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This is mpfr.info, produced by makeinfo version 5.2 from mpfr.texi.
This manual documents how to install and use the Multiple Precision
Floating-Point Reliable Library, version 3.1.3.
Copyright 1991, 1993-2015 Free Software Foundation, Inc.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2 or
any later version published by the Free Software Foundation; with no
Invariant Sections, with no Front-Cover Texts, and with no Back-Cover
Texts. A copy of the license is included in *note GNU Free
Documentation License::.
INFO-DIR-SECTION Software libraries
START-INFO-DIR-ENTRY
* mpfr: (mpfr). Multiple Precision Floating-Point Reliable Library.
END-INFO-DIR-ENTRY

File: mpfr.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir)
GNU MPFR
********
This manual documents how to install and use the Multiple Precision
Floating-Point Reliable Library, version 3.1.3.
Copyright 1991, 1993-2015 Free Software Foundation, Inc.
Permission is granted to copy, distribute and/or modify this document
under the terms of the GNU Free Documentation License, Version 1.2 or
any later version published by the Free Software Foundation; with no
Invariant Sections, with no Front-Cover Texts, and with no Back-Cover
Texts. A copy of the license is included in *note GNU Free
Documentation License::.
* Menu:
* Copying:: MPFR Copying Conditions (LGPL).
* Introduction to MPFR:: Brief introduction to GNU MPFR.
* Installing MPFR:: How to configure and compile the MPFR library.
* Reporting Bugs:: How to usefully report bugs.
* MPFR Basics:: What every MPFR user should now.
* MPFR Interface:: MPFR functions and macros.
* API Compatibility:: API compatibility with previous MPFR versions.
* Contributors::
* References::
* GNU Free Documentation License::
* Concept Index::
* Function and Type Index::

File: mpfr.info, Node: Copying, Next: Introduction to MPFR, Prev: Top, Up: Top
MPFR Copying Conditions
***********************
The GNU MPFR library (or MPFR for short) is "free"; this means that
everyone is free to use it and free to redistribute it on a free basis.
The library is not in the public domain; it is copyrighted and there are
restrictions on its distribution, but these restrictions are designed to
permit everything that a good cooperating citizen would want to do.
What is not allowed is to try to prevent others from further sharing any
version of this library that they might get from you.
Specifically, we want to make sure that you have the right to give
away copies of the library, that you receive source code or else can get
it if you want it, that you can change this library or use pieces of it
in new free programs, and that you know you can do these things.
To make sure that everyone has such rights, we have to forbid you to
deprive anyone else of these rights. For example, if you distribute
copies of the GNU MPFR library, you must give the recipients all the
rights that you have. You must make sure that they, too, receive or can
get the source code. And you must tell them their rights.
Also, for our own protection, we must make certain that everyone
finds out that there is no warranty for the GNU MPFR library. If it is
modified by someone else and passed on, we want their recipients to know
that what they have is not what we distributed, so that any problems
introduced by others will not reflect on our reputation.
The precise conditions of the license for the GNU MPFR library are
found in the Lesser General Public License that accompanies the source
code. See the file COPYING.LESSER.

File: mpfr.info, Node: Introduction to MPFR, Next: Installing MPFR, Prev: Copying, Up: Top
1 Introduction to MPFR
**********************
MPFR is a portable library written in C for arbitrary precision
arithmetic on floating-point numbers. It is based on the GNU MP
library. It aims to provide a class of floating-point numbers with
precise semantics. The main characteristics of MPFR, which make it
differ from most arbitrary precision floating-point software tools, are:
• the MPFR code is portable, i.e., the result of any operation does
not depend on the machine word size ‘mp_bits_per_limb’ (64 on most
current processors);
• the precision in bits can be set _exactly_ to any valid value for
each variable (including very small precision);
• MPFR provides the four rounding modes from the IEEE 754-1985
standard, plus away-from-zero, as well as for basic operations as
for other mathematical functions.
In particular, with a precision of 53 bits, MPFR is able to exactly
reproduce all computations with double-precision machine floating-point
numbers (e.g., ‘double’ type in C, with a C implementation that
rigorously follows Annex F of the ISO C99 standard and ‘FP_CONTRACT’
pragma set to ‘OFF’) on the four arithmetic operations and the square
root, except the default exponent range is much wider and subnormal
numbers are not implemented (but can be emulated).
This version of MPFR is released under the GNU Lesser General Public
License, version 3 or any later version. It is permitted to link MPFR
to most non-free programs, as long as when distributing them the MPFR
source code and a means to re-link with a modified MPFR library is
provided.
1.1 How to Use This Manual
==========================
Everyone should read *note MPFR Basics::. If you need to install the
library yourself, you need to read *note Installing MPFR::, too. To use
the library you will need to refer to *note MPFR Interface::.
The rest of the manual can be used for later reference, although it
is probably a good idea to glance through it.

File: mpfr.info, Node: Installing MPFR, Next: Reporting Bugs, Prev: Introduction to MPFR, Up: Top
2 Installing MPFR
*****************
The MPFR library is already installed on some GNU/Linux distributions,
but the development files necessary to the compilation such as ‘mpfr.h’
are not always present. To check that MPFR is fully installed on your
computer, you can check the presence of the file ‘mpfr.h’ in
‘/usr/include’, or try to compile a small program having ‘#include
<mpfr.h>’ (since ‘mpfr.h’ may be installed somewhere else). For
instance, you can try to compile:
#include <stdio.h>
#include <mpfr.h>
int main (void)
{
printf ("MPFR library: %-12s\nMPFR header: %s (based on %d.%d.%d)\n",
mpfr_get_version (), MPFR_VERSION_STRING, MPFR_VERSION_MAJOR,
MPFR_VERSION_MINOR, MPFR_VERSION_PATCHLEVEL);
return 0;
}
with
cc -o version version.c -lmpfr -lgmp
and if you get errors whose first line looks like
version.c:2:19: error: mpfr.h: No such file or directory
then MPFR is probably not installed. Running this program will give you
the MPFR version.
If MPFR is not installed on your computer, or if you want to install
a different version, please follow the steps below.
2.1 How to Install
==================
Here are the steps needed to install the library on Unix systems (more
details are provided in the ‘INSTALL’ file):
1. To build MPFR, you first have to install GNU MP (version 4.1 or
higher) on your computer. You need a C compiler, preferably GCC,
but any reasonable compiler should work. And you need the standard
Unix ‘make’ command, plus some other standard Unix utility
commands.
Then, in the MPFR build directory, type the following commands.
2. ‘./configure’
This will prepare the build and setup the options according to your
system. You can give options to specify the install directories
(instead of the default ‘/usr/local’), threading support, and so
on. See the ‘INSTALL’ file and/or the output of ‘./configure
--help’ for more information, in particular if you get error
messages.
3. ‘make’
This will compile MPFR, and create a library archive file
‘libmpfr.a’. On most platforms, a dynamic library will be produced
too.
4. ‘make check’
This will make sure that MPFR was built correctly. If any test
fails, information about this failure can be found in the
‘tests/test-suite.log’ file. If you want the contents of this file
to be automatically output in case of failure, you can set the
‘VERBOSE’ environment variable to 1 before running ‘make check’,
for instance by typing:
‘VERBOSE=1 make check’
In case of failure, you may want to check whether the problem is
already known. If not, please report this failure to the MPFR
mailing-list ‘mpfr@inria.fr’. For details, *Note Reporting Bugs::.
5. ‘make install’
This will copy the files ‘mpfr.h’ and ‘mpf2mpfr.h’ to the directory
‘/usr/local/include’, the library files (‘libmpfr.a’ and possibly
others) to the directory ‘/usr/local/lib’, the file ‘mpfr.info’ to
the directory ‘/usr/local/share/info’, and some other documentation
files to the directory ‘/usr/local/share/doc/mpfr’ (or if you
passed the ‘--prefix’ option to ‘configure’, using the prefix
directory given as argument to ‘--prefix’ instead of ‘/usr/local’).
2.2 Other ‘make’ Targets
========================
There are some other useful make targets:
• ‘mpfr.info’ or ‘info’
Create or update an info version of the manual, in ‘mpfr.info’.
This file is already provided in the MPFR archives.
• ‘mpfr.pdf’ or ‘pdf’
Create a PDF version of the manual, in ‘mpfr.pdf’.
• ‘mpfr.dvi’ or ‘dvi’
Create a DVI version of the manual, in ‘mpfr.dvi’.
• ‘mpfr.ps’ or ‘ps’
Create a Postscript version of the manual, in ‘mpfr.ps’.
• ‘mpfr.html’ or ‘html’
Create a HTML version of the manual, in several pages in the
directory ‘doc/mpfr.html’; if you want only one output HTML file,
then type ‘makeinfo --html --no-split mpfr.texi’ from the ‘doc’
directory instead.
• ‘clean’
Delete all object files and archive files, but not the
configuration files.
• ‘distclean’
Delete all generated files not included in the distribution.
• ‘uninstall’
Delete all files copied by ‘make install’.
2.3 Build Problems
==================
In case of problem, please read the ‘INSTALL’ file carefully before
reporting a bug, in particular section “In case of problem”. Some
problems are due to bad configuration on the user side (not specific to
MPFR). Problems are also mentioned in the FAQ
<http://www.mpfr.org/faq.html>.
Please report problems to the MPFR mailing-list ‘mpfr@inria.fr’.
*Note Reporting Bugs::. Some bug fixes are available on the MPFR 3.1.3
web page <http://www.mpfr.org/mpfr-3.1.3/>.
2.4 Getting the Latest Version of MPFR
======================================
The latest version of MPFR is available from
<ftp://ftp.gnu.org/gnu/mpfr/> or <http://www.mpfr.org/>.

File: mpfr.info, Node: Reporting Bugs, Next: MPFR Basics, Prev: Installing MPFR, Up: Top
3 Reporting Bugs
****************
If you think you have found a bug in the MPFR library, first have a look
on the MPFR 3.1.3 web page <http://www.mpfr.org/mpfr-3.1.3/> and the FAQ
<http://www.mpfr.org/faq.html>: perhaps this bug is already known, in
which case you may find there a workaround for it. You might also look
in the archives of the MPFR mailing-list:
<https://sympa.inria.fr/sympa/arc/mpfr>. Otherwise, please investigate
and report it. We have made this library available to you, and it is
not to ask too much from you, to ask you to report the bugs that you
find.
There are a few things you should think about when you put your bug
report together.
You have to send us a test case that makes it possible for us to
reproduce the bug, i.e., a small self-content program, using no other
library than MPFR. Include instructions on how to run the test case.
You also have to explain what is wrong; if you get a crash, or if the
results you get are incorrect and in that case, in what way.
Please include compiler version information in your bug report. This
can be extracted using ‘cc -V’ on some machines, or, if you’re using
GCC, ‘gcc -v’. Also, include the output from ‘uname -a’ and the MPFR
version (the GMP version may be useful too). If you get a failure while
running ‘make’ or ‘make check’, please include the ‘config.log’ file in
your bug report, and in case of test failure, the ‘tests/test-suite.log’
file too.
If your bug report is good, we will do our best to help you to get a
corrected version of the library; if the bug report is poor, we will not
do anything about it (aside of chiding you to send better bug reports).
Send your bug report to the MPFR mailing-list ‘mpfr@inria.fr’.
If you think something in this manual is unclear, or downright
incorrect, or if the language needs to be improved, please send a note
to the same address.

File: mpfr.info, Node: MPFR Basics, Next: MPFR Interface, Prev: Reporting Bugs, Up: Top
4 MPFR Basics
*************
* Menu:
* Headers and Libraries::
* Nomenclature and Types::
* MPFR Variable Conventions::
* Rounding Modes::
* Floating-Point Values on Special Numbers::
* Exceptions::
* Memory Handling::

File: mpfr.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: MPFR Basics, Up: MPFR Basics
4.1 Headers and Libraries
=========================
All declarations needed to use MPFR are collected in the include file
‘mpfr.h’. It is designed to work with both C and C++ compilers. You
should include that file in any program using the MPFR library:
#include <mpfr.h>
Note however that prototypes for MPFR functions with ‘FILE *’
parameters are provided only if ‘<stdio.h>’ is included too (before
‘mpfr.h’):
#include <stdio.h>
#include <mpfr.h>
Likewise ‘<stdarg.h>’ (or ‘<varargs.h>’) is required for prototypes
with ‘va_list’ parameters, such as ‘mpfr_vprintf’.
And for any functions using ‘intmax_t’, you must include ‘<stdint.h>’
or ‘<inttypes.h>’ before ‘mpfr.h’, to allow ‘mpfr.h’ to define
prototypes for these functions. Moreover, users of C++ compilers under
some platforms may need to define ‘MPFR_USE_INTMAX_T’ (and should do it
for portability) before ‘mpfr.h’ has been included; of course, it is
possible to do that on the command line, e.g., with
‘-DMPFR_USE_INTMAX_T’.
Note: If ‘mpfr.h’ and/or ‘gmp.h’ (used by ‘mpfr.h’) are included
several times (possibly from another header file), ‘<stdio.h>’ and/or
‘<stdarg.h>’ (or ‘<varargs.h>’) should be included *before the first
inclusion* of ‘mpfr.h’ or ‘gmp.h’. Alternatively, you can define
‘MPFR_USE_FILE’ (for MPFR I/O functions) and/or ‘MPFR_USE_VA_LIST’ (for
MPFR functions with ‘va_list’ parameters) anywhere before the last
inclusion of ‘mpfr.h’. As a consequence, if your file is a public
header that includes ‘mpfr.h’, you need to use the latter method.
When calling a MPFR macro, it is not allowed to have previously
defined a macro with the same name as some keywords (currently ‘do’,
‘while’ and ‘sizeof’).
You can avoid the use of MPFR macros encapsulating functions by
defining the ‘MPFR_USE_NO_MACRO’ macro before ‘mpfr.h’ is included. In
general this should not be necessary, but this can be useful when
debugging user code: with some macros, the compiler may emit spurious
warnings with some warning options, and macros can prevent some
prototype checking.
All programs using MPFR must link against both ‘libmpfr’ and ‘libgmp’
libraries. On a typical Unix-like system this can be done with ‘-lmpfr
-lgmp’ (in that order), for example:
gcc myprogram.c -lmpfr -lgmp
MPFR is built using Libtool and an application can use that to link
if desired, *note GNU Libtool: (libtool.info)Top.
If MPFR has been installed to a non-standard location, then it may be
necessary to set up environment variables such as ‘C_INCLUDE_PATH’ and
‘LIBRARY_PATH’, or use ‘-I’ and ‘-L’ compiler options, in order to point
to the right directories. For a shared library, it may also be
necessary to set up some sort of run-time library path (e.g.,
‘LD_LIBRARY_PATH’) on some systems. Please read the ‘INSTALL’ file for
additional information.

File: mpfr.info, Node: Nomenclature and Types, Next: MPFR Variable Conventions, Prev: Headers and Libraries, Up: MPFR Basics
4.2 Nomenclature and Types
==========================
A "floating-point number", or "float" for short, is an arbitrary
precision significand (also called mantissa) with a limited precision
exponent. The C data type for such objects is ‘mpfr_t’ (internally
defined as a one-element array of a structure, and ‘mpfr_ptr’ is the C
data type representing a pointer to this structure). A floating-point
number can have three special values: Not-a-Number (NaN) or plus or
minus Infinity. NaN represents an uninitialized object, the result of
an invalid operation (like 0 divided by 0), or a value that cannot be
determined (like +Infinity minus +Infinity). Moreover, like in the IEEE
754 standard, zero is signed, i.e., there are both +0 and −0; the
behavior is the same as in the IEEE 754 standard and it is generalized
to the other functions supported by MPFR. Unless documented otherwise,
the sign bit of a NaN is unspecified.
The "precision" is the number of bits used to represent the significand
of a floating-point number; the corresponding C data type is
‘mpfr_prec_t’. The precision can be any integer between ‘MPFR_PREC_MIN’
and ‘MPFR_PREC_MAX’. In the current implementation, ‘MPFR_PREC_MIN’ is
equal to 2.
Warning! MPFR needs to increase the precision internally, in order
to provide accurate results (and in particular, correct rounding). Do
not attempt to set the precision to any value near ‘MPFR_PREC_MAX’,
otherwise MPFR will abort due to an assertion failure. Moreover, you
may reach some memory limit on your platform, in which case the program
may abort, crash or have undefined behavior (depending on your C
implementation).
The "rounding mode" specifies the way to round the result of a
floating-point operation, in case the exact result can not be
represented exactly in the destination significand; the corresponding C
data type is ‘mpfr_rnd_t’.

File: mpfr.info, Node: MPFR Variable Conventions, Next: Rounding Modes, Prev: Nomenclature and Types, Up: MPFR Basics
4.3 MPFR Variable Conventions
=============================
Before you can assign to an MPFR variable, you need to initialize it by
calling one of the special initialization functions. When you’re done
with a variable, you need to clear it out, using one of the functions
for that purpose. A variable should only be initialized once, or at
least cleared out between each initialization. After a variable has
been initialized, it may be assigned to any number of times. For
efficiency reasons, avoid to initialize and clear out a variable in
loops. Instead, initialize it before entering the loop, and clear it
out after the loop has exited. You do not need to be concerned about
allocating additional space for MPFR variables, since any variable has a
significand of fixed size. Hence unless you change its precision, or
clear and reinitialize it, a floating-point variable will have the same
allocated space during all its life.
As a general rule, all MPFR functions expect output arguments before
input arguments. This notation is based on an analogy with the
assignment operator. MPFR allows you to use the same variable for both
input and output in the same expression. For example, the main function
for floating-point multiplication, ‘mpfr_mul’, can be used like this:
‘mpfr_mul (x, x, x, rnd)’. This computes the square of X with rounding
mode ‘rnd’ and puts the result back in X.

File: mpfr.info, Node: Rounding Modes, Next: Floating-Point Values on Special Numbers, Prev: MPFR Variable Conventions, Up: MPFR Basics
4.4 Rounding Modes
==================
The following five rounding modes are supported:
• ‘MPFR_RNDN’: round to nearest (roundTiesToEven in IEEE 754-2008),
• ‘MPFR_RNDZ’: round toward zero (roundTowardZero in IEEE 754-2008),
• ‘MPFR_RNDU’: round toward plus infinity (roundTowardPositive in
IEEE 754-2008),
• ‘MPFR_RNDD’: round toward minus infinity (roundTowardNegative in
IEEE 754-2008),
• ‘MPFR_RNDA’: round away from zero.
The ‘round to nearest’ mode works as in the IEEE 754 standard: in
case the number to be rounded lies exactly in the middle of two
representable numbers, it is rounded to the one with the least
significant bit set to zero. For example, the number 2.5, which is
represented by (10.1) in binary, is rounded to (10.0)=2 with a precision
of two bits, and not to (11.0)=3. This rule avoids the "drift"
phenomenon mentioned by Knuth in volume 2 of The Art of Computer
Programming (Section 4.2.2).
Most MPFR functions take as first argument the destination variable,
as second and following arguments the input variables, as last argument
a rounding mode, and have a return value of type ‘int’, called the
"ternary value". The value stored in the destination variable is
correctly rounded, i.e., MPFR behaves as if it computed the result with
an infinite precision, then rounded it to the precision of this
variable. The input variables are regarded as exact (in particular,
their precision does not affect the result).
As a consequence, in case of a non-zero real rounded result, the
error on the result is less or equal to 1/2 ulp (unit in the last place)
of that result in the rounding to nearest mode, and less than 1 ulp of
that result in the directed rounding modes (a ulp is the weight of the
least significant represented bit of the result after rounding).
Unless documented otherwise, functions returning an ‘int’ return a
ternary value. If the ternary value is zero, it means that the value
stored in the destination variable is the exact result of the
corresponding mathematical function. If the ternary value is positive
(resp. negative), it means the value stored in the destination variable
is greater (resp. lower) than the exact result. For example with the
‘MPFR_RNDU’ rounding mode, the ternary value is usually positive, except
when the result is exact, in which case it is zero. In the case of an
infinite result, it is considered as inexact when it was obtained by
overflow, and exact otherwise. A NaN result (Not-a-Number) always
corresponds to an exact return value. The opposite of a returned
ternary value is guaranteed to be representable in an ‘int’.
Unless documented otherwise, functions returning as result the value
‘1’ (or any other value specified in this manual) for special cases
(like ‘acos(0)’) yield an overflow or an underflow if that value is not
representable in the current exponent range.

File: mpfr.info, Node: Floating-Point Values on Special Numbers, Next: Exceptions, Prev: Rounding Modes, Up: MPFR Basics
4.5 Floating-Point Values on Special Numbers
============================================
This section specifies the floating-point values (of type ‘mpfr_t’)
returned by MPFR functions (where by “returned” we mean here the
modified value of the destination object, which should not be mixed with
the ternary return value of type ‘int’ of those functions). For
functions returning several values (like ‘mpfr_sin_cos’), the rules
apply to each result separately.
Functions can have one or several input arguments. An input point is
a mapping from these input arguments to the set of the MPFR numbers.
When none of its components are NaN, an input point can also be seen as
a tuple in the extended real numbers (the set of the real numbers with
both infinities).
When the input point is in the domain of the mathematical function,
the result is rounded as described in Section “Rounding Modes” (but see
below for the specification of the sign of an exact zero). Otherwise
the general rules from this section apply unless stated otherwise in the
description of the MPFR function (*note MPFR Interface::).
When the input point is not in the domain of the mathematical
function but is in its closure in the extended real numbers and the
function can be extended by continuity, the result is the obtained
limit. Examples: ‘mpfr_hypot’ on (+Inf,0) gives +Inf. But ‘mpfr_pow’
cannot be defined on (1,+Inf) using this rule, as one can find sequences
(X_N,Y_N) such that X_N goes to 1, Y_N goes to +Inf and X_N to the Y_N
goes to any positive value when N goes to the infinity.
When the input point is in the closure of the domain of the
mathematical function and an input argument is +0 (resp. −0), one
considers the limit when the corresponding argument approaches 0 from
above (resp. below). If the limit is not defined (e.g., ‘mpfr_log’ on
−0), the behavior is specified in the description of the MPFR function.
When the result is equal to 0, its sign is determined by considering
the limit as if the input point were not in the domain: If one
approaches 0 from above (resp. below), the result is +0 (resp. −0); for
example, ‘mpfr_sin’ on +0 gives +0. In the other cases, the sign is
specified in the description of the MPFR function; for example
‘mpfr_max’ on −0 and +0 gives +0.
When the input point is not in the closure of the domain of the
function, the result is NaN. Example: ‘mpfr_sqrt’ on −17 gives NaN.
When an input argument is NaN, the result is NaN, possibly except
when a partial function is constant on the finite floating-point
numbers; such a case is always explicitly specified in *note MPFR
Interface::. Example: ‘mpfr_hypot’ on (NaN,0) gives NaN, but
‘mpfr_hypot’ on (NaN,+Inf) gives +Inf (as specified in *note Special
Functions::), since for any finite input X, ‘mpfr_hypot’ on (X,+Inf)
gives +Inf.

