This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 3.0.1.

Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 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 GNU Free Documentation License.

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.

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.

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

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

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.

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

- 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.

- ‘
`./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. - ‘
`make`’This will compile MPFR, and create a library archive file

`libmpfr.a`. On most platforms, a dynamic library will be produced too. - ‘
`make check`’This will make sure MPFR was built correctly. If you get error messages, please report this to ‘

`mpfr@loria.fr`’. (See Reporting Bugs, for information on what to include in useful bug reports.) - ‘
`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`).

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

`mpfr.html`; if you want only one output HTML file, then type ‘`makeinfo --html --no-split mpfr.texi`’ 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`’.

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 ‘`mpfr@loria.fr`’.
See Reporting Bugs.
Some bug fixes are available on the
MPFR 3.0.1 web page http://www.mpfr.org/mpfr-3.0.1/.

The latest version of MPFR is available from ftp://ftp.gnu.org/gnu/mpfr/ or http://www.mpfr.org/.

If you think you have found a bug in the MPFR library, first have a look on the MPFR 3.0.1 web page http://www.mpfr.org/mpfr-3.0.1/ 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 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: ‘`mpfr@loria.fr`’.

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

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), the
aforementioned standard headers should be included **before** the
first inclusion of `mpfr.h`

or `gmp.h`

. For the time being,
this problem is not avoidable in MPFR without a change in GMP.

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, see GNU Libtool.

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.

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`

.

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`.

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 (experimental).

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.

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 (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 MPFR Interface.
Example: `mpfr_hypot`

on (NaN,0) gives NaN, but `mpfr_hypot`

on (NaN,+Inf) gives +Inf (as specified in Special Functions),
since for any finite input `x`, `mpfr_hypot`

on (`x`,+Inf)
gives +Inf.

MPFR supports 5 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.
- 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 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.

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).

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.

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 beexactlyprecbits 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 precisionprecmust 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 beexactlyprecbits 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 fromx, 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 fromx, 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 fromx, 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

nameas an automatic variable of type`mpfr_t`

, initializes it and sets its precision to beexactlyprecbits and its value to NaN.namemust 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 notcall`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
cannotchange their precision.- You
should notcreate 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). Ifprecis 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

exactlyprecbits, wherepreccan 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.

— Function: mpfr_prec_t **mpfr_get_default_prec** (`void`)

Return the current default MPFR precision in bits.

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); /* precisionexactly256 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

xto beexactlyprecbits, and set its value to NaN. The previous value stored inxis 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 ofxis enough. The precisionpreccan be any integer between`MPFR_PREC_MIN`

and`MPFR_PREC_MAX`

. In case you want to keep the previous value stored inx, use`mpfr_prec_round`

instead.

— 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.

Next: Combined Initialization and Assignment Functions, Previous: Initialization Functions, Up: MPFR Interface

These functions assign new values to already initialized floats (see 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`)

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

Set the value of

ropfromop, rounded toward the given directionrnd. 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).`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 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`)

— Function: int

— Function: int

— Function: int

— Function: int

Set the value of

ropfromopmultiplied by two to the powere, rounded toward the given directionrnd. 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

ropto the value of the stringsin basebase, rounded in the directionrnd. See the documentation of`mpfr_strtofr`

for a detailed description of the valid string formats. Contrary to`mpfr_strtofr`

,`mpfr_set_str`

requires thewholestring to represent a valid floating-point number. This function returns 0 if the entire string up to the final null character is a valid number in basebase; otherwise it returns −1, andropmay have changed. Note: it is preferable to use`mpfr_set_str`

if one wants to distinguish between an infiniteropvalue coming from an infinitesor 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

nptrin basebase, rounded in the directionrnd;basemust be either 0 (to detect the base, as described below) or a number from 2 to 62 (otherwise the behavior is undefined). Ifnptrstarts with valid data, the result is stored inropand`*`

endptrpoints to the character just after the valid data (ifendptris not a null pointer); otherwiseropis set to zero (for consistency with`strtod`

) and the value ofnptris stored in the location referenced byendptr(ifendptris 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, calledbinary 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

baseis 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 ifbase<= 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`)

— Function: void

— Function: void

Set the variable

xto NaN (Not-a-Number), infinity or zero respectively. In`mpfr_set_inf`

or`mpfr_set_zero`

,xis set to plus infinity or plus zero iffsignis 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 values

xandyefficiently. Warning: the precisions are exchanged too; in case the precisions are different,`mpfr_swap`

is thus not equivalent to three`mpfr_set`

calls using a third auxiliary variable.

— 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`)

— Macro: int

— Macro: int

— Macro: int

— Macro: int

— Macro: int

— Macro: int

— Macro: int

Initialize

ropand set its value fromop, rounded in the directionrnd. The precision ofropwill 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

xand set its value from the stringsin basebase, rounded in the directionrnd. See`mpfr_set_str`

.

Next: Basic Arithmetic Functions, Previous: Combined Initialization and Assignment Functions, Up: MPFR Interface

— 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`)

— Function: double

— Function: long double

— Function: _Decimal64

Convert

opto a`float`

(respectively`double`

,`long double`

or`_Decimal64`

), using the rounding modernd. Ifopis NaN, some fixed NaN (either quiet or signaling) or the result of 0.0/0.0 is returned. Ifopis Â±Inf, an infinity of the same sign or the result of Â±1.0/0.0 is returned. Ifopis 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`)

— Function: unsigned long

— Function: intmax_t

— Function: uintmax_t

Convert

opto a`long`

, an`unsigned long`

, an`intmax_t`

or an`uintmax_t`

(respectively) after rounding it with respect tornd. Ifopis NaN, 0 is returned and theerangeflag is set. Ifopis 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; theerangeflag 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`)

— Function: long double

Return

dand setexp(formally, the value pointed to byexp) such that 0.5<=abs(d)<1 anddtimes 2 raised toexpequalsoprounded to double (resp. long double) precision, using the given rounding mode. Ifopis zero, then a zero of the same sign (or an unsigned zero, if the implementation does not have signed zeros) is returned, andexpis set to 0. Ifopis NaN or an infinity, then the corresponding double precision (resp. long-double precision) value is returned, andexpis 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 ofop) intorop, and return the exponentexp(which may be outside the current exponent range) such thatopexactly equalsroptimes 2 raised to the powerexp. Ifopis zero, the minimal exponent`emin`

is returned. Ifopis NaN or an infinity, theerangeflag is set,ropis 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, theerangeflag 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

opto a`mpz_t`

, after rounding it with respect tornd. Ifopis NaN or an infinity, theerangeflag is set,ropis set to 0, and 0 is returned.

— Function: int **mpfr_get_f** (`mpf_t rop, mpfr_t op, mpfr_rnd_t rnd`)

Convert

opto a`mpf_t`

, after rounding it with respect tornd. Theerangeflag is set ifopis NaN or Inf, which do not exist in MPF.

— Function: char * **mpfr_get_str** (`char *str, mpfr_exp_t *expptr, int b, size_t n, mpfr_t op, mpfr_rnd_t rnd`)

Convert

opto a string of digits in baseb, with rounding in the directionrnd, wherenis either zero (see below) or the number of significant digits output in the string; in the latter case,nmust be greater or equal to 2. The base may vary from 2 to 62. If the input number is an ordinary number, the exponent is written through the pointerexpptr(for input 0, the current minimal exponent is written).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. Ifrndis to nearest, andopis 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 ofop. 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

nis 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 ofop. More precisely, in most cases, the chosen precision ofstris the minimal precision m depending only onp= PREC(op) andbthat satisfies the above property, i.e., m = 1 + ceil(p*log(2)/log(b)), withpreplaced byp−1 ifbis a power of 2, but in some very rare cases, it might be m+1 (the smallest case for bases up to 62 is whenpequals 186564318007 for bases 7 and 49).If

stris a null pointer, space for the significand is allocated using the current allocation function, and a pointer to the string is returned. To free the returned string, you must use`mpfr_free_str`

.If

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

— 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: see Section “Custom Allocation” in GNU MP.

— 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`)

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

Return non-zero if

opwould 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 directionrnd.

— 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`)

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

Set

roptoop1+op2rounded in the directionrnd. 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_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`)

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

Set

roptoop1-op2rounded in the directionrnd. 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`)

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

Set

roptoop1timesop2rounded in the directionrnd. 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

ropto the square ofoprounded in the directionrnd.

— 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`)

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

Set

roptoop1/op2rounded in the directionrnd. 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`)

— Function: int

Set

ropto the square root ofoprounded in the directionrnd(setropto −0 ifopis −0, to be consistent with the IEEE 754 standard). Setropto NaN ifopis negative.

— Function: int **mpfr_rec_sqrt** (`mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd`)

Set

ropto the reciprocal square root ofoprounded in the directionrnd. Setropto +Inf ifopis Â±0, +0 ifopis +Inf, and NaN ifopis 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`)

— Function: int

Set

ropto the cubic root (resp. thekth root) ofoprounded in the directionrnd. Forkodd (resp. even) andopnegative (including −Inf), setropto a negative number (resp. NaN). Thekth root of −0 is defined to be −0, whatever the parity ofk.

— 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`)

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

Set

roptoop1raised toop2, rounded in the directionrnd. Special values are handled as described in the ISO C99 and IEEE 754-2008 standards for the`pow`

function:

`pow(Â±0,`

y`)`

returns plus or minus infinity forya negative odd integer.`pow(Â±0,`

y`)`

returns plus infinity forynegative and not an odd integer.`pow(Â±0,`

y`)`

returns plus or minus zero forya positive odd integer.`pow(Â±0,`

y`)`

returns plus zero forypositive and not an odd integer.`pow(-1, Â±Inf)`

returns 1.`pow(+1,`

y`)`

returns 1 for anyy, even a NaN.`pow(`

x`, Â±0)`

returns 1 for anyx, even a NaN.`pow(`

x`,`

y`)`

returns NaN for finite negativexand finite non-integery.`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 forya negative odd integer.`pow(-Inf,`

y`)`

returns plus zero forynegative and not an odd integer.`pow(-Inf,`

y`)`

returns minus infinity forya positive odd integer.`pow(-Inf,`

y`)`

returns plus infinity forypositive and not an odd integer.`pow(+Inf,`

y`)`

returns plus zero forynegative, and plus infinity forypositive.

— 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`)

— Function: int

Set

ropto -opand the absolute value ofoprespectively, rounded in the directionrnd. Just changes or adjusts the sign ifropandopare the same variable, otherwise a rounding might occur if the precision ofropis less than that ofop.

— Function: int **mpfr_dim** (`mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd`)

Set

ropto the positive difference ofop1andop2, i.e.,op1-op2rounded in the directionrndifop1>op2, +0 ifop1<=op2, and NaN ifop1orop2is 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`)

— Function: int

Set

roptoop1times 2 raised toop2rounded in the directionrnd. Just increases the exponent byop2whenropandop1are 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`)

— Function: int

Set

roptoop1divided by 2 raised toop2rounded in the directionrnd. Just decreases the exponent byop2whenropandop1are identical.

