}, and @samp{b}, it is chosen to have
an exact representation with no trailing zeros.
@c Avoid saying "minimum" as this could be confusing with the different
@c possible choices for a/A.
@item With @samp{e} and @samp{E}, it is
@ifnottex
ceil(@var{p}*log(2)/log(@var{10})),
@end ifnottex
@tex
$\left\lceil @var{p} {\log 2 \over \log @var{10}} \right\rceil$,
@end tex
where @var{p} is the precision of the input variable, matching the choice
done for @code{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.
@item With @samp{f}, @samp{F}, @samp{g}, and @samp{G}, it is 6.
@end itemize

@c For the record, concerning the choice for 'e'/'E':
@c   https://sympa.inria.fr/sympa/arc/mpfr/2019-12/msg00000.html
@c   https://sympa.inria.fr/sympa/arc/mpfr/2020-01/msg00000.html
@c   https://gforge.inria.fr/tracker/index.php?func=detail&aid=21816&group_id=136&atid=619

@subsection Functions

For all the following functions, if the number of characters that ought to be
written exceeds the maximum limit @code{INT_MAX} for an @code{int}, nothing is
written in the stream (resp.@: to @code{stdout}, to @var{buf}, to @var{str}),
the function returns @minus{}1, sets the @emph{erange} flag, and @code{errno}
is set to @code{EOVERFLOW} if the @code{EOVERFLOW} macro is defined (such as
on POSIX systems). Note, however, that @code{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).

@deftypefun int mpfr_fprintf (FILE *@var{stream}, const char *@var{template}, @dots{})
@deftypefunx int mpfr_vfprintf (FILE *@var{stream}, const char *@var{template}, va_list @var{ap})
Print to the stream @var{stream} the optional arguments under the control of
the template string @var{template}.
Return the number of characters written or a negative value if an error
occurred.
@end deftypefun

@deftypefun  int mpfr_printf (const char *@var{template}, @dots{})
@deftypefunx int mpfr_vprintf (const char *@var{template}, va_list @var{ap})
Print to @code{stdout} the optional arguments under the control of the
template string @var{template}.
Return the number of characters written or a negative value if an error
occurred.
@end deftypefun

@deftypefun int mpfr_sprintf (char *@var{buf}, const char *@var{template}, @dots{})
@deftypefunx int mpfr_vsprintf (char *@var{buf}, const char *@var{template}, va_list @var{ap})
Form a null-terminated string corresponding to the optional arguments under
the control of the template string @var{template}, and print it in
@var{buf}. No overlap is permitted between
@var{buf} and the other arguments.
Return the number of characters written in the array @var{buf}
@emph{not counting}
the terminating null character or a negative value if an error occurred.
@end deftypefun

@deftypefun int mpfr_snprintf (char *@var{buf}, size_t @var{n}, const char *@var{template}, @dots{})
@deftypefunx int mpfr_vsnprintf (char *@var{buf}, size_t @var{n}, const char *@var{template}, va_list @var{ap})
Form a null-terminated string corresponding to the optional arguments under
the control of the template string @var{template}, and print it in
@var{buf}. If @var{n} is zero, nothing is
written and @var{buf} may be a null pointer, otherwise, the @var{n}@minus{}1
first characters are written in @var{buf} and the @var{n}-th is a null character.
Return the number of characters that would have been written had @var{n} been
sufficiently large, @emph{not counting}
the terminating null character, or a negative value if an error occurred.
@end deftypefun

@deftypefun int mpfr_asprintf (char **@var{str}, const char *@var{template}, @dots{})
@deftypefunx int mpfr_vasprintf (char **@var{str}, const char *@var{template}, va_list @var{ap})
Write their output as a null terminated string in a block of memory allocated
using the allocation function (@pxref{Memory Handling}). A pointer to the
block is stored in
@var{str}. The block of memory must be freed using @code{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 @var{str} are undefined.
@end deftypefun

@node Integer and Remainder Related Functions, Rounding-Related Functions, Formatted Output Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Integer related functions
@cindex Remainder related functions
@section Integer and Remainder Related Functions

@deftypefun int mpfr_rint (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_ceil (mpfr_t @var{rop}, mpfr_t @var{op})
@deftypefunx int mpfr_floor (mpfr_t @var{rop}, mpfr_t @var{op})
@deftypefunx int mpfr_round (mpfr_t @var{rop}, mpfr_t @var{op})
@deftypefunx int mpfr_roundeven (mpfr_t @var{rop}, mpfr_t @var{op})
@deftypefunx int mpfr_trunc (mpfr_t @var{rop}, mpfr_t @var{op})
Set @var{rop} to @var{op} rounded to an integer.
@code{mpfr_rint} rounds to the nearest representable integer in the
given direction @var{rnd}, and the other five functions behave in a
similar way with some fixed rounding mode:
@itemize @bullet
@item @code{mpfr_ceil}: to the next higher or equal representable integer
      (like @code{mpfr_rint} with @code{MPFR_RNDU});
@item @code{mpfr_floor} to the next lower or equal representable integer
      (like @code{mpfr_rint} with @code{MPFR_RNDD});
@item @code{mpfr_round} to the nearest representable integer,
      rounding halfway cases away from zero
      (as in the roundTiesToAway mode of IEEE@tie{}754);
@item @code{mpfr_roundeven} to the nearest representable integer,
      rounding halfway cases with the even-rounding rule
      (like @code{mpfr_rint} with @code{MPFR_RNDN});
@item @code{mpfr_trunc} to the next representable integer toward zero
      (like @code{mpfr_rint} with @code{MPFR_RNDZ}).
@end itemize
When @var{op} is a zero or an infinity, set @var{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 @var{op}, and negative when it is smaller.
More precisely, the return value is 0 when @var{op} is an integer
representable in @var{rop}, 1 or @minus{}1 when @var{op} is an integer
that is not representable in @var{rop}, 2 or @minus{}2 when @var{op} is
not an integer.

When @var{op} is NaN, the NaN flag is set as usual. In the other cases,
the inexact flag is set when @var{rop} differs from @var{op}, following
the ISO C99 rule for the @code{rint} function. If you want the behavior to
be more like IEEE@tie{}754 / ISO TS@tie{}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 @code{mpfr_rint_*} functions instead.

Note that no double rounding is performed; for instance, 10.5 (1010.1 in
binary) is rounded by @code{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.)
@end deftypefun

@deftypefun int mpfr_rint_ceil (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_rint_floor (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_rint_round (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_rint_roundeven (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_rint_trunc (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to @var{op} rounded to an integer:
@itemize @bullet
@item @code{mpfr_rint_ceil}: to the next higher or equal integer;
@item @code{mpfr_rint_floor}: to the next lower or equal integer;
@item @code{mpfr_rint_round}: to the nearest integer,
      rounding halfway cases away from zero;
@item @code{mpfr_rint_roundeven}: to the nearest integer,
      rounding halfway cases to the nearest even integer;
@item @code{mpfr_rint_trunc} to the next integer toward zero.
@end itemize
If the result is not representable, it is rounded in the direction @var{rnd}.
When @var{op} is a zero or an infinity, set @var{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).

Contrary to @code{mpfr_rint}, those functions do perform a double rounding:
first @var{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 @var{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, @code{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.
@end deftypefun

@deftypefun int mpfr_frac (mpfr_t @var{rop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the fractional part of @var{op}, having the same sign as
@var{op}, rounded in the direction @var{rnd} (unlike in @code{mpfr_rint},
@var{rnd} affects only how the exact fractional part is rounded, not how
the fractional part is generated).
When @var{op} is an integer or an infinity, set @var{rop} to zero with
the same sign as @var{op}.
@end deftypefun

@deftypefun int mpfr_modf (mpfr_t @var{iop}, mpfr_t @var{fop}, mpfr_t @var{op}, mpfr_rnd_t @var{rnd})
Set simultaneously @var{iop} to the integral part of @var{op} and @var{fop} to
the fractional part of @var{op}, rounded in the direction @var{rnd} with the
corresponding precision of @var{iop} and @var{fop} (equivalent to
@code{mpfr_trunc(@var{iop}, @var{op}, @var{rnd})} and
@code{mpfr_frac(@var{fop}, @var{op}, @var{rnd})}). The variables @var{iop} and
@var{fop} must be different. Return 0 iff both results are exact (see
@code{mpfr_sin_cos} for a more detailed description of the return value).
@end deftypefun

@deftypefun int mpfr_fmod (mpfr_t @var{r}, mpfr_t @var{x}, mpfr_t @var{y}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_fmod_ui (mpfr_t @var{r}, mpfr_t @var{x}, unsigned long int @var{y}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_fmodquo (mpfr_t @var{r}, long int* @var{q}, mpfr_t @var{x}, mpfr_t @var{y}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_remainder (mpfr_t @var{r}, mpfr_t @var{x}, mpfr_t @var{y}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_remquo (mpfr_t @var{r}, long int* @var{q}, mpfr_t @var{x}, mpfr_t @var{y}, mpfr_rnd_t @var{rnd})
Set @var{r} to the value of @math{@var{x} - @var{n}@var{y}}, rounded
according to the direction @var{rnd}, where @var{n} is the integer quotient
of @var{x} divided by @var{y}, defined as follows: @var{n} is rounded
toward zero for @code{mpfr_fmod}, @code{mpfr_fmod_ui} and @code{mpfr_fmodquo},
and to the nearest integer (ties rounded to even) for @code{mpfr_remainder}
and @code{mpfr_remquo}.

