This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 2.1.0.

Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004 Free Software Foundation, Inc.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.1 or any later version published by the Free Software Foundation; with no Invariant Sections, with the Front-Cover Texts being “A GNU Manual”, and with the Back-Cover Texts being “You have freedom to copy and modify this GNU Manual, like GNU software”. A copy of the license is included in GNU Free Documentation License.

This library 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 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 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 MPFR library are found in the Lesser General Public License that accompanies the source code. See the file COPYING.LIB.

MPFR is a portable library written in C for arbitrary precision arithmetic
on floating-point numbers. It is based on the GNU MP library.
It aims to extend the class of floating-point numbers provided by the
GNU MP library by a precise semantics. The main differences
with the `mpf`

class from GNU MP are:

- the
`mpfr`

code is portable, i.e. the result of any operation does not depend (or should not) on the machine word size`mp_bits_per_limb`

(32 or 64 on most machines); - 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.

In particular, with a precision of 53 bits, `mpfr`

should be able
to exactly reproduce all computations with double-precision machine
floating-point
numbers (`double`

type in C), except the default exponent range
is much wider and subnormal numbers are not implemented.

This version of MPFR is released under the GNU Lesser General Public License. It is permitted to link MPFR to non-free programs, as long as when distributing them the MPFR source code and a means to re-link with a modified MPFR library is provided.

Everyone should read MPFR Basics. If you need to install the library yourself, you need to read Installing MPFR, too.

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

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 a standard Unix make program, plus some other standard Unix utility programs.
- In the MPFR build directory, type
./configure
This will prepare the build and setup the options according to your system. If you get error messages, you might check that you use the same compiler and compile options as for GNU MP (see the INSTALL file).

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

- 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 file libmpfr.a to the directory /usr/local/lib, and the file mpfr.info to the directory /usr/local/info (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 an info version of the manual, in mpfr.info.

- mpfr.dvi or dvi
Create a DVI version of the manual, in mpfr.dvi.

- mpfr.ps
Create a Postscript version of the manual, in mpfr.ps.

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

- distclean
Delete all files not included in the distribution.

- uninstall
Delete all files copied by make install.

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

Please report other problems to mpfr@loria.fr. See Reporting Bugs. Some bug fixes are available on the MPFR web page http://www.mpfr.org/.

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

If you think you have found a bug in the MPFR library, first have a look on the MPFR web page http://www.mpfr.org/: perhaps this bug is already known, in which case you may find there a workaround for it. Otherwise, please investigate and report it. We have made this library available to you, and it is not to ask too much from you, to ask you to report the bugs that you find.

There are a few things you should think about when you put your bug report together.

You have to send us a test case that makes it possible for us to reproduce the bug. Include instructions on how to run the test case.

You also have to explain what is wrong; if you get a crash, or if the results printed are incorrect and in that case, in what way.

Please include compiler version information in your bug report. This can be extracted using cc -V on some machines, or, if you're using gcc, gcc -v. Also, include the output from uname -a.

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

Send your bug report to: 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>

A floating-point number or float for short, is an arbitrary
precision mantissa with a limited precision exponent. The C data type
for such objects is `mpfr_t`

. A floating-point number can have
three special values: Not-a-Number (NaN) or plus or minus Infinity. NaN
represents an uninitialized object, the result of an invalid operation
(like 0 divided by 0), or a value that cannot be determined (like
+Infinity minus +Infinity). Moreover, like in the IEEE 754-1985 standard,
zero is signed, i.e. there are both +0 and −0; the behavior
is the same as in the IEEE 754-1985 standard and it is generalized to
the other functions supported by MPFR.

The precision is the number of bits used to represent the mantissa
of a floating-point number;
the corresponding C data type is `mp_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.

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 mantissa;
the corresponding C data type is `mp_rnd_t`

.

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

.

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

- Functions for floating-point arithmetic, with names beginning with
`mpfr_`

. The associated type is`mpfr_t`

.

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_mode)`

.
This
computes the square of `x` with rounding mode `rnd_mode`

and puts the result back in `x`.

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

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

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

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

The following four rounding modes are supported:

`GMP_RNDN`

: round to nearest`GMP_RNDZ`

: round towards zero`GMP_RNDU`

: round towards plus infinity`GMP_RNDD`

: round towards minus infinity

The round to nearest mode works as in the IEEE 754-1985 standard: in case the number to be rounded lies exactly in the middle of two representable numbers, it is rounded to the one with the least significant bit set to zero. For example, the number 5/2, which is represented by (10.1) in binary, is rounded to (10.0)=2 with a precision of two bits, and not to (11.0)=3. This rule avoids the 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
exactly rounded, i.e. MPFR behaves as if it computed the result with
an infinite precision, then rounded it to the precision of this variable.
The input variables are regarded as exact (in particular, their precision
does not affect the result).

Unless documented otherwise, functions returning an `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 `GMP_RNDU`

rounding mode,
the ternary value is usually positive, except when the result is exact, in
which case it is zero. In the case of an infinite result, it is considered
as inexact when it was obtained by overflow, and exact otherwise. A NaN
result (Not-a-Number) always corresponds to an exact return value.
The opposite of a returned ternary value is guaranteed to be representable
in an `int`

.

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
integer functions. The function prefix for floating-point operations is
`mpfr_`

.

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

The semantics of a calculation in MPFR is specified as follows: Compute the requested operation exactly (with “infinite accuracy”), and round the result to the precision of the destination variable, with the given rounding mode. The MPFR floating-point functions are intended to be a smooth extension of the IEEE 754-1985 arithmetic. The results obtained on one computer should not differ from the results obtained on a computer with a different word size.