File: mpfr.info, Node: Exceptions, Next: Memory Handling, Prev: Floating-Point Values on Special Numbers, Up: MPFR Basics
4.6 Exceptions
==============
MPFR supports 6 exception types:
• Underflow: An underflow occurs when the exact result of a function
is a non-zero real number and the result obtained after the
rounding, assuming an unbounded exponent range (for the rounding),
has an exponent smaller than the minimum value of the current
exponent range. (In the round-to-nearest mode, the halfway case is
rounded toward zero.)
Note: This is not the single possible definition of the underflow.
MPFR chooses to consider the underflow _after_ rounding. The
underflow before rounding can also be defined. For instance,
consider a function that has the exact result 7 multiplied by two
to the power E−4, where E is the smallest exponent (for a
significand between 1/2 and 1), with a 2-bit target precision and
rounding toward plus infinity. The exact result has the exponent
E−1. With the underflow before rounding, such a function call
would yield an underflow, as E−1 is outside the current exponent
range. However, MPFR first considers the rounded result assuming
an unbounded exponent range. The exact result cannot be
represented exactly in precision 2, and here, it is rounded to 0.5
times 2 to E, which is representable in the current exponent range.
As a consequence, this will not yield an underflow in MPFR.
• Overflow: An overflow occurs when the exact result of a function is
a non-zero real number and the result obtained after the rounding,
assuming an unbounded exponent range (for the rounding), has an
exponent larger than the maximum value of the current exponent
range. In the round-to-nearest mode, the result is infinite.
Note: unlike the underflow case, there is only one possible
definition of overflow here.
• Divide-by-zero: An exact infinite result is obtained from finite
inputs.
• NaN: A NaN exception occurs when the result of a function is NaN.
• Inexact: An inexact exception occurs when the result of a function
cannot be represented exactly and must be rounded.
• Range error: A range exception occurs when a function that does not
return a MPFR number (such as comparisons and conversions to an
integer) has an invalid result (e.g., an argument is NaN in
‘mpfr_cmp’, or a conversion to an integer cannot be represented in
the target type).
MPFR has a global flag for each exception, which can be cleared, set
or tested by functions described in *note Exception Related Functions::.
Differences with the ISO C99 standard:
• In C, only quiet NaNs are specified, and a NaN propagation does not
raise an invalid exception. Unless explicitly stated otherwise,
MPFR sets the NaN flag whenever a NaN is generated, even when a NaN
is propagated (e.g., in NaN + NaN), as if all NaNs were signaling.
• An invalid exception in C corresponds to either a NaN exception or
a range error in MPFR.

File: mpfr.info, Node: Memory Handling, Prev: Exceptions, Up: MPFR Basics
4.7 Memory Handling
===================
MPFR functions may create caches, e.g., when computing constants such as
Pi, either because the user has called a function like ‘mpfr_const_pi’
directly or because such a function was called internally by the MPFR
library itself to compute some other function.
At any time, the user can free the various caches with
‘mpfr_free_cache’. It is strongly advised to do that before terminating
a thread, or before exiting when using tools like ‘valgrind’ (to avoid
memory leaks being reported).
MPFR internal data such as flags, the exponent range, the default
precision and rounding mode, and caches (i.e., data that are not
accessed via parameters) are either global (if MPFR has not been
compiled as thread safe) or per-thread (thread local storage, TLS). The
initial values of TLS data after a thread is created entirely depend on
the compiler and thread implementation (MPFR simply does a conventional
variable initialization, the variables being declared with an
implementation-defined TLS specifier).

File: mpfr.info, Node: MPFR Interface, Next: API Compatibility, Prev: MPFR Basics, Up: Top
5 MPFR Interface
****************
The floating-point functions expect arguments of type ‘mpfr_t’.
The MPFR floating-point functions have an interface that is similar
to the GNU MP functions. The function prefix for floating-point
operations is ‘mpfr_’.
The user has to specify the precision of each variable. A
computation that assigns a variable will take place with the precision
of the assigned variable; the cost of that computation should not depend
on the precision of variables used as input (on average).
The semantics of a calculation in MPFR is specified as follows:
Compute the requested operation exactly (with “infinite accuracy”), and
round the result to the precision of the destination variable, with the
given rounding mode. The MPFR floating-point functions are intended to
be a smooth extension of the IEEE 754 arithmetic. The results obtained
on a given computer are identical to those obtained on a computer with a
different word size, or with a different compiler or operating system.
MPFR _does not keep track_ of the accuracy of a computation. This is
left to the user or to a higher layer (for example the MPFI library for
interval arithmetic). As a consequence, if two variables are used to
store only a few significant bits, and their product is stored in a
variable with large precision, then MPFR will still compute the result
with full precision.
The value of the standard C macro ‘errno’ may be set to non-zero by
any MPFR function or macro, whether or not there is an error.
* Menu:
* Initialization Functions::
* Assignment Functions::
* Combined Initialization and Assignment Functions::
* Conversion Functions::
* Basic Arithmetic Functions::
* Comparison Functions::
* Special Functions::
* Input and Output Functions::
* Formatted Output Functions::
* Integer Related Functions::
* Rounding Related Functions::
* Miscellaneous Functions::
* Exception Related Functions::
* Compatibility with MPF::
* Custom Interface::
* Internals::

File: mpfr.info, Node: Initialization Functions, Next: Assignment Functions, Prev: MPFR Interface, Up: MPFR Interface
5.1 Initialization Functions
============================
An ‘mpfr_t’ object must be initialized before storing the first value in
it. The functions ‘mpfr_init’ and ‘mpfr_init2’ are used for that
purpose.
-- Function: void mpfr_init2 (mpfr_t X, mpfr_prec_t PREC)
Initialize X, set its precision to be *exactly* PREC bits and its
value to NaN. (Warning: the corresponding MPF function initializes
to zero instead.)
Normally, a variable should be initialized once only or at least be
cleared, using ‘mpfr_clear’, between initializations. To change
the precision of a variable which has already been initialized, use
‘mpfr_set_prec’. The precision PREC must be an integer between
‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’ (otherwise the behavior is
undefined).
-- Function: void mpfr_inits2 (mpfr_prec_t PREC, mpfr_t X, ...)
Initialize all the ‘mpfr_t’ variables of the given variable
argument ‘va_list’, set their precision to be *exactly* PREC bits
and their value to NaN. See ‘mpfr_init2’ for more details. The
‘va_list’ is assumed to be composed only of type ‘mpfr_t’ (or
equivalently ‘mpfr_ptr’). It begins from X, and ends when it
encounters a null pointer (whose type must also be ‘mpfr_ptr’).
-- Function: void mpfr_clear (mpfr_t X)
Free the space occupied by the significand of X. Make sure to call
this function for all ‘mpfr_t’ variables when you are done with
them.
-- Function: void mpfr_clears (mpfr_t X, ...)
Free the space occupied by all the ‘mpfr_t’ variables of the given
‘va_list’. See ‘mpfr_clear’ for more details. The ‘va_list’ is
assumed to be composed only of type ‘mpfr_t’ (or equivalently
‘mpfr_ptr’). It begins from X, and ends when it encounters a null
pointer (whose type must also be ‘mpfr_ptr’).
Here is an example of how to use multiple initialization functions
(since ‘NULL’ is not necessarily defined in this context, we use
‘(mpfr_ptr) 0’ instead, but ‘(mpfr_ptr) NULL’ is also correct).
{
mpfr_t x, y, z, t;
mpfr_inits2 (256, x, y, z, t, (mpfr_ptr) 0);
…
mpfr_clears (x, y, z, t, (mpfr_ptr) 0);
}
-- Function: void mpfr_init (mpfr_t X)
Initialize X, set its precision to the default precision, and set
its value to NaN. The default precision can be changed by a call to
‘mpfr_set_default_prec’.
Warning! In a given program, some other libraries might change the
default precision and not restore it. Thus it is safer to use
‘mpfr_init2’.
-- Function: void mpfr_inits (mpfr_t X, ...)
Initialize all the ‘mpfr_t’ variables of the given ‘va_list’, set
their precision to the default precision and their value to NaN.
See ‘mpfr_init’ for more details. The ‘va_list’ is assumed to be
composed only of type ‘mpfr_t’ (or equivalently ‘mpfr_ptr’). It
begins from X, and ends when it encounters a null pointer (whose
type must also be ‘mpfr_ptr’).
Warning! In a given program, some other libraries might change the
default precision and not restore it. Thus it is safer to use
‘mpfr_inits2’.
-- Macro: MPFR_DECL_INIT (NAME, PREC)
This macro declares NAME as an automatic variable of type ‘mpfr_t’,
initializes it and sets its precision to be *exactly* PREC bits and
its value to NaN. NAME must be a valid identifier. You must use
this macro in the declaration section. This macro is much faster
than using ‘mpfr_init2’ but has some drawbacks:
• You *must not* call ‘mpfr_clear’ with variables created with
this macro (the storage is allocated at the point of
declaration and deallocated when the brace-level is exited).
• You *cannot* change their precision.
• You *should not* create variables with huge precision with
this macro.
• Your compiler must support ‘Non-Constant Initializers’
(standard in C++ and ISO C99) and ‘Token Pasting’ (standard in
ISO C89). If PREC is not a constant expression, your compiler
must support ‘variable-length automatic arrays’ (standard in
ISO C99). GCC 2.95.3 and above supports all these features.
If you compile your program with GCC in C89 mode and with
‘-pedantic’, you may want to define the ‘MPFR_USE_EXTENSION’
macro to avoid warnings due to the ‘MPFR_DECL_INIT’
implementation.
-- Function: void mpfr_set_default_prec (mpfr_prec_t PREC)
Set the default precision to be *exactly* PREC bits, where PREC can
be any integer between ‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’. The
precision of a variable means the number of bits used to store its
significand. All subsequent calls to ‘mpfr_init’ or ‘mpfr_inits’
will use this precision, but previously initialized variables are
unaffected. The default precision is set to 53 bits initially.
Note: when MPFR is built with the ‘--enable-thread-safe’ configure
option, the default precision is local to each thread. *Note
Memory Handling::, for more information.
-- Function: mpfr_prec_t mpfr_get_default_prec (void)
Return the current default MPFR precision in bits. See the
documentation of ‘mpfr_set_default_prec’.
Here is an example on how to initialize floating-point variables:
{
mpfr_t x, y;
mpfr_init (x); /* use default precision */
mpfr_init2 (y, 256); /* precision _exactly_ 256 bits */
…
/* When the program is about to exit, do ... */
mpfr_clear (x);
mpfr_clear (y);
mpfr_free_cache (); /* free the cache for constants like pi */
}
The following functions are useful for changing the precision during
a calculation. A typical use would be for adjusting the precision
gradually in iterative algorithms like Newton-Raphson, making the
computation precision closely match the actual accurate part of the
numbers.
-- Function: void mpfr_set_prec (mpfr_t X, mpfr_prec_t PREC)
Reset the precision of X to be *exactly* PREC bits, and set its
value to NaN. The previous value stored in X is lost. It is
equivalent to a call to ‘mpfr_clear(x)’ followed by a call to
‘mpfr_init2(x, prec)’, but more efficient as no allocation is done
in case the current allocated space for the significand of X is
enough. The precision PREC can be any integer between
‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’. In case you want to keep the
previous value stored in X, use ‘mpfr_prec_round’ instead.
Warning! You must not use this function if X was initialized with
‘MPFR_DECL_INIT’ or with ‘mpfr_custom_init_set’ (*note Custom
Interface::).
-- Function: mpfr_prec_t mpfr_get_prec (mpfr_t X)
Return the precision of X, i.e., the number of bits used to store
its significand.

File: mpfr.info, Node: Assignment Functions, Next: Combined Initialization and Assignment Functions, Prev: Initialization Functions, Up: MPFR Interface
5.2 Assignment Functions
========================
These functions assign new values to already initialized floats (*note
Initialization Functions::).
-- Function: int mpfr_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_set_ui (mpfr_t ROP, unsigned long int OP,
mpfr_rnd_t RND)
-- Function: int mpfr_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t RND)
-- Function: int mpfr_set_uj (mpfr_t ROP, uintmax_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_set_sj (mpfr_t ROP, intmax_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_set_flt (mpfr_t ROP, float OP, mpfr_rnd_t RND)
-- Function: int mpfr_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND)
-- Function: int mpfr_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t
RND)
-- Function: int mpfr_set_decimal64 (mpfr_t ROP, _Decimal64 OP,
mpfr_rnd_t RND)
-- Function: int mpfr_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND)
Set the value of ROP from OP, rounded toward the given direction
RND. Note that the input 0 is converted to +0 by ‘mpfr_set_ui’,
‘mpfr_set_si’, ‘mpfr_set_uj’, ‘mpfr_set_sj’, ‘mpfr_set_z’,
‘mpfr_set_q’ and ‘mpfr_set_f’, regardless of the rounding mode. If
the system does not support the IEEE 754 standard, ‘mpfr_set_flt’,
‘mpfr_set_d’, ‘mpfr_set_ld’ and ‘mpfr_set_decimal64’ might not
preserve the signed zeros. The ‘mpfr_set_decimal64’ function is
built only with the configure option ‘--enable-decimal-float’,
which also requires ‘--with-gmp-build’, and when the compiler or
system provides the ‘_Decimal64’ data type (recent versions of GCC
support this data type); to use ‘mpfr_set_decimal64’, one should
define the macro ‘MPFR_WANT_DECIMAL_FLOATS’ before including
‘mpfr.h’. ‘mpfr_set_q’ might fail if the numerator (or the
denominator) can not be represented as a ‘mpfr_t’.
Note: If you want to store a floating-point constant to a ‘mpfr_t’,
you should use ‘mpfr_set_str’ (or one of the MPFR constant
functions, such as ‘mpfr_const_pi’ for Pi) instead of
‘mpfr_set_flt’, ‘mpfr_set_d’, ‘mpfr_set_ld’ or
‘mpfr_set_decimal64’. Otherwise the floating-point constant will
be first converted into a reduced-precision (e.g., 53-bit) binary
(or decimal, for ‘mpfr_set_decimal64’) number before MPFR can work
with it.
-- Function: int mpfr_set_ui_2exp (mpfr_t ROP, unsigned long int OP,
mpfr_exp_t E, mpfr_rnd_t RND)
-- Function: int mpfr_set_si_2exp (mpfr_t ROP, long int OP, mpfr_exp_t
E, mpfr_rnd_t RND)
-- Function: int mpfr_set_uj_2exp (mpfr_t ROP, uintmax_t OP, intmax_t
E, mpfr_rnd_t RND)
-- Function: int mpfr_set_sj_2exp (mpfr_t ROP, intmax_t OP, intmax_t E,
mpfr_rnd_t RND)
-- Function: int mpfr_set_z_2exp (mpfr_t ROP, mpz_t OP, mpfr_exp_t E,
mpfr_rnd_t RND)
Set the value of ROP from OP multiplied by two to the power E,
rounded toward the given direction RND. Note that the input 0 is
converted to +0.
-- Function: int mpfr_set_str (mpfr_t ROP, const char *S, int BASE,
mpfr_rnd_t RND)
Set ROP to the value of the string S in base BASE, rounded in the
direction RND. See the documentation of ‘mpfr_strtofr’ for a
detailed description of the valid string formats. Contrary to
‘mpfr_strtofr’, ‘mpfr_set_str’ requires the _whole_ string to
represent a valid floating-point number.
The meaning of the return value differs from other MPFR functions:
it is 0 if the entire string up to the final null character is a
valid number in base BASE; otherwise it is −1, and ROP may have
changed (users interested in the *note ternary value:: should use
‘mpfr_strtofr’ instead).
Note: it is preferable to use ‘mpfr_strtofr’ if one wants to
distinguish between an infinite ROP value coming from an infinite S
or from an overflow.
-- Function: int mpfr_strtofr (mpfr_t ROP, const char *NPTR, char
**ENDPTR, int BASE, mpfr_rnd_t RND)
Read a floating-point number from a string NPTR in base BASE,
rounded in the direction RND; BASE must be either 0 (to detect the
base, as described below) or a number from 2 to 62 (otherwise the
behavior is undefined). If NPTR starts with valid data, the result
is stored in ROP and ‘*ENDPTR’ points to the character just after
the valid data (if ENDPTR is not a null pointer); otherwise ROP is
set to zero (for consistency with ‘strtod’) and the value of NPTR
is stored in the location referenced by ENDPTR (if ENDPTR is not a
null pointer). The usual ternary value is returned.
Parsing follows the standard C ‘strtod’ function with some
extensions. After optional leading whitespace, one has a subject
sequence consisting of an optional sign (‘+’ or ‘-’), and either
numeric data or special data. The subject sequence is defined as
the longest initial subsequence of the input string, starting with
the first non-whitespace character, that is of the expected form.
The form of numeric data is a non-empty sequence of significand
digits with an optional decimal point, and an optional exponent
consisting of an exponent prefix followed by an optional sign and a
non-empty sequence of decimal digits. A significand digit is
either a decimal digit or a Latin letter (62 possible characters),
with ‘A’ = 10, ‘B’ = 11, …, ‘Z’ = 35; case is ignored in bases less
or equal to 36, in bases larger than 36, ‘a’ = 36, ‘b’ = 37, …, ‘z’
= 61. The value of a significand digit must be strictly less than
the base. The decimal point can be either the one defined by the
current locale or the period (the first one is accepted for
consistency with the C standard and the practice, the second one is
accepted to allow the programmer to provide MPFR numbers from
strings in a way that does not depend on the current locale). The
exponent prefix can be ‘e’ or ‘E’ for bases up to 10, or ‘@’ in any
base; it indicates a multiplication by a power of the base. In
bases 2 and 16, the exponent prefix can also be ‘p’ or ‘P’, in
which case the exponent, called _binary exponent_, indicates a
multiplication by a power of 2 instead of the base (there is a
difference only for base 16); in base 16 for example ‘1p2’
represents 4 whereas ‘1@2’ represents 256. The value of an
exponent is always written in base 10.
If the argument BASE is 0, then the base is automatically detected
as follows. If the significand starts with ‘0b’ or ‘0B’, base 2 is
assumed. If the significand starts with ‘0x’ or ‘0X’, base 16 is
assumed. Otherwise base 10 is assumed.
Note: The exponent (if present) must contain at least a digit.
Otherwise the possible exponent prefix and sign are not part of the
number (which ends with the significand). Similarly, if ‘0b’,
‘0B’, ‘0x’ or ‘0X’ is not followed by a binary/hexadecimal digit,
then the subject sequence stops at the character ‘0’, thus 0 is
read.
Special data (for infinities and NaN) can be ‘@inf@’ or
‘@nan@(n-char-sequence-opt)’, and if BASE <= 16, it can also be
‘infinity’, ‘inf’, ‘nan’ or ‘nan(n-char-sequence-opt)’, all case
insensitive. A ‘n-char-sequence-opt’ is a possibly empty string
containing only digits, Latin letters and the underscore (0, 1, 2,
…, 9, a, b, …, z, A, B, …, Z, _). Note: one has an optional sign
for all data, even NaN. For example, ‘-@nAn@(This_Is_Not_17)’ is a
valid representation for NaN in base 17.
-- Function: void mpfr_set_nan (mpfr_t X)
-- Function: void mpfr_set_inf (mpfr_t X, int SIGN)
-- Function: void mpfr_set_zero (mpfr_t X, int SIGN)
Set the variable X to NaN (Not-a-Number), infinity or zero
respectively. In ‘mpfr_set_inf’ or ‘mpfr_set_zero’, X is set to
plus infinity or plus zero iff SIGN is nonnegative; in
‘mpfr_set_nan’, the sign bit of the result is unspecified.
-- Function: void mpfr_swap (mpfr_t X, mpfr_t Y)
Swap the structures pointed to by X and Y. In particular, the
values are exchanged without rounding (this may be different from
three ‘mpfr_set’ calls using a third auxiliary variable).
Warning! Since the precisions are exchanged, this will affect
future assignments. Moreover, since the significand pointers are
also exchanged, you must not use this function if the allocation
method used for X and/or Y does not permit it. This is the case
when X and/or Y were declared and initialized with
‘MPFR_DECL_INIT’, and possibly with ‘mpfr_custom_init_set’ (*note
Custom Interface::).

File: mpfr.info, Node: Combined Initialization and Assignment Functions, Next: Conversion Functions, Prev: Assignment Functions, Up: MPFR Interface
5.3 Combined Initialization and Assignment Functions
====================================================
-- Macro: int mpfr_init_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Macro: int mpfr_init_set_ui (mpfr_t ROP, unsigned long int OP,
mpfr_rnd_t RND)
-- Macro: int mpfr_init_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t
RND)
-- Macro: int mpfr_init_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND)
-- Macro: int mpfr_init_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t
RND)
-- Macro: int mpfr_init_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND)
-- Macro: int mpfr_init_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND)
-- Macro: int mpfr_init_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND)
Initialize ROP and set its value from OP, rounded in the direction
RND. The precision of ROP will be taken from the active default
precision, as set by ‘mpfr_set_default_prec’.
-- Function: int mpfr_init_set_str (mpfr_t X, const char *S, int BASE,
mpfr_rnd_t RND)
Initialize X and set its value from the string S in base BASE,
rounded in the direction RND. See ‘mpfr_set_str’.

File: mpfr.info, Node: Conversion Functions, Next: Basic Arithmetic Functions, Prev: Combined Initialization and Assignment Functions, Up: MPFR Interface
5.4 Conversion Functions
========================
-- Function: float mpfr_get_flt (mpfr_t OP, mpfr_rnd_t RND)
-- Function: double mpfr_get_d (mpfr_t OP, mpfr_rnd_t RND)
-- Function: long double mpfr_get_ld (mpfr_t OP, mpfr_rnd_t RND)
-- Function: _Decimal64 mpfr_get_decimal64 (mpfr_t OP, mpfr_rnd_t RND)
Convert OP to a ‘float’ (respectively ‘double’, ‘long double’ or
‘_Decimal64’), using the rounding mode RND. If OP is NaN, some
fixed NaN (either quiet or signaling) or the result of 0.0/0.0 is
returned. If OP is ±Inf, an infinity of the same sign or the
result of ±1.0/0.0 is returned. If OP is zero, these functions
return a zero, trying to preserve its sign, if possible. The
‘mpfr_get_decimal64’ function is built only under some conditions:
see the documentation of ‘mpfr_set_decimal64’.
-- Function: long mpfr_get_si (mpfr_t OP, mpfr_rnd_t RND)
-- Function: unsigned long mpfr_get_ui (mpfr_t OP, mpfr_rnd_t RND)
-- Function: intmax_t mpfr_get_sj (mpfr_t OP, mpfr_rnd_t RND)
-- Function: uintmax_t mpfr_get_uj (mpfr_t OP, mpfr_rnd_t RND)
Convert OP to a ‘long’, an ‘unsigned long’, an ‘intmax_t’ or an
‘uintmax_t’ (respectively) after rounding it with respect to RND.
If OP is NaN, 0 is returned and the _erange_ flag is set. If OP is
too big for the return type, the function returns the maximum or
the minimum of the corresponding C type, depending on the direction
of the overflow; the _erange_ flag is set too. See also
‘mpfr_fits_slong_p’, ‘mpfr_fits_ulong_p’, ‘mpfr_fits_intmax_p’ and
‘mpfr_fits_uintmax_p’.
-- Function: double mpfr_get_d_2exp (long *EXP, mpfr_t OP, mpfr_rnd_t
RND)
-- Function: long double mpfr_get_ld_2exp (long *EXP, mpfr_t OP,
mpfr_rnd_t RND)
Return D and set EXP (formally, the value pointed to by EXP) such
that 0.5<=abs(D)<1 and D times 2 raised to EXP equals OP rounded to
double (resp. long double) precision, using the given rounding
mode. If OP is zero, then a zero of the same sign (or an unsigned
zero, if the implementation does not have signed zeros) is
returned, and EXP is set to 0. If OP is NaN or an infinity, then
the corresponding double precision (resp. long-double precision)
value is returned, and EXP is undefined.
-- Function: int mpfr_frexp (mpfr_exp_t *EXP, mpfr_t Y, mpfr_t X,
mpfr_rnd_t RND)
Set EXP (formally, the value pointed to by EXP) and Y such that
0.5<=abs(Y)<1 and Y times 2 raised to EXP equals X rounded to the
precision of Y, using the given rounding mode. If X is zero, then
Y is set to a zero of the same sign and EXP is set to 0. If X is
NaN or an infinity, then Y is set to the same value and EXP is
undefined.
-- Function: mpfr_exp_t mpfr_get_z_2exp (mpz_t ROP, mpfr_t OP)
Put the scaled significand of OP (regarded as an integer, with the
precision of OP) into ROP, and return the exponent EXP (which may
be outside the current exponent range) such that OP exactly equals
ROP times 2 raised to the power EXP. If OP is zero, the minimal
exponent ‘emin’ is returned. If OP is NaN or an infinity, the
_erange_ flag is set, ROP is set to 0, and the the minimal exponent
‘emin’ is returned. The returned exponent may be less than the
minimal exponent ‘emin’ of MPFR numbers in the current exponent
range; in case the exponent is not representable in the
‘mpfr_exp_t’ type, the _erange_ flag is set and the minimal value
of the ‘mpfr_exp_t’ type is returned.
-- Function: int mpfr_get_z (mpz_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Convert OP to a ‘mpz_t’, after rounding it with respect to RND. If
OP is NaN or an infinity, the _erange_ flag is set, ROP is set to
0, and 0 is returned.
-- Function: int mpfr_get_f (mpf_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Convert OP to a ‘mpf_t’, after rounding it with respect to RND.
The _erange_ flag is set if OP is NaN or an infinity, which do not
exist in MPF. If OP is NaN, then ROP is undefined. If OP is +Inf
(resp. −Inf), then ROP is set to the maximum (resp. minimum) value
in the precision of the MPF number; if a future MPF version
supports infinities, this behavior will be considered incorrect and
will change (portable programs should assume that ROP is set either
to this finite number or to an infinite number). Note that since
MPFR currently has the same exponent type as MPF (but not with the
same radix), the range of values is much larger in MPF than in
MPFR, so that an overflow or underflow is not possible.
-- Function: char * mpfr_get_str (char *STR, mpfr_exp_t *EXPPTR, int B,
size_t N, mpfr_t OP, mpfr_rnd_t RND)
Convert OP to a string of digits in base B, with rounding in the
direction RND, where N is either zero (see below) or the number of
significant digits output in the string; in the latter case, N must
be greater or equal to 2. The base may vary from 2 to 62;
otherwise the function does nothing and immediately returns a null
pointer. If the input number is an ordinary number, the exponent
is written through the pointer EXPPTR (for input 0, the current
minimal exponent is written); the type ‘mpfr_exp_t’ is large enough
to hold the exponent in all cases.
The generated string is a fraction, with an implicit radix point
immediately to the left of the first digit. For example, the
number −3.1416 would be returned as "−31416" in the string and 1
written at EXPPTR. If RND is to nearest, and OP is exactly in the
middle of two consecutive possible outputs, the one with an even
significand is chosen, where both significands are considered with
the exponent of OP. Note that for an odd base, this may not
correspond to an even last digit: for example with 2 digits in base
7, (14) and a half is rounded to (15) which is 12 in decimal, (16)
and a half is rounded to (20) which is 14 in decimal, and (26) and
a half is rounded to (26) which is 20 in decimal.
If N is zero, the number of digits of the significand is chosen
large enough so that re-reading the printed value with the same
precision, assuming both output and input use rounding to nearest,
will recover the original value of OP. More precisely, in most
cases, the chosen precision of STR is the minimal precision m
depending only on P = PREC(OP) and B that satisfies the above
property, i.e., m = 1 + ceil(P*log(2)/log(B)), with P replaced by
P−1 if B is a power of 2, but in some very rare cases, it might be
m+1 (the smallest case for bases up to 62 is when P equals
186564318007 for bases 7 and 49).
If STR is a null pointer, space for the significand is allocated
using the current allocation function and a pointer to the string
is returned (unless the base is invalid). To free the returned
string, you must use ‘mpfr_free_str’.
If STR is not a null pointer, it should point to a block of storage
large enough for the significand, i.e., at least ‘max(N + 2, 7)’.
The extra two bytes are for a possible minus sign, and for the
terminating null character, and the value 7 accounts for ‘-@Inf@’
plus the terminating null character. The pointer to the string STR
is returned (unless the base is invalid).
Note: The NaN and inexact flags are currently not set when need be;
this will be fixed in future versions. Programmers should
currently assume that whether the flags are set by this function is
unspecified.
-- Function: void mpfr_free_str (char *STR)
Free a string allocated by ‘mpfr_get_str’ using the current
unallocation function. The block is assumed to be ‘strlen(STR)+1’
bytes. For more information about how it is done: *note
(gmp.info)Custom Allocation::.
-- Function: int mpfr_fits_ulong_p (mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_fits_slong_p (mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_fits_uint_p (mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_fits_sint_p (mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_fits_ushort_p (mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_fits_sshort_p (mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_fits_uintmax_p (mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_fits_intmax_p (mpfr_t OP, mpfr_rnd_t RND)
Return non-zero if OP would fit in the respective C data type,
respectively ‘unsigned long’, ‘long’, ‘unsigned int’, ‘int’,
‘unsigned short’, ‘short’, ‘uintmax_t’, ‘intmax_t’, when rounded to
an integer in the direction RND.

File: mpfr.info, Node: Basic Arithmetic Functions, Next: Comparison Functions, Prev: Conversion Functions, Up: MPFR Interface
5.5 Basic Arithmetic Functions
==============================
-- Function: int mpfr_add (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_add_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int
OP2, mpfr_rnd_t RND)
-- Function: int mpfr_add_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_add_d (mpfr_t ROP, mpfr_t OP1, double OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_add_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_add_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2,
mpfr_rnd_t RND)
Set ROP to OP1 + OP2 rounded in the direction RND. For types
having no signed zero, it is considered unsigned (i.e., (+0) + 0 =
(+0) and (−0) + 0 = (−0)). The ‘mpfr_add_d’ function assumes that
the radix of the ‘double’ type is a power of 2, with a precision at
most that declared by the C implementation (macro
‘IEEE_DBL_MANT_DIG’, and if not defined 53 bits).
-- Function: int mpfr_sub (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_ui_sub (mpfr_t ROP, unsigned long int OP1, mpfr_t
OP2, mpfr_rnd_t RND)
-- Function: int mpfr_sub_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int
OP2, mpfr_rnd_t RND)
-- Function: int mpfr_si_sub (mpfr_t ROP, long int OP1, mpfr_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_sub_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_d_sub (mpfr_t ROP, double OP1, mpfr_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_sub_d (mpfr_t ROP, mpfr_t OP1, double OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_z_sub (mpfr_t ROP, mpz_t OP1, mpfr_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_sub_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_sub_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2,
mpfr_rnd_t RND)
Set ROP to OP1 - OP2 rounded in the direction RND. For types
having no signed zero, it is considered unsigned (i.e., (+0) − 0 =
(+0), (−0) − 0 = (−0), 0 − (+0) = (−0) and 0 − (−0) = (+0)). The
same restrictions than for ‘mpfr_add_d’ apply to ‘mpfr_d_sub’ and
‘mpfr_sub_d’.
-- Function: int mpfr_mul (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_mul_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int
OP2, mpfr_rnd_t RND)
-- Function: int mpfr_mul_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_mul_d (mpfr_t ROP, mpfr_t OP1, double OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_mul_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_mul_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2,
mpfr_rnd_t RND)
Set ROP to OP1 times OP2 rounded in the direction RND. When a
result is zero, its sign is the product of the signs of the
operands (for types having no signed zero, it is considered
positive). The same restrictions than for ‘mpfr_add_d’ apply to
‘mpfr_mul_d’.
-- Function: int mpfr_sqr (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the square of OP rounded in the direction RND.
-- Function: int mpfr_div (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_ui_div (mpfr_t ROP, unsigned long int OP1, mpfr_t
OP2, mpfr_rnd_t RND)
-- Function: int mpfr_div_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int
OP2, mpfr_rnd_t RND)
-- Function: int mpfr_si_div (mpfr_t ROP, long int OP1, mpfr_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_div_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_d_div (mpfr_t ROP, double OP1, mpfr_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_div_d (mpfr_t ROP, mpfr_t OP1, double OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_div_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_div_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2,
mpfr_rnd_t RND)
Set ROP to OP1/OP2 rounded in the direction RND. When a result is
zero, its sign is the product of the signs of the operands (for
types having no signed zero, it is considered positive). The same
restrictions than for ‘mpfr_add_d’ apply to ‘mpfr_d_div’ and
‘mpfr_div_d’.
-- Function: int mpfr_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_sqrt_ui (mpfr_t ROP, unsigned long int OP,
mpfr_rnd_t RND)
Set ROP to the square root of OP rounded in the direction RND (set
ROP to −0 if OP is −0, to be consistent with the IEEE 754
standard). Set ROP to NaN if OP is negative.
-- Function: int mpfr_rec_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the reciprocal square root of OP rounded in the
direction RND. Set ROP to +Inf if OP is ±0, +0 if OP is +Inf, and
NaN if OP is negative.
-- Function: int mpfr_cbrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_root (mpfr_t ROP, mpfr_t OP, unsigned long int K,
mpfr_rnd_t RND)
Set ROP to the cubic root (resp. the Kth root) of OP rounded in the
direction RND. For K odd (resp. even) and OP negative (including
−Inf), set ROP to a negative number (resp. NaN). The Kth root of −0
is defined to be −0, whatever the parity of K.
-- Function: int mpfr_pow (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_pow_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int
OP2, mpfr_rnd_t RND)
-- Function: int mpfr_pow_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_pow_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
mpfr_rnd_t RND)
-- Function: int mpfr_ui_pow_ui (mpfr_t ROP, unsigned long int OP1,
unsigned long int OP2, mpfr_rnd_t RND)
-- Function: int mpfr_ui_pow (mpfr_t ROP, unsigned long int OP1, mpfr_t
OP2, mpfr_rnd_t RND)
Set ROP to OP1 raised to OP2, rounded in the direction RND.
Special values are handled as described in the ISO C99 and IEEE
754-2008 standards for the ‘pow’ function:
• ‘pow(±0, Y)’ returns plus or minus infinity for Y a negative
odd integer.
• ‘pow(±0, Y)’ returns plus infinity for Y negative and not an
odd integer.
• ‘pow(±0, Y)’ returns plus or minus zero for Y a positive odd
integer.
• ‘pow(±0, Y)’ returns plus zero for Y positive and not an odd
integer.
• ‘pow(-1, ±Inf)’ returns 1.
• ‘pow(+1, Y)’ returns 1 for any Y, even a NaN.
• ‘pow(X, ±0)’ returns 1 for any X, even a NaN.
• ‘pow(X, Y)’ returns NaN for finite negative X and finite
non-integer Y.
• ‘pow(X, -Inf)’ returns plus infinity for 0 < abs(x) < 1, and
plus zero for abs(x) > 1.
• ‘pow(X, +Inf)’ returns plus zero for 0 < abs(x) < 1, and plus
infinity for abs(x) > 1.
• ‘pow(-Inf, Y)’ returns minus zero for Y a negative odd
integer.
• ‘pow(-Inf, Y)’ returns plus zero for Y negative and not an odd
integer.
• ‘pow(-Inf, Y)’ returns minus infinity for Y a positive odd
integer.
• ‘pow(-Inf, Y)’ returns plus infinity for Y positive and not an
odd integer.
• ‘pow(+Inf, Y)’ returns plus zero for Y negative, and plus
infinity for Y positive.
-- Function: int mpfr_neg (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_abs (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to -OP and the absolute value of OP respectively, rounded
in the direction RND. Just changes or adjusts the sign if ROP and
OP are the same variable, otherwise a rounding might occur if the
precision of ROP is less than that of OP.
-- Function: int mpfr_dim (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
mpfr_rnd_t RND)
Set ROP to the positive difference of OP1 and OP2, i.e., OP1 - OP2
rounded in the direction RND if OP1 > OP2, +0 if OP1 <= OP2, and
NaN if OP1 or OP2 is NaN.
-- Function: int mpfr_mul_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long
int OP2, mpfr_rnd_t RND)
-- Function: int mpfr_mul_2si (mpfr_t ROP, mpfr_t OP1, long int OP2,
mpfr_rnd_t RND)
Set ROP to OP1 times 2 raised to OP2 rounded in the direction RND.
Just increases the exponent by OP2 when ROP and OP1 are identical.
-- Function: int mpfr_div_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long
int OP2, mpfr_rnd_t RND)
-- Function: int mpfr_div_2si (mpfr_t ROP, mpfr_t OP1, long int OP2,
mpfr_rnd_t RND)
Set ROP to OP1 divided by 2 raised to OP2 rounded in the direction
RND. Just decreases the exponent by OP2 when ROP and OP1 are
identical.

File: mpfr.info, Node: Comparison Functions, Next: Special Functions, Prev: Basic Arithmetic Functions, Up: MPFR Interface
5.6 Comparison Functions
========================
-- Function: int mpfr_cmp (mpfr_t OP1, mpfr_t OP2)
-- Function: int mpfr_cmp_ui (mpfr_t OP1, unsigned long int OP2)
-- Function: int mpfr_cmp_si (mpfr_t OP1, long int OP2)
-- Function: int mpfr_cmp_d (mpfr_t OP1, double OP2)
-- Function: int mpfr_cmp_ld (mpfr_t OP1, long double OP2)
-- Function: int mpfr_cmp_z (mpfr_t OP1, mpz_t OP2)
-- Function: int mpfr_cmp_q (mpfr_t OP1, mpq_t OP2)
-- Function: int mpfr_cmp_f (mpfr_t OP1, mpf_t OP2)
Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if
OP1 = OP2, and a negative value if OP1 < OP2. Both OP1 and OP2 are
considered to their full own precision, which may differ. If one
of the operands is NaN, set the _erange_ flag and return zero.
Note: These functions may be useful to distinguish the three
possible cases. If you need to distinguish two cases only, it is
recommended to use the predicate functions (e.g., ‘mpfr_equal_p’
for the equality) described below; they behave like the IEEE 754
comparisons, in particular when one or both arguments are NaN. But
only floating-point numbers can be compared (you may need to do a
conversion first).
-- Function: int mpfr_cmp_ui_2exp (mpfr_t OP1, unsigned long int OP2,
mpfr_exp_t E)
-- Function: int mpfr_cmp_si_2exp (mpfr_t OP1, long int OP2, mpfr_exp_t
E)
Compare OP1 and OP2 multiplied by two to the power E. Similar as
above.
-- Function: int mpfr_cmpabs (mpfr_t OP1, mpfr_t OP2)
Compare |OP1| and |OP2|. Return a positive value if |OP1| > |OP2|,
zero if |OP1| = |OP2|, and a negative value if |OP1| < |OP2|. If
one of the operands is NaN, set the _erange_ flag and return zero.
-- Function: int mpfr_nan_p (mpfr_t OP)
-- Function: int mpfr_inf_p (mpfr_t OP)
-- Function: int mpfr_number_p (mpfr_t OP)
-- Function: int mpfr_zero_p (mpfr_t OP)
-- Function: int mpfr_regular_p (mpfr_t OP)
Return non-zero if OP is respectively NaN, an infinity, an ordinary
number (i.e., neither NaN nor an infinity), zero, or a regular
number (i.e., neither NaN, nor an infinity nor zero). Return zero
otherwise.
-- Macro: int mpfr_sgn (mpfr_t OP)
Return a positive value if OP > 0, zero if OP = 0, and a negative
value if OP < 0. If the operand is NaN, set the _erange_ flag and
return zero. This is equivalent to ‘mpfr_cmp_ui (op, 0)’, but more
efficient.
-- Function: int mpfr_greater_p (mpfr_t OP1, mpfr_t OP2)
-- Function: int mpfr_greaterequal_p (mpfr_t OP1, mpfr_t OP2)
-- Function: int mpfr_less_p (mpfr_t OP1, mpfr_t OP2)
-- Function: int mpfr_lessequal_p (mpfr_t OP1, mpfr_t OP2)
-- Function: int mpfr_equal_p (mpfr_t OP1, mpfr_t OP2)
Return non-zero if OP1 > OP2, OP1 >= OP2, OP1 < OP2, OP1 <= OP2,
OP1 = OP2 respectively, and zero otherwise. Those functions return
zero whenever OP1 and/or OP2 is NaN.
-- Function: int mpfr_lessgreater_p (mpfr_t OP1, mpfr_t OP2)
Return non-zero if OP1 < OP2 or OP1 > OP2 (i.e., neither OP1, nor
OP2 is NaN, and OP1 <> OP2), zero otherwise (i.e., OP1 and/or OP2
is NaN, or OP1 = OP2).
-- Function: int mpfr_unordered_p (mpfr_t OP1, mpfr_t OP2)
Return non-zero if OP1 or OP2 is a NaN (i.e., they cannot be
compared), zero otherwise.

File: mpfr.info, Node: Special Functions, Next: Input and Output Functions, Prev: Comparison Functions, Up: MPFR Interface
5.7 Special Functions
=====================
All those functions, except explicitly stated (for example
‘mpfr_sin_cos’), return a *note ternary value::, i.e., zero for an exact
return value, a positive value for a return value larger than the exact
result, and a negative value otherwise.
Important note: in some domains, computing special functions (either
with correct or incorrect rounding) is expensive, even for small
precision, for example the trigonometric and Bessel functions for large
argument.
-- Function: int mpfr_log (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_log2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_log10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the natural logarithm of OP, log2(OP) or log10(OP),
respectively, rounded in the direction RND. Set ROP to −Inf if OP
is −0 (i.e., the sign of the zero has no influence on the result).
-- Function: int mpfr_exp (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_exp2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_exp10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the exponential of OP, to 2 power of OP or to 10 power
of OP, respectively, rounded in the direction RND.
-- Function: int mpfr_cos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_sin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_tan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the cosine of OP, sine of OP, tangent of OP, rounded in
the direction RND.
-- Function: int mpfr_sin_cos (mpfr_t SOP, mpfr_t COP, mpfr_t OP,
mpfr_rnd_t RND)
Set simultaneously SOP to the sine of OP and COP to the cosine of
OP, rounded in the direction RND with the corresponding precisions
of SOP and COP, which must be different variables. Return 0 iff
both results are exact, more precisely it returns s+4c where s=0 if
SOP is exact, s=1 if SOP is larger than the sine of OP, s=2 if SOP
is smaller than the sine of OP, and similarly for c and the cosine
of OP.
-- Function: int mpfr_sec (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_csc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_cot (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the secant of OP, cosecant of OP, cotangent of OP,
rounded in the direction RND.
-- Function: int mpfr_acos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_asin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_atan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the arc-cosine, arc-sine or arc-tangent of OP, rounded
in the direction RND. Note that since ‘acos(-1)’ returns the
floating-point number closest to Pi according to the given rounding
mode, this number might not be in the output range 0 <= ROP < \pi
of the arc-cosine function; still, the result lies in the image of
the output range by the rounding function. The same holds for
‘asin(-1)’, ‘asin(1)’, ‘atan(-Inf)’, ‘atan(+Inf)’ or for ‘atan(op)’
with large OP and small precision of ROP.
-- Function: int mpfr_atan2 (mpfr_t ROP, mpfr_t Y, mpfr_t X, mpfr_rnd_t
RND)
Set ROP to the arc-tangent2 of Y and X, rounded in the direction
RND: if ‘x > 0’, ‘atan2(y, x) = atan (y/x)’; if ‘x < 0’, ‘atan2(y,
x) = sign(y)*(Pi - atan (abs(y/x)))’, thus a number from -Pi to Pi.
As for ‘atan’, in case the exact mathematical result is +Pi or -Pi,
its rounded result might be outside the function output range.
‘atan2(y, 0)’ does not raise any floating-point exception. Special
values are handled as described in the ISO C99 and IEEE 754-2008
standards for the ‘atan2’ function:
• ‘atan2(+0, -0)’ returns +Pi.
• ‘atan2(-0, -0)’ returns -Pi.
• ‘atan2(+0, +0)’ returns +0.
• ‘atan2(-0, +0)’ returns −0.
• ‘atan2(+0, x)’ returns +Pi for x < 0.
• ‘atan2(-0, x)’ returns -Pi for x < 0.
• ‘atan2(+0, x)’ returns +0 for x > 0.
• ‘atan2(-0, x)’ returns −0 for x > 0.
• ‘atan2(y, 0)’ returns -Pi/2 for y < 0.
• ‘atan2(y, 0)’ returns +Pi/2 for y > 0.
• ‘atan2(+Inf, -Inf)’ returns +3*Pi/4.
• ‘atan2(-Inf, -Inf)’ returns -3*Pi/4.
• ‘atan2(+Inf, +Inf)’ returns +Pi/4.
• ‘atan2(-Inf, +Inf)’ returns -Pi/4.
• ‘atan2(+Inf, x)’ returns +Pi/2 for finite x.
• ‘atan2(-Inf, x)’ returns -Pi/2 for finite x.
• ‘atan2(y, -Inf)’ returns +Pi for finite y > 0.
• ‘atan2(y, -Inf)’ returns -Pi for finite y < 0.
• ‘atan2(y, +Inf)’ returns +0 for finite y > 0.
• ‘atan2(y, +Inf)’ returns −0 for finite y < 0.
-- Function: int mpfr_cosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_sinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_tanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the hyperbolic cosine, sine or tangent of OP, rounded in
the direction RND.
-- Function: int mpfr_sinh_cosh (mpfr_t SOP, mpfr_t COP, mpfr_t OP,
mpfr_rnd_t RND)
Set simultaneously SOP to the hyperbolic sine of OP and COP to the
hyperbolic cosine of OP, rounded in the direction RND with the
corresponding precision of SOP and COP, which must be different
variables. Return 0 iff both results are exact (see ‘mpfr_sin_cos’
for a more detailed description of the return value).
-- Function: int mpfr_sech (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_csch (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_coth (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the hyperbolic secant of OP, cosecant of OP, cotangent
of OP, rounded in the direction RND.
-- Function: int mpfr_acosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_asinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_atanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the inverse hyperbolic cosine, sine or tangent of OP,
rounded in the direction RND.
-- Function: int mpfr_fac_ui (mpfr_t ROP, unsigned long int OP,
mpfr_rnd_t RND)
Set ROP to the factorial of OP, rounded in the direction RND.
-- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the logarithm of one plus OP, rounded in the direction
RND.
-- Function: int mpfr_expm1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the exponential of OP followed by a subtraction by one,
rounded in the direction RND.
-- Function: int mpfr_eint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the exponential integral of OP, rounded in the direction
RND. For positive OP, the exponential integral is the sum of
Euler’s constant, of the logarithm of OP, and of the sum for k from
1 to infinity of OP to the power k, divided by k and factorial(k).
For negative OP, ROP is set to NaN (this definition for negative
argument follows formula 5.1.2 from the Handbook of Mathematical
Functions from Abramowitz and Stegun, a future version might use
another definition).
-- Function: int mpfr_li2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to real part of the dilogarithm of OP, rounded in the
direction RND. MPFR defines the dilogarithm function as the
integral of -log(1-t)/t from 0 to OP.
-- Function: int mpfr_gamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the value of the Gamma function on OP, rounded in the
direction RND. When OP is a negative integer, ROP is set to NaN.
-- Function: int mpfr_lngamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the value of the logarithm of the Gamma function on OP,
rounded in the direction RND. When −2K−1 <= OP <= −2K, K being a
non-negative integer, ROP is set to NaN. See also ‘mpfr_lgamma’.
-- Function: int mpfr_lgamma (mpfr_t ROP, int *SIGNP, mpfr_t OP,
mpfr_rnd_t RND)
Set ROP to the value of the logarithm of the absolute value of the
Gamma function on OP, rounded in the direction RND. The sign (1 or
−1) of Gamma(OP) is returned in the object pointed to by SIGNP.
When OP is an infinity or a non-positive integer, set ROP to +Inf.
When OP is NaN, −Inf or a negative integer, *SIGNP is undefined,
and when OP is ±0, *SIGNP is the sign of the zero.
-- Function: int mpfr_digamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the value of the Digamma (sometimes also called Psi)
function on OP, rounded in the direction RND. When OP is a
negative integer, set ROP to NaN.
-- Function: int mpfr_zeta (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_zeta_ui (mpfr_t ROP, unsigned long OP, mpfr_rnd_t
RND)
Set ROP to the value of the Riemann Zeta function on OP, rounded in
the direction RND.
-- Function: int mpfr_erf (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_erfc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the value of the error function on OP (resp. the
complementary error function on OP) rounded in the direction RND.
-- Function: int mpfr_j0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_j1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_jn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t
RND)
Set ROP to the value of the first kind Bessel function of order 0,
(resp. 1 and N) on OP, rounded in the direction RND. When OP is
NaN, ROP is always set to NaN. When OP is plus or minus Infinity,
ROP is set to +0. When OP is zero, and N is not zero, ROP is set
to +0 or −0 depending on the parity and sign of N, and the sign of
OP.
-- Function: int mpfr_y0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_y1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_yn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t
RND)
Set ROP to the value of the second kind Bessel function of order 0
(resp. 1 and N) on OP, rounded in the direction RND. When OP is
NaN or negative, ROP is always set to NaN. When OP is +Inf, ROP is
set to +0. When OP is zero, ROP is set to +Inf or −Inf depending
on the parity and sign of N.
-- Function: int mpfr_fma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t
OP3, mpfr_rnd_t RND)
-- Function: int mpfr_fms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t
OP3, mpfr_rnd_t RND)
Set ROP to (OP1 times OP2) + OP3 (resp. (OP1 times OP2) - OP3)
rounded in the direction RND.
-- Function: int mpfr_agm (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
mpfr_rnd_t RND)
Set ROP to the arithmetic-geometric mean of OP1 and OP2, rounded in
the direction RND. The arithmetic-geometric mean is the common
limit of the sequences U_N and V_N, where U_0=OP1, V_0=OP2, U_(N+1)
is the arithmetic mean of U_N and V_N, and V_(N+1) is the geometric
mean of U_N and V_N. If any operand is negative, set ROP to NaN.
-- Function: int mpfr_hypot (mpfr_t ROP, mpfr_t X, mpfr_t Y, mpfr_rnd_t
RND)
Set ROP to the Euclidean norm of X and Y, i.e., the square root of
the sum of the squares of X and Y, rounded in the direction RND.
Special values are handled as described in Section F.9.4.3 of the
ISO C99 and IEEE 754-2008 standards: If X or Y is an infinity, then
+Inf is returned in ROP, even if the other number is NaN.
-- Function: int mpfr_ai (mpfr_t ROP, mpfr_t X, mpfr_rnd_t RND)
Set ROP to the value of the Airy function Ai on X, rounded in the
direction RND. When X is NaN, ROP is always set to NaN. When X is
+Inf or −Inf, ROP is +0. The current implementation is not
intended to be used with large arguments. It works with abs(X)
typically smaller than 500. For larger arguments, other methods
should be used and will be implemented in a future version.
-- Function: int mpfr_const_log2 (mpfr_t ROP, mpfr_rnd_t RND)
-- Function: int mpfr_const_pi (mpfr_t ROP, mpfr_rnd_t RND)
-- Function: int mpfr_const_euler (mpfr_t ROP, mpfr_rnd_t RND)
-- Function: int mpfr_const_catalan (mpfr_t ROP, mpfr_rnd_t RND)
Set ROP to the logarithm of 2, the value of Pi, of Euler’s constant
0.577…, of Catalan’s constant 0.915…, respectively, rounded in the
direction RND. These functions cache the computed values to avoid
other calculations if a lower or equal precision is requested. To
free these caches, use ‘mpfr_free_cache’.
-- Function: void mpfr_free_cache (void)
Free various caches used by MPFR internally, in particular the
caches used by the functions computing constants
(‘mpfr_const_log2’, ‘mpfr_const_pi’, ‘mpfr_const_euler’ and
‘mpfr_const_catalan’). You should call this function before
terminating a thread, even if you did not call these functions
directly (they could have been called internally).
-- Function: int mpfr_sum (mpfr_t ROP, mpfr_ptr const TAB[], unsigned
long int N, mpfr_rnd_t RND)
Set ROP to the sum of all elements of TAB, whose size is N, rounded
in the direction RND. Warning: for efficiency reasons, TAB is an
array of pointers to ‘mpfr_t’, not an array of ‘mpfr_t’. If the
returned ‘int’ value is zero, ROP is guaranteed to be the exact
sum; otherwise ROP might be smaller than, equal to, or larger than
the exact sum (in accordance to the rounding mode). However,
‘mpfr_sum’ does guarantee the result is correctly rounded.

File: mpfr.info, Node: Input and Output Functions, Next: Formatted Output Functions, Prev: Special Functions, Up: MPFR Interface
5.8 Input and Output Functions
==============================
This section describes functions that perform input from an input/output
stream, and functions that output to an input/output stream. Passing a
null pointer for a ‘stream’ to any of these functions will make them
read from ‘stdin’ and write to ‘stdout’, respectively.
When using any of these functions, you must include the ‘<stdio.h>’
standard header before ‘mpfr.h’, to allow ‘mpfr.h’ to define prototypes
for these functions.
-- Function: size_t mpfr_out_str (FILE *STREAM, int BASE, size_t N,
mpfr_t OP, mpfr_rnd_t RND)
Output OP on stream STREAM, as a string of digits in base BASE,
rounded in the direction RND. The base may vary from 2 to 62.
Print N significant digits exactly, or if N is 0, enough digits so
that OP can be read back exactly (see ‘mpfr_get_str’).
In addition to the significant digits, a decimal point (defined by
the current locale) at the right of the first digit and a trailing
exponent in base 10, in the form ‘eNNN’, are printed. If BASE is
greater than 10, ‘@’ will be used instead of ‘e’ as exponent
delimiter.
Return the number of characters written, or if an error occurred,
return 0.
-- Function: size_t mpfr_inp_str (mpfr_t ROP, FILE *STREAM, int BASE,
mpfr_rnd_t RND)
Input a string in base BASE from stream STREAM, rounded in the
direction RND, and put the read float in ROP.
This function reads a word (defined as a sequence of characters
between whitespace) and parses it using ‘mpfr_set_str’. See the
documentation of ‘mpfr_strtofr’ for a detailed description of the
valid string formats.
Return the number of bytes read, or if an error occurred, return 0.

File: mpfr.info, Node: Formatted Output Functions, Next: Integer Related Functions, Prev: Input and Output Functions, Up: MPFR Interface
5.9 Formatted Output Functions
==============================
5.9.1 Requirements
------------------
The class of ‘mpfr_printf’ functions provides formatted output in a
similar manner as the standard C ‘printf’. These functions are defined
only if your system supports ISO C variadic functions and the
corresponding argument access macros.
When using any of these functions, you must include the ‘<stdio.h>’
standard header before ‘mpfr.h’, to allow ‘mpfr.h’ to define prototypes
for these functions.
5.9.2 Format String
-------------------
The format specification accepted by ‘mpfr_printf’ is an extension of
the ‘printf’ one. The conversion specification is of the form:
% [flags] [width] [.[precision]] [type] [rounding] conv
‘flags’, ‘width’, and ‘precision’ have the same meaning as for the
standard ‘printf’ (in particular, notice that the ‘precision’ is related
to the number of digits displayed in the base chosen by ‘conv’ and not
related to the internal precision of the ‘mpfr_t’ variable).
‘mpfr_printf’ accepts the same ‘type’ specifiers as GMP (except the
non-standard and deprecated ‘q’, use ‘ll’ instead), namely the length
modifiers defined in the C standard:
‘h’ ‘short’
‘hh’ ‘char’
‘j’ ‘intmax_t’ or ‘uintmax_t’
‘l’ ‘long’ or ‘wchar_t’
‘ll’ ‘long long’
‘L’ ‘long double’
‘t’ ‘ptrdiff_t’
‘z’ ‘size_t’
and the ‘type’ specifiers defined in GMP plus ‘R’ and ‘P’ specific to
MPFR (the second column in the table below shows the type of the
argument read in the argument list and the kind of ‘conv’ specifier to
use after the ‘type’ specifier):
‘F’ ‘mpf_t’, float conversions
‘Q’ ‘mpq_t’, integer conversions
‘M’ ‘mp_limb_t’, integer conversions
‘N’ ‘mp_limb_t’ array, integer conversions
‘Z’ ‘mpz_t’, integer conversions
‘P’ ‘mpfr_prec_t’, integer conversions
‘R’ ‘mpfr_t’, float conversions
The ‘type’ specifiers have the same restrictions as those mentioned
in the GMP documentation: *note (gmp.info)Formatted Output Strings::.
In particular, the ‘type’ specifiers (except ‘R’ and ‘P’) are supported
only if they are supported by ‘gmp_printf’ in your GMP build; this
implies that the standard specifiers, such as ‘t’, must _also_ be
supported by your C library if you want to use them.
The ‘rounding’ field is specific to ‘mpfr_t’ arguments and should not
be used with other types.
With conversion specification not involving ‘P’ and ‘R’ types,
‘mpfr_printf’ behaves exactly as ‘gmp_printf’.
The ‘P’ type specifies that a following ‘d’, ‘i’, ‘o’, ‘u’, ‘x’, or
‘X’ conversion specifier applies to a ‘mpfr_prec_t’ argument. It is
needed because the ‘mpfr_prec_t’ type does not necessarily correspond to
an ‘int’ or any fixed standard type. The ‘precision’ field specifies
the minimum number of digits to appear. The default ‘precision’ is 1.
For example:
mpfr_t x;
mpfr_prec_t p;
mpfr_init (x);
…
p = mpfr_get_prec (x);
mpfr_printf ("variable x with %Pu bits", p);
The ‘R’ type specifies that a following ‘a’, ‘A’, ‘b’, ‘e’, ‘E’, ‘f’,
‘F’, ‘g’, ‘G’, or ‘n’ conversion specifier applies to a ‘mpfr_t’
argument. The ‘R’ type can be followed by a ‘rounding’ specifier
denoted by one of the following characters:
‘U’ round toward plus infinity
‘D’ round toward minus infinity
‘Y’ round away from zero
‘Z’ round toward zero
‘N’ round to nearest (with ties to even)
‘*’ rounding mode indicated by the
‘mpfr_rnd_t’ argument just before the
corresponding ‘mpfr_t’ variable.
The default rounding mode is rounding to nearest. The following
three examples are equivalent:
mpfr_t x;
mpfr_init (x);
…
mpfr_printf ("%.128Rf", x);
mpfr_printf ("%.128RNf", x);
mpfr_printf ("%.128R*f", MPFR_RNDN, x);
Note that the rounding away from zero mode is specified with ‘Y’
because ISO C reserves the ‘A’ specifier for hexadecimal output (see
below).
The output ‘conv’ specifiers allowed with ‘mpfr_t’ parameter are:
‘a’ ‘A’ hex float, C99 style
‘b’ binary output
‘e’ ‘E’ scientific format float
‘f’ ‘F’ fixed point float
‘g’ ‘G’ fixed or scientific float
The conversion specifier ‘b’ which displays the argument in binary is
specific to ‘mpfr_t’ arguments and should not be used with other types.
Other conversion specifiers have the same meaning as for a ‘double’
argument.
In case of non-decimal output, only the significand is written in the
specified base, the exponent is always displayed in decimal. Special
values are always displayed as ‘nan’, ‘-inf’, and ‘inf’ for ‘a’, ‘b’,
‘e’, ‘f’, and ‘g’ specifiers and ‘NAN’, ‘-INF’, and ‘INF’ for ‘A’, ‘E’,
‘F’, and ‘G’ specifiers.
If the ‘precision’ field is not empty, the ‘mpfr_t’ number is rounded
to the given precision in the direction specified by the rounding mode.
If the precision is zero with rounding to nearest mode and one of the
following ‘conv’ specifiers: ‘a’, ‘A’, ‘b’, ‘e’, ‘E’, tie case is
rounded to even when it lies between two consecutive values at the
wanted precision which have the same exponent, otherwise, it is rounded
away from zero. For instance, 85 is displayed as "8e+1" and 95 is
displayed as "1e+2" with the format specification ‘"%.0RNe"’. This also
applies when the ‘g’ (resp. ‘G’) conversion specifier uses the ‘e’
(resp. ‘E’) style. If the precision is set to a value greater than the
maximum value for an ‘int’, it will be silently reduced down to
‘INT_MAX’.
If the ‘precision’ field is empty (as in ‘%Re’ or ‘%.RE’) with ‘conv’
specifier ‘e’ and ‘E’, the number is displayed with enough digits so
that it can be read back exactly, assuming that the input and output
variables have the same precision and that the input and output rounding
modes are both rounding to nearest (as for ‘mpfr_get_str’). The default
precision for an empty ‘precision’ field with ‘conv’ specifiers ‘f’,
‘F’, ‘g’, and ‘G’ is 6.
5.9.3 Functions
---------------
For all the following functions, if the number of characters which ought
to be written appears to exceed the maximum limit for an ‘int’, nothing
is written in the stream (resp. to ‘stdout’, to BUF, to STR), the
function returns −1, sets the _erange_ flag, and (in POSIX system only)
‘errno’ is set to ‘EOVERFLOW’.
-- Function: int mpfr_fprintf (FILE *STREAM, const char *TEMPLATE, …)
-- Function: int mpfr_vfprintf (FILE *STREAM, const char *TEMPLATE,
va_list AP)
Print to the stream STREAM the optional arguments under the control
of the template string TEMPLATE. Return the number of characters
written or a negative value if an error occurred.
-- Function: int mpfr_printf (const char *TEMPLATE, …)
-- Function: int mpfr_vprintf (const char *TEMPLATE, va_list AP)
Print to ‘stdout’ the optional arguments under the control of the
template string TEMPLATE. Return the number of characters written
or a negative value if an error occurred.
-- Function: int mpfr_sprintf (char *BUF, const char *TEMPLATE, …)
-- Function: int mpfr_vsprintf (char *BUF, const char *TEMPLATE,
va_list AP)
Form a null-terminated string corresponding to the optional
arguments under the control of the template string TEMPLATE, and
print it in BUF. No overlap is permitted between BUF and the other
arguments. Return the number of characters written in the array
BUF _not counting_ the terminating null character or a negative
value if an error occurred.
-- Function: int mpfr_snprintf (char *BUF, size_t N, const char
*TEMPLATE, …)
-- Function: int mpfr_vsnprintf (char *BUF, size_t N, const char
*TEMPLATE, va_list AP)
Form a null-terminated string corresponding to the optional
arguments under the control of the template string TEMPLATE, and
print it in BUF. If N is zero, nothing is written and BUF may be a
null pointer, otherwise, the N−1 first characters are written in
BUF and the N-th is a null character. Return the number of
characters that would have been written had N be sufficiently
large, _not counting_ the terminating null character, or a negative
value if an error occurred.
-- Function: int mpfr_asprintf (char **STR, const char *TEMPLATE, …)
-- Function: int mpfr_vasprintf (char **STR, const char *TEMPLATE,
va_list AP)
Write their output as a null terminated string in a block of memory
allocated using the current allocation function. A pointer to the
block is stored in STR. The block of memory must be freed using
‘mpfr_free_str’. The return value is the number of characters
written in the string, excluding the null-terminator, or a negative
value if an error occurred.

File: mpfr.info, Node: Integer Related Functions, Next: Rounding Related Functions, Prev: Formatted Output Functions, Up: MPFR Interface
5.10 Integer and Remainder Related Functions
============================================
-- Function: int mpfr_rint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_ceil (mpfr_t ROP, mpfr_t OP)
-- Function: int mpfr_floor (mpfr_t ROP, mpfr_t OP)
-- Function: int mpfr_round (mpfr_t ROP, mpfr_t OP)
-- Function: int mpfr_trunc (mpfr_t ROP, mpfr_t OP)
Set ROP to OP rounded to an integer. ‘mpfr_rint’ rounds to the
nearest representable integer in the given direction RND,
‘mpfr_ceil’ rounds to the next higher or equal representable
integer, ‘mpfr_floor’ to the next lower or equal representable
integer, ‘mpfr_round’ to the nearest representable integer,
rounding halfway cases away from zero (as in the roundTiesToAway
mode of IEEE 754-2008), and ‘mpfr_trunc’ to the next representable
integer toward zero.
The returned value is zero when the result is exact, positive when
it is greater than the original value of OP, and negative when it
is smaller. More precisely, the returned value is 0 when OP is an
integer representable in ROP, 1 or −1 when OP is an integer that is
not representable in ROP, 2 or −2 when OP is not an integer.
When OP is NaN, the NaN flag is set as usual. In the other cases,
the inexact flag is set when ROP differs from OP, following the ISO
C99 rule for the ‘rint’ function. If you want the behavior to be
more like IEEE 754 / ISO TS 18661-1, i.e., the usual behavior where
the round-to-integer function is regarded as any other mathematical
function, you should use one the ‘mpfr_rint_*’ functions instead
(however it is not possible to round to nearest with the even
rounding rule yet).
Note that ‘mpfr_round’ is different from ‘mpfr_rint’ called with
the rounding to nearest mode (where halfway cases are rounded to an
even integer or significand). Note also that no double rounding is
performed; for instance, 10.5 (1010.1 in binary) is rounded by
‘mpfr_rint’ with rounding to nearest to 12 (1100 in binary) in
2-bit precision, because the two enclosing numbers representable on
two bits are 8 and 12, and the closest is 12. (If one first
rounded to an integer, one would round 10.5 to 10 with even
rounding, and then 10 would be rounded to 8 again with even
rounding.)
-- Function: int mpfr_rint_ceil (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
-- Function: int mpfr_rint_floor (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t
RND)
-- Function: int mpfr_rint_round (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t
RND)
-- Function: int mpfr_rint_trunc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t
RND)
Set ROP to OP rounded to an integer. ‘mpfr_rint_ceil’ rounds to
the next higher or equal integer, ‘mpfr_rint_floor’ to the next
lower or equal integer, ‘mpfr_rint_round’ to the nearest integer,
rounding halfway cases away from zero, and ‘mpfr_rint_trunc’ to the
next integer toward zero. If the result is not representable, it
is rounded in the direction RND. The returned value is the ternary
value associated with the considered round-to-integer function
(regarded in the same way as any other mathematical function).
Contrary to ‘mpfr_rint’, those functions do perform a double
rounding: first OP is rounded to the nearest integer in the
direction given by the function name, then this nearest integer (if
not representable) is rounded in the given direction RND. Thus
these round-to-integer functions behave more like the other
mathematical functions, i.e., the returned result is the correct
rounding of the exact result of the function in the real numbers.
For example, ‘mpfr_rint_round’ with rounding to nearest and a
precision of two bits rounds 6.5 to 7 (halfway cases away from
zero), then 7 is rounded to 8 by the round-even rule, despite the
fact that 6 is also representable on two bits, and is closer to 6.5
than 8.
-- Function: int mpfr_frac (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
Set ROP to the fractional part of OP, having the same sign as OP,
rounded in the direction RND (unlike in ‘mpfr_rint’, RND affects
only how the exact fractional part is rounded, not how the
fractional part is generated).
-- Function: int mpfr_modf (mpfr_t IOP, mpfr_t FOP, mpfr_t OP,
mpfr_rnd_t RND)
Set simultaneously IOP to the integral part of OP and FOP to the
fractional part of OP, rounded in the direction RND with the
corresponding precision of IOP and FOP (equivalent to
‘mpfr_trunc(IOP, OP, RND)’ and ‘mpfr_frac(FOP, OP, RND)’). The
variables IOP and FOP must be different. Return 0 iff both results
are exact (see ‘mpfr_sin_cos’ for a more detailed description of
the return value).
-- Function: int mpfr_fmod (mpfr_t R, mpfr_t X, mpfr_t Y, mpfr_rnd_t
RND)
-- Function: int mpfr_remainder (mpfr_t R, mpfr_t X, mpfr_t Y,
mpfr_rnd_t RND)
-- Function: int mpfr_remquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y,
mpfr_rnd_t RND)
Set R to the value of X - NY, rounded according to the direction
RND, where N is the integer quotient of X divided by Y, defined as
follows: N is rounded toward zero for ‘mpfr_fmod’, and to the
nearest integer (ties rounded to even) for ‘mpfr_remainder’ and
‘mpfr_remquo’.
Special values are handled as described in Section F.9.7.1 of the
ISO C99 standard: If X is infinite or Y is zero, R is NaN. If Y is
infinite and X is finite, R is X rounded to the precision of R. If
R is zero, it has the sign of X. The return value is the ternary
value corresponding to R.
Additionally, ‘mpfr_remquo’ stores the low significant bits from
the quotient N in *Q (more precisely the number of bits in a ‘long’
minus one), with the sign of X divided by Y (except if those low
bits are all zero, in which case zero is returned). Note that X
may be so large in magnitude relative to Y that an exact
representation of the quotient is not practical. The
‘mpfr_remainder’ and ‘mpfr_remquo’ functions are useful for
additive argument reduction.
-- Function: int mpfr_integer_p (mpfr_t OP)
Return non-zero iff OP is an integer.

File: mpfr.info, Node: Rounding Related Functions, Next: Miscellaneous Functions, Prev: Integer Related Functions, Up: MPFR Interface
5.11 Rounding Related Functions
===============================
-- Function: void mpfr_set_default_rounding_mode (mpfr_rnd_t RND)
Set the default rounding mode to RND. The default rounding mode is
to nearest initially.
-- Function: mpfr_rnd_t mpfr_get_default_rounding_mode (void)
Get the default rounding mode.
-- Function: int mpfr_prec_round (mpfr_t X, mpfr_prec_t PREC,
mpfr_rnd_t RND)
Round X according to RND with precision PREC, which must be an
integer between ‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’ (otherwise the
behavior is undefined). If PREC is greater or equal to the
precision of X, then new space is allocated for the significand,
and it is filled with zeros. Otherwise, the significand is rounded
to precision PREC with the given direction. In both cases, the
precision of X is changed to PREC.
Here is an example of how to use ‘mpfr_prec_round’ to implement
Newton’s algorithm to compute the inverse of A, assuming X is
already an approximation to N bits:
mpfr_set_prec (t, 2 * n);
mpfr_set (t, a, MPFR_RNDN); /* round a to 2n bits */
mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to 2n bits */
mpfr_ui_sub (t, 1, t, MPFR_RNDN); /* high n bits cancel with 1 */
mpfr_prec_round (t, n, MPFR_RNDN); /* t is correct to n bits */
mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to n bits */
mpfr_prec_round (x, 2 * n, MPFR_RNDN); /* exact */
mpfr_add (x, x, t, MPFR_RNDN); /* x is correct to 2n bits */
Warning! You must not use this function if X was initialized with
‘MPFR_DECL_INIT’ or with ‘mpfr_custom_init_set’ (*note Custom
Interface::).
-- Function: int mpfr_can_round (mpfr_t B, mpfr_exp_t ERR, mpfr_rnd_t
RND1, mpfr_rnd_t RND2, mpfr_prec_t PREC)
Assuming B is an approximation of an unknown number X in the
direction RND1 with error at most two to the power E(b)-ERR where
E(b) is the exponent of B, return a non-zero value if one is able
to round correctly X to precision PREC with the direction RND2, and
0 otherwise (including for NaN and Inf). This function *does not
modify* its arguments.
If RND1 is ‘MPFR_RNDN’, then the sign of the error is unknown, but
its absolute value is the same, so that the possible range is twice
as large as with a directed rounding for RND1.
Note: if one wants to also determine the correct *note ternary
value:: when rounding B to precision PREC with rounding mode RND, a
useful trick is the following:
if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ,
prec + (rnd == MPFR_RNDN)))
...
Indeed, if RND is ‘MPFR_RNDN’, this will check if one can round to
PREC+1 bits with a directed rounding: if so, one can surely round
to nearest to PREC bits, and in addition one can determine the
correct ternary value, which would not be the case when B is near
from a value exactly representable on PREC bits.
-- Function: mpfr_prec_t mpfr_min_prec (mpfr_t X)
Return the minimal number of bits required to store the significand
of X, and 0 for special values, including 0. (Warning: the
returned value can be less than ‘MPFR_PREC_MIN’.)
The function name is subject to change.
-- Function: const char * mpfr_print_rnd_mode (mpfr_rnd_t RND)
Return a string ("MPFR_RNDD", "MPFR_RNDU", "MPFR_RNDN",
"MPFR_RNDZ", "MPFR_RNDA") corresponding to the rounding mode RND,
or a null pointer if RND is an invalid rounding mode.

File: mpfr.info, Node: Miscellaneous Functions, Next: Exception Related Functions, Prev: Rounding Related Functions, Up: MPFR Interface
5.12 Miscellaneous Functions
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-- Function: void mpfr_nexttoward (mpfr_t X, mpfr_t Y)
If X or Y is NaN, set X to NaN. If X and Y are equal, X is
unchanged. Otherwise, if X is different from Y, replace X by the
next floating-point number (with the precision of X and the current
exponent range) in the direction of Y (the infinite values are seen
as the smallest and largest floating-point numbers). If the result
is