Special values are handled as described in Section@tie{}F.9.7.1 of
the ISO C99 standard:
If @var{x} is infinite or @var{y} is zero, @var{r} is NaN@.
If @var{y} is infinite and @var{x} is finite, @var{r} is @var{x} rounded
to the precision of @var{r}.
If @var{r} is zero, it has the sign of @var{x}.
The return value is the ternary value corresponding to @var{r}.

Additionally, @code{mpfr_fmodquo} and @code{mpfr_remquo} store
the low significant bits from the quotient @var{n} in @var{*q}
(more precisely the number of bits in a @code{long int} minus one),
with the sign of @var{x} divided by @var{y}
(except if those low bits are all zero, in which case zero is returned).
If the result is NaN, the value of @var{*q} is unspecified.
Note that @var{x} may be so large in magnitude relative to @var{y} that an
exact representation of the quotient is not practical.
The @code{mpfr_remainder} and @code{mpfr_remquo} functions are useful for
additive argument reduction.
@end deftypefun

@deftypefun int mpfr_integer_p (mpfr_t @var{op})
Return non-zero iff @var{op} is an integer.
@end deftypefun

@node Rounding-Related Functions, Miscellaneous Functions, Integer and Remainder Related Functions, MPFR Interface
@cindex Rounding mode related functions
@section Rounding-Related Functions

@deftypefun void mpfr_set_default_rounding_mode (mpfr_rnd_t @var{rnd})
Set the default rounding mode to @var{rnd}.
The default rounding mode is to nearest initially.
@end deftypefun

@deftypefun mpfr_rnd_t mpfr_get_default_rounding_mode (void)
Get the default rounding mode.
@end deftypefun

@deftypefun int mpfr_prec_round (mpfr_t @var{x}, mpfr_prec_t @var{prec}, mpfr_rnd_t @var{rnd})
Round @var{x} according to @var{rnd} with precision @var{prec}, which
must be an integer between @code{MPFR_PREC_MIN} and @code{MPFR_PREC_MAX}
(otherwise the behavior is undefined).
If @var{prec} is greater than or equal to the precision of @var{x}, then
new space is allocated for the significand, and it is filled with zeros.
Otherwise, the significand is rounded to precision @var{prec} with the given
direction; no memory reallocation to free the unused limbs is done.
In both cases, the precision of @var{x} is changed to @var{prec}.

Here is an example of how to use @code{mpfr_prec_round} to implement
Newton's algorithm to compute the inverse of @var{a}, assuming @var{x} is
already an approximation to @var{n} bits:
@example
  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 */
@end example

Warning! You must not use this function if @var{x} was initialized
with @code{MPFR_DECL_INIT} or with @code{mpfr_custom_init_set}
(@pxref{Custom Interface}).
@end deftypefun

@deftypefun int mpfr_can_round (mpfr_t @var{b}, mpfr_exp_t @var{err}, mpfr_rnd_t @var{rnd1}, mpfr_rnd_t @var{rnd2}, mpfr_prec_t @var{prec})
Assuming @var{b} is an approximation of an unknown number
@var{x} in the direction @var{rnd1} with error at most two to the power
E(b)-@var{err} where E(b) is the exponent of @var{b}, return a non-zero
value if one is able to round correctly @var{x} to precision @var{prec}
with the direction @var{rnd2} assuming an unbounded exponent range, and
0 otherwise (including for NaN and Inf).
In other words, if the error on @var{b} is bounded by two to the power
@var{k} ulps, and @var{b} has precision @var{prec},
you should give @var{err}=@var{prec}@minus{}@var{k}.
This function @strong{does not modify} its arguments.

If @var{rnd1} is @code{MPFR_RNDN} or @code{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 @var{rnd1} (with the
same value of @var{err}).

When @var{rnd2} is @code{MPFR_RNDF}, let @var{rnd3} be the opposite direction
if @var{rnd1} is a directed rounding, and @code{MPFR_RNDN}
if @var{rnd1} is @code{MPFR_RNDN} or @code{MPFR_RNDF}.
The returned value of @code{mpfr_can_round (b, err, rnd1, MPFR_RNDF, prec)}
is non-zero iff after
the call @code{mpfr_set (y, b, rnd3)} with @var{y} of precision @var{prec},
@var{y} is guaranteed to be a faithful rounding of @var{x}.
@c
@c For rnd1=RNDN, let [u,v] be the interval where x can lie, then mpfr_can_round
@c returns 1 exactly when either:
@c a) [u,v] contains a unique representable number y in precision prec,
@c    and both u and v round to y with RNDN
@c b) [u,v] contains no representable number y in precision prec, then mpfr_set
@c    will return either down(u) or up(v) which are both faithful roundings of x
@c (With rnd2=RNDN instead, mpfr_can_round would only return 1 in case a).
@c
@c For rnd1=RNDU, let [u,b] be the interval where x can lie (we have v=b since
@c rnd1=RNDU implies x <= b), then mpfr_can_round returns 1 exactly when either:
@c a) [u,b] contains a (unique) representable number y in precision prec,
@c    then b rounds to y with rnd3=RNDD
@c b) [u,b] contains no representable number y in precision prec, then mpfr_set
@c    will return down(u) which is a faithful rounding of x
@c (With rnd2 a directed rounding instead of RNDF, mpfr_can_round would only return
@c 1 in case b, apart from special cases where u or b are representable in precision
@c prec).
@c
@c Note: This spec is better than one avoiding "hard cases" (leaving
@c such cases indeterminate to have a mpfr_can_round in constant time).
@c Indeed, the time gained by having a mpfr_can_round without hard cases
@c would be lost by making the user recompute everything in a larger
@c precision if one could have returned non-zero instead of 0.

Note: The @ref{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 @var{prec}, then one can use the following
trick to determine the (non-zero) ternary value in any rounding mode
@var{rnd2} (note that @code{MPFR_RNDZ} below can be replaced by any
directed rounding mode):
@example
if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ,
                    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);
  @}
@end example
Indeed, if @var{rnd2} is @code{MPFR_RNDN}, this will check if one can
round to @var{prec}+1 bits with a directed rounding:
if so, one can surely round to nearest to @var{prec} bits,
and in addition one can determine the correct ternary value, which would not
be the case when @var{b} is near from a value exactly representable on
@var{prec} bits.

A detailed example is available in the @file{examples} subdirectory,
file @file{can_round.c}.
@end deftypefun

@deftypefun mpfr_prec_t mpfr_min_prec (mpfr_t @var{x})
Return the minimal number of bits required to store the significand of
@var{x}, and 0 for special values, including 0.
@c This warning no longer holds now that MPFR_PREC_MIN=1.
@c Warning!  The return value can be less than @code{MPFR_PREC_MIN},
@c in particular for the powers of two.  If you use this function together
@c with @code{mpfr_prec_round} in order to reduce the memory space occupied
@c by a number, you may need to take the maximum of the returned precision
@c and @code{MPFR_PREC_MIN} for @code{mpfr_prec_round}.
@end deftypefun

@deftypefun {const char *} mpfr_print_rnd_mode (mpfr_rnd_t @var{rnd})
Return a string ("MPFR_RNDN", "MPFR_RNDZ", "MPFR_RNDU", "MPFR_RNDD",
"MPFR_RNDA", "MPFR_RNDF") corresponding to the rounding mode @var{rnd},
or a null pointer if @var{rnd} is an invalid rounding mode.
@end deftypefun

@deftypefn Macro int mpfr_round_nearest_away (int (@var{foo})(mpfr_t, type1_t, ..., mpfr_rnd_t), mpfr_t @var{rop}, type1_t @var{op}, ...)
Given a function @var{foo} and one or more values @var{op} (which may be
a @code{mpfr_t}, a @code{long int}, a @code{double}, etc.), put in @var{rop}
the round-to-nearest-away rounding of @code{@var{foo}(@var{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 @var{foo} takes as input, from second to
penultimate argument(s), the argument list given after @var{rop},
a rounding mode as final argument,
puts in its first argument the value @code{@var{foo}(@var{op},...)} rounded
according to this rounding mode, and returns the corresponding ternary value
(which is expected to be correct, otherwise @code{mpfr_round_nearest_away}
will not work as desired).
Due to implementation constraints, this function must not be called when
the minimal exponent @code{emin} is the smallest possible one.
This macro has been made such that the compiler is able to detect
mismatch between the argument list @var{op}
and the function prototype of @var{foo}.
Multiple input arguments @var{op} are supported only with C99 compilers.
Otherwise, for C90 compilers, only one such argument is supported.

Note: this macro is experimental and its interface might change in future
versions.
@example
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);
@end example
@end deftypefn

@node Miscellaneous Functions, Exception Related Functions, Rounding-Related Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Miscellaneous float functions
@section Miscellaneous Functions

@deftypefun void mpfr_nexttoward (mpfr_t @var{x}, mpfr_t @var{y})
If @var{x} or @var{y} is NaN, set @var{x} to NaN; note that the NaN flag
is set as usual.
If @var{x} and @var{y} are equal, @var{x} is unchanged.
Otherwise, if @var{x} is different from @var{y}, replace @var{x} by the
next floating-point number (with the precision of @var{x} and the current
exponent range) in the direction of @var{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.
@c For NaN, the behavior is like IEEE@tie{}754 with sNaN.
@end deftypefun

@deftypefun void mpfr_nextabove (mpfr_t @var{x})
@deftypefunx void mpfr_nextbelow (mpfr_t @var{x})
Equivalent to @code{mpfr_nexttoward} where @var{y} is positive infinity
(resp.@: negative infinity).
@end deftypefun

@deftypefun int mpfr_min (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_max (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
Set @var{rop} to the minimum (resp.@: maximum)
of @var{op1} and @var{op2}. If @var{op1}
and @var{op2} are both NaN, then @var{rop} is set to NaN@. If @var{op1}
or @var{op2} is NaN, then @var{rop} is set to the numeric value. If
@var{op1} and @var{op2} are zeros of different signs, then @var{rop}
is set to @minus{}0 (resp.@: +0).
As usual, the NaN flag is set only when the result is NaN, i.e.,
when both @var{op1} and @var{op2} are NaN@.

Note: These functions correspond to the minimumNumber and maximumNumber
operations of IEEE@tie{}754-2019 for the result. But in MPFR, the NaN flag
is set only when @emph{both} operands are NaN@.
@end deftypefun

@deftypefun int mpfr_urandomb (mpfr_t @var{rop}, gmp_randstate_t @var{state})
Generate a uniformly distributed random float in the interval
@math{0 @le{} @var{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 @var{e} denotes the exponent after normalization, then
the least @math{-@var{e}} significant bits of the significand are always 0).

Return 0, unless the exponent is not in the current exponent range, in
which case @var{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 @code{gmp_randstate_t} structure, which should be
created using the GMP @code{gmp_randinit} function (see the GMP manual).

Note: for a given version of MPFR, the returned value of @var{rop} and the
new value of @var{state} (which controls further random values) do not depend
on the machine word size.
@end deftypefun

@deftypefun int mpfr_urandom (mpfr_t @var{rop}, gmp_randstate_t @var{state}, mpfr_rnd_t @var{rnd})
Generate a uniformly distributed random float.
The floating-point number @var{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 @var{rnd}.

The second argument is a @code{gmp_randstate_t} structure, which should be
created using the GMP @code{gmp_randinit} function (see the GMP manual).

Note: the note for @code{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).
@end deftypefun

@deftypefun int mpfr_nrandom (mpfr_t @var{rop1}, gmp_randstate_t @var{state}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_grandom (mpfr_t @var{rop1}, mpfr_t @var{rop2}, gmp_randstate_t @var{state}, mpfr_rnd_t @var{rnd})
Generate one (possibly two for @code{mpfr_grandom}) random floating-point
number according to a standard normal Gaussian distribution (with mean zero
and variance one). For @code{mpfr_grandom}, if @var{rop2} is a null pointer,
then only one value is generated and stored in @var{rop1}.

The floating-point number @var{rop1} (and @var{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 @var{rnd}.

The @code{gmp_randstate_t} argument should be
created using the GMP @code{gmp_randinit} function (see the GMP manual).

For @code{mpfr_grandom},
the combination of the ternary values is returned like with
@code{mpfr_sin_cos}. If @var{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 @code{mpfr_urandomb} holds too. In addition, the exponent
range and the rounding mode might have a side effect on the next random state.

Note: @code{mpfr_nrandom} is much more efficient than @code{mpfr_grandom},
especially for large precision. Thus @code{mpfr_grandom} is marked as
deprecated and will be removed in a future release.
@end deftypefun

@deftypefun int mpfr_erandom (mpfr_t @var{rop1}, gmp_randstate_t @var{state}, mpfr_rnd_t @var{rnd})
Generate one random floating-point number according to an exponential
distribution, with mean one.
Other characteristics are identical to @code{mpfr_nrandom}.
@end deftypefun

@deftypefun mpfr_exp_t mpfr_get_exp (mpfr_t @var{x})
Return the exponent of @var{x}, assuming that @var{x} is a non-zero ordinary
number and the significand is considered in [1/2,1). For this function,
@var{x} is allowed to be outside of the current range of acceptable values.
The behavior for NaN, infinity or zero is undefined.
@end deftypefun

@deftypefun int mpfr_set_exp (mpfr_t @var{x}, mpfr_exp_t @var{e})
Set the exponent of @var{x} to @var{e} if @var{x} is a non-zero ordinary
number and @var{e} is in the current exponent range, and return 0;
otherwise, return a non-zero value (@var{x} is not changed).
@end deftypefun

@deftypefun int mpfr_signbit (mpfr_t @var{op})
Return a non-zero value iff @var{op} has its sign bit set (i.e., if it is
negative, @minus{}0, or a NaN whose representation has its sign bit set).
@end deftypefun

@deftypefun int mpfr_setsign (mpfr_t @var{rop}, mpfr_t @var{op}, int @var{s}, mpfr_rnd_t @var{rnd})
Set the value of @var{rop} from @var{op}, rounded toward the given
direction @var{rnd}, then set (resp.@: clear) its sign bit if @var{s}
is non-zero (resp.@: zero), even when @var{op} is a NaN@.
@end deftypefun

@deftypefun int mpfr_copysign (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
Set the value of @var{rop} from @var{op1}, rounded toward the given
direction @var{rnd}, then set its sign bit to that of @var{op2} (even
when @var{op1} or @var{op2} is a NaN)@. This function is equivalent to
@code{mpfr_setsign (@var{rop}, @var{op1}, mpfr_signbit (@var{op2}), @var{rnd})}.
@end deftypefun

@c By definition, a C string is always null-terminated, so that we
@c could just say "string" or "null-terminated character array",
@c but "null-terminated string" is not an error and probably better
@c for most users.
@deftypefun {const char *} mpfr_get_version (void)
Return the MPFR version, as a null-terminated string.
@end deftypefun

@defmac MPFR_VERSION
@defmacx MPFR_VERSION_MAJOR
@defmacx MPFR_VERSION_MINOR
@defmacx MPFR_VERSION_PATCHLEVEL
@defmacx MPFR_VERSION_STRING
@code{MPFR_VERSION} is the version of MPFR as a preprocessing constant.
@code{MPFR_VERSION_MAJOR}, @code{MPFR_VERSION_MINOR} and
@code{MPFR_VERSION_PATCHLEVEL} are respectively the major, minor and patch
level of MPFR version, as preprocessing constants.
@code{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 @code{mpfr_get_version} to check at run time
the header file and library used match:
@example
if (strcmp (mpfr_get_version (), MPFR_VERSION_STRING))
  fprintf (stderr, "Warning: header and library do not match\n");
@end example
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).
@end defmac

@deftypefn Macro long MPFR_VERSION_NUM (@var{major}, @var{minor}, @var{patchlevel})
Create an integer in the same format as used by @code{MPFR_VERSION} from the
given @var{major}, @var{minor} and @var{patchlevel}.
Here is an example of how to check the MPFR version at compile time:
@example
#if (!defined(MPFR_VERSION) || (MPFR_VERSION<MPFR_VERSION_NUM(3,0,0)))
# error "Wrong MPFR version."
#endif
@end example
@end deftypefn

@deftypefun {const char *} mpfr_get_patches (void)
Return a null-terminated string containing the ids of the patches applied to
the MPFR library (contents of the @file{PATCHES} file), separated by spaces.
Note: If the program has been compiled with an older MPFR version and is
dynamically linked with a new MPFR library version, the identifiers of the
patches applied to the old (compile-time) MPFR version are not available
(however, this information should not have much interest in general).
@end deftypefun

@deftypefun int mpfr_buildopt_tls_p (void)
Return a non-zero value if MPFR was compiled as thread safe using
compiler-level Thread-Local Storage (that is, MPFR was built with the
@samp{--enable-thread-safe} configure option, see @code{INSTALL} file),
return zero otherwise.
@end deftypefun

@deftypefun int mpfr_buildopt_float128_p (void)
Return a non-zero value if MPFR was compiled with @samp{_Float128} support
(that is, MPFR was built with the @samp{--enable-float128} configure option),
return zero otherwise.
@end deftypefun

@deftypefun int mpfr_buildopt_decimal_p (void)
Return a non-zero value if MPFR was compiled with decimal float support (that
is, MPFR was built with the @samp{--enable-decimal-float} configure option),
return zero otherwise.
@end deftypefun

@deftypefun int mpfr_buildopt_gmpinternals_p (void)
Return a non-zero value if MPFR was compiled with GMP internals
(that is, MPFR was built with either @samp{--with-gmp-build} or
@samp{--enable-gmp-internals} configure option), return zero otherwise.
@end deftypefun

@deftypefun 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 @code{mpfr_const_pi} or
@code{mpfr_const_log2} (that is, MPFR was built with the
@samp{--enable-shared-cache} configure option), return zero otherwise.
If the return value is non-zero, MPFR applications may need to be compiled
with the @samp{-pthread} option.
@end deftypefun

@deftypefun {const char *} mpfr_buildopt_tune_case (void)
Return a string saying which thresholds file has been used at compile time.
This file is normally selected from the processor type.
@end deftypefun

@node Exception Related Functions, Memory Handling Functions, Miscellaneous Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Exception related functions
@section Exception Related Functions

@deftypefun mpfr_exp_t mpfr_get_emin (void)
@deftypefunx mpfr_exp_t mpfr_get_emax (void)
Return the (current) smallest and largest exponents allowed for a
floating-point variable. The smallest positive value of a floating-point
variable is @m{1/2 \times 2^{\rm emin}, one half times 2 raised to the
smallest exponent} and the largest value has the form @m{(1 - \varepsilon)
\times 2^{\rm emax}, (1 - epsilon) times 2 raised to the largest exponent},
where @m{\varepsilon,epsilon} depends on the precision of the considered
variable.
@end deftypefun

@deftypefun int mpfr_set_emin (mpfr_exp_t @var{exp})
@deftypefunx int mpfr_set_emax (mpfr_exp_t @var{exp})
Set the smallest and largest exponents allowed for a floating-point variable.
Return a non-zero value when @var{exp} is not in the range accepted by the
implementation (in that case the smallest or 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 @code{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 @code{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 @math{@code{emin} > @code{emax}} and a floating-point value needs to
be produced as output, the behavior is undefined (@code{mpfr_set_emin}
and @code{mpfr_set_emax} do not check this condition as it might occur
between successive calls to these two functions).
@end deftypefun

@deftypefun mpfr_exp_t mpfr_get_emin_min (void)
@deftypefunx mpfr_exp_t mpfr_get_emin_max (void)
@deftypefunx mpfr_exp_t mpfr_get_emax_min (void)
@deftypefunx mpfr_exp_t mpfr_get_emax_max (void)
Return the minimum and maximum of the exponents
allowed for @code{mpfr_set_emin} and @code{mpfr_set_emax} respectively.
These values are implementation dependent, thus a program using
@code{mpfr_set_emax(mpfr_get_emax_max())}
or @code{mpfr_set_emin(mpfr_get_emin_min())} may not be portable.
@end deftypefun

@deftypefun int mpfr_check_range (mpfr_t @var{x}, int @var{t}, mpfr_rnd_t @var{rnd})
This function assumes that @var{x} is the correctly rounded value of some
real value @var{y} in the direction @var{rnd} and some extended exponent
range, and that @var{t} is the corresponding @ref{ternary value}.
For example, one performed @code{t = mpfr_log (x, u, rnd)}, and @var{y} is the
exact logarithm of @var{u}.
Thus @var{t} is negative if @var{x} is smaller than @var{y},
positive if @var{x} is larger than @var{y}, and zero if @var{x} equals @var{y}.
This function modifies @var{x} if needed
to be in the current range of acceptable values: It
generates an underflow or an overflow if the exponent of @var{x} is
outside the current allowed range; the value of @var{t} may be used
to avoid a double rounding. This function returns zero if the new value of
@var{x} equals the exact one @var{y}, a positive value if that new value
is larger than @var{y}, and a negative value if it is smaller than @var{y}.
Note that unlike most functions,
the new result @var{x} is compared to the (unknown) exact one @var{y},
not the input value @var{x}, i.e., the ternary value is propagated.

Note: If @var{x} is an infinity and @var{t} is different from zero (i.e.,
if the rounded result is an inexact infinity), then the overflow flag is
set. This is useful because @code{mpfr_check_range} is typically called
(at least in MPFR functions) after restoring the flags that could have
been set due to internal computations.
@end deftypefun

@deftypefun int mpfr_subnormalize (mpfr_t @var{x}, int @var{t}, mpfr_rnd_t @var{rnd})
This function rounds @var{x} emulating subnormal number arithmetic:
if @var{x} is outside the subnormal exponent range of the emulated
floating-point system, this function just propagates the
@ref{ternary value} @var{t}; otherwise, if @code{EXP(@var{x})}
denotes the exponent of @var{x}, it rounds @var{x} to precision
@code{EXP(@var{x})-emin+1} according to rounding mode @var{rnd} and previous
ternary value @var{t}, avoiding double rounding problems.
More precisely in the subnormal domain, denoting by @m{e,@var{e}} the value of
@code{emin}, @var{x} is rounded in fixed-point
arithmetic to an integer multiple of @m{2^{e-1}, two to the power
@var{e}@minus{}1}; as a consequence, @m{1.5 \times 2^{e-1},
1.5 multiplied by two to the power @var{e}@minus{}1} when @var{t} is zero
is rounded to @m{2^e, two to the power @var{e}} with rounding to nearest.

The precision @code{PREC(@var{x})} of @var{x} is not modified by
this function. @var{rnd} and @var{t} must be the rounding mode
and the returned ternary value used when computing @var{x}
(as in @code{mpfr_check_range}).
The subnormal exponent range is from @code{emin} to @code{emin+PREC(@var{x})-1}.
If the result cannot be represented in the current exponent range of MPFR
(due to a too small @code{emax}), the behavior is undefined.
Note that unlike most functions, the result is compared to the exact one,
not the input value @var{x}, i.e., the ternary value is propagated.

As usual, if the returned ternary value is non zero, the inexact flag is set.
Moreover, if a second rounding occurred (because the input @var{x} was in the
subnormal range), the underflow flag is set.

Warning! If you change @code{emin} (with @code{mpfr_set_emin}) just before
calling @code{mpfr_subnormalize}, you need to make sure that the value is
in the current exponent range of MPFR@. But it is better to change
@code{emin} before any computation, if possible.
@c Note: not necessarily possible if the user wants to emulate different
@c floating-point systems in the same code.
@end deftypefun

This is an example of how to emulate binary64 IEEE@tie{}754 arithmetic
(a.k.a.@: double precision) using MPFR:

@example
@{
  mpfr_t xa, xb; int i; volatile double a, b;

  mpfr_set_default_prec (53);
  mpfr_set_emin (-1073); mpfr_set_emax (1024);

  mpfr_init (xa); mpfr_init (xb);

  b = 34.3; mpfr_set_d (xb, b, MPFR_RNDN);
  a = 0x1.1235P-1021; mpfr_set_d (xa, a, MPFR_RNDN);

  a /= b;
  i = mpfr_div (xa, xa, xb, MPFR_RNDN);
  i = mpfr_subnormalize (xa, i, MPFR_RNDN); /* new ternary value */

  mpfr_clear (xa); mpfr_clear (xb);
@}
@end example

Note that @code{mpfr_set_emin} and @code{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@tie{}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 @m{2^{-42}, 2 power -42} (using the
fact that the result is at most 1 in absolute value):

@example
@{
  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);
@}
@end example

@deftypefun void mpfr_clear_underflow (void)
@deftypefunx void mpfr_clear_overflow (void)
@deftypefunx void mpfr_clear_divby0 (void)
@deftypefunx void mpfr_clear_nanflag (void)
@deftypefunx void mpfr_clear_inexflag (void)
@deftypefunx void mpfr_clear_erangeflag (void)
Clear (lower) the underflow, overflow, divide-by-zero, invalid,
inexact and @emph{erange} flags.
@end deftypefun

@deftypefun void mpfr_clear_flags (void)
Clear (lower) all global flags (underflow, overflow, divide-by-zero, invalid,
inexact, @emph{erange}). Note: a group of flags can be cleared by using
@code{mpfr_flags_clear}.
@end deftypefun

@deftypefun void mpfr_set_underflow (void)
@deftypefunx void mpfr_set_overflow (void)
@deftypefunx void mpfr_set_divby0 (void)
@deftypefunx void mpfr_set_nanflag (void)
@deftypefunx void mpfr_set_inexflag (void)
@deftypefunx void mpfr_set_erangeflag (void)
Set (raise) the underflow, overflow, divide-by-zero, invalid,
inexact and @emph{erange} flags.
@end deftypefun

@deftypefun int mpfr_underflow_p (void)
@deftypefunx int mpfr_overflow_p (void)
@deftypefunx int mpfr_divby0_p (void)
@deftypefunx int mpfr_nanflag_p (void)
@deftypefunx int mpfr_inexflag_p (void)
@deftypefunx int mpfr_erangeflag_p (void)
Return the corresponding (underflow, overflow, divide-by-zero, invalid,
inexact, @emph{erange}) flag, which is non-zero iff the flag is set.
@end deftypefun

The @code{mpfr_flags_} functions below that take an argument @var{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
@var{mask} is set.  The @code{MPFR_FLAGS_} macros will normally be used
to build this argument.  @xref{Exceptions}.

@deftypefun void mpfr_flags_clear (mpfr_flags_t @var{mask})
Clear (lower) the group of flags specified by @var{mask}.
@end deftypefun

@deftypefun void mpfr_flags_set (mpfr_flags_t @var{mask})
Set (raise) the group of flags specified by @var{mask}.
@end deftypefun

@deftypefun mpfr_flags_t mpfr_flags_test (mpfr_flags_t @var{mask})
Return the flags specified by @var{mask}.  To test whether any flag from
@var{mask} is set, compare the return value to 0.  You can also test
individual flags by AND'ing the result with @code{MPFR_FLAGS_} macros.
Example:
@example
mpfr_flags_t t = mpfr_flags_test (MPFR_FLAGS_UNDERFLOW|
                                  MPFR_FLAGS_OVERFLOW)
@dots{}
if (t)  /* underflow and/or overflow (unlikely) */
  @{
    if (t & MPFR_FLAGS_UNDERFLOW)  @{ /* handle underflow */ @}
    if (t & MPFR_FLAGS_OVERFLOW)   @{ /* handle overflow  */ @}
  @}
@end example
@end deftypefun

@deftypefun mpfr_flags_t mpfr_flags_save (void)
Return all the flags. It is equivalent to
@code{mpfr_flags_test(MPFR_FLAGS_ALL)}.
@end deftypefun

@deftypefun void mpfr_flags_restore (mpfr_flags_t @var{flags}, mpfr_flags_t @var{mask})
Restore the flags specified by @var{mask} to their state represented
in @var{flags}.
@end deftypefun

@node Memory Handling Functions, Compatibility with MPF, Exception Related Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Memory handling functions
@section Memory Handling Functions

These are general functions concerning memory handling
(@pxref{Memory Handling}, for more information).

@deftypefun 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 @code{mpfr_const_*} functions directly (they could have been called
internally).
@end deftypefun

@deftypefun void mpfr_free_cache2 (mpfr_free_cache_t @var{way})
Free various caches and pools used by MPFR internally,
as specified by @var{way}, which is a set of flags:
@itemize @bullet
@item those local to the current thread if flag @code{MPFR_FREE_LOCAL_CACHE}
is set;
@item those shared by all threads if flag @code{MPFR_FREE_GLOBAL_CACHE}
is set.
@end itemize
The other bits of @var{way} are currently ignored and are reserved for
future use; they should be zero.

Note: @code{mpfr_free_cache2(MPFR_FREE_LOCAL_CACHE|MPFR_FREE_GLOBAL_CACHE)}
is currently equivalent to @code{mpfr_free_cache()}.
@end deftypefun

@deftypefun 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 @code{mpfr_free_cache} or @code{mpfr_free_cache2}).
@end deftypefun

@deftypefun int mpfr_mp_memory_cleanup (void)
This function should be called before calling @code{mp_set_memory_functions}.
@xref{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.
@end deftypefun

@node Compatibility with MPF, Custom Interface, Memory Handling Functions, MPFR Interface
@cindex Compatibility with MPF
@section Compatibility With MPF

A header file @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 @code{#include <gmp.h>} line,
@example
#include <mpfr.h>
#include <mpf2mpfr.h>
@end example
@noindent
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 @code{mpfr_set_default_rounding_mode}.

Warning! There are some differences. In particular:
@itemize @bullet
@item 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.
@item The exponent range is also different.
@item The formatted output functions (@code{gmp_printf}, etc.) will not work
for arguments of arbitrary-precision floating-point type (@code{mpf_t}, which
@file{mpf2mpfr.h} redefines as @code{mpfr_t}).
@item The output of @code{mpf_out_str} has a format slightly different from
the one of @code{mpfr_out_str} (concerning the position of the decimal-point
character, trailing zeros and the output of the value 0).
@end itemize

@deftypefun void mpfr_set_prec_raw (mpfr_t @var{x}, mpfr_prec_t @var{prec})
Reset the precision of @var{x} to be @strong{exactly} @var{prec} bits.
The only difference with @code{mpfr_set_prec} is that @var{prec} is assumed to
be small enough so that the significand fits into the current allocated memory
space for @var{x}. Otherwise the behavior is undefined.
@end deftypefun

@deftypefun int mpfr_eq (mpfr_t @var{op1}, mpfr_t @var{op2}, unsigned long int @var{op3})
Return non-zero if @var{op1} and @var{op2} are both non-zero ordinary
numbers with the same exponent and the same first @var{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
@var{op3} larger than 1.
@end deftypefun

@deftypefun void mpfr_reldiff (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mpfr_rnd_t @var{rnd})
Compute the relative difference between @var{op1} and @var{op2}
and store the result in @var{rop}.
This function does not guarantee the correct rounding on the relative
difference; it just computes @math{|@var{op1}-@var{op2}|/@var{op1}}, using the
precision of @var{rop} and the rounding mode @var{rnd} for all operations.
@c VL: say that if op1 and op2 have the same precision and are close to
@c each other, then one gets correct rounding?
@end deftypefun

@deftypefun int mpfr_mul_2exp (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mpfr_rnd_t @var{rnd})
@deftypefunx int mpfr_div_2exp (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mpfr_rnd_t @var{rnd})
These functions are identical to @code{mpfr_mul_2ui} and @code{mpfr_div_2ui}
respectively.
These functions are only kept for compatibility with MPF, one should
prefer @code{mpfr_mul_2ui} and @code{mpfr_div_2ui} otherwise.
@end deftypefun


@node Custom Interface, Internals, Compatibility with MPF, MPFR Interface
@cindex Custom interface
@section 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:

@itemize

@item Either directly store a floating-point number as a @code{mpfr_t}
on the stack.

@item Either store its own representation on the
stack and construct a new temporary @code{mpfr_t} each time it is needed.

@end itemize

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 @code{mpfr_custom_init (s, p)}
uses the macro, while @code{(mpfr_custom_init) (s, p)} uses the function.
Note that the macro may evaluate arguments multiple times (or none).
Moreover, macros implementing functions with the @code{void} return type
may not be used in contexts where an expression is expected, e.g., inside
@code{for(...)} or before a comma operator.
@c These limitations with macros cannot be avoided in a C90 compatible way.
@c In the future, inline functions could be used.

Note 1: MPFR functions may still initialize temporary floating-point numbers
using @code{mpfr_init} and similar functions. See Custom Allocation (GNU MP)@.

Note 2: MPFR functions may use the cached functions (@code{mpfr_const_pi} for
example), even if they are not explicitly called. You have to call
@code{mpfr_free_cache} each time you garbage the memory iff @code{mpfr_init},
through GMP Custom Allocation, allocates its memory on the application stack.

@deftypefun size_t mpfr_custom_get_size (mpfr_prec_t @var{prec})
Return the needed size in bytes to store the significand of a floating-point
number of precision @var{prec}.
@end deftypefun

@deftypefun void mpfr_custom_init (void *@var{significand}, mpfr_prec_t @var{prec})
Initialize a significand of precision @var{prec}, where
@var{significand} must be an area of @code{mpfr_custom_get_size (prec)} bytes
at least and be suitably aligned for an array of @code{mp_limb_t} (GMP type,
@pxref{Internals}).
@c PZ: give an example how to align?
@end deftypefun

@deftypefun void mpfr_custom_init_set (mpfr_t @var{x}, int @var{kind}, mpfr_exp_t @var{exp}, mpfr_prec_t @var{prec}, void *@var{significand})
Perform a dummy initialization of a @code{mpfr_t} and set it to:
@itemize
@item if @GMPabs{@var{kind}} = @code{MPFR_NAN_KIND}, @var{x} is set to NaN;
@item if @GMPabs{@var{kind}} = @code{MPFR_INF_KIND}, @var{x} is set to the
infinity of the same sign as @var{kind};
@item if @GMPabs{@var{kind}} = @code{MPFR_ZERO_KIND}, @var{x} is set to the
zero of the same sign as @var{kind};
@item if @GMPabs{@var{kind}} = @code{MPFR_REGULAR_KIND}, @var{x} is set to
the regular number whose sign is the one of @var{kind}, and whose exponent
and significand are given by @var{exp} and @var{significand}.
@end itemize
In all cases, @var{significand} will be used directly for further computing
involving @var{x}. This function does not allocate anything.
A floating-point number initialized with this function cannot be resized using
@code{mpfr_set_prec} or @code{mpfr_prec_round},
or cleared using @code{mpfr_clear}!
The @var{significand} must have been initialized with @code{mpfr_custom_init}
using the same precision @var{prec}.
@end deftypefun

@deftypefun int mpfr_custom_get_kind (mpfr_t @var{x})
Return the current kind of a @code{mpfr_t} as created by
@code{mpfr_custom_init_set}.
The behavior of this function for any @code{mpfr_t} not initialized
with @code{mpfr_custom_init_set} is undefined.
@end deftypefun

@deftypefun {void *} mpfr_custom_get_significand (mpfr_t @var{x})
Return a pointer to the significand used by a @code{mpfr_t} initialized with
@code{mpfr_custom_init_set}.
The behavior of this function for any @code{mpfr_t} not initialized
with @code{mpfr_custom_init_set} is undefined.
@end deftypefun

@deftypefun mpfr_exp_t mpfr_custom_get_exp (mpfr_t @var{x})
Return the exponent of @var{x}, assuming that @var{x} is a non-zero ordinary
number and the significand is considered in [1/2,1).
But if @var{x} is NaN, infinity or zero, contrary to @code{mpfr_get_exp}
(where the behavior is undefined), the return value is here an unspecified,
valid value of the @code{mpfr_exp_t} type.
The behavior of this function for any @code{mpfr_t} not initialized
with @code{mpfr_custom_init_set} is undefined.
@end deftypefun

@deftypefun void mpfr_custom_move (mpfr_t @var{x}, void *@var{new_position})
Inform MPFR that the significand of @var{x} has moved due to a garbage collect
and update its new position to @code{new_position}.
However, the application has to move the significand and the @code{mpfr_t}
itself.
The behavior of this function for any @code{mpfr_t} not initialized
with @code{mpfr_custom_init_set} is undefined.
@end deftypefun

@node Internals,  , Custom Interface, MPFR Interface
@cindex Internals
@section Internals

@cindex Limb
@c @tindex @code{mp_limb_t}
@noindent
A @dfn{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 @code{mp_limb_t}.

The @code{mpfr_t} type is internally defined as a one-element
array of a structure, and @code{mpfr_ptr} is the C data type representing
a pointer to this structure.
The @code{mpfr_t} type consists of four fields:

@itemize @bullet

@item The @code{_mpfr_prec} field is used to store the precision of
the variable (in bits); this is not less than @code{MPFR_PREC_MIN}.

@item The @code{_mpfr_sign} field is used to store the sign of the variable.

@item The @code{_mpfr_exp} field stores the exponent.
An exponent of 0 means a radix point just above the most significant
limb.  Non-zero values @math{n} are a multiplier @math{2^n} relative to that
point.
A NaN, an infinity and a zero are indicated by special values of the exponent
field.

@item Finally, the @code{_mpfr_d} field is a pointer to the limbs, least
significant limbs stored first.
The number of limbs in use is controlled by @code{_mpfr_prec}, namely
ceil(@code{_mpfr_prec}/@code{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.

@end itemize

@node API Compatibility, MPFR and the IEEE 754 Standard, MPFR Interface, Top
@chapter 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.
@comment This includes undefined 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
(@pxref{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::
@end menu

@node Type and Macro Changes, Added Functions, API Compatibility, API Compatibility
@section Type and Macro Changes

@comment r6789
The official type for exponent values changed from @code{mp_exp_t} to
@code{mpfr_exp_t} in MPFR@tie{}3.0.  The type @code{mp_exp_t} will remain
available as it comes from GMP (with a different meaning).  These types
are currently the same (@code{mpfr_exp_t} is defined as @code{mp_exp_t}
with @code{typedef}), so that programs can still use @code{mp_exp_t};
but this may change in the future.
Alternatively, using the following code after including @file{mpfr.h}
will work with official MPFR versions, as @code{mpfr_exp_t} was never
defined in MPFR@tie{}2.x:
@example
#if MPFR_VERSION_MAJOR < 3
typedef mp_exp_t mpfr_exp_t;
#endif
@end example

The official types for precision values and for rounding modes
respectively changed from @code{mp_prec_t} and @code{mp_rnd_t}
to @code{mpfr_prec_t} and @code{mpfr_rnd_t} in MPFR@tie{}3.0.  This
change was actually done a long time ago in MPFR, at least since
MPFR@tie{}2.2.0, with the following code in @file{mpfr.h}:
@example
#ifndef mp_rnd_t
# define mp_rnd_t  mpfr_rnd_t
#endif
#ifndef mp_prec_t
# define mp_prec_t mpfr_prec_t
#endif
@end example
This means that it is safe to use the new official types
@code{mpfr_prec_t} and @code{mpfr_rnd_t} in your programs.
The types @code{mp_prec_t} and @code{mp_rnd_t} (defined
in MPFR only) may be removed in the future, as the prefix
@code{mp_} is reserved by GMP@.

@comment r6787
The precision type @code{mpfr_prec_t} (@code{mp_prec_t}) was unsigned
before MPFR@tie{}3.0; it is now signed.  @code{MPFR_PREC_MAX} has not
changed, though.  Indeed the MPFR code requires that @code{MPFR_PREC_MAX} be
representable in the exponent type, which may have the same size as
@code{mpfr_prec_t} but has always been signed.
The consequence is that valid code that does not assume anything about
the signedness of @code{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
@code{mpfr_prec_t} is signed may be affected by this problem when
it is built and run against MPFR@tie{}2.x.

The rounding modes @code{GMP_RNDx} were renamed to @code{MPFR_RNDx}
in MPFR@tie{}3.0. However, the old names @code{GMP_RNDx} have been kept for
compatibility (this might change in future versions), using:
@example
#define GMP_RNDN MPFR_RNDN
#define GMP_RNDZ MPFR_RNDZ
#define GMP_RNDU MPFR_RNDU
#define GMP_RNDD MPFR_RNDD
@end example
The rounding mode ``round away from zero'' (@code{MPFR_RNDA}) was added in
MPFR@tie{}3.0 (however, no rounding mode @code{GMP_RNDA} exists).
Faithful rounding (@code{MPFR_RNDF}) was added in MPFR@tie{}4.0, but
currently, it is partially supported.
@c That's sufficient information for now. More should be said in future
@c versions (for instance, a user of 4.1 may want to know if this works
@c in 4.0).

The flags-related macros, whose name starts with @code{MPFR_FLAGS_},
were added in MPFR@tie{}4.0 (for the new functions @code{mpfr_flags_clear},
@code{mpfr_flags_restore}, @code{mpfr_flags_set} and @code{mpfr_flags_test},
in particular).

@node Added Functions, Changed Functions, Type and Macro Changes, API Compatibility
@section Added Functions

We give here in alphabetical order the functions (and function-like macros)
that were added after MPFR@tie{}2.2, and in which MPFR version.

@comment The functions are listed in such a way that if a developer wonders
@comment whether some function existed in some previous version, then he can
@comment find this very quickly.

@itemize @bullet

@item @code{mpfr_acospi} and @code{mpfr_acosu} in MPFR@tie{}4.2.

@item @code{mpfr_add_d} in MPFR@tie{}2.4.

@item @code{mpfr_ai} in MPFR@tie{}3.0 (incomplete, experimental).

@item @code{mpfr_asinpi} and @code{mpfr_asinu} in MPFR@tie{}4.2.

@item @code{mpfr_asprintf} in MPFR@tie{}2.4.

@item @code{mpfr_atanpi} and @code{mpfr_atanu} in MPFR@tie{}4.2.

@item @code{mpfr_atan2pi} and @code{mpfr_atan2u} in MPFR@tie{}4.2.

@item @code{mpfr_beta} in MPFR@tie{}4.0 (incomplete, experimental).

@item @code{mpfr_buildopt_decimal_p} in MPFR@tie{}3.0.

@item @code{mpfr_buildopt_float128_p} in MPFR@tie{}4.0.

@item @code{mpfr_buildopt_gmpinternals_p} in MPFR@tie{}3.1.

@item @code{mpfr_buildopt_sharedcache_p} in MPFR@tie{}4.0.

@item @code{mpfr_buildopt_tls_p} in MPFR@tie{}3.0.

@item @code{mpfr_buildopt_tune_case} in MPFR@tie{}3.1.

@item @code{mpfr_clear_divby0} in MPFR@tie{}3.1
(new divide-by-zero exception).

@item @code{mpfr_cmpabs_ui} in MPFR@tie{}4.1.

@item @code{mpfr_compound_si} in MPFR@tie{}4.2.

@item @code{mpfr_copysign} in MPFR@tie{}2.3.
Note: MPFR@tie{}2.2 had a @code{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)@.

@item @code{mpfr_cospi} and @code{mpfr_cosu} in MPFR@tie{}4.2.

@item @code{mpfr_custom_get_significand} in MPFR@tie{}3.0.
This function was named @code{mpfr_custom_get_mantissa} in previous
versions; @code{mpfr_custom_get_mantissa} is still available via a
macro in @file{mpfr.h}:
@example
#define mpfr_custom_get_mantissa mpfr_custom_get_significand
@end example
Thus code that needs to work with both MPFR@tie{}2.x and MPFR@tie{}3.x should
use @code{mpfr_custom_get_mantissa}.

@item @code{mpfr_d_div} and @code{mpfr_d_sub} in MPFR@tie{}2.4.

@item @code{mpfr_digamma} in MPFR@tie{}3.0.

@item @code{mpfr_divby0_p} in MPFR@tie{}3.1 (new divide-by-zero exception).

@item @code{mpfr_div_d} in MPFR@tie{}2.4.

@item @code{mpfr_dot} in MPFR@tie{}4.1 (incomplete, experimental).

@item @code{mpfr_erandom} in MPFR@tie{}4.0.

@item @code{mpfr_exp2m1} and @code{mpfr_exp10m1} in MPFR@tie{}4.2.

@item @code{mpfr_flags_clear}, @code{mpfr_flags_restore},
@code{mpfr_flags_save}, @code{mpfr_flags_set} and @code{mpfr_flags_test}
in MPFR@tie{}4.0.

@item @code{mpfr_fmma} and @code{mpfr_fmms} in MPFR@tie{}4.0.

@item @code{mpfr_fmod} in MPFR@tie{}2.4.

@item @code{mpfr_fmodquo} in MPFR@tie{}4.0.

@item @code{mpfr_fmod_ui} in MPFR@tie{}4.2.

@item @code{mpfr_fms} in MPFR@tie{}2.3.

@item @code{mpfr_fpif_export} and @code{mpfr_fpif_import} in MPFR@tie{}4.0.

@item @code{mpfr_fprintf} in MPFR@tie{}2.4.

@item @code{mpfr_free_cache2} in MPFR@tie{}4.0.

@item @code{mpfr_free_pool} in MPFR@tie{}4.0.

@item @code{mpfr_frexp} in MPFR@tie{}3.1.

@item @code{mpfr_gamma_inc} in MPFR@tie{}4.0.

@item @code{mpfr_get_decimal128} in MPFR@tie{}4.1.

@item @code{mpfr_get_float128} in MPFR@tie{}4.0 if configured with
@samp{--enable-float128}.

@item @code{mpfr_get_flt} in MPFR@tie{}3.0.

@item @code{mpfr_get_patches} in MPFR@tie{}2.3.

@item @code{mpfr_get_q} in MPFR@tie{}4.0.

@item @code{mpfr_get_str_ndigits} in MPFR@tie{}4.1.

@item @code{mpfr_get_z_2exp} in MPFR@tie{}3.0.
This function was named @code{mpfr_get_z_exp} in previous versions;
@code{mpfr_get_z_exp} is still available via a macro in @file{mpfr.h}:
@example
#define mpfr_get_z_exp mpfr_get_z_2exp
@end example
Thus code that needs to work with both MPFR@tie{}2.x and MPFR@tie{}3.x should
use @code{mpfr_get_z_exp}.

@item @code{mpfr_grandom} in MPFR@tie{}3.1.

@item @code{mpfr_j0}, @code{mpfr_j1} and @code{mpfr_jn} in MPFR@tie{}2.3.

@item @code{mpfr_log2p1} and @code{mpfr_log10p1} in MPFR@tie{}4.2.

@item @code{mpfr_lgamma} in MPFR@tie{}2.3.

@item @code{mpfr_li2} in MPFR@tie{}2.4.

@item @code{mpfr_log_ui} in MPFR@tie{}4.0.

@item @code{mpfr_min_prec} in MPFR@tie{}3.0.

@item @code{mpfr_modf} in MPFR@tie{}2.4.

@item @code{mpfr_mp_memory_cleanup} in MPFR@tie{}4.0.

@item @code{mpfr_mul_d} in MPFR@tie{}2.4.

@item @code{mpfr_nrandom} in MPFR@tie{}4.0.

@item @code{mpfr_powr}, @code{mpfr_pown}, @code{mpfr_pow_sj} and @code{mpfr_pow_uj} in MPFR@tie{}4.2.

@item @code{mpfr_printf} in MPFR@tie{}2.4.

@item @code{mpfr_rec_sqrt} in MPFR@tie{}2.4.

@item @code{mpfr_regular_p} in MPFR@tie{}3.0.

@item @code{mpfr_remainder} and @code{mpfr_remquo} in MPFR@tie{}2.3.

@item @code{mpfr_rint_roundeven} and @code{mpfr_roundeven} in MPFR@tie{}4.0.

@item @code{mpfr_round_nearest_away} in MPFR@tie{}4.0.

@item @code{mpfr_rootn_ui} in MPFR@tie{}4.0.

@item @code{mpfr_set_decimal128} in MPFR@tie{}4.1.

@item @code{mpfr_set_divby0} in MPFR@tie{}3.1 (new divide-by-zero exception).

@item @code{mpfr_set_float128} in MPFR@tie{}4.0 if configured with
@samp{--enable-float128}.

@item @code{mpfr_set_flt} in MPFR@tie{}3.0.

@item @code{mpfr_set_z_2exp} in MPFR@tie{}3.0.

@item @code{mpfr_set_zero} in MPFR@tie{}3.0.

@item @code{mpfr_setsign} in MPFR@tie{}2.3.

@item @code{mpfr_signbit} in MPFR@tie{}2.3.

@item @code{mpfr_sinh_cosh} in MPFR@tie{}2.4.

@item @code{mpfr_sinpi} and @code{mpfr_sinu} in MPFR@tie{}4.2.

@item @code{mpfr_snprintf} and @code{mpfr_sprintf} in MPFR@tie{}2.4.

@item @code{mpfr_sub_d} in MPFR@tie{}2.4.

@item @code{mpfr_tanpi} and @code{mpfr_tanu} in MPFR@tie{}4.2.

@item @code{mpfr_total_order_p} in MPFR@tie{}4.1.

@item @code{mpfr_urandom} in MPFR@tie{}3.0.

@item @code{mpfr_vasprintf}, @code{mpfr_vfprintf}, @code{mpfr_vprintf},
      @code{mpfr_vsprintf} and @code{mpfr_vsnprintf} in MPFR@tie{}2.4.

@item @code{mpfr_y0}, @code{mpfr_y1} and @code{mpfr_yn} in MPFR@tie{}2.3.

@item @code{mpfr_z_sub} in MPFR@tie{}3.1.

@end itemize

@node Changed Functions, Removed Functions, Added Functions, API Compatibility
@section Changed Functions

The following functions have changed after MPFR@tie{}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.

@itemize @bullet

@item The formatted output functions (@code{mpfr_printf}, etc.) have
slightly changed in MPFR@tie{}4.1 in the case where the precision field
is empty: trailing zeros were not output with the conversion specifier
@samp{e} / @samp{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 @samp{f} / @samp{F} / @samp{g} / @samp{G} (this
could partly be regarded as a bug); they are now kept in a way similar
to the formatted output functions from C@. Moreover, the case where the
precision consists only of a period has been fixed in MPFR@tie{}4.2 to
be like @samp{.0} as specified in the ISO C standard (it previously
behaved as a missing precision).
@c https://gforge.inria.fr/tracker/index.php?func=detail&aid=21816&group_id=136&atid=619

@item @code{mpfr_abs}, @code{mpfr_neg} and @code{mpfr_set} changed in
MPFR@tie{}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 @code{mpfr_neg} with
a reused argument: @code{mpfr_neg(x,x,rnd)}.

@item @code{mpfr_check_range} changed in MPFR@tie{}2.3.2 and MPFR@tie{}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@tie{}2.3.2.

@item @code{mpfr_eint} changed in MPFR@tie{}4.0.
This function now returns the value of the E1/eint1 function for
negative argument (before MPFR@tie{}4.0, it was returning NaN)@.

@item @code{mpfr_get_f} changed in MPFR@tie{}3.0.
This function was returning zero, except for NaN and Inf, which do not
exist in MPF@. The @emph{erange} flag is now set in these cases,
and @code{mpfr_get_f} now returns the usual ternary value.

@item @code{mpfr_get_si}, @code{mpfr_get_sj}, @code{mpfr_get_ui}
and @code{mpfr_get_uj} changed in MPFR@tie{}3.0.
In previous MPFR versions, the cases where the @emph{erange} flag
is set were unspecified.

@item @code{mpfr_get_str} changed in MPFR@tie{}4.0.
This function now sets the NaN flag on NaN input (to follow the usual MPFR
rules on NaN and IEEE@tie{}754 recommendations on string conversions
from Subclause@tie{}5.12.1) and sets the inexact flag when the conversion
is inexact.

@item @code{mpfr_get_z} changed in MPFR@tie{}3.0.
The return type was @code{void}; it is now @code{int}, and the usual
ternary value is returned.  Thus programs that need to work with both
MPFR@tie{}2.x and 3.x must not use the return value.  Even in this case,
C code using @code{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@tie{}3.0, but not with MPFR@tie{}2.x:
@example
  bool ? mpfr_get_z(...) : mpfr_add(...);
@end example
On the other hand, the following is correct with MPFR@tie{}2.x, but not
with MPFR@tie{}3.0:
@example
  bool ? mpfr_get_z(...) : (void) mpfr_add(...);
@end example
Portable code should cast @code{mpfr_get_z(...)} to @code{void} to
use the type @code{void} for both terms of the conditional operator,
as in:
@example
  bool ? (void) mpfr_get_z(...) : (void) mpfr_add(...);
@end example
Alternatively, @code{if ... else} can be used instead of the
conditional operator.

Moreover the cases where the @emph{erange} flag is set were unspecified
in MPFR@tie{}2.x.

@item @code{mpfr_get_z_exp} changed in MPFR@tie{}3.0.
In previous MPFR versions, the cases where the @emph{erange} flag
is set were unspecified.
Note: this function has been renamed to @code{mpfr_get_z_2exp}
in MPFR@tie{}3.0, but @code{mpfr_get_z_exp} is still available for
compatibility reasons.

@item @code{mpfr_out_str} changed in MPFR@tie{}4.1.
The argument @var{base} can now be negative (from @minus{}2 to
@minus{}36), in order to follow @code{mpfr_get_str} and GMP's
@code{mpf_out_str} functions.

@item @code{mpfr_set_exp} changed in MPFR@tie{}4.0.
Before MPFR@tie{}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).

@item @code{mpfr_strtofr} changed in MPFR@tie{}2.3.1 and MPFR@tie{}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 @code{0b} or
@code{0x} prefix.  Data corresponding to NaN can now have an optional
sign (such data were previously invalid).

@item @code{mpfr_strtofr} changed in MPFR@tie{}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@tie{}2.3.1 and later,
providing an unsupported base yields an assertion failure (this
behavior may change in the future).

@item @code{mpfr_subnormalize} changed in MPFR@tie{}3.1.
This was actually regarded as a bug fix. The @code{mpfr_subnormalize}
implementation up to MPFR@tie{}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.