— 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`)

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

Compare

op1andop2. Return a positive value ifop1>op2, zero ifop1=op2, and a negative value ifop1<op2. Bothop1andop2are considered to their full own precision, which may differ. If one of the operands is NaN, set theerangeflag 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`)

— Function: int

Compare

op1andop2multiplied by two to the powere. 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 theerangeflag 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`)

— Function: int

— Function: int

— Function: int

— Function: int

Return non-zero if

opis 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 ifop= 0, and a negative value ifop< 0. If the operand is NaN, set theerangeflag 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`)

— Function: int

— Function: int

— Function: int

— Function: int

Return non-zero if

op1>op2,op1>=op2,op1<op2,op1<=op2,op1=op2respectively, and zero otherwise. Those functions return zero wheneverop1and/orop2is NaN.

— Function: int **mpfr_lessgreater_p** (`mpfr_t op1, mpfr_t op2`)

Return non-zero if

op1<op2orop1>op2(i.e., neitherop1, norop2is NaN, andop1<>op2), zero otherwise (i.e.,op1and/orop2is NaN, orop1=op2).

— Function: int **mpfr_unordered_p** (`mpfr_t op1, mpfr_t op2`)

Return non-zero if

op1orop2is a NaN (i.e., they cannot be compared), zero otherwise.

All those functions, except explicitly stated (for example
`mpfr_sin_cos`

), return a ternary value as defined in Section
“Rounding Modes”, 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`)

— Function: int

— Function: int

Set

ropto the natural logarithm ofop, log2(op) or log10(op), respectively, rounded in the directionrnd. Setropto −Inf ifopis −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`)

— Function: int

— Function: int

Set

ropto the exponential ofop, to 2 power ofopor to 10 power ofop, respectively, rounded in the directionrnd.

— 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`)

— Function: int

— Function: int

Set

ropto the cosine ofop, sine ofop, tangent ofop, rounded in the directionrnd.

— Function: int **mpfr_sin_cos** (`mpfr_t sop, mpfr_t cop, mpfr_t op, mpfr_rnd_t rnd`)

Set simultaneously

sopto the sine ofopandcopto the cosine ofop, rounded in the directionrndwith the corresponding precisions ofsopandcop, which must be different variables. Return 0 iff both results are exact, more precisely it returns s+4c where s=0 ifsopis exact, s=1 ifsopis larger than the sine ofop, s=2 ifsopis smaller than the sine ofop, and similarly for c and the cosine ofop.

— 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`)

— Function: int

— Function: int

Set

ropto the secant ofop, cosecant ofop, cotangent ofop, rounded in the directionrnd.

— 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`)

— Function: int

— Function: int

Set

ropto the arc-cosine, arc-sine or arc-tangent ofop, rounded in the directionrnd. 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 largeopand small precision ofrop.

— Function: int **mpfr_atan2** (`mpfr_t rop, mpfr_t y, mpfr_t x, mpfr_rnd_t rnd`)

Set

ropto the arc-tangent2 ofyandx, rounded in the directionrnd: 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`)

— Function: int

— Function: int

Set

ropto the hyperbolic cosine, sine or tangent ofop, rounded in the directionrnd.

— Function: int **mpfr_sinh_cosh** (`mpfr_t sop, mpfr_t cop, mpfr_t op, mpfr_rnd_t rnd`)

Set simultaneously

sopto the hyperbolic sine ofopandcopto the hyperbolic cosine ofop, rounded in the directionrndwith the corresponding precision ofsopandcop, 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`)

— Function: int

— Function: int

Set

ropto the hyperbolic secant ofop, cosecant ofop, cotangent ofop, rounded in the directionrnd.

— 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`)

— Function: int

— Function: int

Set

ropto the inverse hyperbolic cosine, sine or tangent ofop, rounded in the directionrnd.

— Function: int **mpfr_fac_ui** (`mpfr_t rop, unsigned long int op, mpfr_rnd_t rnd`)

Set

ropto the factorial ofop, rounded in the directionrnd.

— Function: int **mpfr_log1p** (`mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd`)

Set

ropto the logarithm of one plusop, rounded in the directionrnd.

— Function: int **mpfr_expm1** (`mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd`)

Set

ropto the exponential ofopfollowed by a subtraction by one, rounded in the directionrnd.

— Function: int **mpfr_eint** (`mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd`)

Set

ropto the exponential integral ofop, rounded in the directionrnd. For positiveop, the exponential integral is the sum of Euler's constant, of the logarithm ofop, and of the sum for k from 1 to infinity ofopto the power k, divided by k and factorial(k). For negativeop,ropis set to NaN.

— Function: int **mpfr_li2** (`mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd`)

Set

ropto real part of the dilogarithm ofop, rounded in the directionrnd. MPFR defines the dilogarithm function as the integral of -log(1-t)/t from 0 toop.

— Function: int **mpfr_gamma** (`mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd`)

Set

ropto the value of the Gamma function onop, rounded in the directionrnd. Whenopis a negative integer,ropis set to NaN.

— Function: int **mpfr_lngamma** (`mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd`)

Set

ropto the value of the logarithm of the Gamma function onop, rounded in the directionrnd. When −2k−1 <=op<= −2k,kbeing a non-negative integer,ropis set to NaN. See also`mpfr_lgamma`

.

— Function: int **mpfr_lgamma** (`mpfr_t rop, int *signp, mpfr_t op, mpfr_rnd_t rnd`)

Set

ropto the value of the logarithm of the absolute value of the Gamma function onop, rounded in the directionrnd. The sign (1 or −1) of Gamma(op) is returned in the object pointed to bysignp. Whenopis an infinity or a non-positive integer, setropto +Inf. Whenopis NaN, −Inf or a negative integer, *signpis undefined, and whenopis Â±0, *signpis the sign of the zero.

— Function: int **mpfr_digamma** (`mpfr_t rop, mpfr_t op, mpfr_rnd_t rnd`)

Set

ropto the value of the Digamma (sometimes also called Psi) function onop, rounded in the directionrnd. Whenopis a negative integer, setropto 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`)

— Function: int

Set

ropto the value of the Riemann Zeta function onop, rounded in the directionrnd.

— 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`)

— Function: int

Set

ropto the value of the error function onop(resp. the complementary error function onop) rounded in the directionrnd.

— 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`)

— Function: int

— Function: int

Set

ropto the value of the first kind Bessel function of order 0, (resp. 1 andn) onop, rounded in the directionrnd. Whenopis NaN,ropis always set to NaN. Whenopis plus or minus Infinity,ropis set to +0. Whenopis zero, andnis not zero,ropis set to +0 or −0 depending on the parity and sign ofn, and the sign ofop.

— 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`)

— Function: int

— Function: int

Set

ropto the value of the second kind Bessel function of order 0 (resp. 1 andn) onop, rounded in the directionrnd. Whenopis NaN or negative,ropis always set to NaN. Whenopis +Inf,ropis set to +0. Whenopis zero,ropis set to +Inf or −Inf depending on the parity and sign ofn.

— 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`)

— Function: int

Set

ropto (op1timesop2) +op3(resp. (op1timesop2) -op3) rounded in the directionrnd.

— Function: int **mpfr_agm** (`mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd`)

Set

ropto the arithmetic-geometric mean ofop1andop2, rounded in the directionrnd. The arithmetic-geometric mean is the common limit of the sequencesu_nandv_n, whereu_0=op1,v_0=op2,u_(n+1) is the arithmetic mean ofu_nandv_n, andv_(n+1) is the geometric mean ofu_nandv_n. If any operand is negative, setropto NaN.

— Function: int **mpfr_hypot** (`mpfr_t rop, mpfr_t x, mpfr_t y, mpfr_rnd_t rnd`)

Set

ropto the Euclidean norm ofxandy, i.e., the square root of the sum of the squares ofxandy, rounded in the directionrnd. Special values are handled as described in Section F.9.4.3 of the ISO C99 and IEEE 754-2008 standards: Ifxoryis an infinity, then +Inf is returned inrop, even if the other number is NaN.

— Function: int **mpfr_ai** (`mpfr_t rop, mpfr_t x, mpfr_rnd_t rnd`)

Set

ropto the value of the Airy function Ai onx, rounded in the directionrnd. Whenxis NaN,ropis always set to NaN. Whenxis +Inf or −Inf,ropis +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`)

— Function: int

— Function: int

— Function: int

Set

ropto the logarithm of 2, the value of Pi, of Euler's constant 0.577..., of Catalan's constant 0.915..., respectively, rounded in the directionrnd. 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

ropto the sum of all elements oftab, whose size isn, rounded in the directionrnd. Warning: for efficiency reasons,tabis an array of pointers to`mpfr_t`

, not an array of`mpfr_t`

. If the returned`int`

value is zero,ropis guaranteed to be the exact sum; otherwiseropmight 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.

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

opon streamstream, as a string of digits in basebase, rounded in the directionrnd. The base may vary from 2 to 62. Printnsignificant digits exactly, or ifnis 0, enough digits so thatopcan 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. Ifbaseis 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

basefrom streamstream, rounded in the directionrnd, and put the read float inrop.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.

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.

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:
see Section “Formatted Output Strings” in GNU MP.
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 ‘`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 `unsigned 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 ‘ *’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.

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`)

— Function: int

Print to the stream

streamthe optional arguments under the control of the template stringtemplate. 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`)

— Function: int

Print to

`stdout`

the optional arguments under the control of the template stringtemplate. 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`)

— Function: int

Form a null-terminated string corresponding to the optional arguments under the control of the template string

template, and print it inbuf. No overlap is permitted betweenbufand the other arguments. Return the number of characters written in the arraybufnot countingthe 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`)

— Function: int

Form a null-terminated string corresponding to the optional arguments under the control of the template string

template, and print it inbuf. Ifnis zero, nothing is written andbufmay be a null pointer, otherwise, then−1 first characters are written inbufand then-th is a null character. Return the number of characters that would have been written hadnbe sufficiently large,not countingthe 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`)

— Function: int

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.

— 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`)

— Function: int

— Function: int

— Function: int

— Function: int

Set

roptooprounded to an integer.`mpfr_rint`

rounds to the nearest representable integer in the given directionrnd,`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 whenopis an integer representable inrop, 1 or −1 whenopis an integer that is not representable inrop, 2 or −2 whenopis not an integer.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`)

— Function: int

— Function: int

— Function: int

Set

roptooprounded 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 directionrnd. 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: firstopis 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 directionrnd. 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

ropto the fractional part ofop, having the same sign asop, rounded in the directionrnd(unlike in`mpfr_rint`

,rndaffects 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

iopto the integral part ofopandfopto the fractional part ofop, rounded in the directionrndwith the corresponding precision ofiopandfop(equivalent to`mpfr_trunc(`

iop`,`

op`,`

rnd`)`

and`mpfr_frac(`

fop`,`

op`,`

rnd`)`

). The variablesiopandfopmust 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`)

— Function: int

— Function: int

Set

rto the value ofx-ny, rounded according to the directionrnd, wherenis the integer quotient ofxdivided byy, defined as follows:nis 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

xis infinite oryis zero,ris NaN. Ifyis infinite andxis finite,risxrounded to the precision ofr. Ifris zero, it has the sign ofx. The return value is the ternary value corresponding tor.Additionally,

`mpfr_remquo`

stores the low significant bits from the quotientnin*q(more precisely the number of bits in a`long`

minus one), with the sign ofxdivided byy(except if those low bits are all zero, in which case zero is returned). Note thatxmay be so large in magnitude relative toythat an exact representation of the quotient is not practical. The`mpfr_remainder`

and`mpfr_remquo`

functions are useful for additive argument reduction.

— 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: int **mpfr_prec_round** (`mpfr_t x, mpfr_prec_t prec, mpfr_rnd_t rnd`)

Round

xaccording torndwith precisionprec, which must be an integer between`MPFR_PREC_MIN`

and`MPFR_PREC_MAX`

(otherwise the behavior is undefined). Ifprecis greater or equal to the precision ofx, then new space is allocated for the significand, and it is filled with zeros. Otherwise, the significand is rounded to precisionprecwith the given direction. In both cases, the precision ofxis changed toprec.Here is an example of how to use

`mpfr_prec_round`

to implement Newton's algorithm to compute the inverse ofa, assumingxis already an approximation tonbits: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 */

— 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

bis an approximation of an unknown numberxin the directionrnd1with error at most two to the power E(b)-errwhere E(b) is the exponent ofb, return a non-zero value if one is able to round correctlyxto precisionprecwith the directionrnd2, and 0 otherwise (including for NaN and Inf). This functiondoes not modifyits arguments.If

rnd1is`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 forrnd1.Note: if one wants to also determine the correct ternary value when rounding

bto precisionprecwith rounding modernd, a useful trick is the following:if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ, prec + (rnd == MPFR_RNDN))) ...Indeed, ifrndis`MPFR_RNDN`

, this will check if one can round toprec+1 bits with a directed rounding: if so, one can surely round to nearest toprecbits, and in addition one can determine the correct ternary value, which would not be the case whenbis near from a value exactly representable onprecbits.

— 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 ifrndis an invalid rounding mode.

— Function: void **mpfr_nexttoward** (`mpfr_t x, mpfr_t y`)

If

xoryis NaN, setxto NaN. Ifxandyare equal,xis unchanged. Otherwise, ifxis different fromy, replacexby the next floating-point number (with the precision ofxand the current exponent range) in the direction ofy(the infinite values are seen as the smallest and largest floating-point numbers). If the result is zero, it keeps the same sign. No underflow or overflow is generated.

— Function: void **mpfr_nextabove** (`mpfr_t x`)

— Function: void**mpfr_nextbelow** (`mpfr_t x`)

— Function: void

Equivalent to

`mpfr_nexttoward`

whereyis plus infinity (resp. minus infinity).

— Function: int **mpfr_min** (`mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd`)

— Function: int**mpfr_max** (`mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd`)

— Function: int

Set

ropto the minimum (resp. maximum) ofop1andop2. Ifop1andop2are both NaN, thenropis set to NaN. Ifop1orop2is NaN, thenropis set to the numeric value. Ifop1andop2are zeros of different signs, thenropis set to −0 (resp. +0).

— Function: int **mpfr_urandomb** (`mpfr_t rop, gmp_randstate_t state`)

Generate a uniformly distributed random float in the interval 0 <=

rop< 1. More precisely, the number can be seen as a float with a random non-normalized significand and exponent 0, which is then normalized (thus ifedenotes the exponent after normalization, then the least -esignificant bits of the significand are always 0).Return 0, unless the exponent is not in the current exponent range, in which case

ropis set to NaN and a non-zero value is returned (this should never happen in practice, except in very specific cases). The second argument is a`gmp_randstate_t`

structure which should be created using the GMP`gmp_randinit`

function (see the GMP manual).

— Function: int **mpfr_urandom** (`mpfr_t rop, gmp_randstate_t state, mpfr_rnd_t rnd`)

Generate a uniformly distributed random float. The floating-point number

ropcan be seen as if a random real number is generated according to the continuous uniform distribution on the interval [0, 1] and then rounded in the directionrnd.The second argument is a

`gmp_randstate_t`

structure which should be created using the GMP`gmp_randinit`

function (see the GMP manual).

— Function: mpfr_exp_t **mpfr_get_exp** (`mpfr_t x`)

Return the exponent of

x, assuming thatxis a non-zero ordinary number and the significand is considered in [1/2,1). The behavior for NaN, infinity or zero is undefined.

— Function: int **mpfr_set_exp** (`mpfr_t x, mpfr_exp_t e`)

Set the exponent of

xifeis in the current exponent range, and return 0 (even ifxis not a non-zero ordinary number); otherwise, return a non-zero value. The significand is assumed to be in [1/2,1).

— Function: int **mpfr_signbit** (`mpfr_t op`)

Return a non-zero value iff

ophas its sign bit set (i.e., if it is negative, −0, or a NaN whose representation has its sign bit set).

— Function: int **mpfr_setsign** (`mpfr_t rop, mpfr_t op, int s, mpfr_rnd_t rnd`)

Set the value of

ropfromop, rounded toward the given directionrnd, then set (resp. clear) its sign bit ifsis non-zero (resp. zero), even whenopis a NaN.

— Function: int **mpfr_copysign** (`mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd`)

Set the value of

ropfromop1, rounded toward the given directionrnd, then set its sign bit to that ofop2(even whenop1orop2is a NaN). This function is equivalent to`mpfr_setsign (`

rop`,`

op1`, mpfr_signbit (`

op2`),`

rnd`)`

.

— Function: const char * **mpfr_get_version** (`void`)

Return the MPFR version, as a null-terminated string.

— Macro: **MPFR_VERSION**

— Macro:**MPFR_VERSION_MAJOR**

— Macro:**MPFR_VERSION_MINOR**

— Macro:**MPFR_VERSION_PATCHLEVEL**

— Macro:**MPFR_VERSION_STRING**

— Macro:

— Macro:

— Macro:

— Macro:

`MPFR_VERSION`

is the version of MPFR as a preprocessing constant.`MPFR_VERSION_MAJOR`

,`MPFR_VERSION_MINOR`

and`MPFR_VERSION_PATCHLEVEL`

are respectively the major, minor and patch level of MPFR version, as preprocessing constants.`MPFR_VERSION_STRING`

is the version (with an optional suffix, used in development and pre-release versions) as a string constant, which can be compared to the result of`mpfr_get_version`

to check at run time the header file and library used match:if (strcmp (mpfr_get_version (), MPFR_VERSION_STRING)) fprintf (stderr, "Warning: header and library do not match\n");Note: Obtaining different strings is not necessarily an error, as in general, a program compiled with some old MPFR version can be dynamically linked with a newer MPFR library version (if allowed by the library versioning system).

— Macro: long **MPFR_VERSION_NUM** (`major, minor, patchlevel`)

Create an integer in the same format as used by

`MPFR_VERSION`

from the givenmajor,minorandpatchlevel. Here is an example of how to check the MPFR version at compile time:#if (!defined(MPFR_VERSION) || (MPFR_VERSION<MPFR_VERSION_NUM(3,0,0))) # error "Wrong MPFR version." #endif

— Function: const char * **mpfr_get_patches** (`void`)

Return a null-terminated string containing the ids of the patches applied to the MPFR library (contents of the

PATCHESfile), separated by spaces. Note: If the program has been compiled with an older MPFR version and is dynamically linked with a new MPFR library version, the identifiers of the patches applied to the old (compile-time) MPFR version are not available (however this information should not have much interest in general).

— Function: int **mpfr_buildopt_tls_p** (`void`)

Return a non-zero value if MPFR was compiled as thread safe using compiler-level Thread Local Storage (that is MPFR was built with the

`--enable-thread-safe`

configure option, see`INSTALL`

file), return zero otherwise.

— Function: int **mpfr_buildopt_decimal_p** (`void`)

Return a non-zero value if MPFR was compiled with decimal float support (that is MPFR was built with the

`--enable-decimal-float`

configure option), return zero otherwise.

— Function: mpfr_exp_t **mpfr_get_emin** (`void`)

— Function: mpfr_exp_t**mpfr_get_emax** (`void`)

— Function: mpfr_exp_t

Return the (current) smallest and largest exponents allowed for a floating-point variable. The smallest positive value of a floating-point variable is one half times 2 raised to the smallest exponent and the largest value has the form (1 - epsilon) times 2 raised to the largest exponent, where epsilon depends on the precision of the considered variable.

— Function: int **mpfr_set_emin** (`mpfr_exp_t exp`)

— Function: int**mpfr_set_emax** (`mpfr_exp_t exp`)

— Function: int

Set the smallest and largest exponents allowed for a floating-point variable. Return a non-zero value when

expis not in the range accepted by the implementation (in that case the smallest or largest exponent is not changed), and zero otherwise. If the user changes the exponent range, it is her/his responsibility to check that all current floating-point variables are in the new allowed range (for example using`mpfr_check_range`

), otherwise the subsequent behavior will be undefined, in the sense of the ISO C standard.

— Function: mpfr_exp_t **mpfr_get_emin_min** (`void`)

— Function: mpfr_exp_t**mpfr_get_emin_max** (`void`)

— Function: mpfr_exp_t**mpfr_get_emax_min** (`void`)

— Function: mpfr_exp_t**mpfr_get_emax_max** (`void`)

— Function: mpfr_exp_t

— Function: mpfr_exp_t

— Function: mpfr_exp_t

Return the minimum and maximum of the exponents allowed for

`mpfr_set_emin`

and`mpfr_set_emax`

respectively. These values are implementation dependent, thus a program using`mpfr_set_emax(mpfr_get_emax_max())`

or`mpfr_set_emin(mpfr_get_emin_min())`

may not be portable.

— Function: int **mpfr_check_range** (`mpfr_t x, int t, mpfr_rnd_t rnd`)

This function assumes that

xis the correctly-rounded value of some real valueyin the directionrndand some extended exponent range, and thattis the corresponding ternary value. For example, one performed`t = mpfr_log (x, u, rnd)`

, andyis the exact logarithm ofu. Thustis negative ifxis smaller thany, positive ifxis larger thany, and zero ifxequalsy. This function modifiesxif needed to be in the current range of acceptable values: It generates an underflow or an overflow if the exponent ofxis outside the current allowed range; the value oftmay be used to avoid a double rounding. This function returns zero if the new value ofxequals the exact oney, a positive value if that new value is larger thany, and a negative value if it is smaller thany. Note that unlike most functions, the new resultxis compared to the (unknown) exact oney, not the input valuex, i.e., the ternary value is propagated.Note: If

xis an infinity andtis different from zero (i.e., if the rounded result is an inexact infinity), then the overflow flag is set. This is useful because`mpfr_check_range`

is typically called (at least in MPFR functions) after restoring the flags that could have been set due to internal computations.

— Function: int **mpfr_subnormalize** (`mpfr_t x, int t, mpfr_rnd_t rnd`)

This function rounds

xemulating subnormal number arithmetic: ifxis outside the subnormal exponent range, it just propagates the ternary valuet; otherwise, it roundsxto precision`EXP(x)-emin+1`

according to rounding moderndand previous ternary valuet, avoiding double rounding problems. More precisely in the subnormal domain, denoting byethe value of`emin`

,xis rounded in fixed-point arithmetic to an integer multiple of two to the powere−1; as a consequence, 1.5 multiplied by two to the powere−1 whentis zero is rounded to two to the powerewith rounding to nearest.

`PREC(x)`

is not modified by this function.rndandtmust be the rounding mode and the returned ternary value used when computingx(as in`mpfr_check_range`

). The subnormal exponent range is from`emin`

to`emin+PREC(x)-1`

. If the result cannot be represented in the current exponent range (due to a too small`emax`

), the behavior is undefined. Note that unlike most functions, the result is compared to the exact one, not the input valuex, i.e., the ternary value is propagated.As usual, if the returned ternary value is non zero, the inexact flag is set. Moreover, if a second rounding occurred (because the input

xwas in the subnormal range), the underflow flag is set.

This is an example of how to emulate binary double IEEE 754 arithmetic (binary64 in IEEE 754-2008) using MPFR:

{ mpfr_t xa, xb; int i; volatile double a, b; mpfr_set_default_prec (53); mpfr_set_emin (-1073); mpfr_set_emax (1024); mpfr_init (xa); mpfr_init (xb); b = 34.3; mpfr_set_d (xb, b, MPFR_RNDN); a = 0x1.1235P-1021; mpfr_set_d (xa, a, MPFR_RNDN); a /= b; i = mpfr_div (xa, xa, xb, MPFR_RNDN); i = mpfr_subnormalize (xa, i, MPFR_RNDN); /* new ternary value */ mpfr_clear (xa); mpfr_clear (xb); }

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

— Function: void **mpfr_clear_underflow** (`void`)

— Function: void**mpfr_clear_overflow** (`void`)

— Function: void**mpfr_clear_nanflag** (`void`)

— Function: void**mpfr_clear_inexflag** (`void`)

— Function: void**mpfr_clear_erangeflag** (`void`)

— Function: void

— Function: void

— Function: void

— Function: void

Clear the underflow, overflow, invalid, inexact and

erangeflags.

— Function: void **mpfr_set_underflow** (`void`)

— Function: void**mpfr_set_overflow** (`void`)

— Function: void**mpfr_set_nanflag** (`void`)

— Function: void**mpfr_set_inexflag** (`void`)

— Function: void**mpfr_set_erangeflag** (`void`)

— Function: void

— Function: void

— Function: void

— Function: void

Set the underflow, overflow, invalid, inexact and

erangeflags.

— Function: void **mpfr_clear_flags** (`void`)

Clear all global flags (underflow, overflow, invalid, inexact,

erange).

— Function: int **mpfr_underflow_p** (`void`)

— Function: int**mpfr_overflow_p** (`void`)

— Function: int**mpfr_nanflag_p** (`void`)

— Function: int**mpfr_inexflag_p** (`void`)

— Function: int**mpfr_erangeflag_p** (`void`)

— Function: int

— Function: int

— Function: int

— Function: int

Return the corresponding (underflow, overflow, invalid, inexact,

erange) flag, which is non-zero iff the flag is set.

A header file `mpf2mpfr.h` is included in the distribution of MPFR for
compatibility with the GNU MP class MPF.
By inserting the following two lines after the `#include <gmp.h>`

line,

#include <mpfr.h> #include <mpf2mpfr.h>any program written for MPF can be compiled directly with MPFR without any changes (except the

`gmp_printf`

functions will not work for arguments of type
`mpfr_t`

).
All operations are then performed with the default MPFR rounding mode,
which can be reset with `mpfr_set_default_rounding_mode`

.
Warning: the `mpf_init`

and `mpf_init2`

functions initialize
to zero, whereas the corresponding MPFR functions initialize to NaN:
this is useful to detect uninitialized values, but is slightly incompatible
with MPF.

— Function: void **mpfr_set_prec_raw** (`mpfr_t x, mpfr_prec_t prec`)

Reset the precision of

xto beexactlyprecbits. The only difference with`mpfr_set_prec`

is thatprecis assumed to be small enough so that the significand fits into the current allocated memory space forx. Otherwise the behavior is undefined.

— Function: int **mpfr_eq** (`mpfr_t op1, mpfr_t op2, unsigned long int op3`)

Return non-zero if

op1andop2are both non-zero ordinary numbers with the same exponent and the same firstop3bits, both zero, or both infinities of the same sign. Return zero otherwise. This function is defined for compatibility with MPF, we do not recommend to use it otherwise. Do not use it either if you want to know whether two numbers are close to each other; for instance, 1.011111 and 1.100000 are regarded as different for any value ofop3larger than 1.

— Function: void **mpfr_reldiff** (`mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_rnd_t rnd`)

Compute the relative difference between

op1andop2and store the result inrop. This function does not guarantee the correct rounding on the relative difference; it just computes |op1-op2|/op1, using the precision ofropand the rounding moderndfor all operations.

— Function: int **mpfr_mul_2exp** (`mpfr_t rop, mpfr_t op1, unsigned long int op2, mpfr_rnd_t rnd`)

— Function: int**mpfr_div_2exp** (`mpfr_t rop, mpfr_t op1, unsigned long int op2, mpfr_rnd_t rnd`)

— Function: int

These functions are identical to

`mpfr_mul_2ui`

and`mpfr_div_2ui`

respectively. These functions are only kept for compatibility with MPF, one should prefer`mpfr_mul_2ui`

and`mpfr_div_2ui`

otherwise.

Some applications use a stack to handle the memory and their objects. However, the MPFR memory design is not well suited for such a thing. So that such applications are able to use MPFR, an auxiliary memory interface has been created: the Custom Interface.

The following interface allows one to use MPFR in two ways:

- Either directly store a floating-point number as a
`mpfr_t`

on the stack. - Either store its own representation on the
stack and construct a new temporary
`mpfr_t`

each time it is needed.

Each function in this interface is also implemented as a macro for
efficiency reasons: for example `mpfr_custom_init (s, p)`

uses the macro, while `(mpfr_custom_init) (s, p)`

uses the function.

Note 1: MPFR functions may still initialize temporary floating-point numbers
using `mpfr_init`

and similar functions. See Custom Allocation (GNU MP).

Note 2: MPFR functions may use the cached functions (`mpfr_const_pi`

for
example), even if they are not explicitly called. You have to call
`mpfr_free_cache`

each time you garbage the memory iff `mpfr_init`

,
through GMP Custom Allocation, allocates its memory on the application stack.

— Function: size_t **mpfr_custom_get_size** (`mpfr_prec_t prec`)

Return the needed size in bytes to store the significand of a floating-point number of precision

prec.

— Function: void **mpfr_custom_init** (`void *significand, mpfr_prec_t prec`)

Initialize a significand of precision

prec, wheresignificandmust be an area of`mpfr_custom_get_size (prec)`

bytes at least and be suitably aligned for an array of`mp_limb_t`

(GMP type, see Internals).

— Function: void **mpfr_custom_init_set** (`mpfr_t x, int kind, mpfr_exp_t exp, mpfr_prec_t prec, void *significand`)

Perform a dummy initialization of a

`mpfr_t`

and set it to:In all cases, it uses

- if
`ABS(kind) == MPFR_NAN_KIND`

,xis set to NaN;- if
`ABS(kind) == MPFR_INF_KIND`

,xis set to the infinity of sign`sign(kind)`

;- if
`ABS(kind) == MPFR_ZERO_KIND`

,xis set to the zero of sign`sign(kind)`

;- if
`ABS(kind) == MPFR_REGULAR_KIND`

,xis set to a regular number:`x = sign(kind)*significand*2^exp`

.significanddirectly for further computing involvingx. It will not allocate anything. A floating-point number initialized with this function cannot be resized using`mpfr_set_prec`

or`mpfr_prec_round`

, or cleared using`mpfr_clear`

! Thesignificandmust have been initialized with`mpfr_custom_init`

using the same precisionprec.

— Function: int **mpfr_custom_get_kind** (`mpfr_t x`)

Return the current kind of a

`mpfr_t`

as created by`mpfr_custom_init_set`

. The behavior of this function for any`mpfr_t`

not initialized with`mpfr_custom_init_set`

is undefined.

— Function: void * **mpfr_custom_get_significand** (`mpfr_t x`)

Return a pointer to the significand used by a

`mpfr_t`

initialized with`mpfr_custom_init_set`

. The behavior of this function for any`mpfr_t`

not initialized with`mpfr_custom_init_set`

is undefined.

— Function: mpfr_exp_t **mpfr_custom_get_exp** (`mpfr_t x`)

Return the exponent of

x, assuming thatxis a non-zero ordinary number. The return value for NaN, Infinity or zero is unspecified but does not produce any trap. The behavior of this function for any`mpfr_t`

not initialized with`mpfr_custom_init_set`

is undefined.

— Function: void **mpfr_custom_move** (`mpfr_t x, void *new_position`)

Inform MPFR that the significand of

xhas moved due to a garbage collect and update its new position to`new_position`

. However the application has to move the significand and the`mpfr_t`

itself. The behavior of this function for any`mpfr_t`

not initialized with`mpfr_custom_init_set`

is undefined.

A limb means the part of a multi-precision number that fits in a single
word. Usually a limb contains
32 or 64 bits. The C data type for a limb is `mp_limb_t`

.

The `mpfr_t`

type is internally defined as a one-element
array of a structure, and `mpfr_ptr`

is the C data type representing
a pointer to this structure.
The `mpfr_t`

type consists of four fields:

- The
`_mpfr_prec`

field is used to store the precision of the variable (in bits); this is not less than`MPFR_PREC_MIN`

. - The
`_mpfr_sign`

field is used to store the sign of the variable. - The
`_mpfr_exp`

field stores the exponent. An exponent of 0 means a radix point just above the most significant limb. Non-zero values n are a multiplier 2^n relative to that point. A NaN, an infinity and a zero are indicated by special values of the exponent field. - Finally, the
`_mpfr_d`

field is a pointer to the limbs, least significant limbs stored first. The number of limbs in use is controlled by`_mpfr_prec`

, namely ceil(`_mpfr_prec`

/`mp_bits_per_limb`

). Non-singular (i.e., different from NaN, Infinity or zero) values always have the most significant bit of the most significant limb set to 1. When the precision does not correspond to a whole number of limbs, the excess bits at the low end of the data are zeros.

The goal of this section is to describe some API changes that occurred from one version of MPFR to another, and how to write code that can be compiled and run with older MPFR versions. The minimum MPFR version that is considered here is 2.2.0 (released on 20 September 2005).

API changes can only occur between major or minor versions. Thus the patchlevel (the third number in the MPFR version) will be ignored in the following. If a program does not use MPFR internals, changes in the behavior between two versions differing only by the patchlevel should only result from what was regarded as a bug or unspecified behavior.

As a general rule, a program written for some MPFR version should work with later versions, possibly except at a new major version, where some features (described as obsolete for some time) can be removed. In such a case, a failure should occur during compilation or linking. If a result becomes incorrect because of such a change, please look at the various changes below (they are minimal, and most software should be unaffected), at the FAQ and at the MPFR web page for your version (a bug could have been introduced and be already fixed); and if the problem is not mentioned, please send us a bug report (see Reporting Bugs).

However, a program written for the current MPFR version (as documented by this manual) may not necessarily work with previous versions of MPFR. This section should help developers to write portable code.

Note: Information given here may be incomplete. API changes are also described in the NEWS file (for each version, instead of being classified like here), together with other changes.

The official type for exponent values changed from `mp_exp_t`

to
`mpfr_exp_t`

in MPFR 3.0. The type `mp_exp_t`

will remain
available as it comes from GMP (with a different meaning). These types
are currently the same (`mpfr_exp_t`

is defined as `mp_exp_t`

with `typedef`

), so that programs can still use `mp_exp_t`

;
but this may change in the future.
Alternatively, using the following code after including `mpfr.h`
will work with official MPFR versions, as `mpfr_exp_t`

was never
defined in MPFR 2.x:

#if MPFR_VERSION_MAJOR < 3 typedef mp_exp_t mpfr_exp_t; #endif

The official types for precision values and for rounding modes
respectively changed from `mp_prec_t`

and `mp_rnd_t`

to `mpfr_prec_t`

and `mpfr_rnd_t`

in MPFR 3.0. This
change was actually done a long time ago in MPFR, at least since
MPFR 2.2.0, with the following code in `mpfr.h`:

#ifndef mp_rnd_t # define mp_rnd_t mpfr_rnd_t #endif #ifndef mp_prec_t # define mp_prec_t mpfr_prec_t #endif

This means that it is safe to use the new official types
`mpfr_prec_t`

and `mpfr_rnd_t`

in your programs.
The types `mp_prec_t`

and `mp_rnd_t`

(defined
in MPFR only) may be removed in the future, as the prefix
`mp_`

is reserved by GMP.

The precision type `mpfr_prec_t`

(`mp_prec_t`

) was unsigned
before MPFR 3.0; it is now signed. `MPFR_PREC_MAX`

has not changed,
though. Indeed the MPFR code requires that `MPFR_PREC_MAX`

be
representable in the exponent type, which may have the same size as
`mpfr_prec_t`

but has always been signed.
The consequence is that valid code that does not assume anything about
the signedness of `mpfr_prec_t`

should work with past and new MPFR
versions.
This change was useful as the use of unsigned types tends to convert
signed values to unsigned ones in expressions due to the usual arithmetic
conversions, which can yield incorrect results if a negative value is
converted in such a way.
Warning! A program assuming (intentionally or not) that
`mpfr_prec_t`

is signed may be affected by this problem when
it is built and run against MPFR 2.x.

The rounding modes `GMP_RNDx`

were renamed to `MPFR_RNDx`

in MPFR 3.0. However the old names `GMP_RNDx`

have been kept for
compatibility (this might change in future versions), using:

#define GMP_RNDN MPFR_RNDN #define GMP_RNDZ MPFR_RNDZ #define GMP_RNDU MPFR_RNDU #define GMP_RNDD MPFR_RNDD

The rounding mode “round away from zero” (`MPFR_RNDA`

) was added in
MPFR 3.0 (however no rounding mode `GMP_RNDA`

exists).

We give here in alphabetical order the functions that were added after MPFR 2.2, and in which MPFR version.

`mpfr_add_d`

in MPFR 2.4.`mpfr_ai`

in MPFR 3.0 (incomplete, experimental).`mpfr_asprintf`

in MPFR 2.4.`mpfr_buildopt_decimal_p`

and`mpfr_buildopt_tls_p`

in MPFR 3.0.`mpfr_copysign`

in MPFR 2.3. Note: MPFR 2.2 had a`mpfr_copysign`

function that was available, but not documented, and with a slight difference in the semantics (when the second input operand is a NaN).`mpfr_custom_get_significand`

in MPFR 3.0. This function was named`mpfr_custom_get_mantissa`

in previous versions;`mpfr_custom_get_mantissa`

is still available via a macro in`mpfr.h`:#define mpfr_custom_get_mantissa mpfr_custom_get_significand

Thus code that needs to work with both MPFR 2.x and MPFR 3.x should use

`mpfr_custom_get_mantissa`

.`mpfr_d_div`

and`mpfr_d_sub`

in MPFR 2.4.`mpfr_digamma`

in MPFR 3.0.`mpfr_div_d`

in MPFR 2.4.`mpfr_fmod`

in MPFR 2.4.`mpfr_fms`

in MPFR 2.3.`mpfr_fprintf`

in MPFR 2.4.`mpfr_get_flt`

in MPFR 3.0.`mpfr_get_patches`

in MPFR 2.3.`mpfr_get_z_2exp`

in MPFR 3.0. This function was named`mpfr_get_z_exp`

in previous versions;`mpfr_get_z_exp`

is still available via a macro in`mpfr.h`:#define mpfr_get_z_exp mpfr_get_z_2exp

Thus code that needs to work with both MPFR 2.x and MPFR 3.x should use

`mpfr_get_z_exp`

.`mpfr_j0`

,`mpfr_j1`

and`mpfr_jn`

in MPFR 2.3.`mpfr_lgamma`

in MPFR 2.3.`mpfr_li2`

in MPFR 2.4.`mpfr_modf`

in MPFR 2.4.`mpfr_mul_d`

in MPFR 2.4.`mpfr_printf`

in MPFR 2.4.`mpfr_rec_sqrt`

in MPFR 2.4.`mpfr_regular_p`

in MPFR 3.0.`mpfr_remainder`

and`mpfr_remquo`

in MPFR 2.3.`mpfr_set_flt`

in MPFR 3.0.`mpfr_set_z_2exp`

in MPFR 3.0.`mpfr_set_zero`

in MPFR 3.0.`mpfr_setsign`

in MPFR 2.3.`mpfr_signbit`

in MPFR 2.3.`mpfr_sinh_cosh`

in MPFR 2.4.`mpfr_snprintf`

and`mpfr_sprintf`

in MPFR 2.4.`mpfr_sub_d`

in MPFR 2.4.`mpfr_urandom`

in MPFR 3.0.`mpfr_vasprintf`

,`mpfr_vfprintf`

,`mpfr_vprintf`

,`mpfr_vsprintf`

and`mpfr_vsnprintf`

in MPFR 2.4.`mpfr_y0`

,`mpfr_y1`

and`mpfr_yn`

in MPFR 2.3.

The following functions have changed after MPFR 2.2. Changes can affect the behavior of code written for some MPFR version when built and run against another MPFR version (older or newer), as described below.

`mpfr_check_range`

changed in MPFR 2.3.2 and MPFR 2.4. If the value is an inexact infinity, the overflow flag is now set (in case it was lost), while it was previously left unchanged. This is really what is expected in practice (and what the MPFR code was expecting), so that the previous behavior was regarded as a bug. Hence the change in MPFR 2.3.2.`mpfr_get_f`

changed in MPFR 3.0. This function was returning zero, except for NaN and Inf, which do not exist in MPF. The*erange*flag is now set in these cases, and`mpfr_get_f`

now returns the usual ternary value.`mpfr_get_si`

,`mpfr_get_sj`

,`mpfr_get_ui`

and`mpfr_get_uj`

changed in MPFR 3.0. In previous MPFR versions, the cases where the*erange*flag is set were unspecified.`mpfr_get_z`

changed in MPFR 3.0. The return type was`void`

; it is now`int`

, and the usual ternary value is returned. Thus programs that need to work with both MPFR 2.x and 3.x must not use the return value. Even in this case, C code using`mpfr_get_z`

as the second or third term of a conditional operator may also be affected. For instance, the following is correct with MPFR 3.0, but not with MPFR 2.x:bool ? mpfr_get_z(...) : mpfr_add(...);

On the other hand, the following is correct with MPFR 2.x, but not with MPFR 3.0:

bool ? mpfr_get_z(...) : (void) mpfr_add(...);

Portable code should cast

`mpfr_get_z(...)`

to`void`

to use the type`void`

for both terms of the conditional operator, as in:bool ? (void) mpfr_get_z(...) : (void) mpfr_add(...);

Alternatively,

`if ... else`

can be used instead of the conditional operator.Moreover the cases where the

*erange*flag is set were unspecified in MPFR 2.x.`mpfr_get_z_exp`

changed in MPFR 3.0. In previous MPFR versions, the cases where the*erange*flag is set were unspecified. Note: this function has been renamed to`mpfr_get_z_2exp`

in MPFR 3.0, but`mpfr_get_z_exp`

is still available for compatibility reasons.`mpfr_strtofr`

changed in MPFR 2.3.1 and MPFR 2.4. This was actually a bug fix since the code and the documentation did not match. But both were changed in order to have a more consistent and useful behavior. The main changes in the code are as follows. The binary exponent is now accepted even without the`0b`

or`0x`

prefix. Data corresponding to NaN can now have an optional sign (such data were previously invalid).`mpfr_strtofr`

changed in MPFR 3.0. This function now accepts bases from 37 to 62 (no changes for the other bases). Note: if an unsupported base is provided to this function, the behavior is undefined; more precisely, in MPFR 2.3.1 and later, providing an unsupported base yields an assertion failure (this behavior may change in the future).`mpfr_subnormalize`

changed in MPFR 3.0.1. This was actually regarded as a bug fix. The`mpfr_subnormalize`

implementation up to MPFR 3.0.0 did not change the flags. In particular, it did not follow the generic rule concerning the inexact flag (and no special behavior was specified). The case of the underflow flag was more a lack of specification.

Functions `mpfr_random`

and `mpfr_random2`

have been
removed in MPFR 3.0 (this only affects old code built against
MPFR 3.0 or later).
(The function `mpfr_random`

had been deprecated since at least MPFR 2.2.0,
and `mpfr_random2`

since MPFR 2.4.0.)

For users of a C++ compiler, the way how the availability of `intmax_t`

is detected has changed in MPFR 3.0.
In MPFR 2.x, if a macro `INTMAX_C`

or `UINTMAX_C`

was defined
(e.g. when the `__STDC_CONSTANT_MACROS`

macro had been defined
before `<stdint.h>`

or `<inttypes.h>`

has been included),
`intmax_t`

was assumed to be defined.
However this was not always the case (more precisely, `intmax_t`

can be defined only in the namespace `std`

, as with Boost), so
that compilations could fail.
Thus the check for `INTMAX_C`

or `UINTMAX_C`

is now disabled for
C++ compilers, with the following consequences:

- Programs written for MPFR 2.x that need
`intmax_t`

may no longer be compiled against MPFR 3.0: a`#define MPFR_USE_INTMAX_T`

may be necessary before`mpfr.h`is included. - The compilation of programs that work with MPFR 3.0 may fail with
MPFR 2.x due to the problem described above. Workarounds are possible,
such as defining
`intmax_t`

and`uintmax_t`

in the global namespace, though this is not clean.

The main developers of MPFR are Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier, Philippe Théveny and Paul Zimmermann.

Sylvie Boldo from ENS-Lyon, France,
contributed the functions `mpfr_agm`

and `mpfr_log`

.
Emmanuel Jeandel, from ENS-Lyon too,
contributed the generic hypergeometric code,
as well as the internal function `mpfr_exp3`

,
a first implementation of the sine and cosine,
and improved versions of
`mpfr_const_log2`

and `mpfr_const_pi`

.
Mathieu Dutour contributed the functions `mpfr_atan`

and `mpfr_asin`

,
and a previous version of `mpfr_gamma`

;
David Daney contributed the hyperbolic and inverse hyperbolic functions,
the base-2 exponential, and the factorial function. Fabrice Rouillier
contributed the `mpfr_xxx_z`

and `mpfr_xxx_q`

functions,
and helped to the Microsoft Windows porting.
Jean-Luc Rémy contributed the `mpfr_zeta`

code.
Ludovic Meunier helped in the design of the `mpfr_erf`

code.
Damien Stehlé contributed the `mpfr_get_ld_2exp`

function.
Sylvain Chevillard contributed the `mpfr_ai`

function.

We would like to thank Jean-Michel Muller and Joris van der Hoeven for very fruitful discussions at the beginning of that project, Torbjörn Granlund and Kevin Ryde for their help about design issues, and Nathalie Revol for her careful reading of a previous version of this documentation. In particular Kevin Ryde did a tremendous job for the portability of MPFR in 2002-2004.

The development of the MPFR library would not have been possible without the continuous support of INRIA, and of the LORIA (Nancy, France) and LIP (Lyon, France) laboratories. In particular the main authors were or are members of the PolKA, Spaces, Cacao and Caramel project-teams at LORIA and of the Arénaire project-team at LIP. This project was started during the Fiable (reliable in French) action supported by INRIA, and continued during the AOC action. The development of MPFR was also supported by a grant (202F0659 00 MPN 121) from the Conseil Régional de Lorraine in 2002, from INRIA by an "associate engineer" grant (2003-2005), an "opération de développement logiciel" grant (2007-2009), and the post-doctoral grant of Sylvain Chevillard in 2009-2010.

- Richard Brent and Paul Zimmermann, "Modern Computer Arithmetic", Cambridge University Press (to appear), also available from the authors' web pages.
- Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier and Paul Zimmermann, "MPFR: A Multiple-Precision Binary Floating-Point Library With Correct Rounding", ACM Transactions on Mathematical Software, volume 33, issue 2, article 13, 15 pages, 2007, http://doi.acm.org/10.1145/1236463.1236468.
- Torbjörn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic Library", version 5.0.1, 2010, https://gmplib.org.
- IEEE standard for binary floating-point arithmetic, Technical Report ANSI-IEEE Standard 754-1985, New York, 1985. Approved March 21, 1985: IEEE Standards Board; approved July 26, 1985: American National Standards Institute, 18 pages.
- IEEE Standard for Floating-Point Arithmetic, ANSI-IEEE Standard 754-2008, 2008. Revision of ANSI-IEEE Standard 754-1985, approved June 12, 2008: IEEE Standards Board, 70 pages.
- Donald E. Knuth, "The Art of Computer Programming", vol 2, "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981.
- Jean-Michel Muller, "Elementary Functions, Algorithms and Implementation", Birkhäuser, Boston, 2nd edition, 2006.
- Jean-Michel Muller, Nicolas Brisebarre, Florent de Dinechin, Claude-Pierre Jeannerod, Vincent Lefèvre, Guillaume Melquiond, Nathalie Revol, Damien Stehlé and Serge Torrès, "Handbook of Floating-Point Arithmetic", Birkhäuser, Boston, 2009.

Version 1.2, November 2002

Copyright © 2000,2001,2002 Free Software Foundation, Inc. 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed.

- PREAMBLE
The purpose of this License is to make a manual, textbook, or other functional and useful document free in the sense of freedom: to assure everyone the effective freedom to copy and redistribute it, with or without modifying it, either commercially or noncommercially. Secondarily, this License preserves for the author and publisher a way to get credit for their work, while not being considered responsible for modifications made by others.

This License is a kind of “copyleft”, which means that derivative works of the document must themselves be free in the same sense. It complements the GNU General Public License, which is a copyleft license designed for free software.

We have designed this License in order to use it for manuals for free software, because free software needs free documentation: a free program should come with manuals providing the same freedoms that the software does. But this License is not limited to software manuals; it can be used for any textual work, regardless of subject matter or whether it is published as a printed book. We recommend this License principally for works whose purpose is instruction or reference.

- APPLICABILITY AND DEFINITIONS
This License applies to any manual or other work, in any medium, that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. Such a notice grants a world-wide, royalty-free license, unlimited in duration, to use that work under the conditions stated herein. The “Document”, below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as “you”. You accept the license if you copy, modify or distribute the work in a way requiring permission under copyright law.

A “Modified Version” of the Document means any work containing the Document or a portion of it, either copied verbatim, or with modifications and/or translated into another language.

A “Secondary Section” is a named appendix or a front-matter section of the Document that deals exclusively with the relationship of the publishers or authors of the Document to the Document's overall subject (or to related matters) and contains nothing that could fall directly within that overall subject. (Thus, if the Document is in part a textbook of mathematics, a Secondary Section may not explain any mathematics.) The relationship could be a matter of historical connection with the subject or with related matters, or of legal, commercial, philosophical, ethical or political position regarding them.

The “Invariant Sections” are certain Secondary Sections whose titles are designated, as being those of Invariant Sections, in the notice that says that the Document is released under this License. If a section does not fit the above definition of Secondary then it is not allowed to be designated as Invariant. The Document may contain zero Invariant Sections. If the Document does not identify any Invariant Sections then there are none.

The “Cover Texts” are certain short passages of text that are listed, as Front-Cover Texts or Back-Cover Texts, in the notice that says that the Document is released under this License. A Front-Cover Text may be at most 5 words, and a Back-Cover Text may be at most 25 words.

A “Transparent” copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, that is suitable for revising the document straightforwardly with generic text editors or (for images composed of pixels) generic paint programs or (for drawings) some widely available drawing editor, and that is suitable for input to text formatters or for automatic translation to a variety of formats suitable for input to text formatters. A copy made in an otherwise Transparent file format whose markup, or absence of markup, has been arranged to thwart or discourage subsequent modification by readers is not Transparent. An image format is not Transparent if used for any substantial amount of text. A copy that is not “Transparent” is called “Opaque”.

Examples of suitable formats for Transparent copies include plain ascii without markup, Texinfo input format, LaTeX input format, SGML or XML using a publicly available DTD, and standard-conforming simple HTML, PostScript or PDF designed for human modification. Examples of transparent image formats include PNG, XCF and JPG. Opaque formats include proprietary formats that can be read and edited only by proprietary word processors, SGML or XML for which the DTD and/or processing tools are not generally available, and the machine-generated HTML, PostScript or PDF produced by some word processors for output purposes only.

The “Title Page” means, for a printed book, the title page itself, plus such following pages as are needed to hold, legibly, the material this License requires to appear in the title page. For works in formats which do not have any title page as such, “Title Page” means the text near the most prominent appearance of the work's title, preceding the beginning of the body of the text.

A section “Entitled XYZ” means a named subunit of the Document whose title either is precisely XYZ or contains XYZ in parentheses following text that translates XYZ in another language. (Here XYZ stands for a specific section name mentioned below, such as “Acknowledgements”, “Dedications”, “Endorsements”, or “History”.) To “Preserve the Title” of such a section when you modify the Document means that it remains a section “Entitled XYZ” according to this definition.

The Document may include Warranty Disclaimers next to the notice which states that this License applies to the Document. These Warranty Disclaimers are considered to be included by reference in this License, but only as regards disclaiming warranties: any other implication that these Warranty Disclaimers may have is void and has no effect on the meaning of this License.

- VERBATIM COPYING
You may copy and distribute the Document in any medium, either commercially or noncommercially, provided that this License, the copyright notices, and the license notice saying this License applies to the Document are reproduced in all copies, and that you add no other conditions whatsoever to those of this License. You may not use technical measures to obstruct or control the reading or further copying of the copies you make or distribute. However, you may accept compensation in exchange for copies. If you distribute a large enough number of copies you must also follow the conditions in section 3.

You may also lend copies, under the same conditions stated above, and you may publicly display copies.

- COPYING IN QUANTITY
If you publish printed copies (or copies in media that commonly have printed covers) of the Document, numbering more than 100, and the Document's license notice requires Cover Texts, you must enclose the copies in covers that carry, clearly and legibly, all these Cover Texts: Front-Cover Texts on the front cover, and Back-Cover Texts on the back cover. Both covers must also clearly and legibly identify you as the publisher of these copies. The front cover must present the full title with all words of the title equally prominent and visible. You may add other material on the covers in addition. Copying with changes limited to the covers, as long as they preserve the title of the Document and satisfy these conditions, can be treated as verbatim copying in other respects.

If the required texts for either cover are too voluminous to fit legibly, you should put the first ones listed (as many as fit reasonably) on the actual cover, and continue the rest onto adjacent pages.

If you publish or distribute Opaque copies of the Document numbering more than 100, you must either include a machine-readable Transparent copy along with each Opaque copy, or state in or with each Opaque copy a computer-network location from which the general network-using public has access to download using public-standard network protocols a complete Transparent copy of the Document, free of added material. If you use the latter option, you must take reasonably prudent steps, when you begin distribution of Opaque copies in quantity, to ensure that this Transparent copy will remain thus accessible at the stated location until at least one year after the last time you distribute an Opaque copy (directly or through your agents or retailers) of that edition to the public.

It is requested, but not required, that you contact the authors of the Document well before redistributing any large number of copies, to give them a chance to provide you with an updated version of the Document.

- MODIFICATIONS
You may copy and distribute a Modified Version of the Document under the conditions of sections 2 and 3 above, provided that you release the Modified Version under precisely this License, with the Modified Version filling the role of the Document, thus licensing distribution and modification of the Modified Version to whoever possesses a copy of it. In addition, you must do these things in the Modified Version:

- Use in the Title Page (and on the covers, if any) a title distinct from that of the Document, and from those of previous versions (which should, if there were any, be listed in the History section of the Document). You may use the same title as a previous version if the original publisher of that version gives permission.
- List on the Title Page, as authors, one or more persons or entities responsible for authorship of the modifications in the Modified Version, together with at least five of the principal authors of the Document (all of its principal authors, if it has fewer than five), unless they release you from this requirement.
- State on the Title page the name of the publisher of the Modified Version, as the publisher.
- Preserve all the copyright notices of the Document.
- Add an appropriate copyright notice for your modifications adjacent to the other copyright notices.
- Include, immediately after the copyright notices, a license notice giving the public permission to use the Modified Version under the terms of this License, in the form shown in the Addendum below.
- Preserve in that license notice the full lists of Invariant Sections and required Cover Texts given in the Document's license notice.
- Include an unaltered copy of this License.
- Preserve the section Entitled “History”, Preserve its Title, and add to it an item stating at least the title, year, new authors, and publisher of the Modified Version as given on the Title Page. If there is no section Entitled “History” in the Document, create one stating the title, year, authors, and publisher of the Document as given on its Title Page, then add an item describing the Modified Version as stated in the previous sentence.
- Preserve the network location, if any, given in the Document for public access to a Transparent copy of the Document, and likewise the network locations given in the Document for previous versions it was based on. These may be placed in the “History” section. You may omit a network location for a work that was published at least four years before the Document itself, or if the original publisher of the version it refers to gives permission.
- For any section Entitled “Acknowledgements” or “Dedications”, Preserve the Title of the section, and preserve in the section all the substance and tone of each of the contributor acknowledgements and/or dedications given therein.
- Preserve all the Invariant Sections of the Document, unaltered in their text and in their titles. Section numbers or the equivalent are not considered part of the section titles.
- Delete any section Entitled “Endorsements”. Such a section may not be included in the Modified Version.
- Do not retitle any existing section to be Entitled “Endorsements” or to conflict in title with any Invariant Section.
- Preserve any Warranty Disclaimers.

If the Modified Version includes new front-matter sections or appendices that qualify as Secondary Sections and contain no material copied from the Document, you may at your option designate some or all of these sections as invariant. To do this, add their titles to the list of Invariant Sections in the Modified Version's license notice. These titles must be distinct from any other section titles.

You may add a section Entitled “Endorsements”, provided it contains nothing but endorsements of your Modified Version by various parties—for example, statements of peer review or that the text has been approved by an organization as the authoritative definition of a standard.

You may add a passage of up to five words as a Front-Cover Text, and a passage of up to 25 words as a Back-Cover Text, to the end of the list of Cover Texts in the Modified Version. Only one passage of Front-Cover Text and one of Back-Cover Text may be added by (or through arrangements made by) any one entity. If the Document already includes a cover text for the same cover, previously added by you or by arrangement made by the same entity you are acting on behalf of, you may not add another; but you may replace the old one, on explicit permission from the previous publisher that added the old one.

The author(s) and publisher(s) of the Document do not by this License give permission to use their names for publicity for or to assert or imply endorsement of any Modified Version.

- COMBINING DOCUMENTS
You may combine the Document with other documents released under this License, under the terms defined in section 4 above for modified versions, provided that you include in the combination all of the Invariant Sections of all of the original documents, unmodified, and list them all as Invariant Sections of your combined work in its license notice, and that you preserve all their Warranty Disclaimers.

The combined work need only contain one copy of this License, and multiple identical Invariant Sections may be replaced with a single copy. If there are multiple Invariant Sections with the same name but different contents, make the title of each such section unique by adding at the end of it, in parentheses, the name of the original author or publisher of that section if known, or else a unique number. Make the same adjustment to the section titles in the list of Invariant Sections in the license notice of the combined work.

In the combination, you must combine any sections Entitled “History” in the various original documents, forming one section Entitled “History”; likewise combine any sections Entitled “Acknowledgements”, and any sections Entitled “Dedications”. You must delete all sections Entitled “Endorsements.”

- COLLECTIONS OF DOCUMENTS
You may make a collection consisting of the Document and other documents released under this License, and replace the individual copies of this License in the various documents with a single copy that is included in the collection, provided that you follow the rules of this License for verbatim copying of each of the documents in all other respects.

You may extract a single document from such a collection, and distribute it individually under this License, provided you insert a copy of this License into the extracted document, and follow this License in all other respects regarding verbatim copying of that document.

- AGGREGATION WITH INDEPENDENT WORKS
A compilation of the Document or its derivatives with other separate and independent documents or works, in or on a volume of a storage or distribution medium, is called an “aggregate” if the copyright resulting from the compilation is not used to limit the legal rights of the compilation's users beyond what the individual works permit. When the Document is included in an aggregate, this License does not apply to the other works in the aggregate which are not themselves derivative works of the Document.

If the Cover Text requirement of section 3 is applicable to these copies of the Document, then if the Document is less than one half of the entire aggregate, the Document's Cover Texts may be placed on covers that bracket the Document within the aggregate, or the electronic equivalent of covers if the Document is in electronic form. Otherwise they must appear on printed covers that bracket the whole aggregate.

- TRANSLATION
Translation is considered a kind of modification, so you may distribute translations of the Document under the terms of section 4. Replacing Invariant Sections with translations requires special permission from their copyright holders, but you may include translations of some or all Invariant Sections in addition to the original versions of these Invariant Sections. You may include a translation of this License, and all the license notices in the Document, and any Warranty Disclaimers, provided that you also include the original English version of this License and the original versions of those notices and disclaimers. In case of a disagreement between the translation and the original version of this License or a notice or disclaimer, the original version will prevail.

If a section in the Document is Entitled “Acknowledgements”, “Dedications”, or “History”, the requirement (section 4) to Preserve its Title (section 1) will typically require changing the actual title.

- TERMINATION
You may not copy, modify, sublicense, or distribute the Document except as expressly provided for under this License. Any other attempt to copy, modify, sublicense or distribute the Document is void, and will automatically terminate your rights under this License. However, parties who have received copies, or rights, from you under this License will not have their licenses terminated so long as such parties remain in full compliance.

- FUTURE REVISIONS OF THIS LICENSE
The Free Software Foundation may publish new, revised versions of the GNU Free Documentation License from time to time. Such new versions will be similar in spirit to the present version, but may differ in detail to address new problems or concerns. See http://www.gnu.org/copyleft/.

Each version of the License is given a distinguishing version number. If the Document specifies that a particular numbered version of this License “or any later version” applies to it, you have the option of following the terms and conditions either of that specified version or of any later version that has been published (not as a draft) by the Free Software Foundation. If the Document does not specify a version number of this License, you may choose any version ever published (not as a draft) by the Free Software Foundation.

To use this License in a document you have written, include a copy of the License in the document and put the following copyright and license notices just after the title page:

Copyright (C)yearyour name. 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, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled ``GNU Free Documentation License''.

If you have Invariant Sections, Front-Cover Texts and Back-Cover Texts, replace the “with...Texts.” line with this:

with the Invariant Sections beinglist their titles, with the Front-Cover Texts beinglist, and with the Back-Cover Texts beinglist.

If you have Invariant Sections without Cover Texts, or some other combination of the three, merge those two alternatives to suit the situation.

If your document contains nontrivial examples of program code, we recommend releasing these examples in parallel under your choice of free software license, such as the GNU General Public License, to permit their use in free software.

- Accuracy: MPFR Interface
- Arithmetic functions: Basic Arithmetic Functions
- Assignment functions: Assignment Functions
- Basic arithmetic functions: Basic Arithmetic Functions
- Combined initialization and assignment functions: Combined Initialization and Assignment Functions
- Comparison functions: Comparison Functions
- Compatibility with MPF: Compatibility with MPF
- Conditions for copying MPFR: Copying
- Conversion functions: Conversion Functions
- Copying conditions: Copying
- Custom interface: Custom Interface
- Exception related functions: Exception Related Functions
- Float arithmetic functions: Basic Arithmetic Functions
- Float comparisons functions: Comparison Functions
- Float functions: MPFR Interface
- Float input and output functions: Input and Output Functions
- Float output functions: Formatted Output Functions
- Floating-point functions: MPFR Interface
- Floating-point number: MPFR Basics
- GNU Free Documentation License: GNU Free Documentation License
- I/O functions: Formatted Output Functions
- I/O functions: Input and Output Functions
- Initialization functions: Initialization Functions
- Input functions: Input and Output Functions
- Installation: Installing MPFR
- Integer related functions: Integer Related Functions
- Internals: Internals
`intmax_t`

: MPFR Basics`inttypes.h`

: MPFR Basics`libmpfr`

: MPFR Basics- Libraries: MPFR Basics
- Libtool: MPFR Basics
- Limb: Internals
- Linking: MPFR Basics
- Miscellaneous float functions: Miscellaneous Functions
`mpfr.h`: MPFR Basics- Output functions: Formatted Output Functions
- Output functions: Input and Output Functions
- Precision: MPFR Interface
- Precision: MPFR Basics
- Reporting bugs: Reporting Bugs
- Rounding mode related functions: Rounding Related Functions
- Rounding Modes: MPFR Basics
- Special functions: Special Functions
`stdarg.h`

: MPFR Basics`stdint.h`

: MPFR Basics`stdio.h`

: MPFR Basics`uintmax_t`

: MPFR Basics

`mpfr_abs`

: Basic Arithmetic Functions`mpfr_acos`

: Special Functions`mpfr_acosh`

: Special Functions`mpfr_add`

: Basic Arithmetic Functions`mpfr_add_d`

: Basic Arithmetic Functions`mpfr_add_q`

: Basic Arithmetic Functions`mpfr_add_si`

: Basic Arithmetic Functions`mpfr_add_ui`

: Basic Arithmetic Functions`mpfr_add_z`

: Basic Arithmetic Functions`mpfr_agm`

: Special Functions`mpfr_ai`

: Special Functions`mpfr_asin`

: Special Functions`mpfr_asinh`

: Special Functions`mpfr_asprintf`

: Formatted Output Functions`mpfr_atan`

: Special Functions`mpfr_atan2`

: Special Functions`mpfr_atanh`

: Special Functions`mpfr_buildopt_decimal_p`

: Miscellaneous Functions`mpfr_buildopt_tls_p`

: Miscellaneous Functions`mpfr_can_round`

: Rounding Related Functions`mpfr_cbrt`

: Basic Arithmetic Functions`mpfr_ceil`

: Integer Related Functions`mpfr_check_range`

: Exception Related Functions`mpfr_clear`

: Initialization Functions`mpfr_clear_erangeflag`

: Exception Related Functions`mpfr_clear_flags`

: Exception Related Functions`mpfr_clear_inexflag`

: Exception Related Functions`mpfr_clear_nanflag`

: Exception Related Functions`mpfr_clear_overflow`

: Exception Related Functions`mpfr_clear_underflow`

: Exception Related Functions`mpfr_clears`

: Initialization Functions`mpfr_cmp`

: Comparison Functions`mpfr_cmp_d`

: Comparison Functions`mpfr_cmp_f`

: Comparison Functions`mpfr_cmp_ld`

: Comparison Functions`mpfr_cmp_q`

: Comparison Functions`mpfr_cmp_si`

: Comparison Functions`mpfr_cmp_si_2exp`

: Comparison Functions`mpfr_cmp_ui`

: Comparison Functions`mpfr_cmp_ui_2exp`

: Comparison Functions`mpfr_cmp_z`

: Comparison Functions`mpfr_cmpabs`

: Comparison Functions`mpfr_const_catalan`

: Special Functions`mpfr_const_euler`

: Special Functions`mpfr_const_log2`

: Special Functions`mpfr_const_pi`

: Special Functions`mpfr_copysign`

: Miscellaneous Functions`mpfr_cos`

: Special Functions`mpfr_cosh`

: Special Functions`mpfr_cot`

: Special Functions`mpfr_coth`

: Special Functions`mpfr_csc`

: Special Functions`mpfr_csch`

: Special Functions`mpfr_custom_get_exp`

: Custom Interface`mpfr_custom_get_kind`

: Custom Interface`mpfr_custom_get_significand`

: Custom Interface`mpfr_custom_get_size`

: Custom Interface`mpfr_custom_init`

: Custom Interface`mpfr_custom_init_set`

: Custom Interface`mpfr_custom_move`

: Custom Interface`mpfr_d_div`

: Basic Arithmetic Functions`mpfr_d_sub`

: Basic Arithmetic Functions`MPFR_DECL_INIT`

: Initialization Functions`mpfr_digamma`

: Special Functions`mpfr_dim`

: Basic Arithmetic Functions`mpfr_div`

: Basic Arithmetic Functions`mpfr_div_2exp`

: Compatibility with MPF`mpfr_div_2si`

: Basic Arithmetic Functions`mpfr_div_2ui`

: Basic Arithmetic Functions`mpfr_div_d`

: Basic Arithmetic Functions`mpfr_div_q`

: Basic Arithmetic Functions`mpfr_div_si`

: Basic Arithmetic Functions`mpfr_div_ui`

: Basic Arithmetic Functions`mpfr_div_z`

: Basic Arithmetic Functions`mpfr_eint`

: Special Functions`mpfr_eq`

: Compatibility with MPF`mpfr_equal_p`

: Comparison Functions`mpfr_erangeflag_p`

: Exception Related Functions`mpfr_erf`

: Special Functions`mpfr_erfc`

: Special Functions`mpfr_exp`

: Special Functions`mpfr_exp10`

: Special Functions`mpfr_exp2`

: Special Functions`mpfr_expm1`

: Special Functions`mpfr_fac_ui`

: Special Functions`mpfr_fits_intmax_p`

: Conversion Functions`mpfr_fits_sint_p`

: Conversion Functions`mpfr_fits_slong_p`

: Conversion Functions`mpfr_fits_sshort_p`

: Conversion Functions`mpfr_fits_uint_p`

: Conversion Functions`mpfr_fits_uintmax_p`

: Conversion Functions`mpfr_fits_ulong_p`

: Conversion Functions`mpfr_fits_ushort_p`

: Conversion Functions`mpfr_floor`

: Integer Related Functions`mpfr_fma`

: Special Functions`mpfr_fmod`

: Integer Related Functions`mpfr_fms`

: Special Functions`mpfr_fprintf`

: Formatted Output Functions`mpfr_frac`

: Integer Related Functions`mpfr_free_cache`

: Special Functions`mpfr_free_str`

: Conversion Functions`mpfr_gamma`

: Special Functions`mpfr_get_d`

: Conversion Functions`mpfr_get_d_2exp`

: Conversion Functions`mpfr_get_decimal64`

: Conversion Functions`mpfr_get_default_prec`

: Initialization Functions`mpfr_get_default_rounding_mode`

: Rounding Related Functions`mpfr_get_emax`

: Exception Related Functions`mpfr_get_emax_max`

: Exception Related Functions`mpfr_get_emax_min`

: Exception Related Functions`mpfr_get_emin`

: Exception Related Functions`mpfr_get_emin_max`

: Exception Related Functions`mpfr_get_emin_min`

: Exception Related Functions`mpfr_get_exp`

: Miscellaneous Functions`mpfr_get_f`

: Conversion Functions`mpfr_get_flt`

: Conversion Functions`mpfr_get_ld`

: Conversion Functions`mpfr_get_ld_2exp`

: Conversion Functions`mpfr_get_patches`

: Miscellaneous Functions`mpfr_get_prec`

: Initialization Functions`mpfr_get_si`

: Conversion Functions`mpfr_get_sj`

: Conversion Functions`mpfr_get_str`

: Conversion Functions`mpfr_get_ui`

: Conversion Functions`mpfr_get_uj`

: Conversion Functions`mpfr_get_version`

: Miscellaneous Functions`mpfr_get_z`

: Conversion Functions`mpfr_get_z_2exp`

: Conversion Functions`mpfr_greater_p`

: Comparison Functions`mpfr_greaterequal_p`

: Comparison Functions`mpfr_hypot`

: Special Functions`mpfr_inexflag_p`

: Exception Related Functions`mpfr_inf_p`

: Comparison Functions`mpfr_init`

: Initialization Functions`mpfr_init2`

: Initialization Functions`mpfr_init_set`

: Combined Initialization and Assignment Functions`mpfr_init_set_d`

: Combined Initialization and Assignment Functions`mpfr_init_set_f`

: Combined Initialization and Assignment Functions`mpfr_init_set_ld`

: Combined Initialization and Assignment Functions`mpfr_init_set_q`

: Combined Initialization and Assignment Functions`mpfr_init_set_si`

: Combined Initialization and Assignment Functions`mpfr_init_set_str`

: Combined Initialization and Assignment Functions`mpfr_init_set_ui`

: Combined Initialization and Assignment Functions`mpfr_init_set_z`

: Combined Initialization and Assignment Functions`mpfr_inits`

: Initialization Functions`mpfr_inits2`

: Initialization Functions`mpfr_inp_str`

: Input and Output Functions`mpfr_integer_p`

: Integer Related Functions`mpfr_j0`

: Special Functions`mpfr_j1`

: Special Functions`mpfr_jn`

: Special Functions`mpfr_less_p`

: Comparison Functions`mpfr_lessequal_p`

: Comparison Functions`mpfr_lessgreater_p`

: Comparison Functions`mpfr_lgamma`

: Special Functions`mpfr_li2`

: Special Functions`mpfr_lngamma`

: Special Functions`mpfr_log`

: Special Functions`mpfr_log10`

: Special Functions`mpfr_log1p`

: Special Functions`mpfr_log2`

: Special Functions`mpfr_max`

: Miscellaneous Functions`mpfr_min`

: Miscellaneous Functions`mpfr_min_prec`

: Rounding Related Functions`mpfr_modf`

: Integer Related Functions`mpfr_mul`

: Basic Arithmetic Functions`mpfr_mul_2exp`

: Compatibility with MPF`mpfr_mul_2si`

: Basic Arithmetic Functions`mpfr_mul_2ui`

: Basic Arithmetic Functions`mpfr_mul_d`

: Basic Arithmetic Functions`mpfr_mul_q`

: Basic Arithmetic Functions`mpfr_mul_si`

: Basic Arithmetic Functions`mpfr_mul_ui`

: Basic Arithmetic Functions`mpfr_mul_z`

: Basic Arithmetic Functions`mpfr_nan_p`

: Comparison Functions`mpfr_nanflag_p`

: Exception Related Functions`mpfr_neg`

: Basic Arithmetic Functions`mpfr_nextabove`

: Miscellaneous Functions`mpfr_nextbelow`

: Miscellaneous Functions`mpfr_nexttoward`

: Miscellaneous Functions`mpfr_number_p`

: Comparison Functions`mpfr_out_str`

: Input and Output Functions`mpfr_overflow_p`

: Exception Related Functions`mpfr_pow`

: Basic Arithmetic Functions`mpfr_pow_si`

: Basic Arithmetic Functions`mpfr_pow_ui`

: Basic Arithmetic Functions`mpfr_pow_z`

: Basic Arithmetic Functions`mpfr_prec_round`

: Rounding Related Functions`mpfr_prec_t`

: MPFR Basics`mpfr_print_rnd_mode`

: Rounding Related Functions`mpfr_printf`

: Formatted Output Functions`mpfr_rec_sqrt`

: Basic Arithmetic Functions`mpfr_regular_p`

: Comparison Functions`mpfr_reldiff`

: Compatibility with MPF`mpfr_remainder`

: Integer Related Functions`mpfr_remquo`

: Integer Related Functions`mpfr_rint`

: Integer Related Functions`mpfr_rint_ceil`

: Integer Related Functions`mpfr_rint_floor`

: Integer Related Functions`mpfr_rint_round`

: Integer Related Functions`mpfr_rint_trunc`

: Integer Related Functions`mpfr_rnd_t`

: MPFR Basics`mpfr_root`

: Basic Arithmetic Functions`mpfr_round`

: Integer Related Functions`mpfr_sec`

: Special Functions`mpfr_sech`

: Special Functions`mpfr_set`

: Assignment Functions`mpfr_set_d`

: Assignment Functions`mpfr_set_decimal64`

: Assignment Functions`mpfr_set_default_prec`

: Initialization Functions`mpfr_set_default_rounding_mode`

: Rounding Related Functions`mpfr_set_emax`

: Exception Related Functions`mpfr_set_emin`

: Exception Related Functions`mpfr_set_erangeflag`

: Exception Related Functions`mpfr_set_exp`

: Miscellaneous Functions`mpfr_set_f`

: Assignment Functions`mpfr_set_flt`

: Assignment Functions`mpfr_set_inexflag`

: Exception Related Functions`mpfr_set_inf`

: Assignment Functions`mpfr_set_ld`

: Assignment Functions`mpfr_set_nan`

: Assignment Functions`mpfr_set_nanflag`

: Exception Related Functions`mpfr_set_overflow`

: Exception Related Functions`mpfr_set_prec`

: Initialization Functions`mpfr_set_prec_raw`

: Compatibility with MPF`mpfr_set_q`

: Assignment Functions`mpfr_set_si`

: Assignment Functions`mpfr_set_si_2exp`

: Assignment Functions`mpfr_set_sj`

: Assignment Functions`mpfr_set_sj_2exp`

: Assignment Functions`mpfr_set_str`

: Assignment Functions`mpfr_set_ui`

: Assignment Functions`mpfr_set_ui_2exp`

: Assignment Functions`mpfr_set_uj`

: Assignment Functions`mpfr_set_uj_2exp`

: Assignment Functions`mpfr_set_underflow`

: Exception Related Functions`mpfr_set_z`

: Assignment Functions`mpfr_set_z_2exp`

: Assignment Functions`mpfr_set_zero`

: Assignment Functions`mpfr_setsign`

: Miscellaneous Functions`mpfr_sgn`

: Comparison Functions`mpfr_si_div`

: Basic Arithmetic Functions`mpfr_si_sub`

: Basic Arithmetic Functions`mpfr_signbit`

: Miscellaneous Functions`mpfr_sin`

: Special Functions`mpfr_sin_cos`

: Special Functions`mpfr_sinh`

: Special Functions`mpfr_sinh_cosh`

: Special Functions`mpfr_snprintf`

: Formatted Output Functions`mpfr_sprintf`

: Formatted Output Functions`mpfr_sqr`

: Basic Arithmetic Functions`mpfr_sqrt`

: Basic Arithmetic Functions`mpfr_sqrt_ui`

: Basic Arithmetic Functions`mpfr_strtofr`

: Assignment Functions`mpfr_sub`

: Basic Arithmetic Functions`mpfr_sub_d`

: Basic Arithmetic Functions`mpfr_sub_q`

: Basic Arithmetic Functions`mpfr_sub_si`

: Basic Arithmetic Functions`mpfr_sub_ui`

: Basic Arithmetic Functions`mpfr_sub_z`

: Basic Arithmetic Functions`mpfr_subnormalize`

: Exception Related Functions`mpfr_sum`

: Special Functions`mpfr_swap`

: Assignment Functions`mpfr_t`

: MPFR Basics`mpfr_tan`

: Special Functions`mpfr_tanh`

: Special Functions`mpfr_trunc`

: Integer Related Functions`mpfr_ui_div`

: Basic Arithmetic Functions`mpfr_ui_pow`

: Basic Arithmetic Functions`mpfr_ui_pow_ui`

: Basic Arithmetic Functions`mpfr_ui_sub`

: Basic Arithmetic Functions`mpfr_underflow_p`

: Exception Related Functions`mpfr_unordered_p`

: Comparison Functions`mpfr_urandom`

: Miscellaneous Functions`mpfr_urandomb`

: Miscellaneous Functions`mpfr_vasprintf`

: Formatted Output Functions`MPFR_VERSION`

: Miscellaneous Functions`MPFR_VERSION_MAJOR`

: Miscellaneous Functions`MPFR_VERSION_MINOR`

: Miscellaneous Functions`MPFR_VERSION_NUM`

: Miscellaneous Functions`MPFR_VERSION_PATCHLEVEL`

: Miscellaneous Functions`MPFR_VERSION_STRING`

: Miscellaneous Functions`mpfr_vfprintf`

: Formatted Output Functions`mpfr_vprintf`

: Formatted Output Functions`mpfr_vsnprintf`

: Formatted Output Functions`mpfr_vsprintf`

: Formatted Output Functions`mpfr_y0`

: Special Functions`mpfr_y1`

: Special Functions`mpfr_yn`

: Special Functions`mpfr_zero_p`

: Comparison Functions`mpfr_zeta`

: Special Functions`mpfr_zeta_ui`

: Special Functions