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

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, mp_prec_t prec`)

Initialize

x, set its precision to beexactlyprecbits and its value to NaN. (Warning: the corresponding`mpf`

functions initialize 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_clear** (`mpfr_t x`)

Free the space occupied by

x. Make sure to call this function for all`mpfr_t`

variables when you are done with them.

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

Initialize

xand set its value to NaN.Normally, a variable should be initialized once only or at least be cleared, using

`mpfr_clear`

, between initializations. The precision ofxis the default precision, which can be changed by a call to`mpfr_set_default_prec`

.

— Function: void **mpfr_set_default_prec** (`mp_prec_t prec`)

Set the default precision to be

exactlyprecbits. The precision of a variable means the number of bits used to store its mantissa. All subsequent calls to`mpfr_init`

will use this precision, but previously initialized variables are unaffected. This default precision is set to 53 bits initially. The precision can be any integer between`MPFR_PREC_MIN`

and`MPFR_PREC_MAX`

.

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); }

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, mp_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 mantissa 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 in

x, use`mpfr_prec_round`

instead.

— Function: mp_prec_t **mpfr_get_prec** (`mpfr_t x`)

Return the precision actually used for assignments of

x, i.e. the number of bits used to store its mantissa.

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

These functions assign new values to already initialized floats
(see Initialization Functions). When using 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.

— Function: int **mpfr_set** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_set_ui** (`mpfr_t rop, unsigned long int op, mp_rnd_t rnd`)

— Function: int**mpfr_set_si** (`mpfr_t rop, long int op, mp_rnd_t rnd`)

— Function: int**mpfr_set_uj** (`mpfr_t rop, uintmax_t op, mp_rnd_t rnd`)

— Function: int**mpfr_set_sj** (`mpfr_t rop, intmax_t op, mp_rnd_t rnd`)

— Function: int**mpfr_set_d** (`mpfr_t rop, double op, mp_rnd_t rnd`)

— Function: int**mpfr_set_ld** (`mpfr_t rop, long double op, mp_rnd_t rnd`)

— Function: int**mpfr_set_z** (`mpfr_t rop, mpz_t op, mp_rnd_t rnd`)

— Function: int**mpfr_set_q** (`mpfr_t rop, mpq_t op, mp_rnd_t rnd`)

— Function: int**mpfr_set_f** (`mpfr_t rop, mpf_t op, mp_rnd_t rnd`)

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

— Function: int

Set the value of

ropfromop, rounded towards the given directionrnd. Note that the input 0 is converted to +0 by`mpfr_set_ui`

,`mpfr_set_si`

,`mpfr_set_sj`

,`mpfr_set_uj`

,`mpfr_set_z`

,`mpfr_set_q`

and`mpfr_set_f`

, regardless of the rounding mode. If the system doesn't support the IEEE-754 standard,`mpfr_set_d`

and`mpfr_set_ld`

might not preserve the signed zeros.

— Function: int **mpfr_set_ui_2exp** (`mpfr_t rop, unsigned long int op, mp_exp_t e, mp_rnd_t rnd`)

— Function: int**mpfr_set_si_2exp** (`mpfr_t rop, long int op, mp_exp_t e, mp_rnd_t rnd`)

— Function: int**mpfr_set_uj_2exp** (`mpfr_t rop, uintmax_t op, intmax_t e, mp_rnd_t rnd`)

— Function: int**mpfr_set_sj_2exp** (`mpfr_t rop, intmax_t op, intmax_t e, mp_rnd_t rnd`)

— Function: int

— Function: int

— Function: int

Set the value of

ropfromopmultiplied by two to the powere, rounded towards 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, mp_rnd_t rnd`)

Set

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

for a detailed description of the valid string formats. 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.

— Function: int **mpfr_strtofr** (`mpfr_t rop, const char *nptr, char **endptr, int base, mp_rnd_t rnd`)

Read a floating point number from a string

nptrin basebase, rounded in the directionrnd. If successful, the result is stored inropand`*`

endptrpoints to the character just after those parsed. Ifstrdoesn't start with a valid number thenropis set to zero and the value ofnptris stored in the location referenced byendptr.Parsing follows the standard C

`strtod`

function. This means optional leading whitespace, an optional`+`

or`-`

, mantissa digits, and an optional exponent consisting of an`e`

or`E`

(ifbase<= 10) or`@`

, an optional sign, and digits. A hex mantissa can be given with a leading`0x`

or`0X`

, in which case`p`

or`P`

may introduce an optional binary exponent. A binary mantissa can be given with a leading`0b`

or`0B`

, in which case`e`

,`E`

,`p`

,`P`

or`@`

may introduce the binary exponent.In addition

`infinity`

,`inf`

(ifbase<= 10) or`@inf@`

with an optional sign, or`nan`

,`nan(n-char-sequence)`

(ifbase<= 10),`@nan@`

or`@nan@(n-char-sequence)`

all non case significant, can be given. A`n-char-sequence`

is a string containing only digits (0, 1, 2, ..., 9) and non-digits (_, a, b, ..., z, A, B, ..., Z).There must be at least one digit in the mantissa for the number to be valid. If an exponent has no digits it's ignored and parsing stops after the mantissa. If an

`0x`

,`0X`

,`0b`

or`0B`

is not followed by hexadecimal/binary digits, parsing stops after the first`0`

: the subject sequence is defined as the longest initial subsequence of the input string, starting with the first non-white-space character, that is of the expected form. The subject sequence contains no characters if the input string is not of the expected form.Note that in the hex format the exponent

`P`

represents a power of 2, whereas`@`

represents a power of the base (i.e. 16).If the argument

baseis different from 0, it must be in the range 2 to 36. Case is ignored; upper-case and lower-case letters have the same valueIf

`base`

is 0, then it tries to identify the used base: if the mantissa begins with the`0x`

prefix, it assumes thatbaseis 16. If it begins with`0b`

, it assumes thatbaseis 2. Otherwise, it assumes it is 10.It returns a usual ternary value. If

endptris not a null pointer, a pointer to the character after the last character used in the conversion is stored in the location referenced byendptr.

— Function: void **mpfr_set_inf** (`mpfr_t x, int sign`)

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

— Function: void

Set the variable

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

,xis set to plus infinity iffsignis nonnegative.

— 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, mp_rnd_t rnd`)

— Macro: int**mpfr_init_set_ui** (`mpfr_t rop, unsigned long int op, mp_rnd_t rnd`)

— Macro: int**mpfr_init_set_si** (`mpfr_t rop, signed long int op, mp_rnd_t rnd`)

— Macro: int**mpfr_init_set_d** (`mpfr_t rop, double op, mp_rnd_t rnd`)

— Macro: int**mpfr_init_set_ld** (`mpfr_t rop, long double op, mp_rnd_t rnd`)

— Macro: int**mpfr_init_set_z** (`mpfr_t rop, mpz_t op, mp_rnd_t rnd`)

— Macro: int**mpfr_init_set_q** (`mpfr_t rop, mpq_t op, mp_rnd_t rnd`)

— Macro: int**mpfr_init_set_f** (`mpfr_t rop, mpf_t op, mp_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, mp_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: double **mpfr_get_d** (`mpfr_t op, mp_rnd_t rnd`)

— Function: long double**mpfr_get_ld** (`mpfr_t op, mp_rnd_t rnd`)

— Function: long double

Convert

opto a`double`

(respectively`long double`

), using the rounding modernd. If the system doesn't support IEEE 754 standard, this function might not preserve the signed zeros.

— Function: double **mpfr_get_d_2exp** (`long *exp, mpfr_t op, mp_rnd_t rnd`)

Return

dand setexpsuch that 0.5<=abs(d)<1 anddtimes 2 raised toexpequalsoprounded to double precision, using the given rounding mode.

— Function: long **mpfr_get_si** (`mpfr_t op, mp_rnd_t rnd`)

— Function: unsigned long**mpfr_get_ui** (`mpfr_t op, mp_rnd_t rnd`)

— Function: intmax_t**mpfr_get_sj** (`mpfr_t op, mp_rnd_t rnd`)

— Function: uintmax_t**mpfr_get_uj** (`mpfr_t op, mp_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, the result is undefined. Ifopis too big for the return type, it returns the maximum or the minimum of the corresponding C type, depending on the direction of the overflow. The flag erange is set too. See also`mpfr_fits_slong_p`

,`mpfr_fits_ulong_p`

,`mpfr_fits_intmax_p`

and`mpfr_fits_uintmax_p`

.

— Function: mp_exp_t **mpfr_get_z_exp** (`mpz_t rop, mpfr_t op`)

Put the scaled mantissa 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 equalsropmultiplied by two exponentexp. If the exponent is not representable in the`mp_exp_t`

type, the behavior is undefined.

— Function: void **mpfr_get_z** (`mpz_t rop, mpfr_t op, mp_rnd_t rnd`)

Convert

opto a`mpz_t`

, after rounding it with respect tornd. Ifopis NaN or Inf, the result is undefined.

— Function: char * **mpfr_get_str** (`char *str, mp_exp_t *expptr, int base, size_t n, mpfr_t op, mp_rnd_t rnd`)

Convert

opto a string of digits in basebase, with rounding in the directionrnd. The base may vary from 2 to 36.The generated string is a fraction, with an implicit radix point immediately to the left of the first digit. For example, the number -3.1416 would be returned as "-31416" in the string and 1 written at

expptr. Ifrndis to nearest, andopis exactly in the middle of two possible outputs, the one with an even last digit is chosen (for an odd base, this may not correspond to an even mantissa).If

nis zero, the number of digits of the mantissa is determined automatically from the precision ofopand the value ofbase. Warning: this functionality may disappear or change in future versions. Otherwise generate exactlynsignificant digits, which must be at least 2.If

stris a null pointer, space for the mantissa 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 mantissa, i.e., at least`max(`

n`+ 2, 7)`

. The extra two bytes are for a possible minus sign, and for the terminating null character.If

nis 0, note that the space requirements forstrin this case will be impossible for the user to predetermine. Therefore, one needs to pass a null pointer for the string argument whenevernis 0.If the input number is an ordinary number, the exponent is written through the pointer

expptr(the current minimal exponent for 0).A pointer to the string is returned, unless there is an error, in which case a null pointer is returned.

— Function: void **mpfr_free_str** (`char *str`)

Free a string allocated by

`mpfr_get_str`

using the current unallocation function (preliminary interface). The block is assumed to be`strlen(`

str`)+1`

bytes. For more information about how it is done: see Custom Allocation (GNU MP).

— Function: int **mpfr_fits_ulong_p** (`mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_fits_slong_p** (`mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_fits_uint_p** (`mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_fits_sint_p** (`mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_fits_ushort_p** (`mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_fits_sshort_p** (`mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_fits_intmax_p** (`mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_fits_uintmax_p** (`mpfr_t op, mp_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, when rounded to an integer in the directionrnd.

— Function: int **mpfr_add** (`mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd`)

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

— Function: int**mpfr_add_si** (`mpfr_t rop, mpfr_t op1, long int op2, mp_rnd_t rnd`)

— Function: int**mpfr_add_z** (`mpfr_t rop, mpfr_t op1, mpz_t op2, mp_rnd_t rnd`)

— Function: int**mpfr_add_q** (`mpfr_t rop, mpfr_t op1, mpq_t op2, mp_rnd_t rnd`)

— 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)).

— Function: int **mpfr_sub** (`mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd`)

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

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

— Function: int**mpfr_si_sub** (`mpfr_t rop, long int op1, mpfr_t op2, mp_rnd_t rnd`)

— Function: int**mpfr_sub_si** (`mpfr_t rop, mpfr_t op1, long int op2, mp_rnd_t rnd`)

— Function: int**mpfr_sub_z** (`mpfr_t rop, mpfr_t op1, mpz_t op2, mp_rnd_t rnd`)

— Function: int**mpfr_sub_q** (`mpfr_t rop, mpfr_t op1, mpq_t op2, mp_rnd_t rnd`)

— 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)).

— Function: int **mpfr_mul** (`mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd`)

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

— Function: int**mpfr_mul_si** (`mpfr_t rop, mpfr_t op1, long int op2, mp_rnd_t rnd`)

— Function: int**mpfr_mul_z** (`mpfr_t rop, mpfr_t op1, mpz_t op2, mp_rnd_t rnd`)

— Function: int**mpfr_mul_q** (`mpfr_t rop, mpfr_t op1, mpq_t op2, mp_rnd_t rnd`)

— 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).

— Function: int **mpfr_sqr** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

Set

ropto the square ofoprounded in the directionrnd.

— Function: int **mpfr_div** (`mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd`)

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

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

— Function: int**mpfr_si_div** (`mpfr_t rop, long int op1, mpfr_t op2, mp_rnd_t rnd`)

— Function: int**mpfr_div_si** (`mpfr_t rop, mpfr_t op1, long int op2, mp_rnd_t rnd`)

— Function: int**mpfr_div_z** (`mpfr_t rop, mpfr_t op1, mpz_t op2, mp_rnd_t rnd`)

— Function: int**mpfr_div_q** (`mpfr_t rop, mpfr_t op1, mpq_t op2, mp_rnd_t rnd`)

— 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).

— Function: int **mpfr_sqrt** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_sqrt_ui** (`mpfr_t rop, unsigned long int op, mp_rnd_t rnd`)

— Function: int

Set

ropto the square root ofoprounded in the directionrnd. Return −0 ifropis −0 (to be consistent with the IEEE 754-1985 standard). Setropto NaN ifopis negative.

— Function: int **mpfr_cbrt** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

Set

ropto the cubic root (defined over the real numbers) ofoprounded in the directionrnd.

— Function: int **mpfr_pow** (`mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd`)

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

— Function: int**mpfr_pow_si** (`mpfr_t rop, mpfr_t op1, long int op2, mp_rnd_t rnd`)

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

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

— Function: int

— Function: int

— Function: int

— Function: int

Set

roptoop1raised toop2, rounded in the directionrnd. Special values are currently handled as described in the ISO C99 standard for the`pow`

function (note this may change in future versions):

`pow(±0,`

y`)`

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

y`)`

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

y`)`

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

y`)`

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

returns 1.`pow(+1,`

y`)`

returns 1 for anyy, 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, mp_rnd_t rnd`)

Set

ropto -oprounded in the directionrnd. Just changes the sign ifropandopare the same variable.

— Function: int **mpfr_abs** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

Set

ropto the absolute value ofop, rounded in the directionrnd. Just changes the sign ifropandopare the same variable.

— Function: int **mpfr_mul_2ui** (`mpfr_t rop, mpfr_t op1, unsigned long int op2, mp_rnd_t rnd`)

— Function: int**mpfr_mul_2si** (`mpfr_t rop, mpfr_t op1, long int op2, mp_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, mp_rnd_t rnd`)

— Function: int**mpfr_div_2si** (`mpfr_t rop, mpfr_t op1, long int op2, mp_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, signed 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 (Not-a-Number), return zero and set the erange flag.

— Function: int **mpfr_cmp_ui_2exp** (`mpfr_t op1, unsigned long int op2, mp_exp_t e`)

— Function: int**mpfr_cmp_si_2exp** (`mpfr_t op1, long int op2, mp_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 (Not-a-Number), return zero and set the erange flag.

— 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

— Function: int

— Function: int

Return non-zero if

opis respectively Not-a-Number (NaN), an infinity, an ordinary number (i.e. neither Not-a-Number nor an infinity) or 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. Its result is undefined whenopis NaN (Not-a-Number).

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

Return non-zero if

op1>op2, zero otherwise.

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

Return non-zero if

op1>=op2, zero otherwise.

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

Return non-zero if

op1<=op2, zero otherwise.

— 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/orop2are NaN, orop1=op2).

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

Return non-zero if

op1=op2, zero otherwise (i.e.op1and/orop2are 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, return zero for an exact return value, a positive value for a return value larger than the exact result, and a negative value otherwise.

— Function: int **mpfr_log** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_log2** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_log10** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int

— Function: int

Set

ropto the natural logarithm ofop, log2(op) or log10(op), respectively, rounded in the directionrnd.

— Function: int **mpfr_exp** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_exp2** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_exp10** (`mpfr_t rop, mpfr_t op, mp_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, mp_rnd_t rnd`)

— Function: int**mpfr_sin** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_tan** (`mpfr_t rop, mpfr_t op, mp_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, mp_rnd_t rnd`)

Set simultaneously

sopto the sine ofopandcopto the cosine ofop, rounded in the directionrndwith the corresponding precisions ofsopandcop. Return 0 iff both results are exact.

— Function: int **mpfr_acos** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_asin** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_atan** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int

— Function: int

Set

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

— Function: int **mpfr_cosh** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_sinh** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_tanh** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int

— Function: int

Set

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

— Function: int **mpfr_acosh** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_asinh** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_atanh** (`mpfr_t rop, mpfr_t op, mp_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, mp_rnd_t rnd`)

Set

ropto the factorial of the`unsigned long int`

op, rounded in the directionrnd.

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

Set

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

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

Set

ropto the exponential ofopminus one, rounded in the directionrnd.

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

Set

ropto the value of the Gamma function onop, rounded in the directionrnd.

— Function: int **mpfr_zeta** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

Set

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

— Function: int **mpfr_erf** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

Set

ropto the value of the error function onop, rounded in the directionrnd.

— Function: int **mpfr_fma** (`mpfr_t rop, mpfr_t op1, mpfr_t op2, mpfr_t op3, mp_rnd_t rnd`)

Set

roptoop1timesop2+op3, rounded in the directionrnd.

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

Set

ropto the arithmetic-geometric mean ofop1andop2, rounded in the directionrnd. The arithmetic-geometric mean is the common limit of the sequences u[n] and v[n], where u[0]=op1, v[0]=op2, u[n+1] is the arithmetic mean of u[n] and v[n], and v[n+1] is the geometric mean of u[n] and v[n]. If any operand is negative, the return value is NaN.

— Function: int **mpfr_const_log2** (`mpfr_t rop, mp_rnd_t rnd`)

— Function: int**mpfr_const_pi** (`mpfr_t rop, mp_rnd_t rnd`)

— Function: int**mpfr_const_euler** (`mpfr_t rop, mp_rnd_t rnd`)

— Function: int

— Function: int

Set

ropto the logarithm of 2, the value of Pi, the value of Euler's constant 0.577..., 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 the cache used by the functions computing constants if needed (currently

`mpfr_const_log2`

,`mpfr_const_pi`

and`mpfr_const_euler`

).

— Function: int **mpfr_sum** (`mpfr_t rop, const mpfr_t* const tab`[]`, unsigned long n, mp_rnd_t rnd`)

Set

retto the sum of all elements oftabwhose size isn, rounded in the directionrnd. Warning,tabis a table of pointers to mpfr_t, not a table of mpfr_t (preliminary interface).

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` argument 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, mp_rnd_t rnd`)

Output

opon streamstream, as a string of digits in basebase, rounded in the directionrnd. The base may vary from 2 to 36. Printnsignificant digits exactly, or ifnis 0, the maximum number of digits accurately representable byop(this feature may disappear).In addition to the significant digits, a decimal point at the right of the first digit and a trailing exponent in base 10, in the form eNNN, are printed. If

baseis greater than 10, @ will be used instead of e as exponent delimiter.Return the number of bytes written, or if an error occurred, return 0.

— Function: size_t **mpfr_inp_str** (`mpfr_t rop, FILE *stream, int base, mp_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`

(it may change). 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.

— Function: int **mpfr_rint** (`mpfr_t rop, mpfr_t op, mp_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 rounding mode,`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, and`mpfr_trunc`

to the next representable integer towards zero.The returned value is zero when the result is exact, positive when it is greater than the original value of

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 the nearest mode (where halfway cases are rounded to an even integer or mantissa). Note also that no double rounding is performed; for instance, 4.5 (100.1 in binary) is rounded by`mpfr_round`

to 4 (100 in binary) in 2-bit precision, though`round(4.5)`

is equal to 5 and 5 (101 in binary) is rounded to 6 (110 in binary) in 2-bit precision.

— Function: int **mpfr_rint_ceil** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_rint_floor** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_rint_round** (`mpfr_t rop, mpfr_t op, mp_rnd_t rnd`)

— Function: int**mpfr_rint_trunc** (`mpfr_t rop, mpfr_t op, mp_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 towards 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).

— Function: int **mpfr_frac** (`mpfr_t rop, mpfr_t op, mp_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: void **mpfr_nexttoward** (`mpfr_t x, mpfr_t y`)

If

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

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

Set

ropto the minimum 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.

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

Set

ropto the 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.

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

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

rop< 1. Return 0, unless the exponent is not in the current exponent range, in which caseropis set to NaN and a non-zero value is returned.

— Function: void **mpfr_random** (`mpfr_t rop`)

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

rop< 1. This function is deprecated;`mpfr_urandomb`

should be used instead.

— Function: void **mpfr_random2** (`mpfr_t rop, mp_size_t size, mp_exp_t exp`)

Generate a random float of at most

sizelimbs, with long strings of zeros and ones in the binary representation. The exponent of the number is in the interval −exptoexp. This function is useful for testing functions and algorithms, since this kind of random numbers have proven to be more likely to trigger corner-case bugs. Negative random numbers are generated whensizeis negative. Put +0 inropwhen size if zero.

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

Get the exponent of

x, assuming thatxis a non-zero ordinary number. The return value for Nan, Infinity or Zero is undefined.

— Function: int **mpfr_set_exp** (`mpfr_t x, mp_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.

— 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:

— 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.

— 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(2,1,0))) # error "Wrong MPFR version." #endif

— Function: void **mpfr_set_default_rounding_mode** (`mp_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, mp_prec_t prec, mp_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 mantissa, and it is filled with zeros. Otherwise, the mantissa is rounded to precisionprecwith the given direction. In both cases, the precision ofxis changed toprec.

— Function: int **mpfr_round_prec** (`mpfr_t x, mp_rnd_t rnd, mp_prec_t prec`)

[This function is obsolete. Please use

`mpfr_prec_round`

instead.]

— Function: const char * **mpfr_print_rnd_mode** (`mp_rnd_t rnd`)

Return the input string (GMP_RNDD, GMP_RNDU, GMP_RNDN, GMP_RNDZ) corresponding to the rounding mode

rndor a null pointer ifrndis an invalid rounding mode.

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

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

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

— Function: mp_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.

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

— Function: int**mpfr_set_emax** (`mp_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: mp_exp_t **mpfr_get_emin_min** (`void`)

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

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

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

— Function: mp_exp_t

— Function: mp_exp_t

— Function: mp_exp_t

Return the minimum and maximum of the smallest and largest exponents allowed for

`mpfr_set_emin`

and`mpfr_set_emax`

. These values are implementation dependent; it is possible to create a non portable program by writing`mpfr_set_emax(mpfr_get_emax_max())`

and`mpfr_set_emin(mpfr_get_emin_min())`

since the values of the smallest and largest exponents become implementation dependent.

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

This function forces

xto be in the current range of acceptable values,tbeing the current ternary value: negative ifxis smaller than the exact value, positive ifxis larger than the exact value and zero ifxis exact (before the call). 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 rounded result is equal to the exact one, a positive value if the rounded result is larger than the exact one, a negative value if the rounded result is smaller than the exact one. Note that unlike most functions, the result is compared to the exact one, not the input valuex, i.e. the ternary value is propagated.

— 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 erange flags.

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

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

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

— 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
can notchange 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). If
precis not a compiler constant, your compiler must support Variable-length automatic arrays (standard in ISO C99). GCC 2.95.3 supports all these features.

— Function: void **mpfr_inits** (`mpfr_t x, ...`)

Initialize all the

`mpfr_t`

variables of the given`va_list`

, set their precision to be 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`

. It begins fromx. It ends when it encounters a null pointer.

— Function: void **mpfr_inits2** (`mp_prec_t prec, mpfr_t x, ...`)

Initialize all the

`mpfr_t`

variables of the given`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`

. It begins fromx. It ends when it encounters a null pointer.

— 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`

. It begins fromx. It ends when it encounters a null pointer.

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

{ mpfr_t x, y, z, t; mpfr_inits2 (256, x, y, z, t, (void *) 0); ... mpfr_clears (x, y, z, t, (void *) 0); }

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

line,

#include <mpfr.h> #include <mpf2mpfr.h>any program written for MPF can be compiled directly with MPFR without any changes. All operations are then performed with the default MPFR rounding mode, which can be reset with

`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, mp_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 mantissa 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`

, but does not make much sense.

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

Compute the relative difference between

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

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

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

— Function: int

See

`mpfr_mul_2ui`

and`mpfr_div_2ui`

. These functions are only kept for compatibility with MPF.

The following types and
functions were mainly designed for the implementation of `mpfr`

,
but may be useful for users too.
However no upward compatibility is guaranteed.
You may need to include mpfr-impl.h to use them.

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 a special value of the exponent. - Finally, the
`_mpfr_d`

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 values always have the most significant bit of the most significant limb set to 1. When the precision does not correspond to a whole number of limbs, the excess bits at the low end of the data are zero.

— Function: int **mpfr_can_round** (`mpfr_t b, mp_exp_t err, mp_rnd_t rnd1, mp_rnd_t rnd2, mp_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 exactlyxto precisionprecwith the directionrnd2, and 0 otherwise (including for NaN and Inf). This functiondoes not modifyits arguments.

— Function: double **mpfr_get_d1** (`mpfr_t op`)

Convert

opto a`double`

, using the default MPFR rounding mode (see function`mpfr_set_default_rounding_mode`

). This function is obsolete.

The main developers consist of Guillaume Hanrot, Vincent Lefèvre, Patrick Pélissier and Paul Zimmermann.

We would like to thank Jean-Michel Muller and Joris van der Hoeven for very fruitful discussions at the beginning of that project, 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. Kevin Ryde did a tremendous job for the portability of MPFR, and integrating it into GMP 4.x; alas the GMP developers decided in January 2004 not to include MPFR any more.

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

and `mpfr_log`

.
Emmanuel Jeandel, from ENS-Lyon too,
contributed the generic hypergeometric code in
`generic.c`

, as well as the `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`

,
David Daney contributed the hyperbolic and inverse hyperbolic functions,
the base-2 exponential, and the factorial function. Fabrice Rouillier
contributed the original version of mul_ui.c, the gmp_op.c
file, and helped to the Windows porting.
Jean-Luc Rémy contributed the `mpfr_zeta`

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

code.

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

- Torbjörn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic Library", version 4.1.2, 2002.
- 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.
- 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", Birkhauser, Boston, 1997.

Version 1.1, March 2000

Copyright © 2000 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307, 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 written 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 that contains a notice placed by the copyright holder saying it can be distributed under the terms of this License. The “Document”, below, refers to any such manual or work. Any member of the public is a licensee, and is addressed as “you”.

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. (For example, 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.

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 “Transparent” copy of the Document means a machine-readable copy, represented in a format whose specification is available to the general public, whose contents can be viewed and edited directly and 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 has been designed to thwart or discourage subsequent modification by readers is not Transparent. 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 designed for human modification. Opaque formats include PostScript, PDF, 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 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.

- 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 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 publicly-accessible computer-network location containing a complete Transparent copy of the Document, free of added material, which the general network-using public has access to download anonymously at no charge using public-standard network protocols. 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 less than five).
- 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”, and 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.
- In any section entitled “Acknowledgments” or “Dedications”, preserve the section's title, and preserve in the section all the substance and tone of each of the contributor acknowledgments 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 as “Endorsements” or to conflict in title with any Invariant Section.

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.

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 “Acknowledgments”, 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, does not as a whole count as a Modified Version of the Document, provided no compilation copyright is claimed for the compilation. Such a compilation is called an “aggregate”, and this License does not apply to the other self-contained works thus compiled with the Document, on account of their being thus compiled, if they 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 quarter of the entire aggregate, the Document's Cover Texts may be placed on covers that surround only the Document within the aggregate. Otherwise they must appear on covers around 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 provided that you also include the original English version of this License. In case of a disagreement between the translation and the original English version of this License, the original English version will prevail.

- 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 draf>mpfr_const_piThe Free Softwariant Secti. -251">

## A.1 ADWITDUM:is Le permissi rules of thisly terindepende15 InternaTpermissi rules of treprect a singlies to ik, or o, must eitheou inse version of tng of each of he authou shou option o from theiunmodins in the lis radix mmediatelot have anon functions: sumed

(C) e>

Cycode>rt

Cy terthes of

The Free Softwar (void *)us compilsome or all Invariant of > Cdified, his, add nt of

Cdifint of

Cdifint of aant SectA Transparent lion of the may not be inclu may add a secti``ion(void *)ovf the GNU Free Documenta'' not allowedages. to it, ng those of Invariank, o this Lics cot, ng those of Invaridquo;Histosthat ransnse nos verst the and actions as s. to it,e passage of Froniank, o this Lict, passage of Fronidquo;Histosthat raion entiover Texts: Front-nt of >

Cdifint of dquo;History&rdquofCover Texts or Back cIfly terindependmatters) andch ov othe.

ion: a f_erfanke d book. Wr docuon of fsehe. end omed

## Conon Licul up -->

Ae anacyf="#: cerns. S#="p" href="#MPFR-Interface">MPFR IAdvanced F: cerns. S#="p" href="#Advanced-Functions">Advanced FAdvanced F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced FAsur oNU Fnf">Advanced F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced FAdvanced F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced FAdvanced F: cerns. S#Cice of -Io detectree-Dont -Asur oNU F-="#AdvancedCice of Io detectree-D-Cove-sur oNU Fns">Advanced FAdvanced F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced FCompatibility : cerns. S#Cicithref="#Compatibility-with-MPF">Compatibility Advanced F: cerns. S#Citr any li="#AdvancedCitr any lns">Advanced FEe Docrbolity : cerns. S#Ee Docrbol">Ee Docrbolity Appendix A GNU Free Documentatiy : cerns. S#="n" href="#GNU-Free-Documentation-License">GNU Free Documentationnfor la Precisionf">Advanced F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fnfor lcicitreditsnf">Advanced F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced Fnfor lf">Advanced F: cerns. S#="p" href="#MPFR-Interface">MPFR IAdvanced F: cerns. S#Imat,ont -}) -->i="#AdvancedImat, he D}) --> s">Advanced FAdvanced F: cerns. S#="p" href="#MPFR-Interface">MPFR IGNU Free DocumentationAdvanced F: cerns. S#Imat,ont -}) -->i="#AdvancedImat, he D}) --> s">Advanced FAdvanced F: cerns. S#Im detectree-Do="#AdvancedIm detectree-D-s">Advanced FAdvanced F: cerns. S#Imat,ont -}) -->i="#AdvancedImat, he D}) --> s">Advanced FAdvanced F: cerns. S#Iotegla-Rct or f="#AdvancedImtegla Rct or ns">Advanced F5.15 F: cerns. S#Iotea name=ction">5.15 FAdvanced F: cerns. S#=isceedrnenex-="#AdvancedMisceedrnenexds">Advanced Fmpfr-iit F: cerns. S#="p" BcuocsFR-InteBcuocsFR IAdvanced F: cerns. S#Imat,ont -}) -->i="#AdvancedImat, he D}) --> s">Advanced FMPFR IAdvanced F: cerns. S#Ss requ ="#AdvancedSs requss">Advanced F

, Previous: Conceptbutors, Up: Top

## Conon LiLicul up -->

Advanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced F5.15 FAdvanced FAdvanced FEe Docrbolity Advanced Fmpfed F: cerns. S#Ee Docrbol">Ee Docrbolity mpfed F: cerns. S#Ee Docrbol">Ee Docrbolity mpfed F: cerns. S#Ee Docrbol">Ee Docrbolity mpfed F: cerns. S#Ee Docrbol">Ee Docrbolity mpfed F: cerns. S#Ee Docrbol">Ee Docrbolity mpfed F: cerns. S#Ee Docrbol">Ee Docrbolity Advanced FAdvanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced FAdvanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced FAdvanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced FAdvanced Fmpfed F: cerns. S#Ss requ ="#AdvancedSs requss">Advanced Fmpfed F: cerns. S#Ss requ ="#AdvancedSs requss">Advanced Fmpfed F: cerns. S#Ss requ ="#AdvancedSs requss">Advanced FAdvanced FAdvanced Fmpfed F: cerns. S#="p" href="#Advanced-Functions">Advanced FAdvanced Fmpfed F: cerns. S#Cicithref="#Compatibility-with-MPF">Compatibility mpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced FCompatibility mpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced Fmpfed F: cerns. S#Ee Docrbol">Ee Docrbolity Advanced FAdvanced FAdvanced FAdvanced FAdvanced Fmpfed F: cerns. S#Ss requ ="#AdvancedSs requss">Advanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced Fmpfed F: cerns. S#Ss requ ="#AdvancedSs requss">Advanced Fmpfed F: cerns. S#Citr any li="#AdvancedCitr any lns">Advanced FAdvanced Fmpfed F: cerns. S#Citr any li="#AdvancedCitr any lns">Advanced Fmpfed F: cerns. S#Iotea name=ction">5.15 FAdvanced FAdvanced Fmp-212itfrecmpe>e>mpfed F: cerns. S#,et_defa-Model"#Rfault MPModeled Fmpfed F: cerns. S#Ee Docrbol">Ee Docrbolity Ee Docrbolity Ee Docrbolity mpfred F: cerns. S#Ee Docrbol">Ee Docrbolity Ee Docrbolity Ee Docrbolity mpfed F: cerns. S#=isceedrnenex-="#AdvancedMisceedrnenexds">Advanced Fmpfed F: cerns. S#Citr any li="#AdvancedCitr any lns">Advanced Fe>mpfed F: cerns. S#Io detectree-Do="#AdvancedIm detectree-D-s">Advanced Fmpfed F: cerns. S#Citr any li="#AdvancedCitr any lns">Advanced Fmpfed F: cerns. S#Citr any li="#AdvancedCitr any lns">Advanced Fmpfed F: cerns. S#Citr any li="#AdvancedCitr any lns">Advanced Fmpfed F: cerns. S#Citr any li="#AdvancedCitr any lns">Advanced Fmpfed F: cerns. S#Citr any li="#AdvancedCitr any lns">Advanced Fmpfed F: cerns. S#=isceedrnenex-="#AdvancedMisceedrnenexds">Advanced Fmpfed F: cerns. S#Citr any li="#AdvancedCitr any lns">Advanced FAdvanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced Fmpfed F: cerns. S#Ee Docrbol">Ee Docrbolity mpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced FAdvanced FAdvanced Fmpfed F: cerns. S#Cice of -Io detectree-Dont -Asur oNU F-="#AdvancedCice of Io detectree-D-Cove-sur oNU Fns">Advanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced Fmpfed F: cerns. S#Imat,ont -}) -->i="#AdvancedImat, he D}) --> s">Advanced Fmpfed F: cerns. S#Iotegla-Rct or f="#AdvancedImtegla Rct or ns">Advanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced FAdvanced Fmpfed F: cerns. S#Cicithref="#Compatibility-with-MPF">Compatibility mpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced FAdvanced Fmpfed F: cerns. S#Ee Docrbol">Ee Docrbolity Advanced FAdvanced FAdvanced FAdvanced FAdvanced Fmpfed F: cerns. S#Imat,ont -}) -->i="#AdvancedImat, he D}) --> s">Advanced Fmpfed F: cerns. S#Ee Docrbol">Ee Docrbolity Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmp-215itfrecmpe>e>mpfed F: cerns. S#,et_defa-Model"#Rfault MPModeled FAdvanced FAdvanced FCompatibility Advanced Fmpfed F: cerns. S#Iotegla-Rct or f="#AdvancedImtegla Rct or ns">Advanced Fmpfed F: cerns. S#Iotegla-Rct or f="#AdvancedImtegla Rct or ns">Advanced Fmpfed F: cerns. S#Iotegla-Rct or f="#AdvancedImtegla Rct or ns">Advanced FA-190itfrecAode>mpfed F: cerns. S#Iotegla-Rct or f="#AdvancedImtegla Rct or ns">Advanced FAdvanced Fe>mpfed F: cerns. S#,et_defa-Model"#Rfault MPModeled FAdvanced Fmpfed F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced FAdvanced Fmp-211itfrecmpe>e>mpfed F: cerns. S#,et_defa-Model"#Rfault MPModeled Fmpfed F: cerns. S#Ee Docrbol">Ee Docrbolity mpfred F: cerns. S#Ee Docrbol">Ee Docrbolity mpfed F: cerns. S#=isceedrnenex-="#AdvancedMisceedrnenexds">Advanced Fmpfed F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced Fmpfed F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced Fmpfed F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced Fmpfred F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced Fe>mpfed F: cerns. S#Io detectree-Do="#AdvancedIm detectree-D-s">Advanced Fmpfred F: cerns. S#Cicithref="#Compatibility-with-MPF">Compatibility mpfed F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced Fmpfed F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced FAdvanced Fmpfed F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced FAdvanced Fmpfed F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced Fmpfed F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced FAdvanced Fmpfed F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced FAdvanced Fmpfed F: cerns. S#Asur oNU F-="#Advanced-sur oNU Fns">Advanced FAdvanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced FAdvanced Fmpfed F: cerns. S#Ss requ ="#AdvancedSs requss">Advanced FAdvanced FAdvanced FAdvanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced FAdvanced FAdvanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced FAdvanced FAdvanced Fmpfed F: cerns. S#="p" BcuocsFR-InteBcuocsFR IAdvanced FAdvanced FA-186itfrecAode>mpfed F: cerns. S#Iotegla-Rct or f="#AdvancedImtegla Rct or ns">Advanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfred F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced FAdvanced Fmpfed F: cerns. S#Bcuoc Ating-poin-="#AdvancedBcuocle Precisions">Advanced Fmpfed F: cerns. S#Ee Docrbol">Ee Docrbolity mpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced FAdvanced Fmpfed F: cerns. S#=isceedrnenex-=" #advancedmisceedrnenexds">Advanced Fmpfed F: cerns. S#=isceedrnenex-="#AdvancedMisceedrnenexds">Advanced Fmpfed F: cerns. S#=isceedrnenex-="#AdvancedMisceedrnenexds">Advanced Fmpfed F: cerns. S#=isceedrnenex-="#AdvancedMisceedrnenexds">Advanced Fmpfed F: cerns. S#=isceedrnenex-="#AdvancedMisceedrnenexds">Advanced Fmpfed F: cerns. S#Cicitrediti="#AdvancedCicitreditns">Advanced FAdvanced F