\input texinfo    @c -*-texinfo-*-
@c %**start of header
@setfilename mpfr.info
@documentencoding ISO-8859-1
@set VERSION 2.3.1
@set UPDATED-MONTH January 2008
@settitle MPFR @value{VERSION}
@synindex tp fn
@iftex
@afourpaper
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@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, 2004, 2005, 2006, 2007, 2008 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

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@ifnottex
@node Top, Copying, (dir), (dir)
@top MPFR
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@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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@menu
* Copying::                     MPFR 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.
* MPFR Interface::              MPFR functions and macros.
* 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,
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@c  Give either \times or the word "times".
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times
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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 but can be emulated.

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

@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 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.

@item
In the MPFR build directory, type
@samp{./configure}

This will prepare the build and setup the options according to your system.
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}.
A dynamic library may be produced too (see configure).

@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/share/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

@section Other `make' Targets

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 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.mpfr.org/}.

@section Getting the Latest Version of MPFR

The latest version of MPFR is available from @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/}: 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} and the MPFR
version (the GMP version may be useful 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 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, MPFR Interface, 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 significand (also called mantissa) with a limited precision
exponent. The C data type
for such objects is @code{mpfr_t} (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). 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 @minus{}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 significand
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.

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. Moreover, you
may reach some memory limit on your platform, in which case the program
may abort, crash or have undefined behavior (depending on your C
implementation).

@cindex Rounding Modes
@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 significand;
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 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.

@section Rounding Modes

The following four rounding modes are supported:

@itemize @bullet
@item @code{GMP_RNDN}: round to nearest
@item @code{GMP_RNDZ}: round toward zero
@item @code{GMP_RNDU}: round toward plus infinity
@item @code{GMP_RNDD}: round toward 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
correctly rounded, i.e.@: MPFR behaves as if it computed the result with
an infinite precision, then rounded it to the precision of this variable.
The input variables are regarded as exact (in particular, their precision
does not affect the result).

As a consequence, in case of a non-zero real rounded result, the error
on the result is less or equal to 1/2 ulp (unit in the last place) of
the target in the rounding to nearest mode, and less than 1 ulp of the
target in the directed rounding modes (a ulp is the weight of the least
significant represented bit of the target 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{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.
The opposite of a returned ternary value is guaranteed to be representable
in an @code{int}.

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

@section Floating-Point Values on Special Numbers

This section specifies the floating-point values (of type @code{mpfr_t})
returned by MPFR 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 Section ``Rounding Modes'' (but see
below for the specification of the sign of an exact zero). Otherwise
the general rules from this section apply unless stated otherwise in
the description of the MPFR function (@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 the limit is not defined (e.g., @code{mpfr_log} on
@minus{}0), the behavior must be specified in the description of the
MPFR function.

When the result is equal to 0, its sign is determined by considering the
limit as if the input point were not in the domain: If one approaches 0
from above (resp.@: below), the result is +0 (resp.@: @minus{}0). In the
other cases, the sign must be specified in the description of the MPFR
function. Example: @code{mpfr_sin} on +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{Special Functions}),
since for any finite input @var{x}, @code{mpfr_hypot} on (@var{x},+Inf)
gives +Inf.

@section Exceptions

MPFR supports 5 exception types:

@itemize @bullet

@item 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 exponent of the current range. In the round-to-nearest
mode, the halfway case is rounded toward zero.

Note: This is not the single definition of the underflow. MPFR chooses
to consider the underflow after rounding. The underflow before rounding
can also be defined. For instance, consider a function that has the
exact result @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) in the current
range, with a 2-bit target precision and rounding toward plus 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:
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 exponent of the current range. In the round-to-nearest
mode, the result is infinite.

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

@item Inexact:
An inexact exception occurs when the result of a function cannot be
represented exactly and must be rounded.

@item Range error:
A range exception occurs when a function that does not return a MPFR
number (such as comparisons and conversions to an integer) has an
invalid result (e.g. an argument is NaN in @code{mpfr_cmp} or in a
conversion to an integer).

@end itemize

MPFR has a global flag for each exception, which can be cleared, set
or tested by functions described in @ref{Exception Related Functions}.

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 MPFR Interface, Contributors, MPFR Basics, Top
@comment  node-name,  next,  previous,  up
@chapter MPFR Interface
@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.

The value of the standard C macro @code{errno} may be set to non-zero by
any MPFR function or macro, whether or not there is an error.

@menu
* Initialization Functions::
* Assignment Functions::
* Combined Initialization and Assignment Functions::
* Conversion Functions::
* Basic Arithmetic Functions::
* Comparison Functions::
* Special Functions::
* Input and Output Functions::
* Integer Related Functions::
* Miscellaneous Functions::
* Rounding Mode Related Functions::
* Exception Related Functions::
* Advanced 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}, mp_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
@code{mpf} functions initialize 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 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

@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_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 significand.
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)
Return the default MPFR precision in bits.
@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 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.
@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 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}). When using 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.

@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_uj (mpfr_t @var{rop}, uintmax_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_set_sj (mpfr_t @var{rop}, intmax_t @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_decimal64 (mpfr_t @var{rop}, _Decimal64 @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
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_sj}, @code{mpfr_set_uj},
@code{mpfr_set_z}, @code{mpfr_set_q} and
@code{mpfr_set_f}, regardless of the rounding mode.
If the system doesn't support the IEEE-754 standard, @code{mpfr_set_d},
@code{mpfr_set_ld} and
@code{mpfr_set_decimal64} might not preserve the signed zeros.
The @code{mpfr_set_decimal64} function is built only with the configure
option @samp{--enable-decimal-float}, which also requires
@samp{--with-gmp-build}, and when the compiler or
system provides the @samp{_Decimal64} data type
(GCC version 4.2.0 is known to support this data type,
but only when configured with @samp{--enable-decimal-float} too).
@code{mpfr_set_q} might not be able to work if the numerator (or the
denominator) can not be representable as a @code{mpfr_t}.

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_d},
@code{mpfr_set_ld} or @code{mpfr_set_decimal64}.
Otherwise the floating-point constant will be first
converted into a reduced-precision (e.g., 53-bit) binary number before
MPFR can work with it.
@end deftypefun

@deftypefun int mpfr_set_ui_2exp (mpfr_t @var{rop}, unsigned long int @var{op}, mp_exp_t @var{e}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_set_si_2exp (mpfr_t @var{rop}, long int @var{op}, mp_exp_t @var{e}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_set_uj_2exp (mpfr_t @var{rop}, uintmax_t @var{op}, intmax_t @var{e}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_set_sj_2exp (mpfr_t @var{rop}, intmax_t @var{op}, intmax_t @var{e}, mp_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}, mp_rnd_t @var{rnd})
Set @var{rop} to the value of the whole 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.
@c Additionally, special values
@c @code{@@NaN@@}, @code{@@Inf@@}, @code{+@@Inf@@} and @code{-@@Inf@@},
@c all case insensitive, without leading whitespace and possibly followed by
@c other characters, are accepted too (it may change).
This function returns 0 if the entire string up to the final null character
is a valid number in base @var{base}; otherwise it returns @minus{}1, and
@var{rop} may have changed.
@end deftypefun

@deftypefun int mpfr_strtofr (mpfr_t @var{rop}, const char *@var{nptr}, char **@var{endptr}, int @var{base}, mp_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 36 (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 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.
Case is ignored. After optional leading whitespace, one has a subject
sequence consisting of an optional sign (@code{+} or @code{-}), and either
numeric data or special data. The subject sequence is defined as the
longest initial subsequence of the input string, starting with the first
non-whitespace character, that is of the expected form.

The form of numeric data is a non-empty sequence of significand digits
with an optional decimal point, and an optional exponent consisting of
an exponent prefix followed by an optional sign and a non-empty sequence
of decimal digits. A significand digit is either a decimal digit or a
Latin letter (62 possible characters), with @code{a} = 10, @code{b} = 11,
@dots{}, @code{z} = 36; its value must be strictly less than the base.
The decimal point can be either the one defined by the current locale or
the period (the first one is accepted for consistency with the C standard
and the practice, the second one is accepted to allow the programmer to
provide MPFR numbers from strings in a way that does not depend on the
current locale).
The exponent prefix can be @code{e} or @code{E} for bases up to 10, or
@code{@@} in any base; it indicates a multiplication by a power of the
base. In bases 2 and 16, the exponent prefix can also be @code{p} or
@code{P}, in which case it introduces a binary exponent: it indicates a
multiplication by a power of 2 (there is a difference only for base 16).
The value of an exponent is always written in base 10.
In base 2, the significand can start with @code{0b} or @code{0B}, and
in base 16, it can start with @code{0x} or @code{0X}.

If the argument @var{base} is 0, then the base is automatically detected
as follows. If the significand starts with @code{0b} or @code{0B}, base 2
is assumed. If the significand starts with @code{0x} or @code{0X}, base 16
is assumed. Otherwise base 10 is assumed.

Note: The exponent 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 @code{0b}, @code{0B}, @code{0x} or @code{0X}
is not followed by a binary/hexadecimal digit, then the subject sequence
stops at the character @code{0}.

Special data (for infinities and NaN) can be @code{@@inf@@} or
@code{@@nan@@(n-char-sequence)}, and if @math{@var{base} @le{} 16},
it can also be @code{infinity}, @code{inf}, @code{nan} or
@code{nan(n-char-sequence)}, all case insensitive.
A @code{n-char-sequence} is a non-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.

@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, 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}, 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, MPFR Interface
@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})
@deftypefunx _Decimal64 mpfr_get_decimal64 (mpfr_t @var{op}, mp_rnd_t @var{rnd})
Convert @var{op} to a @code{double} (respectively @code{_Decimal64} or
@code{long double}), using the rounding mode @var{rnd}.
If @var{op} is NaN, some fixed NaN (either quiet or signaling) or the result
of 0.0/0.0 is returned. 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_decimal64} function is built only under some conditions:
see the documentation of @code{mpfr_set_decimal64}.
@end deftypefun

@deftypefun double mpfr_get_d_2exp (long *@var{exp}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx {long double} mpfr_get_ld_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 (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 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{rnd})
@deftypefunx intmax_t mpfr_get_sj (mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx uintmax_t mpfr_get_uj (mpfr_t @var{op}, mp_rnd_t @var{rnd})
Convert @var{op} to a @code{long}, an @code{unsigned long},
an @code{intmax_t} or an @code{uintmax_t} (respectively) after rounding
it with respect to @var{rnd}.
If @var{op} is NaN, the result is undefined.
If @var{op} is too big for the return type, it returns the maximum
or the minimum of the corresponding C type, depending on the direction
of the overflow. The flag erange is set too.
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 mp_exp_t mpfr_get_z_exp (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
@ifnottex
@var{rop} multiplied by two exponent @var{exp}.
@end ifnottex
@tex
$rop \times 2^{\rm exp}$.
@end tex
If the exponent is not representable in the @code{mp_exp_t} type, the
behavior is undefined.
@end deftypefun

@deftypefun void mpfr_get_z (mpz_t @var{rop}, mpfr_t @var{op}, mp_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 Inf, the result is undefined.
@end deftypefun

@deftypefun int mpfr_get_f (mpf_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Convert @var{op} to a @code{mpf_t}, after rounding it with respect to
@var{rnd}. Return zero iff no error occurred,
in particular a non-zero value is returned if
@var{op} is NaN or Inf, which do not exist in @code{mpf}.
@end deftypefun

@deftypefun {char *} mpfr_get_str (char *@var{str}, mp_exp_t *@var{expptr}, int @var{b}, size_t @var{n}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Convert @var{op} to a string of digits in base @var{b}, with rounding in
the direction @var{rnd}, where @var{n} is either zero (see below) or the
number of significant digits; in the latter case, @var{n} must be greater
or equal to 2. 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 @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
possible outputs, the one with an even last digit is chosen
(for an odd base, this may not correspond to an even significand).

If @var{n} is zero, the number of digits of the significand is chosen
large enough so that re-reading the printed value with the same precision,
assuming both output and input use rounding to nearest, will recover
the original value of @var{op}.
More precisely, in most cases, the chosen precision of @var{str} is
the minimal precision depending on @var{n} and @var{b} only that
satisfies the above property, i.e.,
@ifnottex
m = 1 + ceil(@var{n}*log(2)/log(@var{b})),
@end ifnottex
@tex
$m = 1 + \lceil n {\log 2 \over \log b} \rceil$,
@end tex
but in some very rare cases, it might be @math{m+1}.

If @var{str} is a null pointer, space for the significand is allocated using
the current allocation function, and a pointer to the string is returned.
To free the returned string, you must use @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, i.e., at least @code{max(@var{n} + 2, 7)}.
The extra two bytes are for a possible minus sign, and for the terminating null
character.

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 void mpfr_free_str (char *@var{str})
Free a string allocated by @code{mpfr_get_str} using the current unallocation
function (preliminary interface).
The block is assumed to be @code{strlen(@var{str})+1} bytes.
For more information about how it is done:
@ifnothtml
@pxref{Custom Allocation,,, gmp,GNU MP}.
@end ifnothtml
@ifhtml
see Custom Allocation (GNU MP).
@end ifhtml
@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})
@deftypefunx int mpfr_fits_intmax_p (mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_fits_uintmax_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, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Basic arithmetic functions
@cindex Float arithmetic functions
@cindex Arithmetic functions
@section Basic 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_si (mpfr_t @var{rop}, mpfr_t @var{op1}, 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}. For types having no signed zero, it is considered unsigned
(i.e.@: (+0) + 0 = (+0) and (@minus{}0) + 0 = (@minus{}0)).
@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_si_sub (mpfr_t @var{rop}, long int @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_sub_si (mpfr_t @var{rop}, mpfr_t @var{op1}, 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}. For types having no signed zero, it 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)).
@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_si (mpfr_t @var{rop}, mpfr_t @var{op1}, 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}.
When a result is zero, its sign is the product of the signs of the operands
(for types having no signed zero, it is considered positive).
@end deftypefun

@deftypefun int mpfr_sqr (mpfr_t @var{rop}, mpfr_t @var{op}, mp_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}, 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_si_div (mpfr_t @var{rop}, long int @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_div_si (mpfr_t @var{rop}, mpfr_t @var{op1}, 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}.
When a result is zero, its sign is the product of the signs of the operands
(for types having no signed zero, it is considered positive).
@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{op} 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})
@deftypefunx int mpfr_root (mpfr_t @var{rop}, mpfr_t @var{op}, unsigned long int @var{k}, mp_rnd_t @var{rnd})
Set @var{rop} to the cubic root (resp.@: the @var{k}th root)
of @var{op} rounded in the direction @var{rnd}.
An odd (resp.@: even) root of a negative number (including @minus{}Inf)
returns a negative number (resp.@: NaN).
The @var{k}th root of @minus{}0 is defined to be @minus{}0,
whatever the parity of @var{k}.
@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_pow_z (mpfr_t @var{rop}, mpfr_t @var{op1}, mpz_t @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 (note this may change in future versions):
@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{y}, 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}.
Just changes the sign if @var{rop} and @var{op} are the same variable.
@end deftypefun

@deftypefun int mpfr_dim (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_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}}, and +0 otherwise.
Returns NaN when @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}, 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})
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.
@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})
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.
@end deftypefun

@node Comparison Functions, Special Functions, Basic Arithmetic Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Float comparisons functions
@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}, signed 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 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-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}, 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, set the 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})
Return non-zero if @var{op} is respectively NaN, an infinity, an ordinary
number (i.e.@: neither NaN nor an infinity) or 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 erange flag and return zero.
@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, MPFR Interface
@cindex Special functions
@section Special Functions

All those functions, except explicitly 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.

Important note: in some domains, computing special functions (either with
correct or incorrect rounding) is expensive, even for small precision,
for example the trigonometric and Bessel functions for large argument.

@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}.
Return @minus{}Inf if @var{op} is @minus{}0 (i.e.@: the sign of the zero
has no influence on the result).
@end deftypefun

@deftypefun int mpfr_exp (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_exp2 (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_exp10 (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the exponential of @var{op},
 to @m{2^{op}, 2 power of @var{op}}
or to @m{10^{op}, 10 power of @var{op}}, respectively,
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_sec (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_csc (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_cot (mpfr_t @var{rop}, mpfr_t @var{op}, mp_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_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}, which must be different variables.
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}.
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 @math{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)}.
@end deftypefun

@deftypefun int mpfr_atan2 (mpfr_t @var{rop}, mpfr_t @var{y}, mpfr_t @var{x}, mp_rnd_t @var{rnd})
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}))}.
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.

@code{atan2(y, 0)} does not raise any floating-point exception.
Special values are currently handled as described in the ISO C99 standard
for the @code{atan2} function (note this may change in future versions):
@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}, 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_sech (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_csch (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_coth (mpfr_t @var{rop}, mpfr_t @var{op}, mp_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}, 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_eint (mpfr_t @var{y}, mpfr_t @var{x}, mp_rnd_t @var{rnd})
Set @var{y} to the exponential integral of @var{x},
rounded in the direction @var{rnd}.
For positive @var{x},
the exponential integral is the sum of Euler's constant, of the logarithm
of @var{x}, and of the sum for k from 1 to infinity of
@ifnottex
@var{x} to the power k, divided by k and factorial(k).
@end ifnottex
@tex
$x^k/k/k!$.
@end tex
For negative @var{x}, the returned value is NaN.
@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}. When @var{op} is a negative integer, NaN is returned.
@end deftypefun

@deftypefun int mpfr_lngamma (mpfr_t @var{rop}, mpfr_t @var{op}, mp_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 @math{@minus{}2@var{k}@minus{}1 @le{} @var{x} @le{} @minus{}2@var{k}},
@var{k} being a non-negative integer, NaN is returned.
See also @code{mpfr_lgamma}.
@end deftypefun

@deftypefun int mpfr_lgamma (mpfr_t @var{rop}, int *@var{signp}, mpfr_t @var{op}, mp_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 an infinity or a non-positive integer,
+Inf is returned. 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_zeta (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_zeta_ui (mpfr_t @var{rop}, unsigned long @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_erfc (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the value of 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}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_j1 (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_jn (mpfr_t @var{rop}, long @var{n}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the value of the first order Bessel function of order 0, 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 minus Infinity,
@var{rop} is set to +0. When @var{op} is zero, and @var{n} is not zero,
@var{rop} is +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}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_y1 (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_yn (mpfr_t @var{rop}, long @var{n}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the value of the second order Bessel function of order 0, 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 +0. When @var{op} is zero,
@var{rop} is +Inf or @minus{}Inf depending on the parity and sign of @var{n}.
@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_fms (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].
If any operand is negative, the return value is NaN.
@end deftypefun

@deftypefun int mpfr_hypot (mpfr_t @var{rop}, mpfr_t @var{x}, mpfr_t @var{y}, mp_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 currently handled as described in Section F.9.4.3 of
the ISO C99 standard, for the @code{hypot} function (note this may change
in future versions): If @var{x} or @var{y} is an infinity, then plus
infinity is returned in @var{rop}, even if the other number is NaN.
@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})
@deftypefunx int mpfr_const_catalan (mpfr_t @var{rop}, mp_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}.
@end deftypefun

@deftypefun void mpfr_free_cache (void)
Free various caches used by MPFR internally, in particular the
caches used by the functions computing constants (currently
@code{mpfr_const_log2}, @code{mpfr_const_pi},
@code{mpfr_const_euler} and @code{mpfr_const_catalan}).
You should call this function when terminating a thread.
@end deftypefun

@deftypefun int mpfr_sum (mpfr_t @var{rop}, mpfr_ptr const @var{tab}[], unsigned long @var{n}, mp_rnd_t @var{rnd})
Set @var{ret} to the sum of all elements of @var{tab} whose size is @var{n},
rounded in the direction @var{rnd}. Warning, @var{tab} is a table of pointers
to mpfr_t, not a table of mpfr_t (preliminary interface). The returned
@code{int} value is zero when the computed value is the exact value,
and non-zero when this cannot be guaranteed, without giving the
direction of the error as the other functions do.
@end deftypefun

@node Input and Output Functions, Integer Related Functions, Special Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Float input and output functions
@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 @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 the direction @var{rnd}.
The base may vary from 2 to 36.  Print @var{n} significant digits exactly,
or if @var{n} is 0, enough digits so that @var{op} can be read back
exactly (see @code{mpfr_get_str}).

In addition to the significant digits, a decimal point (defined by the
current locale) at the right of the
first digit and a trailing exponent in base 10, in the form @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 the direction @var{rnd}, and put the
read float in @var{rop}.
@c The argument @var{base} must be in the range 2 to 36.

@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} (it may change).
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 @code{@@NaN@@}, @code{@@Inf@@}, @code{+@@Inf@@} and @code{-@@Inf@@},
@c possibly followed by other characters; if the base is smaller or equal
@c to 16, the following strings are accepted too: @code{NaN}, @code{Inf},
@c @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 Integer Related Functions, Miscellaneous Functions, Input and Output Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Integer related functions
@section Integer Related 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 toward 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 nearest mode (where halfway cases are rounded to an even
integer or significand). 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_rint_ceil (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_rint_floor (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_rint_round (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_rint_trunc (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to @var{op} rounded to an integer.
@code{mpfr_rint_ceil} rounds to the next higher or equal integer,
@code{mpfr_rint_floor} to the next lower or equal integer,
@code{mpfr_rint_round} to the nearest integer, rounding halfway cases away
from zero, and @code{mpfr_rint_trunc} to the next integer toward zero.
If the result is not representable, it is rounded in the direction @var{rnd}.
The returned value is the ternary value associated with the considered
round-to-integer function (regarded in the same way as any other
mathematical function).
@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_remainder (mpfr_t @var{r}, mpfr_t @var{x}, mpfr_t @var{y}, mp_rnd_t @var{rnd})
@deftypefunx int mpfr_remquo (mpfr_t @var{r}, long* @var{q}, mpfr_t @var{x}, mpfr_t @var{y}, mp_rnd_t @var{rnd})
Set @var{r} to the remainder of the division of @var{x} by @var{y}, with
quotient rounded to the nearest integer (ties rounded to even), and
@var{r} rounded according to the direction @var{rnd}.
If @var{r} is zero, it has the sign of @var{x}.
The return value is the inexact flag corresponding to @var{r}.
Additionally, @code{mpfr_remquo} stores
the low significant bits from the quotient in @var{*q}
(more precisely the number of bits in a @code{long} 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).
Note that @var{x} may be so large in magnitude relative to @var{y} that an
exact representation of the quotient is not practical.
These 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 Miscellaneous Functions, Rounding Mode Related Functions, Integer 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. 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_min (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
Set @var{rop} to the minimum 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.
@end deftypefun

@deftypefun int mpfr_max (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
Set @var{rop} to the 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 +0.
@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. 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.
@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.
Put +0 in @var{rop} when size if zero. The internal state of the default
pseudorandom number generator is modified by a call to this function (the
same one as GMP if MPFR was built using @samp{--with-gmp-build}).
@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 and the significand is chosen in [1/2,1). The behavior for NaN,
infinity or zero is undefined.
@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 a non-zero value.
The significand is assumed to be in [1/2,1).
@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}, mp_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}, mp_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(2,1,0)))
# 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 ids 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

@node Rounding Mode Related Functions, Exception Related Functions, Miscellaneous Functions, MPFR Interface
@cindex Rounding mode related functions
@section Rounding Mode Related Functions

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

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

@deftypefun int mpfr_prec_round (mpfr_t @var{x}, mp_prec_t @var{prec}, mp_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 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. 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 {const char *} mpfr_print_rnd_mode (mp_rnd_t @var{rnd})
Return 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 Exception Related Functions, Advanced Functions, Rounding Mode Related Functions, MPFR Interface
@comment  node-name,  next,  previous,  up
@cindex Exception related functions
@section Exception Related Functions

@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.
@c It is also her/his responsibility to check that @m {emin <= emax}.
@end deftypefun

@deftypefun mp_exp_t mpfr_get_emin_min (void)
@deftypefunx mp_exp_t mpfr_get_emin_max (void)
@deftypefunx mp_exp_t mpfr_get_emax_min (void)
@deftypefunx mp_exp_t mpfr_get_emax_max (void)
Return the minimum and maximum of the smallest and largest exponents
allowed for @code{mpfr_set_emin} and @code{mpfr_set_emax}. These values
are implementation dependent; it is possible to create a non
portable program by writing @code{mpfr_set_emax(mpfr_get_emax_max())}
and @code{mpfr_set_emin(mpfr_get_emin_min())} since the values
of the smallest and largest exponents become implementation dependent.
@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 int mpfr_subnormalize (mpfr_t @var{x}, int @var{t}, mp_rnd_t @var{rnd})
This function rounds @var{x} emulating subnormal number arithmetic:
if @var{x} is outside the subnormal exponent range, it just propagates the
ternary value @var{t}; otherwise, it rounds @var{x} to precision
@code{EXP(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 @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.

@code{PREC(x)} is not modified by this function.
@var{rnd} and @var{t} must be the used rounding mode for computing @var{x}
and the returned ternary value when computing @var{x}.
The subnormal exponent range is from @code{emin} to @code{emin+PREC(x)-1}.
If the result cannot be represented in the current exponent range
(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.
This is a preliminary interface.
@end deftypefun

This is an example of how to emulate double IEEE-754 arithmetic
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, GMP_RNDN);
  a = 0x1.1235P-1021; mpfr_set_d (xa, a, GMP_RNDN);

  a /= b;
  i = mpfr_div (xa, xa, xb, GMP_RNDN);
  i = mpfr_subnormalize (xa, i, GMP_RNDN);

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

Warning: this emulates a double IEEE-754 arithmetic with correct rounding
in the subnormal range, which may not be the case for your hardware.

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

@deftypefun void mpfr_set_underflow (void)
@deftypefunx void mpfr_set_overflow (void)
@deftypefunx void mpfr_set_nanflag (void)
@deftypefunx void mpfr_set_inexflag (void)
@deftypefunx void mpfr_set_erangeflag (void)
Set the underflow, overflow, invalid, inexact and erange flags.
@end deftypefun

@deftypefun void mpfr_clear_flags (void)
Clear all global flags (underflow, overflow, inexact, invalid, erange).
@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)
@deftypefunx int mpfr_erangeflag_p (void)
Return the corresponding (underflow, overflow, invalid, inexact, erange)
flag, which is non-zero iff the flag is set.
@end deftypefun

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

All the given interfaces are preliminary. They might change incompatibly in
future revisions.

@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{can not} 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 C89). If @var{prec} is not a compiler constant, your compiler
must support @samp{Variable-length automatic arrays} (standard in ISO C99).
@samp{GCC 2.95.3} supports all these features. If you compile your program
with gcc in c89 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_inits (mpfr_t @var{x}, ...)
Initialize all the @code{mpfr_t} variables of the given @code{va_list},
set their precision to be 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}.
It begins from @var{x}. It ends when it encounters a null pointer.
@end deftypefun


@deftypefun void mpfr_inits2 (mp_prec_t @var{prec}, mpfr_t @var{x}, ...)
Initialize all the @code{mpfr_t} variables of the given @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}.
It begins from @var{x}. It ends when it encounters a null pointer.
@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}.
It begins from @var{x}. It ends when it encounters a null pointer.
@end deftypefun

Here is an example of how to use multiple initialization functions:

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


@node Compatibility with MPF, Custom Interface, Advanced 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.
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}.

Warning: the @code{mpf_init} and @code{mpf_init2} functions initialize
to zero, whereas the corresponding @code{mpfr} functions initialize to NaN:
this is useful to detect uninitialized values, but is slightly incompatible
with @code{mpf}.

@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 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 @code{mpf}. Do not use it 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}, 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 correct 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

@deftypefun int mpfr_mul_2exp (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned 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})
See @code{mpfr_mul_2ui} and @code{mpfr_div_2ui}. These functions are only kept
for compatibility with MPF.
@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 them to use MPFR in two ways:
@itemize
@item Either they directly store the MPFR FP number as a @code{mpfr_t}
on the stack.
@item Either they store their own representation of a FP number 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 FP numbers except garbaging the used
memory: all the memory stuff (allocating, destroying, garbaging) is kept 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 1: MPFR functions may still initialize temporary FP numbers
using standard mpfr_init. See Custom Allocation (GNU MP).

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

Note 3: This interface is preliminary.

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

@deftypefun void mpfr_custom_init (void *@var{significand}, mp_prec_t @var{prec})
Initialize a significand of precision @var{prec}.
@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}.
@end deftypefun

@deftypefun void mpfr_custom_init_set (mpfr_t @var{x}, int @var{kind}, mp_exp_t @var{exp}, mp_prec_t @var{prec}, void *@var{significand})
Perform a dummy initialization of a @code{mpfr_t} and set it to:
@itemize
@item if @code{ABS(kind) == MPFR_NAN_KIND}, @var{x} is set to NaN;
@item if @code{ABS(kind) == MPFR_INF_KIND}, @var{x} is set to the infinity
of sign @code{sign(kind)};
@item if @code{ABS(kind) == MPFR_ZERO_KIND}, @var{x} is set to the zero of
sign @code{sign(kind)};
@item if @code{ABS(kind) == MPFR_REGULAR_KIND}, @var{x} is set to a regular
number: @code{x = sign(kind)*significand*2^exp}
@end itemize
In all cases, it uses @var{significand} directly for further computing
involving @var{x}. It will not allocate anything.
A FP number initialized with this function cannot be resized using
@code{mpfr_set_prec}, or cleared using @code{mpfr_clear}!
@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 used 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_mantissa (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 mp_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. The return value for NaN, Infinity or Zero is unspecified but doesn't
produce any trap.
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 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

See the test suite for examples.

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

@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 a special value of the exponent.

@item 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{mp_bits_per_limb}).
Non-singular 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.

@end itemize

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

@c @deftypefun int mpfr_sub_one_ulp (mpfr_t @var{x}, mp_rnd_t @var{rnd})
@c Subtract one ulp to @var{x} if @var{x} is finite and positive, add one
@c ulp if @var{x} is finite and negative; otherwise, @var{x} is not changed.
@c The return value is zero unless an underflow occurs, in which case the
@c @code{mpfr_sub_one_ulp} function behaves like a conventional subtraction.
@c @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 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},
and 0 otherwise (including for NaN and Inf).
This function @strong{does not modify} its arguments.

Note: if one wants to also determine the correct ternary value when rounding
@var{b} to precision @var{prec}, a useful trick is the following:
@verbatim
   if (mpfr_can_round (b, err, rnd1, GMP_RNDZ, prec + (rnd2 == GMP_RNDN)))
      ...
@end verbatim
Indeed, if @var{rnd2} is @code{GMP_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.
@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


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

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

@node Contributors, References, MPFR Interface, Top
@comment  node-name,  next,  previous,  up
@unnumbered Contributors

The main developers consist of Guillaume Hanrot, Vincent Lef@`evre,
Patrick P@'elissier 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 Nathalie Revol for her careful reading of a previous version of
this documentation.
Kevin Ryde did a tremendous job for the portability of MPFR,
and integrating it into GMP 4.x;
alas the GMP developers decided in January 2004 not to include MPFR any more.

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},
and a previous version of @code{mpfr_gamma};
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.
Damien Stehl@'e contributed the @code{mpfr_get_ld_2exp} function.

The development of the MPFR library would not have been possible without the
continuous support of INRIA, and of the LORIA and LIP laboratories.
In particular the main authors were or are members of the
PolKA, Spaces, Cacao project-teams at LORIA (Nancy, France)
and of the Arenaire project-team at LIP (Lyon, France).
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:
