path: root/mpfr.texi

\input texinfo    @c -*-texinfo-*-
@c %**start of header
@setfilename mpfr.info
@settitle MPFR 1.0
@synindex tp fn
@iftex
@afourpaper
@end iftex
@comment %**end of header

@ifinfo
@format
START-INFO-DIR-ENTRY
* mpfr: (mpfr.info).               Multiple Precision Floating-Point Reliable Library.
END-INFO-DIR-ENTRY
@end format
@end ifinfo

@c smallbook

@iftex
@finalout
@end iftex

@c Note: the edition number is listed in *three* places; please update
@c all three.  Also, update the month and year where appropriate.

@c ==> Update edition number for settitle and subtitle, and in the
@c ==> following paragraph; update date, too.


@ifinfo
This file documents MPFR, a library for reliable multiple precision floating-point arithmetic

Copyright (C) 1999, PolKA team, INRIA Lorraine and LORIA, France

Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
are preserved on all copies.

@ignore
Permission is granted to process this file through TeX and print the
results, provided the printed document carries copying permission
notice identical to this one except for the removal of this paragraph
(this paragraph not being relevant to the printed manual).

@end ignore
Permission is granted to copy and distribute modified versions of this
manual under the conditions for verbatim copying, provided that the entire
resulting derived work is distributed under the terms of a permission
notice identical to this one.

Permission is granted to copy and distribute translations of this manual
into another language, under the above conditions for modified versions,
except that this permission notice may be stated in a translation approved
by the Foundation.
@end ifinfo

@setchapternewpage on
@titlepage
@c  use the new format for titles

@title MPFR
@subtitle The Multiple Precision Floating-Point Reliable Library
@subtitle Edition 1.0
@subtitle June 1999

@author the PolKA project, INRIA Lorraine and LORIA

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

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

@page
@vskip 0pt plus 1filll
Copyright @copyright{} 1999 The PolKA project

@sp 2

Published by the Free Software Foundation @*
59 Temple Place - Suite 330 @*
Boston, MA 02111-1307, USA @*

Permission is granted to make and distribute verbatim copies of
this manual provided the copyright notice and this permission notice
are preserved on all copies.

Permission is granted to copy and distribute modified versions of this
manual under the conditions for verbatim copying, provided that the entire
resulting derived work is distributed under the terms of a permission
notice identical to this one.

Permission is granted to copy and distribute translations of this manual
into another language, under the above conditions for modified versions,
except that this permission notice may be stated in a translation approved
by the Foundation.
@end titlepage
@headings double

@ifinfo
@node Top, Copying, (dir), (dir)

@top MPFR

This manual documents how to install and use the Multiple Precision
Floating-Point Reliable Library, version 1.0

@end ifinfo

@menu
* Copying::                   GMP Copying Conditions (LGPL).
* Introduction to MPFR::      Brief introduction to MPFR.
* Installing MPFR::           How to configure and compile the MPFR library.
* MPFR Basics::               What every MPFR user should now.
* Reporting Bugs::            How to usefully report bugs.
* Floating-point Functions::  Functions for arithmetic on floats.

* Contributors::
* References::
* Concept Index::
* Function Index::
@end menu

@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
Library General Public License that accompany the source code.
See the file COPYING.@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 reliable 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 @dfn{reliable} floating-point numbers. It may replace
the GNU MP floating-point numbers in a future release. The main differences
with the @code{mpf} class 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 positive value 
	for each variable (including one bit of precision);
@item @code{mpfr} provides the four rounding modes from the IEEE 754
	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 range of values is much wider
(the exponent has at least 32 bits instead of 11).

This version of MPFR is released under the GNU Library General Public
License.
It is permitted to link MPFR to non-free programs, as long as MPFR
source code is provided when distributing the non-free program.


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

To build MPFR, you first have to install GNU MP 
(version 3.0 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.

Here are the steps needed to install the library on Unix systems:

@enumerate
@item
In most cases, @samp{./configure --with-gmp=/usr/local/gmp} should work,
where @samp{/usr/local/gmp} is the directory where you have installed
GNU MP.
When you install GNU MP, you have to copy the files
@samp{config.h},
@samp{gmp-impl.h}, @samp{gmp-mparam.h}, 
and @samp{longlong.h} from the GNU MP source directory 
to @samp{/usr/local/gmp/include};
these additional files are needed by MPFR.
An alternative way is 
@samp{./configure --with-gmp-include=dir1 --with-gmp-lib=dir2}, where
@samp{dir1} contains the GNU MP header files, and
@samp{dir2} the GNU MP library files.
If you get error messages, your machine might not be supported.

If you want to compile in a separate object directory, cd to that directory,
and prefix the configure command with the path to the MPFR source directory.
Not all @samp{make} programs have the necessary features to support this.  In
particular, SunOS and Slowaris @samp{make} have bugs that make them unable to
build from a separate object directory.  Use GNU @samp{make} instead.
Also, the generated makefiles use some pattern matching rules that are not
supported by all @samp{make} programs. Again, use GNU @samp{make} instead.

@item
@samp{make}

This will compile MPFR, and create a library archive file @file{libmpfr.a} 
in the working directory.

@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{libmpfr.a}, 
to @file{/usr/local/include} and @file{/usr/local/lib}
respectively (or if you passed the @samp{--prefix} option to
@file{configure}, to the directory given as argument to @samp{--prefix}).
@end enumerate

There are some other useful make targets:

@itemize @bullet
@item
@samp{mpfr.dvi}

Create a DVI version of the manual, in @file{mpfr.dvi}.
@c and a set of info files, in @file{mpfr.info}, @file{mpfr.info-1}, @file{mpfr.info-2}, etc.

@item
@samp{mpfr.ps}

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

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

@item
@samp{clean}

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

@item
@samp{distclean}

Delete all files not included in the distribution.

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


@section Known Build Problems

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

Please report other problems to @samp{mpfr@@loria.fr}.
@xref{Reporting Bugs}.


@node MPFR Basics, Reporting Bugs, Installing MPFR, 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:

@code{#include "mpfr.h"}

@section Nomenclature and Types

@cindex Floating-point number
@tindex @code{mpfr_t}
@noindent
@dfn{Floating point number} or @dfn{Float} for short, is an arbitrary precision
mantissa with a limited precision exponent.  The C data type for such objects
is @code{mpfr_t}.

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

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

@cindex Limb
@c @tindex @code{mp_limb_t}
@noindent
A @dfn{limb} means the part of a multi-precision number that fits in a single
word.  (We chose this word because a limb of the human body is analogous to a
digit, only larger, and containing several digits.)  Normally a limb contains
32 or 64 bits.  The C data type for a limb is @code{mp_limb_t}.

@section Function Classes

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

@enumerate
@item
Functions for floating-point arithmetic, with names beginning with
@code{mpfr_}.  The associated type is @code{mpfr_t}.  There are about 50
functions is this class.
@end enumerate


@section MPFR Variable Conventions

As a general rule, all MPFR functions expect output arguments before input
arguments.  This notation is based on an analogy with the assignment operator.

MPFR allows you to use the same variable for both input and output in the same
expression.  For example, the main function for floating-point multiplication,
@code{mpfr_mul}, can be used like this: @code{mpfr_mul (x, x, x, rnd_mode)}.  
This
computes the square of @var{x} with rounding mode @code{rnd_mode}
and puts the result back in @var{x}.

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

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

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

You don't need to be concerned about allocating additional space for MPFR
variables, since any variable has a mantissa of fixed size.
Hence unless you change its precision, or clear and reinitialize it, 
a floating-point variable will have the same allocated space during all its
life.

@section Getting the Latest Version of MP

The latest version of the MPFR library is available from the web page
of the PolKA team at
@samp{http://www.loria.fr/equipes/polka}. 

@node Reporting Bugs, Floating-point Functions, MPFR Basics, 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, please investigate it 
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{what `which cc`}, or,
if you're using gcc, @samp{gcc -v}.  Also, include the output from @samp{uname
-a}.

If your bug report is good, we will do our best to help you to get a corrected
version of the library; if the bug report is poor, we won't do anything about
it (aside of chiding you to send better bug reports).

Send your bug report to: @samp{mpfr@@loria.fr}.

If you think something in this manual is unclear, or downright incorrect, or if
the language needs to be improved, please send a note to the same address.

@node Floating-point Functions, Initializing Floats, Reporting Bugs, Top
@comment  node-name,  next,  previous,  up
@chapter Floating-point Functions
@cindex Floating-point functions
@cindex Float functions

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

The MPFR floating-point functions have an interface that is similar to the 
GNU MP
integer functions.  The function prefix for floating-point operations is
@code{mpfr_}.

There is one significant characteristic of floating-point numbers that has
motivated a difference between this function class and other MPFR function
classes: the inherent inexactness of floating point arithmetic.  The user has
to specify the precision of each variable.  A computation that assigns a
variable will take place with the precision of the assigned variable; the
cost of that computation should not depend from the 
precision of variables used as input on average.

@cindex User-defined precision
The precision of a calculation is defined as follows: Compute the requested
operation exactly (with ``infinite precision''), and round the result to
the destination variable precision with the given rounding mode.
Even if the user has asked for a very
high precision, MP will not calculate with superfluous digits.  For example,
if two low-precision numbers of nearly equal magnitude are added, the
precision of the result will be limited to what is required to represent the
result accurately.

The MPFR floating-point functions are intended to be a smooth extension
of the IEEE P754 arithmetic. The results obtained on one
computer should not differ from the results obtained on a computer with a
different word size.

@menu
* Rounding Modes
* Initializing Floats::
* Assigning Floats::
* Simultaneous Float Init & Assign::
* Converting Floats::
* Float Arithmetic::
* Float Comparison::
* I/O of Floats::
* Miscellaneous Float Functions::
@end menu

@cindex Rouding modes
@section Global Variables and Rounding Modes

@deftypevr {Global Variable} {mp_prec_t} __gmp_default_fp_bit_precision
The default precision in bits (53 initially).
@end deftypevr

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

The following four rounding modes are supported:
@itemize @bullet
@item @code{GMP_RNDN}: round to nearest
@item @code{GMP_RNDZ}: round towards zero
@item @code{GMP_RNDU}: round towards plus infinity
@item @code{GMP_RNDD}: round towards minus infinity
@end itemize
The @samp{round to nearest} mode works as in the IEEE P754 standard: in case
the number to be rounded lies exactly in the middle of two reprensentable
numbers, it is rounded to the one with the least significant bit set to zero.
For example, the number 5, which is represented by (101) in binary, is rounded
to (100)=4 with a precision of two bits, and not to (110)=6.
This rule avoids the @dfn{drift} phenomena mentioned by Knuth in volume 2
of The Art of Computer Programming (section 4.2.2, pages 221-222).

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

@deftypefun void mpfr_round (mpfr_t @var{x}, mp_rnd_t @var{rnd}, mp_prec_t @var{prec})
Round @var{x} according to @var{rnd} with precision @var{prec}, which
may be different from that of @var{x}. 
If @var{prec} is greater or equal to the precision of @var{x}, then new
space is allocated for the mantissa, and it is filled with zeroes.
Otherwise, the mantissa is rounded to precision @var{prec} with the given
direction. In both cases, the precision of @var{x} is changed to @var{prec}.
@end deftypefun

@deftypefun void mpfr_set_machine_rnd_mode (mp_rnd_t @var{rnd})
Set the machine rounding mode to @var{rnd}.
This function is useful for debugging purpose, but
also as a common interface to all different ways of setting the
rounding mode, which unfortunately differ from one operating system to
another one.
@end deftypefun

@deftypefun {char *} mpfr_print_rnd_mode (mp_rnd_t @var{rnd})
Returns the input string (GMP_RNDD, GMP_RNDU, GMP_RNDN, GMP_RNDZ)
corresponding to the rounding mode @var{rnd}.
@end deftypefun

@node Initializing Floats, Assigning Floats, Floating-point Functions, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Initialization and Assignment Functions

@deftypefun void mpfr_set_default_prec (mp_prec_t @var{prec})
Set the default precision to be @strong{exactly} @var{prec} bits.  The
precision of a variable means the number of bits used to store its mantissa.
All
subsequent calls to @code{mpf_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 positive integer, even a precision of 1 is possible.
@end deftypefun

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

@deftypefun void mpfr_init (mpfr_t @var{x})
Initialize @var{x}, and set its value to zero.
Normally, a variable should be initialized once only
or at least be cleared, using @code{mpfr_clear}, between initializations.  The
precision of @var{x} is the default precision, which can be changed
by a call to @code{mpfr_set_default_prec}.
@end deftypefun

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

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

@need 2000
Here is an example on how to initialize floating-point variables:
@example
@{
  mpfr_t x, y;
  mpfr_init (x);			/* use default precision */
  mpfr_init2 (y, 256);		/* precision @emph{exactly} 256 bits */
  @dots{}
  /* Unless the program is about to exit, do ... */
  mpfr_clear (x);
  mpfr_clear (y);
@}
@end example

The following two 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.
The previous value stored in @var{x} is lost. It is equivalent to
a call to @code{mpfr_clear(x)} followed by a call to 
@code{mpfr_init2(x, prec)}, but more efficient as no allocation is done in
case the current allocated space for the mantissa of @var{x} is enough.

In case you want to keep the previous value stored in @var{x},
use @code{mpfr_round} instead.
@end deftypefun

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

@deftypefun void mpfr_set_prec_raw (mpf_t @var{x}, unsigned long int @var{p})
Reset the precision of @var{x} to be @strong{exactly} @var{prec} bits.
The only difference with @code{mpfr_set_prec} is that @var{p} is assumed to
be small enough so that the mantissa fits into the current allocated memory
space for @var{x}. Otherwise an error will occur.
@end deftypefun

@node Assigning Floats, Simultaneous Float Init & Assign, Initializing Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@subsection Assignment Functions
@cindex Float assignment functions

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

@deftypefun void mpfr_set (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx void mpfr_set_ui (mpfr_t @var{rop}, unsigned long int @var{op}, mp_rnd_t @var{rnd})
@deftypefunx void mpfr_set_si (mpfr_t @var{rop}, long int @var{op}, mp_rnd_t @var{rnd}
@deftypefunx void mpfr_set_d (mpfr_t @var{rop}, double @var{op}, mp_rnd_t @var{rnd})
@deftypefunx void mpfr_set_z (mpfr_t @var{rop}, mpz_t @var{op}, mp_rnd_t @var{rnd})
@deftypefunx void mpfr_set_q (mpfr_t @var{rop}, mpq_t @var{op}, mp_rnd_t @var{rnd})
Set the value of @var{rop} from @var{op}, rounded to the precision of @var{rop}
towards the given direction @var{rnd}.
Please note that even a @code{long int} may have to be rounded,
if the destination precision is less than the machine word width.
@end deftypefun

@deftypefun int mpfr_set_str (mpfr_t @var{x}, char *@var{s}, int @var{base}, mp_rnd_t @var{rnd})
Set @var{x} to the value of the string @var{s} in base @var{base}, 
rounded in direction @var{rnd} to the precision of @var{x}.
The exponent is read in decimal.
This function returns 0 if the entire string up to the final '\0' is a valid
number in base @var{base}, and @minus{}1 otherwise.
@end deftypefun

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

@deftypefun void mpfr_set_f (mpfr_t @var{x}, mpf_t @var{y}, mp_rnd_t @var{rnd})
Set @var{x} to the GNU MP floating-point number
@var{y}, rounded with the @var{rnd} mode and the precision
of @var{x}.
@end deftypefun

@deftypefun void mpfr_swap (mpfr_t @var{x}, mpf_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 Simultaneous Float Init & Assign, Converting Floats, Assigning Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@subsection Combined Initialization and Assignment Functions
@cindex Initialization and assignment functions

@deftypefn Macro void mpfr_init_set (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
@deftypefnx Macro void mpfr_init_set_ui (mpfr_t @var{rop}, unsigned long int @var{op}, mp_rnd_t @var{rnd})
@deftypefnx Macro void mpfr_init_set_si (mpfr_t @var{rop}, signed long int @var{op}, mp_rnd_t @var{rnd})
@deftypefnx Macro void mpfr_init_set_d (mpfr_t @var{rop}, double @var{op}, mp_rnd_t @var{rnd})
Initialize @var{rop} and set its value from @var{op}, rounded to 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}, char *@var{s}, int @var{base}, mp_rnd_t @var{rnd})
Initialize @var{x} and sets its value from
the string @var{s} in base @var{base}, 
rounded to direction @var{rnd}.
See @code{mpfr_set_str}.
@end deftypefun

@node Converting Floats, Float Arithmetic, Simultaneous Float Init & Assign, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Conversion Functions
@cindex Conversion functions

@deftypefun double mpfr_get_d (mpfr_t @var{op})
Convert @var{op} to a double, using the current @emph{machine} rounding mode.
@end deftypefun

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

If @var{n_digits} is 0, it prints the maximum possible number of digits 
giving an exact rounding in the given base @var{base} with the direction
@var{rnd}. In other words, if @var{op} was the exact rounding 
of a real number in direction @var{rnd}, then the printed value is also
an exact rounding in base @var{base} of that real number with the same
precision. An error occurs when one is unable to determine the leading
digit, which can happen especially if the precision of @var{op} is small.

If @var{str} is NULL, space for the mantissa is allocated using the default
allocation function, and a pointer to the string is returned.
In that case, the user should her/himself free the corresponding memory
with @code{(*_mp_free_func)(s, strlen(s) + k)} where @code{k} is 1 when
the number is nonnegative, and 2 when there is a minus sign.

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

The exponent is written through the pointer @var{expptr}.

If @var{n_digits} is 0, note that the space
requirements for @var{str} in this case will be impossible for the user to
predetermine.  Therefore, you need to pass NULL for the string argument
whenever @var{n_digits} is 0.

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


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

@deftypefun void mpfr_add (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx void mpfr_add_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
@ifinfo
Set @var{rop} to @var{op1} + @var{op2} rounded in the direction @var{rnd}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $@var{op1} + @var{op2}$ rounded in the direction @var{rnd}.
@end tex
@end iftex
@end deftypefun

@deftypefun void mpfr_sub (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx void mpfr_ui_sub (mpfr_t @var{rop}, unsigned long int @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx void mpfr_sub_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
Set @var{rop} to @var{op1} @minus{} @var{op2} rounded in the direction @var{rnd}.
@end deftypefun

@deftypefun void mpfr_mul (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx void mpfr_mul_ui (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
@ifinfo
Set @var{rop} to @var{op1} times @var{op2} rounded in the direction @var{rnd}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $@var{op1} \times @var{op2}$ rounded in the direction @var{rnd}.
@end tex
@end iftex
@end deftypefun

Division is undefined if the divisor is zero, and passing a zero divisor to
the divide functions will make these functions intentionally divide by zero.
This gives the user the possibility to handle arithmetic exceptions in these
functions in the same manner as other arithmetic exceptions.

@deftypefun void mpfr_div (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
@deftypefunx void 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})
Set @var{rop} to @var{op1}/@var{op2} rounded in the direction @var{rnd}.
@code{mpfr_div_ui} returns 0 if the division is exact, 
a non-zero value otherwise.
@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})
@ifinfo
Set @var{rop} to the square root of @var{op} rounded in the direction @var{rnd}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $\sqrt{@var{op}}$ rounded in the direction @var{rnd}.
@end tex
@end iftex
Set @var{rop} to NaN if @var{op} is negative.
Return 0 if the operation is exact, a non-zero value otherwise.
@end deftypefun

@deftypefun 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_ui_pow_ui (mpfr_t @var{rop}, unsigned long int @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
Set @var{rop} to @var{op1} raised to @var{op2}. The computation is done by
binary exponentiation, each multiplication being rounded to direction 
@var{rnd}. Hence the relative error with respect to the exact result may be
larger than one unit in last place (ulp). In other words, this function does
not provide an exact rounding.
The returned value @var{k} is such that the error in ulps on the result
is less than 2 power @var{k}.
@end deftypefun

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

@deftypefun void mpfr_mul_2exp (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
@ifinfo
Set @var{rop} to @var{op1} times 2 raised to @var{op2}
@end ifinfo
@iftex
@tex
Set @var{rop} to $@var{op1} \times 2^{op2}$
@end tex
@end iftex
rounded to the direction @var{rnd}. Just increases the exponent by @var{op2}
when @var{rop} and @var{op1} are identical.
@end deftypefun

@deftypefun void mpfr_div_2exp (mpfr_t @var{rop}, mpfr_t @var{op1}, unsigned long int @var{op2}, mp_rnd_t @var{rnd})
@ifinfo
Set @var{rop} to @var{op1} divided by 2 raised to @var{op2}.
@end ifinfo
@iftex
@tex
Set @var{rop} to $@var{op1}/2^{op2}$.
@end tex
@end iftex
rounded to the direction @var{rnd}. Just decreases the exponent by @var{op2}
when @var{rop} and @var{op1} are identical.
@end deftypefun

@node Float Comparison, I/O of Floats, Float Arithmetic, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Comparison Functions
@cindex Float comparisons functions
@cindex Comparison functions

@deftypefun int mpfr_cmp (mpfr_t @var{op1}, mpfr_t @var{op2})
@deftypefunx int mpfr_cmp_ui (mpfr_t @var{op1}, unsigned long int @var{op2})
@deftypefunx int mpfr_cmp_si (mpfr_t @var{op1}, signed long int @var{op2})
@ifinfo
Compare @var{op1} and @var{op2}.  Return a positive value if @var{op1} >
@var{op2}, zero if @var{op1} = @var{op2}, and a negative value if @var{op1} <
@var{op2}.
@end ifinfo
@iftex
@tex
Compare @var{op1} and @var{op2}.  Return a positive value if $@var{op1} >
@var{op2}$, zero if $@var{op1} = @var{op2}$, and a negative value if $@var{op1}
< @var{op2}$.
@end tex
@end iftex
Both @var{op1} and @var{op2} are considered to their full own precision,
which may differ. In case @var{op1} and @var{op2} are of same sign but 
different, the absolute value returned is
one plus the absolute difference of their exponents.
@end deftypefun

@deftypefun int mpfr_cmp_ui_2exp (mpfr_t @var{op1}, unsigned long int @var{op2}, int @var{e})
@deftypefunx int mpfr_cmp_si_2exp (mpfr_t @var{op1}, long int @var{op2}, int @var{e})
Compare @var{op1} and @var{op2} multiplied by two to the power @var{e}.
@end deftypefun

@deftypefun int mpfr_eq (mpfr_t @var{op1}, mpfr_t @var{op2}, unsigned long int op3)
Return non-zero if the first @var{op3} bits of @var{op1} and @var{op2} are
equal, zero otherwise.  I.e., test of @var{op1} and @var{op2} are
approximately equal.
@end deftypefun

@deftypefun void mpfr_reldiff (mpfr_t @var{rop}, mpfr_t @var{op1}, mpfr_t @var{op2}, mp_rnd_t @var{rnd})
Compute the relative difference between @var{op1} and @var{op2} 
and store the result in @var{rop}.
This function does not guarantee the exact rounding on the relative difference;
it just computes @var{abs(op1-op2)/op1}, using the rounding mode @var{rnd} for
all operations.
@end deftypefun

@deftypefn Macro int mpfr_sgn (mpfr_t @var{op})
@ifinfo
Return +1 if @var{op} > 0, 0 if @var{op} = 0, and @minus{}1 if @var{op} < 0.
@end ifinfo
@iftex
@tex
Return $+1$ if $@var{op} > 0$, $0$ if $@var{op} = 0$, 
and $-1$ if $@var{op} < 0$.
@end tex
@end iftex
This function is actually implemented as a macro.  It evaluates its
arguments multiple times.
It is provided for compatibility with @var{mpf}.
@end deftypefn

@deftypefn Macro int SIGN (mpfr_t @var{op})
@ifinfo
Return +1 if @var{op} > 0, and @minus{}1 if @var{op} < 0.
@end ifinfo
@iftex
@tex
Return $+1$ if $@var{op} > 0$, and $-1$ if $@var{op} < 0$.
@end tex
@end iftex
This function is actually implemented as a macro.  It evaluates its
arguments multiple times.
As in the IEEE 754 standard, there is a positive zero and a negative one, 
therefore the @var{SIGN} macro does not distinguish zero from non-zero
numbers.
@end deftypefn

@deftypefn Macro int MPFR_NOTZERO (mpfr_t @var{op})
Returns zero when its argument is zero, and a non-zero value otherwise.
@end deftypefn

@section Special Functions
@cindex Special functions

@deftypefun int mpfr_log (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the natural logarithm of @var{op},
rounded to the direction @var{rnd} with the precision of @var{rop}.
Return 0 iff the result is exact (this occurs in fact only when @var{op} is 1
i.e. the result is 0).
@end deftypefun

@deftypefun int mpfr_exp (mpfr_t @var{rop}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Set @var{rop} to the exponential of @var{op},
rounded to the direction @var{rnd} with the precision of @var{rop}.
Return 0 iff the result is exact (this occurs in fact only when @var{op} is 0
i.e. the result is 1).
@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 @var{sop} to the sinus of @var{op}, @var{cop} to the cosinus of @var{op},
rounded to the direction @var{rnd} with the precision of @var{rop}.
Return 0 iff the results are exact (this occurs in fact only when @var{op} is 0
i.e. the sine is 0 and the cosine is 1).
@end deftypefun

@deftypefun void 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 to the direction @var{rnd} with the precision of @var{rop}.
@end deftypefun

@deftypefun void mpfr_const_log2 (mpfr_t @var{rop}, mp_rnd_t @var{rnd})
Set @var{rop} to the logarithm of 2 rounded to the direction @var{rnd}
with the precision of @var{rop}. This function stores the computed
value to avoid another calculation if a lower or equal precision is
requested.
@end deftypefun

@deftypefun void mpfr_const_pi (mpfr_t @var{rop}, mp_rnd_t @var{rnd})
Set @var{rop} to the value of Pi rounded to the direction @var{rnd}
with the precision of @var{rop}. This function uses the Borwein, Borwein,
Plouffe formula which directly gives the expansion of Pi in base 16.
@end deftypefun

@node I/O of Floats, Miscellaneous Float Functions, Float Comparison, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Input and Output Functions
@cindex Float input and output functions
@cindex Input functions
@cindex Output functions
@cindex I/O functions

Functions that perform input from a stdio stream, and functions that output to
a stdio 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, it is a good idea to include @file{stdio.h}
before @file{mpfr.h}, since that will 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_digits}, mpfr_t @var{op}, mp_rnd_t @var{rnd})
Output @var{op} on stdio stream @var{stream}, as a string of digits in
base @var{base}, rounded to direction @var{rnd}.
The base may vary from 2 to 36.  Print at most
@var{n_digits} significant digits, or if @var{n_digits} is 0, the maximum
number of digits accurately representable by @var{op}.

In addition to the significant digits, a decimal point at the right of the
first digit and a
trailing exponent, 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 stdio stream @var{stream},
rounded in direction @var{rnd}, and put the
read float in @var{rop}.  The string is of the form @samp{M@@N} or, if the
base is 10 or less, alternatively @samp{MeN} or @samp{MEN}.
@samp{M} is the mantissa and
@samp{N} is the exponent.  The mantissa is always in the specified base.  The
exponent is 
@c either in the specified base or, if @var{base} is negative, 
in decimal.

The argument @var{base} may be in the ranges 2 to 36.
@c , or @minus{}36 to
@c @minus{}2.  Negative values are used to specify that the exponent is in
@c decimal.

Unlike the corresponding @code{mpz} function, the base will not be determined
from the leading characters of the string if @var{base} is 0.  This is so that
numbers like @samp{0.23} are not interpreted as octal.

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

@deftypefun void mpfr_print_raw (mpfr_t @var{float})
Output @var{float} on stdout
in raw binary format (the exponent is in decimal, yet).
The last bits from the least significant limb which do not belong to
the mantissa are printed between square brackets;
they should always be zero.
@end deftypefun

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


@node Miscellaneous Float Functions, Contributors, I/O of Floats, Floating-point Functions
@comment  node-name,  next,  previous,  up
@section Miscellaneous Functions
@cindex Miscellaneous float functions

@deftypefun void mpfr_ceil (mpfr_t @var{rop}, mpfr_t @var{op})
@deftypefunx void mpfr_floor (mpfr_t @var{rop}, mpfr_t @var{op})
@deftypefunx void mpfr_trunc (mpfr_t @var{rop}, mpfr_t @var{op})
Set @var{rop} to @var{op} rounded to an integer. @code{mpfr_ceil} rounds
to the next higher integer, @code{mpfr_floor} to the next lower,
and @code{mpfr_trunc} to the integer towards zero.
@end deftypefun

@deftypefun void mpfr_urandomb (mpfr_t @var{rop}, gmp_randstate_t @var{state})
Generate a uniformly distributed random float in the interval 0 <= X < 1.
@end deftypefun

@deftypefun void mpfr_random (mpfr_t @var{rop})
Put in @var{rop} a random float in the interval [0,1[.
Random limbs are generated using the
@code{random} system function.
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{max_size} is negative.
@end deftypefun

@deftypefun void mpfr_srandom (unsigned long int @var{seed})
Set the random seed used by @code{mpfr_random} to @var{seed}.
@end deftypefun

@deftypefun void mpfr_random2 (mpfr_t @var{rop}, mp_size_t @var{max_size}, mp_exp_t @var{max_exp})
Generate a random float of at most @var{max_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}.
Negative random numbers are generated when @var{max_size} is negative.
@end deftypefun

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

@section Internal Functions

These functions were mainly designed for the implementation of @code{mpfr},
but may be useful for users too.

@deftypefun Macro int MPFR_IS_NAN (mpfr_t x)
Returns a non-zero value iff its argument is @samp{Not a Number}.
@end deftypefun

@deftypefun Macro int MPFR_SET_NAN (mpfr_t x)
Sets its argument to @samp{Not a Number}.
@end deftypefun

@deftypefun Macro int MPFR_ABSSIZE (mpfr_t x)
Returns the number of limb the mantissa of @var{x} points to.
@end deftypefun

@deftypefun Macro int MPFR_EXP (mpfr_t x)
Returns the exponent of @var{x}.
@end deftypefun

@deftypefun Macro mp_limb_t* MPFR_MANT (mpfr_t x)
Returns a pointer to the mantissa of @var{x}.
@end deftypefun

@deftypefun Macro mp_prec_t MPFR_PREC (mpfr_t x)
Returns the precision of @var{x}.
@end deftypefun

@deftypefun Macro int MPFR_CHANGE_SIGN (mpfr_t x)
Changes the sign of @var{x}.
@end deftypefun

@deftypefun int mpfr_add_one_ulp (mpfr_t @var{x})
Add one unit in last place (ulp) to the mantissa of @var{x} 
if it is positive or zero, and subtracts one ulp otherwise.
Always return 0 (result is exact).
@end deftypefun

@deftypefun int mpfr_sub_one_ulp (mpfr_t @var{x})
Subtract one ulp to @var{x} if it is positive or zero, 
and adds one ulp otherwise.
Always return 0 (result is exact).
@end deftypefun

@deftypefun int mpfr_round_raw (mp_limb_t* @var{y}, mp_limb_t* @var{x}, mp_prec_t @var{xprec}, int @var{neg}, mp_prec_t @var{yprec}, mp_rnd_t @var{rnd})
Puts in mantissa beginning at @var{y} the value of that 
beginning at @var{x} (with precision @var{xprec} bits and negative iff 
@var{neg} is not zero)
rounded with mode @var{rnd} to precision @var{yprec}.
@end deftypefun

@deftypefun int mpfr_round_raw2 (mp_limb_t* @var{x}, mp_prec_t @var{xn}, int @var{neg}, mp_rnd_t @var{rnd}, mp_prec_t @var{prec})
Returns 0 if the multiple precision number stored in the @var{xn} limbs
starting at @var{xp} (least significant limbs first) with sign @var{neg}
(positive iff @var{neg} equals zero)
is rounded similarly with direction @var{rnd} and precision @var{prec}
than towards zero, and 1 otherwise.
In other words, the bit returned is what has to be added to bit @var{prec}
to round to mode @var{rnd}. This function @strong{does not modify} the
mantissa stored in @var{xp}.
@end deftypefun

@deftypefun int mpfr_can_round (mpfr_t @var{b}, mp_prec_t @var{err}, mp_rnd_t @var{rnd1}, mp_rnd_t @var{rnd2}, mp_prec_t @var{prec})
Assuming @var{b} is an approximation of an unknown number
@var{x} in direction @var{rnd1} with error at most two to the power 
E(b)-@var{err} where E(b) is the exponent of 
@var{b}, returns 1 if one is able to round exactly @var{x} to precision 
@var{prec} with direction @var{rnd2},
   and 0 otherwise. This function @strong{does not modify} its arguments.
@end deftypefun

@node Contributors, References, Miscellaneous Float Functions, Top
@comment  node-name,  next,  previous,  up
@unnumbered Contributors

We would like to thank Jean-Michel Muller and Joris van der Hoeven for very
fruitful discussions at the beginning of that project, Torbjorn Granlund
for his help about design issues 
and his suggestions for a easy integration into GNU MP.

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},
the @code{mpfr_sin_cos} functions, and improved versions of
@code{mpfr_const_log2} and @code{mpfr_const_pi}.

Thom Mulders (ETH Z\"urich) contributed the routines for short
multiplication.

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

@itemize @bullet

@item
Torbjorn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic Library", 
  version 3.1, 2000.

@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.

@end itemize

@node Concept Index, , , Top
@comment  node-name,  next,  previous,  up
@unnumbered Concept Index
@printindex cp

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

@contents
@bye