@item @code{mpfr_sum} changed in MPFR@tie{}4.0.
The @code{mpfr_sum} function has completely been rewritten for MPFR@tie{}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 @code{mpfr_sum} implementation could also take all the memory
and crash on inputs of very different magnitude.

@item @code{mpfr_urandom} and @code{mpfr_urandomb} changed in MPFR@tie{}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
@code{gmp_randinit_default} is used).  As a consequence, the returned values
can be different between MPFR@tie{}3.1 and previous MPFR versions.
Note: as the reproducibility of these functions was not specified
before MPFR@tie{}3.1, the MPFR@tie{}3.1 behavior is @emph{not} regarded as
backward incompatible with previous versions.

@item @code{mpfr_urandom} changed in MPFR@tie{}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@tie{}4.0
and previous MPFR versions.

@end itemize

@node Removed Functions, Other Changes, Changed Functions, API Compatibility
@section Removed Functions

Functions @code{mpfr_random} and @code{mpfr_random2} have been
removed in MPFR@tie{}3.0 (this only affects old code built against
MPFR@tie{}3.0 or later).
(The function @code{mpfr_random} had been deprecated since at least
MPFR@tie{}2.2.0, and @code{mpfr_random2} since MPFR@tie{}2.4.0.)

Macros @code{mpfr_add_one_ulp} and @code{mpfr_sub_one_ulp} have been
removed in MPFR@tie{}4.0. They were no longer documented since
MPFR@tie{}2.1.0 and were announced as deprecated since MPFR@tie{}3.1.0.

Function @code{mpfr_grandom} is marked as deprecated in MPFR@tie{}4.0.
It will be removed in a future release.

@node Other Changes, , Removed Functions, API Compatibility
@section Other Changes

@comment r6699
For users of a C++ compiler, the way how the availability of @code{intmax_t}
is detected has changed in MPFR@tie{}3.0.
In MPFR@tie{}2.x, if a macro @code{INTMAX_C} or @code{UINTMAX_C} was defined
(e.g. when the @code{__STDC_CONSTANT_MACROS} macro had been defined
before @code{<stdint.h>} or @code{<inttypes.h>} has been included),
@code{intmax_t} was assumed to be defined.
However, this was not always the case (more precisely, @code{intmax_t}
can be defined only in the namespace @code{std}, as with Boost), so
that compilations could fail.
Thus the check for @code{INTMAX_C} or @code{UINTMAX_C} is now disabled for
C++ compilers, with the following consequences:

@itemize

@item Programs written for MPFR@tie{}2.x that need @code{intmax_t} may no
longer be compiled against MPFR@tie{}3.0: a @code{#define MPFR_USE_INTMAX_T}
may be necessary before @file{mpfr.h} is included.

@item The compilation of programs that work with MPFR@tie{}3.0 may fail with
MPFR@tie{}2.x due to the problem described above.  Workarounds are possible,
such as defining @code{intmax_t} and @code{uintmax_t} in the global
namespace, though this is not clean.

@end itemize

The divide-by-zero exception is new in MPFR@tie{}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@tie{}3.1, the @file{mpfr.h} header can be included several times,
while still supporting optional functions (@pxref{Headers and Libraries}).

The way memory is allocated by MPFR should be regarded as well-specified
only as of MPFR@tie{}4.0.

@node MPFR and the IEEE 754 Standard, Contributors, API Compatibility, Top
@chapter MPFR and the IEEE 754 Standard

This section describes differences between MPFR and the IEEE@tie{}754
standard, and behaviors that are not specified yet in IEEE@tie{}754.

The MPFR numbers do not include subnormals. The reason is that subnormals
are less useful than in IEEE@tie{}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 @code{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@tie{}754,
some operations are quiet (even on a signaling NaN)@.

The @code{mpfr_rec_sqrt} function differs from IEEE@tie{}754 on @minus{}0,
where it gives +Inf (like for +0), following the usual limit rules,
instead of @minus{}Inf.

The @code{mpfr_root} function predates IEEE@tie{}754-2008, where rootn was
introduced, and behaves differently from the IEEE@tie{}754 rootn operation.
It is deprecated and @code{mpfr_rootn_ui} should be used instead.

@c The following paragraph should cover functions like mpfr_div_ui and
@c mpfr_log_ui. There are no issues with mpfr_pow_{ui,si,z} because the
@c IEEE 754 pown operation agrees with mpfr_pow.
@c Discussions:
@c   https://sympa.inria.fr/sympa/arc/mpfr/2017-04/msg00019.html
@c   https://sympa.inria.fr/sympa/arc/mpfr/2017-11/msg00009.html
@c   https://sympa.inria.fr/sympa/arc/mpfr/2017-12/msg00008.html

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 @code{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
@code{mpfr_set_ui} or @code{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 (@code{mpfr_add_ui}, etc.), but for
these functions, only the sign of a zero result would be affected, with
+0 and @minus{}0 considered equal.
Such operations are currently out of the scope of the IEEE@tie{}754 standard,
and at the time of specification in MPFR, the Floating-Point Working Group
in charge of the revision of IEEE@tie{}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 @code{mpfr_set}, @code{mpfr_neg} or @code{mpfr_abs}, if the destination
variable has a smaller precision.

@node Contributors, References, MPFR and the IEEE 754 Standard, Top
@comment  node-name,  next,  previous,  up
@unnumbered Contributors

The main developers of MPFR are Guillaume Hanrot, Vincent Lef@`evre,
Patrick P@'elissier, Philippe Th@'eveny and Paul Zimmermann.

Sylvie Boldo from ENS-Lyon, France,
contributed the functions @code{mpfr_agm} and @code{mpfr_log}.
Sylvain Chevillard contributed the @code{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 @code{mpfr_get_str}
function.
Mathieu Dutour contributed the functions @code{mpfr_acos}, @code{mpfr_asin}
and @code{mpfr_atan}, and a previous version of @code{mpfr_gamma}.
Laurent Fousse contributed the original version of the @code{mpfr_sum}
function (used up to MPFR@tie{}3.1).
Emmanuel Jeandel, from ENS-Lyon too,
contributed the generic hypergeometric code,
as well as the internal function @code{mpfr_exp3},
a first implementation of the sine and cosine,
and improved versions of
@code{mpfr_const_log2} and @code{mpfr_const_pi}.
Ludovic Meunier helped in the design of the @code{mpfr_erf} code.
Jean-Luc R@'emy contributed the @code{mpfr_zeta} code.
Fabrice Rouillier contributed the @code{mpfr_xxx_z} and @code{mpfr_xxx_q}
functions, and helped to the Microsoft Windows porting.
Damien Stehl@'e contributed the @code{mpfr_get_ld_2exp} function.
Charles Karney contributed the @code{mpfr_nrandom} and @code{mpfr_erandom}
functions.

We would like to thank Jean-Michel Muller and Joris van der Hoeven for very
fruitful discussions at the beginning of that project, Torbj@"orn 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, Caramel and Caramba
project-teams at LORIA and of the
Ar@'enaire 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 R@'egional de Lorraine in 2002,
from INRIA by an "associate engineer" grant (2003-2005),
an "op@'eration de d@'eveloppement 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.
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,
Fredrik Johansson a faster version of @code{mpfr_const_euler},
and Jianyang Pan a formally proven version of the @code{mpfr_add1sp1}
internal routine.

@node References, GNU Free Documentation License, Contributors, Top
@comment  node-name,  next,  previous,  up
@unnumbered References

@itemize @bullet

@item
Richard Brent and Paul Zimmermann,
"Modern Computer Arithmetic",
Cambridge University Press,
Cambridge Monographs on Applied and Computational Mathematics,
Number 18, 2010.
Electronic version freely available at
@url{https://members.loria.fr/PZimmermann/mca/pub226.html}.

@item
Laurent Fousse, Guillaume Hanrot, Vincent Lef@`evre,
Patrick P@'elissier 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,
@url{https://doi.org/10.1145/1236463.1236468}.

@item
Torbj@"orn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic Library",
  version 6.1.2, 2016, @url{https://gmplib.org/}.

@item
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.

@item
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.

@item
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.

@item
Donald E.@: Knuth, "The Art of Computer Programming", vol 2,
"Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981.

@item
Jean-Michel Muller, "Elementary Functions, Algorithms and Implementation",
Birkh@"auser, Boston, 3rd edition, 2016.

@item
Jean-Michel Muller, Nicolas Brisebarre, Florent de Dinechin,
Claude-Pierre Jeannerod, Vincent Lef@`evre, Guillaume Melquiond,
Nathalie Revol, Damien Stehl@'e and Serge Torr@`es,
"Handbook of Floating-Point Arithmetic",
Birkh@"auser, Boston, 2009.

@end itemize


@node GNU Free Documentation License, Concept Index, References, Top
@appendix GNU Free Documentation License
@cindex GNU Free Documentation License
@include fdl.texi


@node Concept Index, Function and Type Index, GNU Free Documentation License, Top
@comment  node-name,  next,  previous,  up
@unnumbered Concept Index
@printindex cp

@node Function and Type Index,  , Concept Index, Top
@comment  node-name,  next,  previous,  up
@unnumbered Function and Type Index
@printindex fn

@bye

@c Local variables:
@c fill-column: 78
@c End: