path: root/mpfr/mpfr.texi

\input texinfo    @c -*-texinfo-*-
@c %**start of header
@setfilename mpfr.info
@documentencoding ISO-8859-1
@set VERSION 2.0.2
@set UPDATED-MONTH October 2003
@settitle MPFR @value{VERSION}
@synindex tp fn
@iftex
@afourpaper
@end iftex
@comment %**end of header

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

Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002,
2003 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 1.1 or any later
version published by the Free Software Foundation; with no Invariant Sections,
with the Front-Cover Texts being ``A GNU Manual'', and with the Back-Cover
Texts being ``You have freedom to copy and modify this GNU Manual, like GNU
software''.  A copy of the license is included in @ref{GNU Free Documentation
License}.
@end copying


@c  Texinfo version 4.2 or up will be needed to process this file.
@c
@c  A suitable texinfo.tex is supplied, a newer one should work
@c  equally well.
@c
@c  The edition number is in the VERSION variable above and should be
@c  updated where appropriate.  Also, update the month and year in
@c  UPDATED-MONTH.


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

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

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

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

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

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

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

@menu
* Copying::                     GMP Copying Conditions (LGPL).
* Introduction to MPFR::        Brief introduction to 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.
* Floating-point Functions::    Functions for arithmetic on floats.
* Contributors::
* References::
* GNU Free Documentation License::
* Concept Index::
* Function 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
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@macro m {T,N}
@math{\N\}
@end macro
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@c  Usage: @GMPabs{x}
@c  Give either |x| in tex, or abs(x) in info or html.
@tex
\gdef\GMPabs#1{|#1|}
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@macro GMPabs {X}
@abs{}(\X\)
@end macro
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@c  Usage: @GMPtimes{}
@c  Give either \times or the word "times".
@tex
\gdef\GMPtimes{\times}
@end tex
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@macro GMPtimes
times
@end macro
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@c  @abs{} can be used in both tex and info, or just \abs in tex.
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\gdef\abs{\mathop{\rm abs}}
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abs
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*
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@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
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@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

This library 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
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 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 MPFR library are found in the
Lesser General Public License that accompanies the source code.
See the file COPYING.LIB.@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 extend the class of floating-point numbers provided by the
GNU MP library by a precise semantics. The main differences
with the @code{mpf} class from GNU MP are:

@itemize @bullet
@item the @code{mpfr} code is portable, i.e.@: the result of any operation
does not depend (or should not) on the machine word size
@code{mp_bits_per_limb} (32 or 64 on most machines);
@item the precision in bits can be set exactly to any valid value
for each variable (including very small precision);
@item @code{mpfr} provides the four rounding modes from the IEEE 754-1985
standard.
@end itemize

In particular, with a precision of 53 bits, @code{mpfr} should be able
to exactly reproduce all computations with double-precision machine
floating-point
numbers (@code{double} type in C), except the default exponent range
is much wider and subnormal numbers are not implemented.

This version of MPFR is released under the GNU Lesser General Public
License.
It is permitted to link MPFR to 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.

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

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 4.1 or higher) on your computer.
You need a C compiler, preferably GCC, but any reasonable compiler should
work.  And you need a standard Unix @samp{make} program, plus some other
standard Unix utility programs.
MPFR needs some internal GMP header files that are not installed.
So, keep the GMP build directory as is, at least until you have
built MPFR.

@item
In the MPFR build directory, type
@samp{./configure --with-gmp-include=GMPBUILD --with-gmp-lib=GMPINSTALL/lib}
where @samp{GMPBUILD} is the GMP build directory and @samp{GMPINSTALL} the
directory where you have installed GMP. Because of the internal header files
required by MPFR, the option @samp{--with-gmp=GMPINSTALL} is not sufficient
and should not be used.
If you get error messages, you might check that you use the same compiler
and compile options as for GNU MP (see the @file{INSTALL} file).

@item
@samp{make}

This will compile MPFR, and create a library archive file @file{libmpfr.a}
in the working directory. No dynamic library is provided yet.

@item
@samp{make check}

This will make sure MPFR was built correctly.
If you get error messages, please
report this to @samp{mpfr@@loria.fr}.  (@xref{Reporting Bugs}, for
information on what to include in useful bug reports.)

@item
@samp{make install}

This will copy the files @file{mpfr.h} and @file{mpf2mpfr.h} to the directory
@file{/usr/local/include}, the file @file{libmpfr.a} to the directory
@file{/usr/local/lib}, and the file @file{mpfr.info} to the directory
@file{/usr/local/info} (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

There are some other useful make targets:

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

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

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

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

@item
@samp{mpfr.ps}

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

@c @item
@c @samp{html}
@c Create a HTML version of the manual, in @file{mpfr.html}.

@item
@samp{clean}

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

@item
@samp{distclean}

Delete all files not included in the distribution.

@item
@samp{uninstall}

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


@section Known Build Problems

MPFR suffers from all bugs from the GNU MP library, plus many many more.

Please report other problems to @samp{mpfr@@loria.fr}.
@xref{Reporting Bugs}.
Some bug fixes are available on the MPFR web page
@url{http://www.loria.fr/projets/mpfr/} or @url{http://www.mpfr.org/}.


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

If you think you have found a bug in the MPFR library, first have a look on the
MPFR web page @url{http://www.mpfr.org/} or
@url{http://www.loria.fr/projets/mpfr/}: perhaps this bug is already known,
in which case you may find there a workaround for it.
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.  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
printed 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're using gcc, @samp{gcc -v}.  Also, include the output from @samp{uname
-a}.

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 won't do anything about
it (aside of chiding you to send better bug reports).

Send your bug report to: @samp{mpfr@@loria.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, Floating-point Functions, Reporting Bugs, Top
@comment  node-name,  next,  previous,  up
@chapter MPFR Basics


@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:
@verbatim
#include <mpfr.h>
@end verbatim

@section Nomenclature and Types

@cindex Floating-point number
@tindex @code{mpfr_t}
@noindent
A @dfn{floating-point number} or @dfn{float} for short, is an arbitrary
precision mantissa with a limited precision exponent. The C data type
for such objects is @code{mpfr_t}. A floating-point number can have
three special values: Not-a-Number (NaN) or plus or minus Infinity. NaN
represents an uninitialized object, the result of an invalid operation
(like 0 divided by 0), or a value that cannot be determined (like
+Infinity minus +Infinity). Moreover, like in the IEEE 754-1985
standard, zero is signed, i.e.@: there are both +0 and -0; the behavior
is the same as in the IEEE 754-1985 standard and it is generalized to
the other functions supported by MPFR.


@cindex Precision
@tindex @code{mp_prec_t}
@noindent
The @dfn{Precision} is the number of bits used to represent the mantissa
of a floating-point number;
the corresponding C data type is @code{mp_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 2 and @code{MPFR_PREC_MAX} is equal to @code{ULONG_MAX}/2.

@cindex Rounding Mode
@tindex @code{mp_rnd_t}
@noindent
The @dfn{rounding mode} specifies the way to round the result of a
floating-point operation, in case the exact result can not be represented
exactly in the destination mantissa;
the corresponding C data type is @code{mp_rnd_t}.

@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.  (We chose this word because a limb of the human body is analogous to a
digit, only larger, and containing several digits.)  Normally a limb contains
32 or 64 bits.  The C data type for a limb is @code{mp_limb_t}.

@section Function Classes

There is only one class of functions in the MPFR library:

@enumerate
@item
Functions for floating-point arithmetic, with names beginning with
@code{mpfr_}.  The associated type is @code{mpfr_t}.
@end enumerate


@section MPFR Variable Conventions

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_mode)}.
This
computes the square of @var{x} with rounding mode @code{rnd_mode}
and puts the result back in @var{x}.

Before you can assign to an MPFR variable, you need to initialize it by calling
one of the special initialization functions.  When you're done with a
variable, you need to clear it out, using one of the functions for that
purpose.

A variable should only be initialized once, or at least cleared out between
each initialization.  After a variable has been initialized, it may be
assigned to any number of times.

For efficiency reasons, avoid to initialize and clear out a variable in loops.
Instead, initialize it before entering the loop, and clear it out after the
loop has exited.

You don't need to be concerned about allocating additional space for MPFR
variables, since any variable has a mantissa 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.

@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.
After inserting the following two lines after the @code{#include <gmp.h>}
line,
@verbatim
#include <mpfr.h>
#include <mpf2mpfr.h>
@end verbatim
@noindent
any program written for
MPF can be compiled directly with 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}.

@deftypevr {Global Variable} {mp_rnd_t} __gmpfr_default_rounding_mode
The default rounding mode (to nearest initially).
@end deftypevr

@section Getting the Latest Version of MPFR

The latest version of MPFR is available from @url{http://www.mpfr.org/}
or @url{http://www.loria.fr/projets/mpfr/}.

@node Floating-point Functions, Contributors, MPFR Basics, Top
@comment  node-name,  next,  previous,  up
@chapter Floating-point Functions
@cindex Floating-point functions
@cindex Float 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
integer functions.  The function prefix for floating-point operations is
@code{mpfr_}.

There is one significant characteristic of floating-point numbers that has
motivated a difference between this function class and other GNU MP function
classes: the inherent inexactness of floating-point arithmetic.  The user has
to specify the precision for 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 from 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 754-1985 arithmetic. The results obtained on one computer should
not differ from the results obtained on a computer with a different word size.

@cindex Accuracy
MPFR does not keep track of the accuracy of a computation. This is left
to the user or to a higher layer.
As a consequence, if two variables are used to store
only a few significant bits, and their product is stored in a variable with large
precision, then MPFR will still compute the result with full precision.

@menu
* Rounding Modes::
* Exceptions::
* Initialization Functions::
* Assignment Functions::
* Combined Initialization and Assignment Functions::
* Conversion Functions::
* Basic Arithmetic Functions::
* Comparison Functions::
* Special Functions::
* Input and Output Functions::
* Miscellaneous Functions::
* Internals::
@end menu

@node Rounding Modes, Exceptions, Floating-point Functions, Floating-point Functions
@section Rounding Modes
@cindex Rounding modes

The following four rounding modes are supported:

@itemize @bullet
@item @code{GMP_RNDN}: round to nearest
@item @code{GMP_RNDZ}: round towards zero
@item @code{GMP_RNDU}: round towards plus infinity
@item @code{GMP_RNDD}: round towards minus infinity
@end itemize

The @samp{round to nearest} mode works as in the IEEE 754-1985 standard: in
case the number to be rounded lies exactly in the middle of two representable
numbers, it is rounded to the one with the least significant bit set to zero.
For example, the number 5/2, 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 4.2.2).

Most MPFR functions take as first argument the destination variable, as
second and following arguments the input variables, as last argument a
rounding mode, and have a return value of type @code{int}, called the
@dfn{ternary value}. The value stored in the destination variable is
exactly 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).

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{GMP_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.

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

@deftypefun int mpfr_prec_round (mpfr_t @var{x}, mp_prec_t @var{prec}, mp_rnd_t @var{rnd})
Rounds @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 or equal to the precision of @var{x}, then new
space is allocated for the mantissa, and it is filled with zeros.
Otherwise, the mantissa is rounded to precision @var{prec} with the given
direction. In both cases, the precision of @var{x} is changed to @var{prec}.
@end deftypefun

@deftypefun int mpfr_round_prec (mpfr_t @var{x}, mp_rnd_t @var{rnd}, mp_prec_t @var{prec})
[This function is obsolete. Please use @code{mpfr_prec_round} instead.]
@end deftypefun

@deftypefun {char *} mpfr_print_rnd_mode (mp_rnd_t @var{rnd})
Returns the input string (GMP_RNDD, GMP_RNDU, GMP_RNDN, GMP_RNDZ)
corresponding to the rounding mode @var{rnd} or a null pointer if
@var{rnd} is an invalid rounding mode.
@end deftypefun

@node Exceptions, Initialization Functions, Rounding Modes, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Exceptions
@cindex Exceptions

Note: Overflow handling is still experimental and currently implemented
very partially. If an overflow occurs internally at the wrong place,
anything can happen (crash, wrong results, etc).

@deftypefun mp_exp_t mpfr_get_emin (void)
@deftypefunx mp_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}.
@end deftypefun

@deftypefun int mpfr_set_emin (mp_exp_t @var{exp})
@deftypefunx int mpfr_set_emax (mp_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.
If the user changes the exponent range, it is her/his responsibility to check
that all current floating-point variables are in the new allowed range
(for example using @code{mpfr_check_range}),
otherwise the subsequent
behavior will be undefined, in the sense of the ISO C standard.
@end deftypefun

@deftypefun int mpfr_check_range (mpfr_t @var{x}, int @var{t}, mp_rnd_t @var{rnd})
This function forces @var{x} to be in the current range of acceptable
values, @var{t} being the current ternary value: negative if @var{x}
is smaller than the exact value, positive if @var{x} is larger than
the exact value and zero if @var{x} is exact (before the call). 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 rounded
result is equal to the exact one, a positive value if the rounded
result is larger than the exact one, a negative value if the rounded
result is smaller than the exact one. 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.
@end deftypefun

@deftypefun void mpfr_clear_underflow (void)
@deftypefunx void mpfr_clear_overflow (void)
@deftypefunx void mpfr_clear_nanflag (void)
@deftypefunx void mpfr_clear_inexflag (void)
Clear the underflow, overflow, invalid, and inexact flags.
@end deftypefun

@deftypefun void mpfr_clear_flags (void)
Clear all global flags (underflow, overflow, inexact, invalid).
@end deftypefun

@deftypefun int mpfr_underflow_p (void)
@deftypefunx int mpfr_overflow_p (void)
@deftypefunx int mpfr_nanflag_p (void)
@deftypefunx int mpfr_inexflag_p (void)
Return the corresponding (underflow, overflow, invalid, inexact) flag,
which is non-zero iff the flag is set.
@end deftypefun

@node Initialization Functions, Assignment Functions, Exceptions, Floating-point Functions
@comment  node-name,  next,  previous,  up
@cindex Initialization functions
@section Initialization Functions

@deftypefun void mpfr_set_default_prec (mp_prec_t @var{prec})
Set the default precision to be @strong{exactly} @var{prec} bits.  The
precision of a variable means the number of bits used to store its mantissa.
All
subsequent calls to @code{mpfr_init} will use this precision, but previously
initialized variables are unaffected.
This default precision is set to 53 bits initially.
The precision can be any integer between @code{MPFR_PREC_MIN} and
@code{MPFR_PREC_MAX}.
@end deftypefun

@deftypefun mp_prec_t mpfr_get_default_prec (void)
Returns the default MPFR precision in bits.
@end deftypefun

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_init (mpfr_t @var{x})
Initialize @var{x} and set its value to NaN.

Normally, a variable should be initialized once only
or at least be cleared, using @code{mpfr_clear}, between initializations.  The
precision of @var{x} is the default precision, which can be changed
by a call to @code{mpfr_set_default_prec}.
@end deftypefun

@deftypefun void mpfr_init2 (mpfr_t @var{x}, mp_prec_t @var{prec})
Initialize @var{x}, set its precision to be @strong{exactly}
@var{prec} bits and its value to NaN.

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 which has already been initialized,
use @code{mpfr_set_prec}.
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_clear (mpfr_t @var{x})
Free the space occupied by @var{x}.  Make sure to call this function for all
@code{mpfr_t} variables when you are done with them.
@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);
@}
@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}, mp_prec_t @var{prec})
Reset 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 mantissa 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.
@end deftypefun

@deftypefun mp_prec_t mpfr_get_prec (mpfr_t @var{x})
Return the precision actually used for assignments of @var{x}, i.e.@: the
number of bits used to store its mantissa.
@end deftypefun

@deftypefun void mpfr_set_prec_raw (mpfr_t @var{x}, mp_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 mantissa fits into the current allocated memory
space for @var{x}. Otherwise the behavior is undefined.
@end deftypefun

@node Assignment Functions, Combined Initialization and Assignment Functions, Initialization Functions, Floating-point Functions
@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}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_set_ui (mpfr_t @var{rop}, unsigned long int @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_set_si (mpfr_t @var{rop}, long int @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_set_d (mpfr_t @var{rop}, double @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_set_ld (mpfr_t @var{rop}, long double @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_set_z (mpfr_t @var{rop}, mpz_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_set_q (mpfr_t @var{rop}, mpq_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_set_f (mpfr_t @var{rop}, mpf_t @var{op}, mp_rnd_t @var{rnd})
Set the value of @var{rop} from @var{op}, rounded
towards the given direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_set_str (mpfr_t @var{x}, const char *@var{s}, int @var{base}, mp_rnd_t @var{rnd})
Set @var{x} to the value of the whole string @var{s} in base @var{base}
(between 2 and 36), rounded in direction @var{rnd}.
See the documentation of @code{mpfr_inp_str} for a detailed description
of the valid string formats.
This function returns 0 if the entire string up to the final @code{\0} is a
valid number in base @var{base}; otherwise it returns @minus{}1.
@end deftypefun

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

@deftypefun void mpfr_swap (mpfr_t @var{x}, mpfr_t @var{y})
Swap the values @var{x} and @var{y} efficiently. Warning: the
precisions are exchanged too; in case the precisions are different,
@code{mpfr_swap} is thus not equivalent to three @code{mpfr_set} calls
using a third auxiliary variable.
@end deftypefun

@node Combined Initialization and Assignment Functions, Conversion Functions, Assignment Functions, Floating-point Functions
@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}, mp_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_ui (mpfr_t @var{rop}, unsigned long int @var{op}, mp_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_si (mpfr_t @var{rop}, signed long int @var{op}, mp_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_d (mpfr_t @var{rop}, double @var{op}, mp_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_ld (mpfr_t @var{rop}, long double @var{op}, mp_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_z (mpfr_t @var{rop}, mpz_t @var{op}, mp_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_q (mpfr_t @var{rop}, mpq_t @var{op}, mp_rnd_t @var{rnd})
@deftypefnx Macro int mpfr_init_set_f (mpfr_t @var{rop}, mpf_t @var{op}, mp_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}, mp_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, Basic Arithmetic Functions, Combined Initialization and Assignment Functions, Floating-point Functions
@comment  node-name,  next,  previous,  up
@cindex Conversion functions
@section Conversion Functions

@deftypefun double mpfr_get_d (mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx {long double} mpfr_get_ld (mpfr_t @var{op}, mp_rnd_t @var{rnd})
Convert @var{op} to a @code{double} (respectively @code{long double}),
using the rounding mode @var{rnd}.
@end deftypefun

@deftypefun double mpfr_get_d1 (mpfr_t @var{op})
Convert @var{op} to a @code{double}, using the default MPFR rounding mode
(see function @code{mpfr_set_default_rounding_mode}). This function is
obsolete.
@end deftypefun

@deftypefun double mpfr_get_d_2exp (long *@var{exp}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Return @var{d} and set @var{exp} such that @math{0.5@le{}@GMPabs{@var{d}}<1}
and @m{@var{d}\times 2^{exp}, @var{d} times 2 raised to @var{exp}} equals
@var{op} rounded to double precision, using the given rounding mode.
@end deftypefun

@deftypefun long mpfr_get_si (mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx {unsigned long} mpfr_get_ui (mpfr_t @var{op}, mp_rnd_t @var{op})
Convert @var{op} to a @code{long} or @code{unsigned long}, after rounding
it with respect to @var{rnd}.
If @var{op} is NaN or Inf, or too big for the return type,
the result is undefined.

See also @code{mpfr_fits_slong_p} and @code{mpfr_fits_ulong_p}.
@end deftypefun

@deftypefun mp_exp_t mpfr_get_z_exp (mpz_t @var{z}, mpfr_t @var{op})
Put the scaled mantissa of @var{op} (regarded as an integer, with the
precision of @var{op}) into @var{z}, and return the exponent @var{exp}
(which may be outside the current exponent range) such that @var{op}
exactly equals
@ifnottex
@var{z} multiplied by two exponent @var{exp}.
@end ifnottex
@tex
$z \times 2^{\rm exp}$.
@end tex
If the exponent is not representable in the @code{mp_exp_t} type, the
behavior is undefined.
@end deftypefun

@deftypefun {char *} mpfr_get_str (char *@var{str}, mp_exp_t *@var{expptr}, int @var{base}, size_t @var{n}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Convert @var{op} to a string of digits in base @var{base}, with rounding in
direction @var{rnd}. The base may vary
from 2 to 36.

The generated string is a fraction, with an implicit radix point immediately
to the left of the first digit.  For example, the number 3.1416 would be
returned as "31416" in the string and 1 written at @var{expptr}.

If @var{n} is zero, the number of digits of the mantissa is determined
automatically from the precision of @var{op} and the value of @var{base}.
Warning: this functionality may disappear or change in future versions.
Otherwise generate exactly @var{n} significant digits, which must be at
least 2.

If @var{str} is a null pointer, space for the mantissa is allocated using
the current allocation function, and a pointer to the string is returned.
The block will be @code{strlen(s)+1} bytes. For more information on how
this block is allocated and how to free it: @pxref{Custom Allocation,,, gmp,
GNU MP}.

If @var{str} is not a null pointer, it should point to a block of storage
large enough for the mantissa, i.e., at least @var{n} + 2. The extra
two bytes are for a possible minus sign, and for the terminating null
character.

If @var{n} is 0, note that the space requirements for @var{str}
in this case will be impossible for the user to predetermine. Therefore,
one needs to pass a null pointer for the string argument whenever
@var{n} is 0.

If the input number is an ordinary number, the exponent is written through
the pointer @var{expptr} (the current minimal exponent for 0).

A pointer to the string is returned, unless there is an error, in which
case a null pointer is returned.
@end deftypefun

@deftypefun int mpfr_fits_ulong_p (mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_slong_p (mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_uint_p (mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_sint_p (mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_ushort_p (mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_sshort_p (mpfr_t @var{op}, mp_rnd_t @var{rnd})
Return non-zero if @var{op} would fit in the respective C data type, when
rounded to an integer in the direction @var{rnd}.
@end deftypefun

@node Basic Arithmetic Functions, Comparison Functions, Conversion Functions, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Basic Arithmetic Functions
@cindex Basic arithmetic functions
@cindex Float arithmetic functions
@cindex Arithmetic functions

@deftypefun int mpfr_add (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_add_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_add_z (mpfr_t @var{rop}, mpfr_t @var{op1}, mpz_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_add_q (mpfr_t @var{rop}, mpfr_t @var{op1}, mpq_t @var{op2}, mp_rnd_t @var{rnd})
Set @var{rop} to @math{@var{op1} + @var{op2}} rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_sub (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_ui_sub (mpfr_t @var{rop}, unsigned long int @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_sub_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_sub_z (mpfr_t @var{rop}, mpfr_t @var{op1}, mpz_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_sub_q (mpfr_t @var{rop}, mpfr_t @var{op1}, mpq_t @var{op2}, mp_rnd_t @var{rnd})
Set @var{rop} to @math{@var{op1} - @var{op2}} rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_mul (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_mul_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_mul_z (mpfr_t @var{rop}, mpfr_t @var{op1}, mpz_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_mul_q (mpfr_t @var{rop}, mpfr_t @var{op1}, mpq_t @var{op2}, mp_rnd_t @var{rnd})
Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2}} rounded in the
direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_div (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_ui_div (mpfr_t @var{rop}, unsigned long int @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_div_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_div_z (mpfr_t @var{rop}, mpfr_t @var{op1}, mpz_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_div_q (mpfr_t @var{rop}, mpfr_t @var{op1}, mpq_t @var{op2}, mp_rnd_t @var{rnd})
Set @var{rop} to @math{@var{op1}/@var{op2}} rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_sqrt (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_sqrt_ui (mpfr_t @var{rop}, unsigned long int @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to @m{\sqrt{@var{op}}, the square root of @var{op}}
rounded in the direction @var{rnd}. Return @minus{}0 if @var{rop} is
@minus{}0 (to be consistent with the IEEE 754-1985 standard).
Set @var{rop} to NaN if @var{op} is negative.
@end deftypefun

@deftypefun int mpfr_cbrt (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the cubic root (defined over the real numbers) of @var{op}
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_pow (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_pow_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_pow_si (mpfr_t @var{rop}, mpfr_t @var{op1}, long int @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_ui_pow_ui (mpfr_t @var{rop}, unsigned long int @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_ui_pow (mpfr_t @var{rop}, unsigned long int @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
Set @var{rop} to @m{@var{op1}^{op2}, @var{op1} raised to @var{op2}},
rounded in the direction @var{rnd}.
Special values are currently handled as described in the ISO C99 standard
for the @code{pow} function:
@itemize @bullet
@item @code{pow(@pom{}0, @var{y})} returns plus or minus infinity for @var{y} a negative odd integer.
@item @code{pow(@pom{}0, @var{y})} returns plus 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{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 plus 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 plus 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 minus infinity for @var{y} a positive odd integer.
@item @code{pow(-inf, @var{y})} returns plus infinity for @var{y} positive and not an odd integer.
@item @code{pow(+inf, @var{y})} returns plus zero for @var{y} negative, and plus infinity for @var{y} positive.
@end itemize
@end deftypefun

@deftypefun int mpfr_neg (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to @math{-@var{op}} rounded in the direction @var{rnd}.
Just changes the sign
if @var{rop} and @var{op} are the same variable.
@end deftypefun

@deftypefun int mpfr_abs (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the absolute value of @var{op},
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_mul_2ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_mul_2si (mpfr_t @var{rop}, mpfr_t @var{op1}, long int @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_mul_2exp (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
Set @var{rop} to @m{@var{op1} \times 2^{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.
[@code{mpfr_mul_2exp} is kept for upward compatibility.]
@end deftypefun

@deftypefun int mpfr_div_2ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_div_2si (mpfr_t @var{rop}, mpfr_t @var{op1}, long int @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_div_2exp (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
Set @var{rop} to @m{@var{op1}/2^{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.
[@code{mpfr_div_2exp} is kept for upward compatibility.]
@end deftypefun

@node Comparison Functions, Special Functions, Basic Arithmetic Functions, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions
@cindex Float comparisons functions
@cindex 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}, signed long int @var{op2})
@deftypefunx int mpfr_cmp_d (mpfr_t @var{op1}, double @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 (Not-a-Number), the behavior is undefined.
@end deftypefun

@deftypefun int mpfr_cmp_ui_2exp (mpfr_t @var{op1}, unsigned long int @var{op2}, mp_exp_t @var{e})
@deftypefunx int mpfr_cmp_si_2exp (mpfr_t @var{op1}, long int @var{op2}, mp_exp_t @var{e})
Compare @var{op1} and @m{@var{op2} \times 2^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})
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 (Not-a-Number), 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 @code{mpf}, but does not
make much sense.
@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})
Return non-zero if @var{op} is respectively Not-a-Number (NaN),
an infinity, an ordinary number (i.e.@: neither Not-a-Number nor
an infinity). Return zero otherwise.
@end deftypefun

@deftypefun void mpfr_reldiff (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_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 exact rounding on the relative difference;
it just computes @math{|@var{op1}-@var{op2}|/@var{op1}}, using the
rounding mode @var{rnd} for all operations and the precision of @var{rop}.
@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}.
Its result is undefined when @var{op} is NaN (Not-a-Number).

This function is actually implemented as a macro.  It may evaluate its
argument multiple times.
@end deftypefn

@deftypefun int mpfr_greater_p (mpfr_t @var{op1}, mpfr_t @var{op2})
Return non-zero if @math{@var{op1} > @var{op2}}, zero otherwise.
@end deftypefun

@deftypefun int mpfr_greaterequal_p (mpfr_t @var{op1}, mpfr_t @var{op2})
Return non-zero if @math{@var{op1} @ge{} @var{op2}}, zero otherwise.
@end deftypefun

@deftypefun int mpfr_less_p (mpfr_t @var{op1}, mpfr_t @var{op2})
Return non-zero if @math{@var{op1} < @var{op2}}, zero otherwise.
@end deftypefun

@deftypefun int mpfr_lessequal_p (mpfr_t @var{op1}, mpfr_t @var{op2})
Return non-zero if @math{@var{op1} @le{} @var{op2}}, zero otherwise.
@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} are NaN, or @math{@var{op1} = @var{op2}}).
@end deftypefun

@deftypefun int mpfr_equal_p (mpfr_t @var{op1}, mpfr_t @var{op2})
Return non-zero if @math{@var{op1} = @var{op2}}, zero otherwise
(i.e.@: @var{op1} and/or @var{op2} are NaN, or
@math{@var{op1} @ne{} @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

@node Special Functions, Input and Output Functions, Comparison Functions, Floating-point Functions
@section Special Functions
@cindex Special functions

All those functions, except explicitely stated, return zero for an
exact return value, a positive value for a return value larger than the 
exact result, and a negative value otherwise.

@deftypefun int mpfr_log (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_log2 (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_log10 (mpfr_t @var{rop}, mpfr_t @var{op}, mp_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}.
@end deftypefun

@deftypefun int mpfr_exp (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the exponential of @var{op},
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_exp2 (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to @m{2^{op}, 2 power of @var{op}},
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_cos (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_sin (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_tan (mpfr_t @var{rop}, mpfr_t @var{op}, mp_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_sin_cos (mpfr_t @var{sop}, mpfr_t @var{cop}, mpfr_t @var{op}, mp_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}.
Return 0 iff both results are exact.
@end deftypefun

@deftypefun int mpfr_acos (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_asin (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_atan (mpfr_t @var{rop}, mpfr_t @var{op}, mp_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}.
@end deftypefun

@deftypefun int mpfr_cosh (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_sinh (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_tanh (mpfr_t @var{rop}, mpfr_t @var{op}, mp_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_acosh (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_asinh (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_atanh (mpfr_t @var{rop}, mpfr_t @var{op}, mp_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_fac_ui (mpfr_t @var{rop}, unsigned long int  @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the factorial of the @code{unsigned long int} @var{op},
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_log1p (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the logarithm of one plus @var{op},
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_expm1 (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the exponential of @var{op} minus one,
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_gamma (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the value of the Gamma function on @var{op},
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_zeta (mpfr_t @var{rop}, mpfr_t @var{op}, mp_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}, mp_rnd_t @var{rnd})
Set @var{rop} to the value of the error function on @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}, mp_rnd_t @var{rnd})
Set @var{rop} to @math{@var{op1} @GMPtimes{} @var{op2} + @var{op3}},
rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun int mpfr_agm (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_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
u[n] and v[n], where u[0]=@var{op1}, v[0]=@var{op2}, u[n+1] is the
arithmetic mean of u[n] and v[n], and v[n+1] is the geometric mean of
u[n] and v[n].
@end deftypefun

@deftypefun int mpfr_const_log2 (mpfr_t @var{rop}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_const_pi (mpfr_t @var{rop}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_const_euler (mpfr_t @var{rop}, mp_rnd_t @var{rnd})
Set @var{rop} to the logarithm of 2, the value of @m{\pi,Pi}, the value
of Euler's constant 0.577@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. There is
currently no way to free the cache.
@end deftypefun

@node Input and Output Functions, Miscellaneous Functions, Special Functions, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Input and Output Functions
@cindex Float input and output functions
@cindex Input functions
@cindex Output functions
@cindex I/O 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 @var{stream} argument to any of
these functions will make them read from @code{stdin} and write to
@code{stdout}, respectively.

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.

@deftypefun size_t mpfr_out_str (FILE *@var{stream}, int @var{base}, size_t @var{n}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Output @var{op} on stream @var{stream}, as a string of digits in
base @var{base}, rounded in direction @var{rnd}.
The base may vary from 2 to 36.  Print @var{n} significant digits exactly,
or if @var{n} is 0, the maximum number of digits accurately representable
by @var{op} (this feature may disappear).

In addition to the significant digits, a decimal point at the right of the
first digit and a trailing exponent in base 10, in the form @samp{eNNN},
are printed. If @var{base} is greater than 10, @samp{@@} will be used
instead of @samp{e} as exponent delimiter.

Return the number of bytes 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}, mp_rnd_t @var{rnd})
Input a string in base @var{base} from stream @var{stream},
rounded in direction @var{rnd}, and put the
read float in @var{rop}.  The string is of the form @samp{M@@N} or, if the
base is 10 or less, alternatively @samp{MeN} or @samp{MEN}, or, if the base
is 16, alternatively @samp{MpB} or @samp{MPB}.
@samp{M} is the mantissa in the specified base, @samp{N} is the exponent
written in decimal for the specified base, and in base 16, @samp{B} is the
binary exponent written in decimal (i.e.@: it indicates the power of 2 by
which the mantissa is to be scaled).
The argument @var{base} may be in the range 2 to 36.

Special values can be read as follows (the case does not matter):
@code{@@NaN@@}, @code{@@Inf@@}, @code{+@@Inf@@} and @code{-@@Inf@@},
possibly followed by other characters; if the base is smaller or equal
to 16, the following strings are accepted too: @code{NaN}, @code{Inf},
@code{+Inf} and @code{-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

@node Miscellaneous Functions, Internals, Input and Output Functions, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Miscellaneous Functions
@cindex Miscellaneous float functions

@deftypefun int mpfr_rint (mpfr_t @var{rop}, mpfr_t @var{op}, mp_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_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 rounding mode, @code{mpfr_ceil} rounds
to the next higher or equal representable integer, @code{mpfr_floor} to
the next lower or equal representable integer, @code{mpfr_round} to the
nearest representable integer, rounding halfway cases away from zero,
and @code{mpfr_trunc} to the next representable integer towards zero.

The returned 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 returned 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.

Note that @code{mpfr_round} is different from @code{mpfr_rint} called with
the rounding to the nearest mode (where halfway cases are rounded to an even
integer or mantissa). Note also that no double rounding is performed; for
instance, 4.5 (100.1 in binary) is rounded by @code{mpfr_round} to 4 (100
in binary) in 2-bit precision, though @code{round(4.5)} is equal to 5 and
5 (101 in binary) is rounded to 6 (110 in binary) in 2-bit precision.
@end deftypefun

@deftypefun int mpfr_frac (mpfr_t @var{rop}, mpfr_t @var{op}, mp_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).
@end deftypefun

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

@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. 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}, if there is one
(the infinite values are seen as the smallest and largest floating-point
numbers). If the result is zero, it keeps the same sign. No underflow or
overflow is generated.
@end deftypefun

@deftypefun void mpfr_nextabove (mpfr_t @var{x})
Equivalent to @code{mpfr_nexttoward} where @var{y} is plus infinity.
@end deftypefun

@deftypefun void mpfr_nextbelow (mpfr_t @var{x})
Equivalent to @code{mpfr_nexttoward} where @var{y} is minus infinity.
@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}.
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.
@end deftypefun

@deftypefun void mpfr_random (mpfr_t @var{rop})
Generate a uniformly distributed random float in the interval
@math{0 @le{} @var{rop} < 1}.
This function is deprecated; @code{mpfr_urandomb} should be used instead.
@end deftypefun

@deftypefun void mpfr_random2 (mpfr_t @var{rop}, mp_size_t @var{size}, mp_exp_t @var{exp})
Generate a random float of at most @var{size} limbs, with long strings of
zeros and ones in the binary representation. The exponent of the number is in
the interval @minus{}@var{exp} to @var{exp}.
This function is useful for
testing functions and algorithms, since this kind of random numbers have
proven to be more likely to trigger corner-case bugs.
Negative random numbers are generated when @var{size} is negative.
@end deftypefun

@c @deftypefun size_t mpfr_size (mpfr_t @var{op})
@c Return the size of @var{op} measured in number of limbs.  If @var{op} is
@c zero, the returned value will be zero.  (@xref{Nomenclature}, for an
@c explanation of the concept @dfn{limb}.)
@c
@c @strong{This function is obsolete.  It will disappear from future MP
@c releases.}
@c @end deftypefun

@node Internals,  , Miscellaneous Functions, Floating-point Functions
@section Internals
@cindex Internals

The following types and
functions were mainly designed for the implementation of @code{mpfr},
but may be useful for users too.
However no upward compatibility is guaranteed.
You may need to include @file{mpfr-impl.h} to use them.

The @code{mpfr_t} type consists of four fields.
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}.

The @code{_mpfr_size} field is used to store the number of
allocated limbs, with the high bits reserved to store
the sign (bit 31), the NaN flag (bit 30),
and the Infinity flag (bit 29);
thus bits 0 to 28 remain for the number of allocated limbs, with a maximal
value of 536870911.
A NaN is indicated by the NaN flag set, and the other fields are
undefined.
An Infinity is indicated by the NaN flag clear and the Infinity flag set;
the sign bit of an Infinity indicates the sign, the limb data
and the exponent are undefined.

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.

Finally, the @code{_mpfr_d} 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{BITS_PER_MP_LIMB}).
Zeros are represented by the most significant limb being zero, other
limb data and the exponent are undefined (this implies that the
corresponding objects may contain invalid values, thus should not be
evaluated even if they are not taken into account).
Non-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 zero.
When the precision has been lowered by @code{mpfr_set_prec}, the space
allocated at @code{_mpfr_d} remains as given by @code{_mpfr_size}, but
@code{_mpfr_prec} indicates how much of that space is actually used.

@deftypefun int mpfr_add_one_ulp (mpfr_t @var{x}, mp_rnd_t @var{rnd})
Add one unit in last place (ulp) to @var{x} if @var{x} is finite
and positive, subtract one ulp if @var{x} is finite and negative;
otherwise, @var{x} is not changed.
The return value is zero unless an overflow occurs, in which case the
@code{mpfr_add_one_ulp} function behaves like a conventional addition.
@end deftypefun

@deftypefun int mpfr_sub_one_ulp (mpfr_t @var{x}, mp_rnd_t @var{rnd})
Subtract one ulp to @var{x} if @var{x} is finite and positive, add one
ulp if @var{x} is finite and negative; otherwise, @var{x} is not changed.
The return value is zero unless an underflow occurs, in which case the
@code{mpfr_sub_one_ulp} function behaves like a conventional subtraction.
@end deftypefun

@deftypefun int mpfr_can_round (mpfr_t @var{b}, mp_exp_t @var{err}, mp_rnd_t @var{rnd1}, mp_rnd_t @var{rnd2}, mp_prec_t @var{prec})
Assuming @var{b} is an approximation of an unknown number
@var{x} in direction @var{rnd1} with error at most two to the power
E(b)-@var{err} where E(b) is the exponent of @var{b}, returns a non-zero
value if one is able to round exactly @var{x} to precision
@var{prec} with direction @var{rnd2},
and 0 otherwise (including for NaN and Inf).
This function @strong{does not modify} its arguments.
@end deftypefun

@deftypefun mp_exp_t mpfr_get_exp (mpfr_t @var{x})
Get the exponent of @var{x}, assuming that @var{x} is a non-zero ordinary
number.
@end deftypefun

@deftypefun int mpfr_set_exp (mpfr_t @var{x}, mp_exp_t @var{e})
Set the exponent of @var{x} if @var{e} is in the current exponent range,
and return 0 (even if @var{x} is not a non-zero ordinary number);
otherwise, return 1.
@end deftypefun

@deftypefun void mpfr_set_str_binary (mpfr_t @var{x}, const char *@var{s})
Set @var{x} to the value of the binary number in string @var{s}, which has to
be of the
form +/-xxxx.xxxxxxEyy. The exponent is read in decimal, but is interpreted
as the power of two to be multiplied by the mantissa.
The mantissa length of @var{s} has to be less or equal to the precision of
@var{x}, otherwise an error occurs.
If @var{s} starts with @code{N}, it is interpreted as NaN (Not-a-Number);
if it starts with @code{I} after the sign, it is interpreted as infinity,
with the corresponding sign.
@end deftypefun

@deftypefun void mpfr_print_binary (mpfr_t @var{float})
Output @var{float} on stdout
in raw binary format (the exponent is written in decimal, yet).
@end deftypefun

@node Contributors, References, Floating-point Functions, Top
@comment  node-name,  next,  previous,  up
@unnumbered Contributors

The main developers consist of Guillaume Hanrot, Vincent Lef@`evre,
Kevin Ryde and Paul Zimmermann.

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 their suggestions for an easy integration into GNU MP,
and Nathalie Revol for her careful reading of a previous version of
this documentation.

Sylvie Boldo from ENS-Lyon, France,
contributed the functions @code{mpfr_agm} and @code{mpfr_log}.
Emmanuel Jeandel, from ENS-Lyon too,
contributed the generic hypergeometric code in
@code{generic.c}, as well as the @code{mpfr_exp3},
a first implementation of the sine and cosine,
and improved versions of
@code{mpfr_const_log2} and @code{mpfr_const_pi}.
Mathieu Dutour contributed the functions @code{mpfr_atan} and @code{mpfr_asin},
David Daney contributed the hyperbolic and inverse hyperbolic functions,
the base-2 exponential, and the factorial function. Fabrice Rouillier
contributed the original version of @file{mul_ui.c}, the @file{gmp_op.c}
file, and helped to the Windows porting.
Jean-Luc R@'emy contributed the @code{mpfr_zeta} code.
Ludovic Meunier helped in the design of the @code{mpfr_erf} code.

The development of the MPFR library would not have been possible without the
continuous support of LORIA, INRIA and INRIA Lorraine.
The development of MPFR was also supported by a grant 
(202F0659 00 MPN 121) from the Conseil R@'egional de Lorraine in 2002.

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

@itemize @bullet

@item
Torbj@"orn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic Library",
  version 4.1.2, 2002.

@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
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",
Birkhauser, Boston, 1997.

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

@node Function 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: