gfiber / toolchains / bruno / f0dcd7f7efd6f7b917902cadf4d492dcb3676093 / . / usr / share / info / mpfr.info

This is mpfr.info, produced by makeinfo version 5.2 from mpfr.texi. | |

This manual documents how to install and use the Multiple Precision | |

Floating-Point Reliable Library, version 3.1.3. | |

Copyright 1991, 1993-2015 Free Software Foundation, Inc. | |

Permission is granted to copy, distribute and/or modify this document | |

under the terms of the GNU Free Documentation License, Version 1.2 or | |

any later version published by the Free Software Foundation; with no | |

Invariant Sections, with no Front-Cover Texts, and with no Back-Cover | |

Texts. A copy of the license is included in *note GNU Free | |

Documentation License::. | |

INFO-DIR-SECTION Software libraries | |

START-INFO-DIR-ENTRY | |

* mpfr: (mpfr). Multiple Precision Floating-Point Reliable Library. | |

END-INFO-DIR-ENTRY | |

File: mpfr.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir) | |

GNU MPFR | |

******** | |

This manual documents how to install and use the Multiple Precision | |

Floating-Point Reliable Library, version 3.1.3. | |

Copyright 1991, 1993-2015 Free Software Foundation, Inc. | |

Permission is granted to copy, distribute and/or modify this document | |

under the terms of the GNU Free Documentation License, Version 1.2 or | |

any later version published by the Free Software Foundation; with no | |

Invariant Sections, with no Front-Cover Texts, and with no Back-Cover | |

Texts. A copy of the license is included in *note GNU Free | |

Documentation License::. | |

* Menu: | |

* Copying:: MPFR Copying Conditions (LGPL). | |

* Introduction to MPFR:: Brief introduction to GNU MPFR. | |

* Installing MPFR:: How to configure and compile the MPFR library. | |

* Reporting Bugs:: How to usefully report bugs. | |

* MPFR Basics:: What every MPFR user should now. | |

* MPFR Interface:: MPFR functions and macros. | |

* API Compatibility:: API compatibility with previous MPFR versions. | |

* Contributors:: | |

* References:: | |

* GNU Free Documentation License:: | |

* Concept Index:: | |

* Function and Type Index:: | |

File: mpfr.info, Node: Copying, Next: Introduction to MPFR, Prev: Top, Up: Top | |

MPFR Copying Conditions | |

*********************** | |

The GNU MPFR library (or MPFR for short) is "free"; this means that | |

everyone is free to use it and free to redistribute it on a free basis. | |

The library is not in the public domain; it is copyrighted and there are | |

restrictions on its distribution, but these restrictions are designed to | |

permit everything that a good cooperating citizen would want to do. | |

What is not allowed is to try to prevent others from further sharing any | |

version of this library that they might get from you. | |

Specifically, we want to make sure that you have the right to give | |

away copies of the library, that you receive source code or else can get | |

it if you want it, that you can change this library or use pieces of it | |

in new free programs, and that you know you can do these things. | |

To make sure that everyone has such rights, we have to forbid you to | |

deprive anyone else of these rights. For example, if you distribute | |

copies of the GNU MPFR library, you must give the recipients all the | |

rights that you have. You must make sure that they, too, receive or can | |

get the source code. And you must tell them their rights. | |

Also, for our own protection, we must make certain that everyone | |

finds out that there is no warranty for the GNU MPFR library. If it is | |

modified by someone else and passed on, we want their recipients to know | |

that what they have is not what we distributed, so that any problems | |

introduced by others will not reflect on our reputation. | |

The precise conditions of the license for the GNU MPFR library are | |

found in the Lesser General Public License that accompanies the source | |

code. See the file COPYING.LESSER. | |

File: mpfr.info, Node: Introduction to MPFR, Next: Installing MPFR, Prev: Copying, Up: Top | |

1 Introduction to MPFR | |

********************** | |

MPFR is a portable library written in C for arbitrary precision | |

arithmetic on floating-point numbers. It is based on the GNU MP | |

library. It aims to provide a class of floating-point numbers with | |

precise semantics. The main characteristics of MPFR, which make it | |

differ from most arbitrary precision floating-point software tools, are: | |

• the MPFR code is portable, i.e., the result of any operation does | |

not depend on the machine word size ‘mp_bits_per_limb’ (64 on most | |

current processors); | |

• the precision in bits can be set _exactly_ to any valid value for | |

each variable (including very small precision); | |

• MPFR provides the four rounding modes from the IEEE 754-1985 | |

standard, plus away-from-zero, as well as for basic operations as | |

for other mathematical functions. | |

In particular, with a precision of 53 bits, MPFR is able to exactly | |

reproduce all computations with double-precision machine floating-point | |

numbers (e.g., ‘double’ type in C, with a C implementation that | |

rigorously follows Annex F of the ISO C99 standard and ‘FP_CONTRACT’ | |

pragma set to ‘OFF’) on the four arithmetic operations and the square | |

root, except the default exponent range is much wider and subnormal | |

numbers are not implemented (but can be emulated). | |

This version of MPFR is released under the GNU Lesser General Public | |

License, version 3 or any later version. It is permitted to link MPFR | |

to most non-free programs, as long as when distributing them the MPFR | |

source code and a means to re-link with a modified MPFR library is | |

provided. | |

1.1 How to Use This Manual | |

========================== | |

Everyone should read *note MPFR Basics::. If you need to install the | |

library yourself, you need to read *note Installing MPFR::, too. To use | |

the library you will need to refer to *note MPFR Interface::. | |

The rest of the manual can be used for later reference, although it | |

is probably a good idea to glance through it. | |

File: mpfr.info, Node: Installing MPFR, Next: Reporting Bugs, Prev: Introduction to MPFR, Up: Top | |

2 Installing MPFR | |

***************** | |

The MPFR library is already installed on some GNU/Linux distributions, | |

but the development files necessary to the compilation such as ‘mpfr.h’ | |

are not always present. To check that MPFR is fully installed on your | |

computer, you can check the presence of the file ‘mpfr.h’ in | |

‘/usr/include’, or try to compile a small program having ‘#include | |

<mpfr.h>’ (since ‘mpfr.h’ may be installed somewhere else). For | |

instance, you can try to compile: | |

#include <stdio.h> | |

#include <mpfr.h> | |

int main (void) | |

{ | |

printf ("MPFR library: %-12s\nMPFR header: %s (based on %d.%d.%d)\n", | |

mpfr_get_version (), MPFR_VERSION_STRING, MPFR_VERSION_MAJOR, | |

MPFR_VERSION_MINOR, MPFR_VERSION_PATCHLEVEL); | |

return 0; | |

} | |

with | |

cc -o version version.c -lmpfr -lgmp | |

and if you get errors whose first line looks like | |

version.c:2:19: error: mpfr.h: No such file or directory | |

then MPFR is probably not installed. Running this program will give you | |

the MPFR version. | |

If MPFR is not installed on your computer, or if you want to install | |

a different version, please follow the steps below. | |

2.1 How to Install | |

================== | |

Here are the steps needed to install the library on Unix systems (more | |

details are provided in the ‘INSTALL’ file): | |

1. To build MPFR, you first have to install GNU MP (version 4.1 or | |

higher) on your computer. You need a C compiler, preferably GCC, | |

but any reasonable compiler should work. And you need the standard | |

Unix ‘make’ command, plus some other standard Unix utility | |

commands. | |

Then, in the MPFR build directory, type the following commands. | |

2. ‘./configure’ | |

This will prepare the build and setup the options according to your | |

system. You can give options to specify the install directories | |

(instead of the default ‘/usr/local’), threading support, and so | |

on. See the ‘INSTALL’ file and/or the output of ‘./configure | |

--help’ for more information, in particular if you get error | |

messages. | |

3. ‘make’ | |

This will compile MPFR, and create a library archive file | |

‘libmpfr.a’. On most platforms, a dynamic library will be produced | |

too. | |

4. ‘make check’ | |

This will make sure that MPFR was built correctly. If any test | |

fails, information about this failure can be found in the | |

‘tests/test-suite.log’ file. If you want the contents of this file | |

to be automatically output in case of failure, you can set the | |

‘VERBOSE’ environment variable to 1 before running ‘make check’, | |

for instance by typing: | |

‘VERBOSE=1 make check’ | |

In case of failure, you may want to check whether the problem is | |

already known. If not, please report this failure to the MPFR | |

mailing-list ‘mpfr@inria.fr’. For details, *Note Reporting Bugs::. | |

5. ‘make install’ | |

This will copy the files ‘mpfr.h’ and ‘mpf2mpfr.h’ to the directory | |

‘/usr/local/include’, the library files (‘libmpfr.a’ and possibly | |

others) to the directory ‘/usr/local/lib’, the file ‘mpfr.info’ to | |

the directory ‘/usr/local/share/info’, and some other documentation | |

files to the directory ‘/usr/local/share/doc/mpfr’ (or if you | |

passed the ‘--prefix’ option to ‘configure’, using the prefix | |

directory given as argument to ‘--prefix’ instead of ‘/usr/local’). | |

2.2 Other ‘make’ Targets | |

======================== | |

There are some other useful make targets: | |

• ‘mpfr.info’ or ‘info’ | |

Create or update an info version of the manual, in ‘mpfr.info’. | |

This file is already provided in the MPFR archives. | |

• ‘mpfr.pdf’ or ‘pdf’ | |

Create a PDF version of the manual, in ‘mpfr.pdf’. | |

• ‘mpfr.dvi’ or ‘dvi’ | |

Create a DVI version of the manual, in ‘mpfr.dvi’. | |

• ‘mpfr.ps’ or ‘ps’ | |

Create a Postscript version of the manual, in ‘mpfr.ps’. | |

• ‘mpfr.html’ or ‘html’ | |

Create a HTML version of the manual, in several pages in the | |

directory ‘doc/mpfr.html’; if you want only one output HTML file, | |

then type ‘makeinfo --html --no-split mpfr.texi’ from the ‘doc’ | |

directory instead. | |

• ‘clean’ | |

Delete all object files and archive files, but not the | |

configuration files. | |

• ‘distclean’ | |

Delete all generated files not included in the distribution. | |

• ‘uninstall’ | |

Delete all files copied by ‘make install’. | |

2.3 Build Problems | |

================== | |

In case of problem, please read the ‘INSTALL’ file carefully before | |

reporting a bug, in particular section “In case of problem”. Some | |

problems are due to bad configuration on the user side (not specific to | |

MPFR). Problems are also mentioned in the FAQ | |

<http://www.mpfr.org/faq.html>. | |

Please report problems to the MPFR mailing-list ‘mpfr@inria.fr’. | |

*Note Reporting Bugs::. Some bug fixes are available on the MPFR 3.1.3 | |

web page <http://www.mpfr.org/mpfr-3.1.3/>. | |

2.4 Getting the Latest Version of MPFR | |

====================================== | |

The latest version of MPFR is available from | |

<ftp://ftp.gnu.org/gnu/mpfr/> or <http://www.mpfr.org/>. | |

File: mpfr.info, Node: Reporting Bugs, Next: MPFR Basics, Prev: Installing MPFR, Up: Top | |

3 Reporting Bugs | |

**************** | |

If you think you have found a bug in the MPFR library, first have a look | |

on the MPFR 3.1.3 web page <http://www.mpfr.org/mpfr-3.1.3/> and the FAQ | |

<http://www.mpfr.org/faq.html>: perhaps this bug is already known, in | |

which case you may find there a workaround for it. You might also look | |

in the archives of the MPFR mailing-list: | |

<https://sympa.inria.fr/sympa/arc/mpfr>. Otherwise, please investigate | |

and report it. We have made this library available to you, and it is | |

not to ask too much from you, to ask you to report the bugs that you | |

find. | |

There are a few things you should think about when you put your bug | |

report together. | |

You have to send us a test case that makes it possible for us to | |

reproduce the bug, i.e., a small self-content program, using no other | |

library than MPFR. Include instructions on how to run the test case. | |

You also have to explain what is wrong; if you get a crash, or if the | |

results you get are incorrect and in that case, in what way. | |

Please include compiler version information in your bug report. This | |

can be extracted using ‘cc -V’ on some machines, or, if you’re using | |

GCC, ‘gcc -v’. Also, include the output from ‘uname -a’ and the MPFR | |

version (the GMP version may be useful too). If you get a failure while | |

running ‘make’ or ‘make check’, please include the ‘config.log’ file in | |

your bug report, and in case of test failure, the ‘tests/test-suite.log’ | |

file too. | |

If your bug report is good, we will do our best to help you to get a | |

corrected version of the library; if the bug report is poor, we will not | |

do anything about it (aside of chiding you to send better bug reports). | |

Send your bug report to the MPFR mailing-list ‘mpfr@inria.fr’. | |

If you think something in this manual is unclear, or downright | |

incorrect, or if the language needs to be improved, please send a note | |

to the same address. | |

File: mpfr.info, Node: MPFR Basics, Next: MPFR Interface, Prev: Reporting Bugs, Up: Top | |

4 MPFR Basics | |

************* | |

* Menu: | |

* Headers and Libraries:: | |

* Nomenclature and Types:: | |

* MPFR Variable Conventions:: | |

* Rounding Modes:: | |

* Floating-Point Values on Special Numbers:: | |

* Exceptions:: | |

* Memory Handling:: | |

File: mpfr.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: MPFR Basics, Up: MPFR Basics | |

4.1 Headers and Libraries | |

========================= | |

All declarations needed to use MPFR are collected in the include file | |

‘mpfr.h’. It is designed to work with both C and C++ compilers. You | |

should include that file in any program using the MPFR library: | |

#include <mpfr.h> | |

Note however that prototypes for MPFR functions with ‘FILE *’ | |

parameters are provided only if ‘<stdio.h>’ is included too (before | |

‘mpfr.h’): | |

#include <stdio.h> | |

#include <mpfr.h> | |

Likewise ‘<stdarg.h>’ (or ‘<varargs.h>’) is required for prototypes | |

with ‘va_list’ parameters, such as ‘mpfr_vprintf’. | |

And for any functions using ‘intmax_t’, you must include ‘<stdint.h>’ | |

or ‘<inttypes.h>’ before ‘mpfr.h’, to allow ‘mpfr.h’ to define | |

prototypes for these functions. Moreover, users of C++ compilers under | |

some platforms may need to define ‘MPFR_USE_INTMAX_T’ (and should do it | |

for portability) before ‘mpfr.h’ has been included; of course, it is | |

possible to do that on the command line, e.g., with | |

‘-DMPFR_USE_INTMAX_T’. | |

Note: If ‘mpfr.h’ and/or ‘gmp.h’ (used by ‘mpfr.h’) are included | |

several times (possibly from another header file), ‘<stdio.h>’ and/or | |

‘<stdarg.h>’ (or ‘<varargs.h>’) should be included *before the first | |

inclusion* of ‘mpfr.h’ or ‘gmp.h’. Alternatively, you can define | |

‘MPFR_USE_FILE’ (for MPFR I/O functions) and/or ‘MPFR_USE_VA_LIST’ (for | |

MPFR functions with ‘va_list’ parameters) anywhere before the last | |

inclusion of ‘mpfr.h’. As a consequence, if your file is a public | |

header that includes ‘mpfr.h’, you need to use the latter method. | |

When calling a MPFR macro, it is not allowed to have previously | |

defined a macro with the same name as some keywords (currently ‘do’, | |

‘while’ and ‘sizeof’). | |

You can avoid the use of MPFR macros encapsulating functions by | |

defining the ‘MPFR_USE_NO_MACRO’ macro before ‘mpfr.h’ is included. In | |

general this should not be necessary, but this can be useful when | |

debugging user code: with some macros, the compiler may emit spurious | |

warnings with some warning options, and macros can prevent some | |

prototype checking. | |

All programs using MPFR must link against both ‘libmpfr’ and ‘libgmp’ | |

libraries. On a typical Unix-like system this can be done with ‘-lmpfr | |

-lgmp’ (in that order), for example: | |

gcc myprogram.c -lmpfr -lgmp | |

MPFR is built using Libtool and an application can use that to link | |

if desired, *note GNU Libtool: (libtool.info)Top. | |

If MPFR has been installed to a non-standard location, then it may be | |

necessary to set up environment variables such as ‘C_INCLUDE_PATH’ and | |

‘LIBRARY_PATH’, or use ‘-I’ and ‘-L’ compiler options, in order to point | |

to the right directories. For a shared library, it may also be | |

necessary to set up some sort of run-time library path (e.g., | |

‘LD_LIBRARY_PATH’) on some systems. Please read the ‘INSTALL’ file for | |

additional information. | |

File: mpfr.info, Node: Nomenclature and Types, Next: MPFR Variable Conventions, Prev: Headers and Libraries, Up: MPFR Basics | |

4.2 Nomenclature and Types | |

========================== | |

A "floating-point number", or "float" for short, is an arbitrary | |

precision significand (also called mantissa) with a limited precision | |

exponent. The C data type for such objects is ‘mpfr_t’ (internally | |

defined as a one-element array of a structure, and ‘mpfr_ptr’ is the C | |

data type representing a pointer to this structure). A floating-point | |

number can have three special values: Not-a-Number (NaN) or plus or | |

minus Infinity. NaN represents an uninitialized object, the result of | |

an invalid operation (like 0 divided by 0), or a value that cannot be | |

determined (like +Infinity minus +Infinity). Moreover, like in the IEEE | |

754 standard, zero is signed, i.e., there are both +0 and −0; the | |

behavior is the same as in the IEEE 754 standard and it is generalized | |

to the other functions supported by MPFR. Unless documented otherwise, | |

the sign bit of a NaN is unspecified. | |

The "precision" is the number of bits used to represent the significand | |

of a floating-point number; the corresponding C data type is | |

‘mpfr_prec_t’. The precision can be any integer between ‘MPFR_PREC_MIN’ | |

and ‘MPFR_PREC_MAX’. In the current implementation, ‘MPFR_PREC_MIN’ is | |

equal to 2. | |

Warning! MPFR needs to increase the precision internally, in order | |

to provide accurate results (and in particular, correct rounding). Do | |

not attempt to set the precision to any value near ‘MPFR_PREC_MAX’, | |

otherwise MPFR will abort due to an assertion failure. Moreover, you | |

may reach some memory limit on your platform, in which case the program | |

may abort, crash or have undefined behavior (depending on your C | |

implementation). | |

The "rounding mode" specifies the way to round the result of a | |

floating-point operation, in case the exact result can not be | |

represented exactly in the destination significand; the corresponding C | |

data type is ‘mpfr_rnd_t’. | |

File: mpfr.info, Node: MPFR Variable Conventions, Next: Rounding Modes, Prev: Nomenclature and Types, Up: MPFR Basics | |

4.3 MPFR Variable Conventions | |

============================= | |

Before you can assign to an MPFR variable, you need to initialize it by | |

calling one of the special initialization functions. When you’re done | |

with a variable, you need to clear it out, using one of the functions | |

for that purpose. A variable should only be initialized once, or at | |

least cleared out between each initialization. After a variable has | |

been initialized, it may be assigned to any number of times. For | |

efficiency reasons, avoid to initialize and clear out a variable in | |

loops. Instead, initialize it before entering the loop, and clear it | |

out after the loop has exited. You do not need to be concerned about | |

allocating additional space for MPFR variables, since any variable has a | |

significand of fixed size. Hence unless you change its precision, or | |

clear and reinitialize it, a floating-point variable will have the same | |

allocated space during all its life. | |

As a general rule, all MPFR functions expect output arguments before | |

input arguments. This notation is based on an analogy with the | |

assignment operator. MPFR allows you to use the same variable for both | |

input and output in the same expression. For example, the main function | |

for floating-point multiplication, ‘mpfr_mul’, can be used like this: | |

‘mpfr_mul (x, x, x, rnd)’. This computes the square of X with rounding | |

mode ‘rnd’ and puts the result back in X. | |

File: mpfr.info, Node: Rounding Modes, Next: Floating-Point Values on Special Numbers, Prev: MPFR Variable Conventions, Up: MPFR Basics | |

4.4 Rounding Modes | |

================== | |

The following five rounding modes are supported: | |

• ‘MPFR_RNDN’: round to nearest (roundTiesToEven in IEEE 754-2008), | |

• ‘MPFR_RNDZ’: round toward zero (roundTowardZero in IEEE 754-2008), | |

• ‘MPFR_RNDU’: round toward plus infinity (roundTowardPositive in | |

IEEE 754-2008), | |

• ‘MPFR_RNDD’: round toward minus infinity (roundTowardNegative in | |

IEEE 754-2008), | |

• ‘MPFR_RNDA’: round away from zero. | |

The ‘round to nearest’ mode works as in the IEEE 754 standard: in | |

case the number to be rounded lies exactly in the middle of two | |

representable numbers, it is rounded to the one with the least | |

significant bit set to zero. For example, the number 2.5, which is | |

represented by (10.1) in binary, is rounded to (10.0)=2 with a precision | |

of two bits, and not to (11.0)=3. This rule avoids the "drift" | |

phenomenon mentioned by Knuth in volume 2 of The Art of Computer | |

Programming (Section 4.2.2). | |

Most MPFR functions take as first argument the destination variable, | |

as second and following arguments the input variables, as last argument | |

a rounding mode, and have a return value of type ‘int’, called the | |

"ternary value". The value stored in the destination variable is | |

correctly rounded, i.e., MPFR behaves as if it computed the result with | |

an infinite precision, then rounded it to the precision of this | |

variable. The input variables are regarded as exact (in particular, | |

their precision does not affect the result). | |

As a consequence, in case of a non-zero real rounded result, the | |

error on the result is less or equal to 1/2 ulp (unit in the last place) | |

of that result in the rounding to nearest mode, and less than 1 ulp of | |

that result in the directed rounding modes (a ulp is the weight of the | |

least significant represented bit of the result after rounding). | |

Unless documented otherwise, functions returning an ‘int’ return a | |

ternary value. If the ternary value is zero, it means that the value | |

stored in the destination variable is the exact result of the | |

corresponding mathematical function. If the ternary value is positive | |

(resp. negative), it means the value stored in the destination variable | |

is greater (resp. lower) than the exact result. For example with the | |

‘MPFR_RNDU’ rounding mode, the ternary value is usually positive, except | |

when the result is exact, in which case it is zero. In the case of an | |

infinite result, it is considered as inexact when it was obtained by | |

overflow, and exact otherwise. A NaN result (Not-a-Number) always | |

corresponds to an exact return value. The opposite of a returned | |

ternary value is guaranteed to be representable in an ‘int’. | |

Unless documented otherwise, functions returning as result the value | |

‘1’ (or any other value specified in this manual) for special cases | |

(like ‘acos(0)’) yield an overflow or an underflow if that value is not | |

representable in the current exponent range. | |

File: mpfr.info, Node: Floating-Point Values on Special Numbers, Next: Exceptions, Prev: Rounding Modes, Up: MPFR Basics | |

4.5 Floating-Point Values on Special Numbers | |

============================================ | |

This section specifies the floating-point values (of type ‘mpfr_t’) | |

returned by MPFR functions (where by “returned” we mean here the | |

modified value of the destination object, which should not be mixed with | |

the ternary return value of type ‘int’ of those functions). For | |

functions returning several values (like ‘mpfr_sin_cos’), the rules | |

apply to each result separately. | |

Functions can have one or several input arguments. An input point is | |

a mapping from these input arguments to the set of the MPFR numbers. | |

When none of its components are NaN, an input point can also be seen as | |

a tuple in the extended real numbers (the set of the real numbers with | |

both infinities). | |

When the input point is in the domain of the mathematical function, | |

the result is rounded as described in Section “Rounding Modes” (but see | |

below for the specification of the sign of an exact zero). Otherwise | |

the general rules from this section apply unless stated otherwise in the | |

description of the MPFR function (*note MPFR Interface::). | |

When the input point is not in the domain of the mathematical | |

function but is in its closure in the extended real numbers and the | |

function can be extended by continuity, the result is the obtained | |

limit. Examples: ‘mpfr_hypot’ on (+Inf,0) gives +Inf. But ‘mpfr_pow’ | |

cannot be defined on (1,+Inf) using this rule, as one can find sequences | |

(X_N,Y_N) such that X_N goes to 1, Y_N goes to +Inf and X_N to the Y_N | |

goes to any positive value when N goes to the infinity. | |

When the input point is in the closure of the domain of the | |

mathematical function and an input argument is +0 (resp. −0), one | |

considers the limit when the corresponding argument approaches 0 from | |

above (resp. below). If the limit is not defined (e.g., ‘mpfr_log’ on | |

−0), the behavior is specified in the description of the MPFR function. | |

When the result is equal to 0, its sign is determined by considering | |

the limit as if the input point were not in the domain: If one | |

approaches 0 from above (resp. below), the result is +0 (resp. −0); for | |

example, ‘mpfr_sin’ on +0 gives +0. In the other cases, the sign is | |

specified in the description of the MPFR function; for example | |

‘mpfr_max’ on −0 and +0 gives +0. | |

When the input point is not in the closure of the domain of the | |

function, the result is NaN. Example: ‘mpfr_sqrt’ on −17 gives NaN. | |

When an input argument is NaN, the result is NaN, possibly except | |

when a partial function is constant on the finite floating-point | |

numbers; such a case is always explicitly specified in *note MPFR | |

Interface::. Example: ‘mpfr_hypot’ on (NaN,0) gives NaN, but | |

‘mpfr_hypot’ on (NaN,+Inf) gives +Inf (as specified in *note Special | |

Functions::), since for any finite input X, ‘mpfr_hypot’ on (X,+Inf) | |

gives +Inf. | |

File: mpfr.info, Node: Exceptions, Next: Memory Handling, Prev: Floating-Point Values on Special Numbers, Up: MPFR Basics | |

4.6 Exceptions | |

============== | |

MPFR supports 6 exception types: | |

• Underflow: An underflow occurs when the exact result of a function | |

is a non-zero real number and the result obtained after the | |

rounding, assuming an unbounded exponent range (for the rounding), | |

has an exponent smaller than the minimum value of the current | |

exponent range. (In the round-to-nearest mode, the halfway case is | |

rounded toward zero.) | |

Note: This is not the single possible definition of the underflow. | |

MPFR chooses to consider the underflow _after_ rounding. The | |

underflow before rounding can also be defined. For instance, | |

consider a function that has the exact result 7 multiplied by two | |

to the power E−4, where E is the smallest exponent (for a | |

significand between 1/2 and 1), with a 2-bit target precision and | |

rounding toward plus infinity. The exact result has the exponent | |

E−1. With the underflow before rounding, such a function call | |

would yield an underflow, as E−1 is outside the current exponent | |

range. However, MPFR first considers the rounded result assuming | |

an unbounded exponent range. The exact result cannot be | |

represented exactly in precision 2, and here, it is rounded to 0.5 | |

times 2 to E, which is representable in the current exponent range. | |

As a consequence, this will not yield an underflow in MPFR. | |

• Overflow: An overflow occurs when the exact result of a function is | |

a non-zero real number and the result obtained after the rounding, | |

assuming an unbounded exponent range (for the rounding), has an | |

exponent larger than the maximum value of the current exponent | |

range. In the round-to-nearest mode, the result is infinite. | |

Note: unlike the underflow case, there is only one possible | |

definition of overflow here. | |

• Divide-by-zero: An exact infinite result is obtained from finite | |

inputs. | |

• NaN: A NaN exception occurs when the result of a function is NaN. | |

• Inexact: An inexact exception occurs when the result of a function | |

cannot be represented exactly and must be rounded. | |

• Range error: A range exception occurs when a function that does not | |

return a MPFR number (such as comparisons and conversions to an | |

integer) has an invalid result (e.g., an argument is NaN in | |

‘mpfr_cmp’, or a conversion to an integer cannot be represented in | |

the target type). | |

MPFR has a global flag for each exception, which can be cleared, set | |

or tested by functions described in *note Exception Related Functions::. | |

Differences with the ISO C99 standard: | |

• In C, only quiet NaNs are specified, and a NaN propagation does not | |

raise an invalid exception. Unless explicitly stated otherwise, | |

MPFR sets the NaN flag whenever a NaN is generated, even when a NaN | |

is propagated (e.g., in NaN + NaN), as if all NaNs were signaling. | |

• An invalid exception in C corresponds to either a NaN exception or | |

a range error in MPFR. | |

File: mpfr.info, Node: Memory Handling, Prev: Exceptions, Up: MPFR Basics | |

4.7 Memory Handling | |

=================== | |

MPFR functions may create caches, e.g., when computing constants such as | |

Pi, either because the user has called a function like ‘mpfr_const_pi’ | |

directly or because such a function was called internally by the MPFR | |

library itself to compute some other function. | |

At any time, the user can free the various caches with | |

‘mpfr_free_cache’. It is strongly advised to do that before terminating | |

a thread, or before exiting when using tools like ‘valgrind’ (to avoid | |

memory leaks being reported). | |

MPFR internal data such as flags, the exponent range, the default | |

precision and rounding mode, and caches (i.e., data that are not | |

accessed via parameters) are either global (if MPFR has not been | |

compiled as thread safe) or per-thread (thread local storage, TLS). The | |

initial values of TLS data after a thread is created entirely depend on | |

the compiler and thread implementation (MPFR simply does a conventional | |

variable initialization, the variables being declared with an | |

implementation-defined TLS specifier). | |

File: mpfr.info, Node: MPFR Interface, Next: API Compatibility, Prev: MPFR Basics, Up: Top | |

5 MPFR Interface | |

**************** | |

The floating-point functions expect arguments of type ‘mpfr_t’. | |

The MPFR floating-point functions have an interface that is similar | |

to the GNU MP functions. The function prefix for floating-point | |

operations is ‘mpfr_’. | |

The user has to specify the precision of each variable. A | |

computation that assigns a variable will take place with the precision | |

of the assigned variable; the cost of that computation should not depend | |

on the precision of variables used as input (on average). | |

The semantics of a calculation in MPFR is specified as follows: | |

Compute the requested operation exactly (with “infinite accuracy”), and | |

round the result to the precision of the destination variable, with the | |

given rounding mode. The MPFR floating-point functions are intended to | |

be a smooth extension of the IEEE 754 arithmetic. The results obtained | |

on a given computer are identical to those obtained on a computer with a | |

different word size, or with a different compiler or operating system. | |

MPFR _does not keep track_ of the accuracy of a computation. This is | |

left to the user or to a higher layer (for example the MPFI library for | |

interval arithmetic). As a consequence, if two variables are used to | |

store only a few significant bits, and their product is stored in a | |

variable with large precision, then MPFR will still compute the result | |

with full precision. | |

The value of the standard C macro ‘errno’ may be set to non-zero by | |

any MPFR function or macro, whether or not there is an error. | |

* Menu: | |

* Initialization Functions:: | |

* Assignment Functions:: | |

* Combined Initialization and Assignment Functions:: | |

* Conversion Functions:: | |

* Basic Arithmetic Functions:: | |

* Comparison Functions:: | |

* Special Functions:: | |

* Input and Output Functions:: | |

* Formatted Output Functions:: | |

* Integer Related Functions:: | |

* Rounding Related Functions:: | |

* Miscellaneous Functions:: | |

* Exception Related Functions:: | |

* Compatibility with MPF:: | |

* Custom Interface:: | |

* Internals:: | |

File: mpfr.info, Node: Initialization Functions, Next: Assignment Functions, Prev: MPFR Interface, Up: MPFR Interface | |

5.1 Initialization Functions | |

============================ | |

An ‘mpfr_t’ object must be initialized before storing the first value in | |

it. The functions ‘mpfr_init’ and ‘mpfr_init2’ are used for that | |

purpose. | |

-- Function: void mpfr_init2 (mpfr_t X, mpfr_prec_t PREC) | |

Initialize X, set its precision to be *exactly* PREC bits and its | |

value to NaN. (Warning: the corresponding MPF function initializes | |

to zero instead.) | |

Normally, a variable should be initialized once only or at least be | |

cleared, using ‘mpfr_clear’, between initializations. To change | |

the precision of a variable which has already been initialized, use | |

‘mpfr_set_prec’. The precision PREC must be an integer between | |

‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’ (otherwise the behavior is | |

undefined). | |

-- Function: void mpfr_inits2 (mpfr_prec_t PREC, mpfr_t X, ...) | |

Initialize all the ‘mpfr_t’ variables of the given variable | |

argument ‘va_list’, set their precision to be *exactly* PREC bits | |

and their value to NaN. See ‘mpfr_init2’ for more details. The | |

‘va_list’ is assumed to be composed only of type ‘mpfr_t’ (or | |

equivalently ‘mpfr_ptr’). It begins from X, and ends when it | |

encounters a null pointer (whose type must also be ‘mpfr_ptr’). | |

-- Function: void mpfr_clear (mpfr_t X) | |

Free the space occupied by the significand of X. Make sure to call | |

this function for all ‘mpfr_t’ variables when you are done with | |

them. | |

-- Function: void mpfr_clears (mpfr_t X, ...) | |

Free the space occupied by all the ‘mpfr_t’ variables of the given | |

‘va_list’. See ‘mpfr_clear’ for more details. The ‘va_list’ is | |

assumed to be composed only of type ‘mpfr_t’ (or equivalently | |

‘mpfr_ptr’). It begins from X, and ends when it encounters a null | |

pointer (whose type must also be ‘mpfr_ptr’). | |

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

(since ‘NULL’ is not necessarily defined in this context, we use | |

‘(mpfr_ptr) 0’ instead, but ‘(mpfr_ptr) NULL’ is also correct). | |

{ | |

mpfr_t x, y, z, t; | |

mpfr_inits2 (256, x, y, z, t, (mpfr_ptr) 0); | |

… | |

mpfr_clears (x, y, z, t, (mpfr_ptr) 0); | |

} | |

-- Function: void mpfr_init (mpfr_t X) | |

Initialize X, set its precision to the default precision, and set | |

its value to NaN. The default precision can be changed by a call to | |

‘mpfr_set_default_prec’. | |

Warning! In a given program, some other libraries might change the | |

default precision and not restore it. Thus it is safer to use | |

‘mpfr_init2’. | |

-- Function: void mpfr_inits (mpfr_t X, ...) | |

Initialize all the ‘mpfr_t’ variables of the given ‘va_list’, set | |

their precision to the default precision and their value to NaN. | |

See ‘mpfr_init’ for more details. The ‘va_list’ is assumed to be | |

composed only of type ‘mpfr_t’ (or equivalently ‘mpfr_ptr’). It | |

begins from X, and ends when it encounters a null pointer (whose | |

type must also be ‘mpfr_ptr’). | |

Warning! In a given program, some other libraries might change the | |

default precision and not restore it. Thus it is safer to use | |

‘mpfr_inits2’. | |

-- Macro: MPFR_DECL_INIT (NAME, PREC) | |

This macro declares NAME as an automatic variable of type ‘mpfr_t’, | |

initializes it and sets its precision to be *exactly* PREC bits and | |

its value to NaN. NAME must be a valid identifier. You must use | |

this macro in the declaration section. This macro is much faster | |

than using ‘mpfr_init2’ but has some drawbacks: | |

• You *must not* call ‘mpfr_clear’ with variables created with | |

this macro (the storage is allocated at the point of | |

declaration and deallocated when the brace-level is exited). | |

• You *cannot* change their precision. | |

• You *should not* create variables with huge precision with | |

this macro. | |

• Your compiler must support ‘Non-Constant Initializers’ | |

(standard in C++ and ISO C99) and ‘Token Pasting’ (standard in | |

ISO C89). If PREC is not a constant expression, your compiler | |

must support ‘variable-length automatic arrays’ (standard in | |

ISO C99). GCC 2.95.3 and above supports all these features. | |

If you compile your program with GCC in C89 mode and with | |

‘-pedantic’, you may want to define the ‘MPFR_USE_EXTENSION’ | |

macro to avoid warnings due to the ‘MPFR_DECL_INIT’ | |

implementation. | |

-- Function: void mpfr_set_default_prec (mpfr_prec_t PREC) | |

Set the default precision to be *exactly* PREC bits, where PREC can | |

be any integer between ‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’. The | |

precision of a variable means the number of bits used to store its | |

significand. All subsequent calls to ‘mpfr_init’ or ‘mpfr_inits’ | |

will use this precision, but previously initialized variables are | |

unaffected. The default precision is set to 53 bits initially. | |

Note: when MPFR is built with the ‘--enable-thread-safe’ configure | |

option, the default precision is local to each thread. *Note | |

Memory Handling::, for more information. | |

-- Function: mpfr_prec_t mpfr_get_default_prec (void) | |

Return the current default MPFR precision in bits. See the | |

documentation of ‘mpfr_set_default_prec’. | |

Here is an example on how to initialize floating-point variables: | |

{ | |

mpfr_t x, y; | |

mpfr_init (x); /* use default precision */ | |

mpfr_init2 (y, 256); /* precision _exactly_ 256 bits */ | |

… | |

/* When the program is about to exit, do ... */ | |

mpfr_clear (x); | |

mpfr_clear (y); | |

mpfr_free_cache (); /* free the cache for constants like pi */ | |

} | |

The following functions are useful for changing the precision during | |

a calculation. A typical use would be for adjusting the precision | |

gradually in iterative algorithms like Newton-Raphson, making the | |

computation precision closely match the actual accurate part of the | |

numbers. | |

-- Function: void mpfr_set_prec (mpfr_t X, mpfr_prec_t PREC) | |

Reset the precision of X to be *exactly* PREC bits, and set its | |

value to NaN. The previous value stored in X is lost. It is | |

equivalent to a call to ‘mpfr_clear(x)’ followed by a call to | |

‘mpfr_init2(x, prec)’, but more efficient as no allocation is done | |

in case the current allocated space for the significand of X is | |

enough. The precision PREC can be any integer between | |

‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’. In case you want to keep the | |

previous value stored in X, use ‘mpfr_prec_round’ instead. | |

Warning! You must not use this function if X was initialized with | |

‘MPFR_DECL_INIT’ or with ‘mpfr_custom_init_set’ (*note Custom | |

Interface::). | |

-- Function: mpfr_prec_t mpfr_get_prec (mpfr_t X) | |

Return the precision of X, i.e., the number of bits used to store | |

its significand. | |

File: mpfr.info, Node: Assignment Functions, Next: Combined Initialization and Assignment Functions, Prev: Initialization Functions, Up: MPFR Interface | |

5.2 Assignment Functions | |

======================== | |

These functions assign new values to already initialized floats (*note | |

Initialization Functions::). | |

-- Function: int mpfr_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_set_ui (mpfr_t ROP, unsigned long int OP, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_set_uj (mpfr_t ROP, uintmax_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_set_sj (mpfr_t ROP, intmax_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_set_flt (mpfr_t ROP, float OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t | |

RND) | |

-- Function: int mpfr_set_decimal64 (mpfr_t ROP, _Decimal64 OP, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND) | |

Set the value of ROP from OP, rounded toward the given direction | |

RND. Note that the input 0 is converted to +0 by ‘mpfr_set_ui’, | |

‘mpfr_set_si’, ‘mpfr_set_uj’, ‘mpfr_set_sj’, ‘mpfr_set_z’, | |

‘mpfr_set_q’ and ‘mpfr_set_f’, regardless of the rounding mode. If | |

the system does not support the IEEE 754 standard, ‘mpfr_set_flt’, | |

‘mpfr_set_d’, ‘mpfr_set_ld’ and ‘mpfr_set_decimal64’ might not | |

preserve the signed zeros. The ‘mpfr_set_decimal64’ function is | |

built only with the configure option ‘--enable-decimal-float’, | |

which also requires ‘--with-gmp-build’, and when the compiler or | |

system provides the ‘_Decimal64’ data type (recent versions of GCC | |

support this data type); to use ‘mpfr_set_decimal64’, one should | |

define the macro ‘MPFR_WANT_DECIMAL_FLOATS’ before including | |

‘mpfr.h’. ‘mpfr_set_q’ might fail if the numerator (or the | |

denominator) can not be represented as a ‘mpfr_t’. | |

Note: If you want to store a floating-point constant to a ‘mpfr_t’, | |

you should use ‘mpfr_set_str’ (or one of the MPFR constant | |

functions, such as ‘mpfr_const_pi’ for Pi) instead of | |

‘mpfr_set_flt’, ‘mpfr_set_d’, ‘mpfr_set_ld’ or | |

‘mpfr_set_decimal64’. Otherwise the floating-point constant will | |

be first converted into a reduced-precision (e.g., 53-bit) binary | |

(or decimal, for ‘mpfr_set_decimal64’) number before MPFR can work | |

with it. | |

-- Function: int mpfr_set_ui_2exp (mpfr_t ROP, unsigned long int OP, | |

mpfr_exp_t E, mpfr_rnd_t RND) | |

-- Function: int mpfr_set_si_2exp (mpfr_t ROP, long int OP, mpfr_exp_t | |

E, mpfr_rnd_t RND) | |

-- Function: int mpfr_set_uj_2exp (mpfr_t ROP, uintmax_t OP, intmax_t | |

E, mpfr_rnd_t RND) | |

-- Function: int mpfr_set_sj_2exp (mpfr_t ROP, intmax_t OP, intmax_t E, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_set_z_2exp (mpfr_t ROP, mpz_t OP, mpfr_exp_t E, | |

mpfr_rnd_t RND) | |

Set the value of ROP from OP multiplied by two to the power E, | |

rounded toward the given direction RND. Note that the input 0 is | |

converted to +0. | |

-- Function: int mpfr_set_str (mpfr_t ROP, const char *S, int BASE, | |

mpfr_rnd_t RND) | |

Set ROP to the value of the string S in base BASE, rounded in the | |

direction RND. See the documentation of ‘mpfr_strtofr’ for a | |

detailed description of the valid string formats. Contrary to | |

‘mpfr_strtofr’, ‘mpfr_set_str’ requires the _whole_ string to | |

represent a valid floating-point number. | |

The meaning of the return value differs from other MPFR functions: | |

it is 0 if the entire string up to the final null character is a | |

valid number in base BASE; otherwise it is −1, and ROP may have | |

changed (users interested in the *note ternary value:: should use | |

‘mpfr_strtofr’ instead). | |

Note: it is preferable to use ‘mpfr_strtofr’ if one wants to | |

distinguish between an infinite ROP value coming from an infinite S | |

or from an overflow. | |

-- Function: int mpfr_strtofr (mpfr_t ROP, const char *NPTR, char | |

**ENDPTR, int BASE, mpfr_rnd_t RND) | |

Read a floating-point number from a string NPTR in base BASE, | |

rounded in the direction RND; BASE must be either 0 (to detect the | |

base, as described below) or a number from 2 to 62 (otherwise the | |

behavior is undefined). If NPTR starts with valid data, the result | |

is stored in ROP and ‘*ENDPTR’ points to the character just after | |

the valid data (if ENDPTR is not a null pointer); otherwise ROP is | |

set to zero (for consistency with ‘strtod’) and the value of NPTR | |

is stored in the location referenced by ENDPTR (if ENDPTR is not a | |

null pointer). The usual ternary value is returned. | |

Parsing follows the standard C ‘strtod’ function with some | |

extensions. After optional leading whitespace, one has a subject | |

sequence consisting of an optional sign (‘+’ or ‘-’), and either | |

numeric data or special data. The subject sequence is defined as | |

the longest initial subsequence of the input string, starting with | |

the first non-whitespace character, that is of the expected form. | |

The form of numeric data is a non-empty sequence of significand | |

digits with an optional decimal point, and an optional exponent | |

consisting of an exponent prefix followed by an optional sign and a | |

non-empty sequence of decimal digits. A significand digit is | |

either a decimal digit or a Latin letter (62 possible characters), | |

with ‘A’ = 10, ‘B’ = 11, …, ‘Z’ = 35; case is ignored in bases less | |

or equal to 36, in bases larger than 36, ‘a’ = 36, ‘b’ = 37, …, ‘z’ | |

= 61. The value of a significand digit must be strictly less than | |

the base. The decimal point can be either the one defined by the | |

current locale or the period (the first one is accepted for | |

consistency with the C standard and the practice, the second one is | |

accepted to allow the programmer to provide MPFR numbers from | |

strings in a way that does not depend on the current locale). The | |

exponent prefix can be ‘e’ or ‘E’ for bases up to 10, or ‘@’ in any | |

base; it indicates a multiplication by a power of the base. In | |

bases 2 and 16, the exponent prefix can also be ‘p’ or ‘P’, in | |

which case the exponent, called _binary exponent_, indicates a | |

multiplication by a power of 2 instead of the base (there is a | |

difference only for base 16); in base 16 for example ‘1p2’ | |

represents 4 whereas ‘1@2’ represents 256. The value of an | |

exponent is always written in base 10. | |

If the argument BASE is 0, then the base is automatically detected | |

as follows. If the significand starts with ‘0b’ or ‘0B’, base 2 is | |

assumed. If the significand starts with ‘0x’ or ‘0X’, base 16 is | |

assumed. Otherwise base 10 is assumed. | |

Note: The exponent (if present) must contain at least a digit. | |

Otherwise the possible exponent prefix and sign are not part of the | |

number (which ends with the significand). Similarly, if ‘0b’, | |

‘0B’, ‘0x’ or ‘0X’ is not followed by a binary/hexadecimal digit, | |

then the subject sequence stops at the character ‘0’, thus 0 is | |

read. | |

Special data (for infinities and NaN) can be ‘@inf@’ or | |

‘@nan@(n-char-sequence-opt)’, and if BASE <= 16, it can also be | |

‘infinity’, ‘inf’, ‘nan’ or ‘nan(n-char-sequence-opt)’, all case | |

insensitive. A ‘n-char-sequence-opt’ is a possibly empty string | |

containing only digits, Latin letters and the underscore (0, 1, 2, | |

…, 9, a, b, …, z, A, B, …, Z, _). Note: one has an optional sign | |

for all data, even NaN. For example, ‘-@nAn@(This_Is_Not_17)’ is a | |

valid representation for NaN in base 17. | |

-- Function: void mpfr_set_nan (mpfr_t X) | |

-- Function: void mpfr_set_inf (mpfr_t X, int SIGN) | |

-- Function: void mpfr_set_zero (mpfr_t X, int SIGN) | |

Set the variable X to NaN (Not-a-Number), infinity or zero | |

respectively. In ‘mpfr_set_inf’ or ‘mpfr_set_zero’, X is set to | |

plus infinity or plus zero iff SIGN is nonnegative; in | |

‘mpfr_set_nan’, the sign bit of the result is unspecified. | |

-- Function: void mpfr_swap (mpfr_t X, mpfr_t Y) | |

Swap the structures pointed to by X and Y. In particular, the | |

values are exchanged without rounding (this may be different from | |

three ‘mpfr_set’ calls using a third auxiliary variable). | |

Warning! Since the precisions are exchanged, this will affect | |

future assignments. Moreover, since the significand pointers are | |

also exchanged, you must not use this function if the allocation | |

method used for X and/or Y does not permit it. This is the case | |

when X and/or Y were declared and initialized with | |

‘MPFR_DECL_INIT’, and possibly with ‘mpfr_custom_init_set’ (*note | |

Custom Interface::). | |

File: mpfr.info, Node: Combined Initialization and Assignment Functions, Next: Conversion Functions, Prev: Assignment Functions, Up: MPFR Interface | |

5.3 Combined Initialization and Assignment Functions | |

==================================================== | |

-- Macro: int mpfr_init_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Macro: int mpfr_init_set_ui (mpfr_t ROP, unsigned long int OP, | |

mpfr_rnd_t RND) | |

-- Macro: int mpfr_init_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t | |

RND) | |

-- Macro: int mpfr_init_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND) | |

-- Macro: int mpfr_init_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t | |

RND) | |

-- Macro: int mpfr_init_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND) | |

-- Macro: int mpfr_init_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND) | |

-- Macro: int mpfr_init_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND) | |

Initialize ROP and set its value from OP, rounded in the direction | |

RND. The precision of ROP will be taken from the active default | |

precision, as set by ‘mpfr_set_default_prec’. | |

-- Function: int mpfr_init_set_str (mpfr_t X, const char *S, int BASE, | |

mpfr_rnd_t RND) | |

Initialize X and set its value from the string S in base BASE, | |

rounded in the direction RND. See ‘mpfr_set_str’. | |

File: mpfr.info, Node: Conversion Functions, Next: Basic Arithmetic Functions, Prev: Combined Initialization and Assignment Functions, Up: MPFR Interface | |

5.4 Conversion Functions | |

======================== | |

-- Function: float mpfr_get_flt (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: double mpfr_get_d (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: long double mpfr_get_ld (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: _Decimal64 mpfr_get_decimal64 (mpfr_t OP, mpfr_rnd_t RND) | |

Convert OP to a ‘float’ (respectively ‘double’, ‘long double’ or | |

‘_Decimal64’), using the rounding mode RND. If OP is NaN, some | |

fixed NaN (either quiet or signaling) or the result of 0.0/0.0 is | |

returned. If OP is ±Inf, an infinity of the same sign or the | |

result of ±1.0/0.0 is returned. If OP is zero, these functions | |

return a zero, trying to preserve its sign, if possible. The | |

‘mpfr_get_decimal64’ function is built only under some conditions: | |

see the documentation of ‘mpfr_set_decimal64’. | |

-- Function: long mpfr_get_si (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: unsigned long mpfr_get_ui (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: intmax_t mpfr_get_sj (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: uintmax_t mpfr_get_uj (mpfr_t OP, mpfr_rnd_t RND) | |

Convert OP to a ‘long’, an ‘unsigned long’, an ‘intmax_t’ or an | |

‘uintmax_t’ (respectively) after rounding it with respect to RND. | |

If OP is NaN, 0 is returned and the _erange_ flag is set. If OP is | |

too big for the return type, the function returns the maximum or | |

the minimum of the corresponding C type, depending on the direction | |

of the overflow; the _erange_ flag is set too. See also | |

‘mpfr_fits_slong_p’, ‘mpfr_fits_ulong_p’, ‘mpfr_fits_intmax_p’ and | |

‘mpfr_fits_uintmax_p’. | |

-- Function: double mpfr_get_d_2exp (long *EXP, mpfr_t OP, mpfr_rnd_t | |

RND) | |

-- Function: long double mpfr_get_ld_2exp (long *EXP, mpfr_t OP, | |

mpfr_rnd_t RND) | |

Return D and set EXP (formally, the value pointed to by EXP) such | |

that 0.5<=abs(D)<1 and D times 2 raised to EXP equals OP rounded to | |

double (resp. long double) precision, using the given rounding | |

mode. If OP is zero, then a zero of the same sign (or an unsigned | |

zero, if the implementation does not have signed zeros) is | |

returned, and EXP is set to 0. If OP is NaN or an infinity, then | |

the corresponding double precision (resp. long-double precision) | |

value is returned, and EXP is undefined. | |

-- Function: int mpfr_frexp (mpfr_exp_t *EXP, mpfr_t Y, mpfr_t X, | |

mpfr_rnd_t RND) | |

Set EXP (formally, the value pointed to by EXP) and Y such that | |

0.5<=abs(Y)<1 and Y times 2 raised to EXP equals X rounded to the | |

precision of Y, using the given rounding mode. If X is zero, then | |

Y is set to a zero of the same sign and EXP is set to 0. If X is | |

NaN or an infinity, then Y is set to the same value and EXP is | |

undefined. | |

-- Function: mpfr_exp_t mpfr_get_z_2exp (mpz_t ROP, mpfr_t OP) | |

Put the scaled significand of OP (regarded as an integer, with the | |

precision of OP) into ROP, and return the exponent EXP (which may | |

be outside the current exponent range) such that OP exactly equals | |

ROP times 2 raised to the power EXP. If OP is zero, the minimal | |

exponent ‘emin’ is returned. If OP is NaN or an infinity, the | |

_erange_ flag is set, ROP is set to 0, and the the minimal exponent | |

‘emin’ is returned. The returned exponent may be less than the | |

minimal exponent ‘emin’ of MPFR numbers in the current exponent | |

range; in case the exponent is not representable in the | |

‘mpfr_exp_t’ type, the _erange_ flag is set and the minimal value | |

of the ‘mpfr_exp_t’ type is returned. | |

-- Function: int mpfr_get_z (mpz_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Convert OP to a ‘mpz_t’, after rounding it with respect to RND. If | |

OP is NaN or an infinity, the _erange_ flag is set, ROP is set to | |

0, and 0 is returned. | |

-- Function: int mpfr_get_f (mpf_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Convert OP to a ‘mpf_t’, after rounding it with respect to RND. | |

The _erange_ flag is set if OP is NaN or an infinity, which do not | |

exist in MPF. If OP is NaN, then ROP is undefined. If OP is +Inf | |

(resp. −Inf), then ROP is set to the maximum (resp. minimum) value | |

in the precision of the MPF number; if a future MPF version | |

supports infinities, this behavior will be considered incorrect and | |

will change (portable programs should assume that ROP is set either | |

to this finite number or to an infinite number). Note that since | |

MPFR currently has the same exponent type as MPF (but not with the | |

same radix), the range of values is much larger in MPF than in | |

MPFR, so that an overflow or underflow is not possible. | |

-- Function: char * mpfr_get_str (char *STR, mpfr_exp_t *EXPPTR, int B, | |

size_t N, mpfr_t OP, mpfr_rnd_t RND) | |

Convert OP to a string of digits in base B, with rounding in the | |

direction RND, where N is either zero (see below) or the number of | |

significant digits output in the string; in the latter case, N must | |

be greater or equal to 2. The base may vary from 2 to 62; | |

otherwise the function does nothing and immediately returns a null | |

pointer. If the input number is an ordinary number, the exponent | |

is written through the pointer EXPPTR (for input 0, the current | |

minimal exponent is written); the type ‘mpfr_exp_t’ is large enough | |

to hold the exponent in all cases. | |

The generated string is a fraction, with an implicit radix point | |

immediately to the left of the first digit. For example, the | |

number −3.1416 would be returned as "−31416" in the string and 1 | |

written at EXPPTR. If RND is to nearest, and OP is exactly in the | |

middle of two consecutive possible outputs, the one with an even | |

significand is chosen, where both significands are considered with | |

the exponent of OP. Note that for an odd base, this may not | |

correspond to an even last digit: for example with 2 digits in base | |

7, (14) and a half is rounded to (15) which is 12 in decimal, (16) | |

and a half is rounded to (20) which is 14 in decimal, and (26) and | |

a half is rounded to (26) which is 20 in decimal. | |

If N is zero, the number of digits of the significand is chosen | |

large enough so that re-reading the printed value with the same | |

precision, assuming both output and input use rounding to nearest, | |

will recover the original value of OP. More precisely, in most | |

cases, the chosen precision of STR is the minimal precision m | |

depending only on P = PREC(OP) and B that satisfies the above | |

property, i.e., m = 1 + ceil(P*log(2)/log(B)), with P replaced by | |

P−1 if B is a power of 2, but in some very rare cases, it might be | |

m+1 (the smallest case for bases up to 62 is when P equals | |

186564318007 for bases 7 and 49). | |

If STR is a null pointer, space for the significand is allocated | |

using the current allocation function and a pointer to the string | |

is returned (unless the base is invalid). To free the returned | |

string, you must use ‘mpfr_free_str’. | |

If STR is not a null pointer, it should point to a block of storage | |

large enough for the significand, i.e., at least ‘max(N + 2, 7)’. | |

The extra two bytes are for a possible minus sign, and for the | |

terminating null character, and the value 7 accounts for ‘-@Inf@’ | |

plus the terminating null character. The pointer to the string STR | |

is returned (unless the base is invalid). | |

Note: The NaN and inexact flags are currently not set when need be; | |

this will be fixed in future versions. Programmers should | |

currently assume that whether the flags are set by this function is | |

unspecified. | |

-- Function: void mpfr_free_str (char *STR) | |

Free a string allocated by ‘mpfr_get_str’ using the current | |

unallocation function. The block is assumed to be ‘strlen(STR)+1’ | |

bytes. For more information about how it is done: *note | |

(gmp.info)Custom Allocation::. | |

-- Function: int mpfr_fits_ulong_p (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_fits_slong_p (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_fits_uint_p (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_fits_sint_p (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_fits_ushort_p (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_fits_sshort_p (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_fits_uintmax_p (mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_fits_intmax_p (mpfr_t OP, mpfr_rnd_t RND) | |

Return non-zero if OP would fit in the respective C data type, | |

respectively ‘unsigned long’, ‘long’, ‘unsigned int’, ‘int’, | |

‘unsigned short’, ‘short’, ‘uintmax_t’, ‘intmax_t’, when rounded to | |

an integer in the direction RND. | |

File: mpfr.info, Node: Basic Arithmetic Functions, Next: Comparison Functions, Prev: Conversion Functions, Up: MPFR Interface | |

5.5 Basic Arithmetic Functions | |

============================== | |

-- Function: int mpfr_add (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_add_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int | |

OP2, mpfr_rnd_t RND) | |

-- Function: int mpfr_add_si (mpfr_t ROP, mpfr_t OP1, long int OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_add_d (mpfr_t ROP, mpfr_t OP1, double OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_add_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_add_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, | |

mpfr_rnd_t RND) | |

Set ROP to OP1 + OP2 rounded in the direction RND. For types | |

having no signed zero, it is considered unsigned (i.e., (+0) + 0 = | |

(+0) and (−0) + 0 = (−0)). The ‘mpfr_add_d’ function assumes that | |

the radix of the ‘double’ type is a power of 2, with a precision at | |

most that declared by the C implementation (macro | |

‘IEEE_DBL_MANT_DIG’, and if not defined 53 bits). | |

-- Function: int mpfr_sub (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_ui_sub (mpfr_t ROP, unsigned long int OP1, mpfr_t | |

OP2, mpfr_rnd_t RND) | |

-- Function: int mpfr_sub_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int | |

OP2, mpfr_rnd_t RND) | |

-- Function: int mpfr_si_sub (mpfr_t ROP, long int OP1, mpfr_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_sub_si (mpfr_t ROP, mpfr_t OP1, long int OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_d_sub (mpfr_t ROP, double OP1, mpfr_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_sub_d (mpfr_t ROP, mpfr_t OP1, double OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_z_sub (mpfr_t ROP, mpz_t OP1, mpfr_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_sub_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_sub_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, | |

mpfr_rnd_t RND) | |

Set ROP to OP1 - OP2 rounded in the direction RND. For types | |

having no signed zero, it is considered unsigned (i.e., (+0) − 0 = | |

(+0), (−0) − 0 = (−0), 0 − (+0) = (−0) and 0 − (−0) = (+0)). The | |

same restrictions than for ‘mpfr_add_d’ apply to ‘mpfr_d_sub’ and | |

‘mpfr_sub_d’. | |

-- Function: int mpfr_mul (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_mul_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int | |

OP2, mpfr_rnd_t RND) | |

-- Function: int mpfr_mul_si (mpfr_t ROP, mpfr_t OP1, long int OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_mul_d (mpfr_t ROP, mpfr_t OP1, double OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_mul_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_mul_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, | |

mpfr_rnd_t RND) | |

Set ROP to OP1 times OP2 rounded in the direction RND. When a | |

result is zero, its sign is the product of the signs of the | |

operands (for types having no signed zero, it is considered | |

positive). The same restrictions than for ‘mpfr_add_d’ apply to | |

‘mpfr_mul_d’. | |

-- Function: int mpfr_sqr (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the square of OP rounded in the direction RND. | |

-- Function: int mpfr_div (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_ui_div (mpfr_t ROP, unsigned long int OP1, mpfr_t | |

OP2, mpfr_rnd_t RND) | |

-- Function: int mpfr_div_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int | |

OP2, mpfr_rnd_t RND) | |

-- Function: int mpfr_si_div (mpfr_t ROP, long int OP1, mpfr_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_div_si (mpfr_t ROP, mpfr_t OP1, long int OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_d_div (mpfr_t ROP, double OP1, mpfr_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_div_d (mpfr_t ROP, mpfr_t OP1, double OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_div_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_div_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, | |

mpfr_rnd_t RND) | |

Set ROP to OP1/OP2 rounded in the direction RND. When a result is | |

zero, its sign is the product of the signs of the operands (for | |

types having no signed zero, it is considered positive). The same | |

restrictions than for ‘mpfr_add_d’ apply to ‘mpfr_d_div’ and | |

‘mpfr_div_d’. | |

-- Function: int mpfr_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_sqrt_ui (mpfr_t ROP, unsigned long int OP, | |

mpfr_rnd_t RND) | |

Set ROP to the square root of OP rounded in the direction RND (set | |

ROP to −0 if OP is −0, to be consistent with the IEEE 754 | |

standard). Set ROP to NaN if OP is negative. | |

-- Function: int mpfr_rec_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the reciprocal square root of OP rounded in the | |

direction RND. Set ROP to +Inf if OP is ±0, +0 if OP is +Inf, and | |

NaN if OP is negative. | |

-- Function: int mpfr_cbrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_root (mpfr_t ROP, mpfr_t OP, unsigned long int K, | |

mpfr_rnd_t RND) | |

Set ROP to the cubic root (resp. the Kth root) of OP rounded in the | |

direction RND. For K odd (resp. even) and OP negative (including | |

−Inf), set ROP to a negative number (resp. NaN). The Kth root of −0 | |

is defined to be −0, whatever the parity of K. | |

-- Function: int mpfr_pow (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_pow_ui (mpfr_t ROP, mpfr_t OP1, unsigned long int | |

OP2, mpfr_rnd_t RND) | |

-- Function: int mpfr_pow_si (mpfr_t ROP, mpfr_t OP1, long int OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_pow_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_ui_pow_ui (mpfr_t ROP, unsigned long int OP1, | |

unsigned long int OP2, mpfr_rnd_t RND) | |

-- Function: int mpfr_ui_pow (mpfr_t ROP, unsigned long int OP1, mpfr_t | |

OP2, mpfr_rnd_t RND) | |

Set ROP to OP1 raised to OP2, rounded in the direction RND. | |

Special values are handled as described in the ISO C99 and IEEE | |

754-2008 standards for the ‘pow’ function: | |

• ‘pow(±0, Y)’ returns plus or minus infinity for Y a negative | |

odd integer. | |

• ‘pow(±0, Y)’ returns plus infinity for Y negative and not an | |

odd integer. | |

• ‘pow(±0, Y)’ returns plus or minus zero for Y a positive odd | |

integer. | |

• ‘pow(±0, Y)’ returns plus zero for Y positive and not an odd | |

integer. | |

• ‘pow(-1, ±Inf)’ returns 1. | |

• ‘pow(+1, Y)’ returns 1 for any Y, even a NaN. | |

• ‘pow(X, ±0)’ returns 1 for any X, even a NaN. | |

• ‘pow(X, Y)’ returns NaN for finite negative X and finite | |

non-integer Y. | |

• ‘pow(X, -Inf)’ returns plus infinity for 0 < abs(x) < 1, and | |

plus zero for abs(x) > 1. | |

• ‘pow(X, +Inf)’ returns plus zero for 0 < abs(x) < 1, and plus | |

infinity for abs(x) > 1. | |

• ‘pow(-Inf, Y)’ returns minus zero for Y a negative odd | |

integer. | |

• ‘pow(-Inf, Y)’ returns plus zero for Y negative and not an odd | |

integer. | |

• ‘pow(-Inf, Y)’ returns minus infinity for Y a positive odd | |

integer. | |

• ‘pow(-Inf, Y)’ returns plus infinity for Y positive and not an | |

odd integer. | |

• ‘pow(+Inf, Y)’ returns plus zero for Y negative, and plus | |

infinity for Y positive. | |

-- Function: int mpfr_neg (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_abs (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to -OP and the absolute value of OP respectively, rounded | |

in the direction RND. Just changes or adjusts the sign if ROP and | |

OP are the same variable, otherwise a rounding might occur if the | |

precision of ROP is less than that of OP. | |

-- Function: int mpfr_dim (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, | |

mpfr_rnd_t RND) | |

Set ROP to the positive difference of OP1 and OP2, i.e., OP1 - OP2 | |

rounded in the direction RND if OP1 > OP2, +0 if OP1 <= OP2, and | |

NaN if OP1 or OP2 is NaN. | |

-- Function: int mpfr_mul_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long | |

int OP2, mpfr_rnd_t RND) | |

-- Function: int mpfr_mul_2si (mpfr_t ROP, mpfr_t OP1, long int OP2, | |

mpfr_rnd_t RND) | |

Set ROP to OP1 times 2 raised to OP2 rounded in the direction RND. | |

Just increases the exponent by OP2 when ROP and OP1 are identical. | |

-- Function: int mpfr_div_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long | |

int OP2, mpfr_rnd_t RND) | |

-- Function: int mpfr_div_2si (mpfr_t ROP, mpfr_t OP1, long int OP2, | |

mpfr_rnd_t RND) | |

Set ROP to OP1 divided by 2 raised to OP2 rounded in the direction | |

RND. Just decreases the exponent by OP2 when ROP and OP1 are | |

identical. | |

File: mpfr.info, Node: Comparison Functions, Next: Special Functions, Prev: Basic Arithmetic Functions, Up: MPFR Interface | |

5.6 Comparison Functions | |

======================== | |

-- Function: int mpfr_cmp (mpfr_t OP1, mpfr_t OP2) | |

-- Function: int mpfr_cmp_ui (mpfr_t OP1, unsigned long int OP2) | |

-- Function: int mpfr_cmp_si (mpfr_t OP1, long int OP2) | |

-- Function: int mpfr_cmp_d (mpfr_t OP1, double OP2) | |

-- Function: int mpfr_cmp_ld (mpfr_t OP1, long double OP2) | |

-- Function: int mpfr_cmp_z (mpfr_t OP1, mpz_t OP2) | |

-- Function: int mpfr_cmp_q (mpfr_t OP1, mpq_t OP2) | |

-- Function: int mpfr_cmp_f (mpfr_t OP1, mpf_t OP2) | |

Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero if | |

OP1 = OP2, and a negative value if OP1 < OP2. Both OP1 and OP2 are | |

considered to their full own precision, which may differ. If one | |

of the operands is NaN, set the _erange_ flag and return zero. | |

Note: These functions may be useful to distinguish the three | |

possible cases. If you need to distinguish two cases only, it is | |

recommended to use the predicate functions (e.g., ‘mpfr_equal_p’ | |

for the equality) described below; they behave like the IEEE 754 | |

comparisons, in particular when one or both arguments are NaN. But | |

only floating-point numbers can be compared (you may need to do a | |

conversion first). | |

-- Function: int mpfr_cmp_ui_2exp (mpfr_t OP1, unsigned long int OP2, | |

mpfr_exp_t E) | |

-- Function: int mpfr_cmp_si_2exp (mpfr_t OP1, long int OP2, mpfr_exp_t | |

E) | |

Compare OP1 and OP2 multiplied by two to the power E. Similar as | |

above. | |

-- Function: int mpfr_cmpabs (mpfr_t OP1, mpfr_t OP2) | |

Compare |OP1| and |OP2|. Return a positive value if |OP1| > |OP2|, | |

zero if |OP1| = |OP2|, and a negative value if |OP1| < |OP2|. If | |

one of the operands is NaN, set the _erange_ flag and return zero. | |

-- Function: int mpfr_nan_p (mpfr_t OP) | |

-- Function: int mpfr_inf_p (mpfr_t OP) | |

-- Function: int mpfr_number_p (mpfr_t OP) | |

-- Function: int mpfr_zero_p (mpfr_t OP) | |

-- Function: int mpfr_regular_p (mpfr_t OP) | |

Return non-zero if OP is respectively NaN, an infinity, an ordinary | |

number (i.e., neither NaN nor an infinity), zero, or a regular | |

number (i.e., neither NaN, nor an infinity nor zero). Return zero | |

otherwise. | |

-- Macro: int mpfr_sgn (mpfr_t OP) | |

Return a positive value if OP > 0, zero if OP = 0, and a negative | |

value if OP < 0. If the operand is NaN, set the _erange_ flag and | |

return zero. This is equivalent to ‘mpfr_cmp_ui (op, 0)’, but more | |

efficient. | |

-- Function: int mpfr_greater_p (mpfr_t OP1, mpfr_t OP2) | |

-- Function: int mpfr_greaterequal_p (mpfr_t OP1, mpfr_t OP2) | |

-- Function: int mpfr_less_p (mpfr_t OP1, mpfr_t OP2) | |

-- Function: int mpfr_lessequal_p (mpfr_t OP1, mpfr_t OP2) | |

-- Function: int mpfr_equal_p (mpfr_t OP1, mpfr_t OP2) | |

Return non-zero if OP1 > OP2, OP1 >= OP2, OP1 < OP2, OP1 <= OP2, | |

OP1 = OP2 respectively, and zero otherwise. Those functions return | |

zero whenever OP1 and/or OP2 is NaN. | |

-- Function: int mpfr_lessgreater_p (mpfr_t OP1, mpfr_t OP2) | |

Return non-zero if OP1 < OP2 or OP1 > OP2 (i.e., neither OP1, nor | |

OP2 is NaN, and OP1 <> OP2), zero otherwise (i.e., OP1 and/or OP2 | |

is NaN, or OP1 = OP2). | |

-- Function: int mpfr_unordered_p (mpfr_t OP1, mpfr_t OP2) | |

Return non-zero if OP1 or OP2 is a NaN (i.e., they cannot be | |

compared), zero otherwise. | |

File: mpfr.info, Node: Special Functions, Next: Input and Output Functions, Prev: Comparison Functions, Up: MPFR Interface | |

5.7 Special Functions | |

===================== | |

All those functions, except explicitly stated (for example | |

‘mpfr_sin_cos’), return a *note ternary value::, i.e., zero for an exact | |

return value, a positive value for a return value larger than the exact | |

result, and a negative value otherwise. | |

Important note: in some domains, computing special functions (either | |

with correct or incorrect rounding) is expensive, even for small | |

precision, for example the trigonometric and Bessel functions for large | |

argument. | |

-- Function: int mpfr_log (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_log2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_log10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the natural logarithm of OP, log2(OP) or log10(OP), | |

respectively, rounded in the direction RND. Set ROP to −Inf if OP | |

is −0 (i.e., the sign of the zero has no influence on the result). | |

-- Function: int mpfr_exp (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_exp2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_exp10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the exponential of OP, to 2 power of OP or to 10 power | |

of OP, respectively, rounded in the direction RND. | |

-- Function: int mpfr_cos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_sin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_tan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the cosine of OP, sine of OP, tangent of OP, rounded in | |

the direction RND. | |

-- Function: int mpfr_sin_cos (mpfr_t SOP, mpfr_t COP, mpfr_t OP, | |

mpfr_rnd_t RND) | |

Set simultaneously SOP to the sine of OP and COP to the cosine of | |

OP, rounded in the direction RND with the corresponding precisions | |

of SOP and COP, which must be different variables. Return 0 iff | |

both results are exact, more precisely it returns s+4c where s=0 if | |

SOP is exact, s=1 if SOP is larger than the sine of OP, s=2 if SOP | |

is smaller than the sine of OP, and similarly for c and the cosine | |

of OP. | |

-- Function: int mpfr_sec (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_csc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_cot (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the secant of OP, cosecant of OP, cotangent of OP, | |

rounded in the direction RND. | |

-- Function: int mpfr_acos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_asin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_atan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the arc-cosine, arc-sine or arc-tangent of OP, rounded | |

in the direction RND. Note that since ‘acos(-1)’ returns the | |

floating-point number closest to Pi according to the given rounding | |

mode, this number might not be in the output range 0 <= ROP < \pi | |

of the arc-cosine function; still, the result lies in the image of | |

the output range by the rounding function. The same holds for | |

‘asin(-1)’, ‘asin(1)’, ‘atan(-Inf)’, ‘atan(+Inf)’ or for ‘atan(op)’ | |

with large OP and small precision of ROP. | |

-- Function: int mpfr_atan2 (mpfr_t ROP, mpfr_t Y, mpfr_t X, mpfr_rnd_t | |

RND) | |

Set ROP to the arc-tangent2 of Y and X, rounded in the direction | |

RND: if ‘x > 0’, ‘atan2(y, x) = atan (y/x)’; if ‘x < 0’, ‘atan2(y, | |

x) = sign(y)*(Pi - atan (abs(y/x)))’, thus a number from -Pi to Pi. | |

As for ‘atan’, in case the exact mathematical result is +Pi or -Pi, | |

its rounded result might be outside the function output range. | |

‘atan2(y, 0)’ does not raise any floating-point exception. Special | |

values are handled as described in the ISO C99 and IEEE 754-2008 | |

standards for the ‘atan2’ function: | |

• ‘atan2(+0, -0)’ returns +Pi. | |

• ‘atan2(-0, -0)’ returns -Pi. | |

• ‘atan2(+0, +0)’ returns +0. | |

• ‘atan2(-0, +0)’ returns −0. | |

• ‘atan2(+0, x)’ returns +Pi for x < 0. | |

• ‘atan2(-0, x)’ returns -Pi for x < 0. | |

• ‘atan2(+0, x)’ returns +0 for x > 0. | |

• ‘atan2(-0, x)’ returns −0 for x > 0. | |

• ‘atan2(y, 0)’ returns -Pi/2 for y < 0. | |

• ‘atan2(y, 0)’ returns +Pi/2 for y > 0. | |

• ‘atan2(+Inf, -Inf)’ returns +3*Pi/4. | |

• ‘atan2(-Inf, -Inf)’ returns -3*Pi/4. | |

• ‘atan2(+Inf, +Inf)’ returns +Pi/4. | |

• ‘atan2(-Inf, +Inf)’ returns -Pi/4. | |

• ‘atan2(+Inf, x)’ returns +Pi/2 for finite x. | |

• ‘atan2(-Inf, x)’ returns -Pi/2 for finite x. | |

• ‘atan2(y, -Inf)’ returns +Pi for finite y > 0. | |

• ‘atan2(y, -Inf)’ returns -Pi for finite y < 0. | |

• ‘atan2(y, +Inf)’ returns +0 for finite y > 0. | |

• ‘atan2(y, +Inf)’ returns −0 for finite y < 0. | |

-- Function: int mpfr_cosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_sinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_tanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the hyperbolic cosine, sine or tangent of OP, rounded in | |

the direction RND. | |

-- Function: int mpfr_sinh_cosh (mpfr_t SOP, mpfr_t COP, mpfr_t OP, | |

mpfr_rnd_t RND) | |

Set simultaneously SOP to the hyperbolic sine of OP and COP to the | |

hyperbolic cosine of OP, rounded in the direction RND with the | |

corresponding precision of SOP and COP, which must be different | |

variables. Return 0 iff both results are exact (see ‘mpfr_sin_cos’ | |

for a more detailed description of the return value). | |

-- Function: int mpfr_sech (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_csch (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_coth (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the hyperbolic secant of OP, cosecant of OP, cotangent | |

of OP, rounded in the direction RND. | |

-- Function: int mpfr_acosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_asinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_atanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the inverse hyperbolic cosine, sine or tangent of OP, | |

rounded in the direction RND. | |

-- Function: int mpfr_fac_ui (mpfr_t ROP, unsigned long int OP, | |

mpfr_rnd_t RND) | |

Set ROP to the factorial of OP, rounded in the direction RND. | |

-- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the logarithm of one plus OP, rounded in the direction | |

RND. | |

-- Function: int mpfr_expm1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the exponential of OP followed by a subtraction by one, | |

rounded in the direction RND. | |

-- Function: int mpfr_eint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the exponential integral of OP, rounded in the direction | |

RND. For positive OP, the exponential integral is the sum of | |

Euler’s constant, of the logarithm of OP, and of the sum for k from | |

1 to infinity of OP to the power k, divided by k and factorial(k). | |

For negative OP, ROP is set to NaN (this definition for negative | |

argument follows formula 5.1.2 from the Handbook of Mathematical | |

Functions from Abramowitz and Stegun, a future version might use | |

another definition). | |

-- Function: int mpfr_li2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to real part of the dilogarithm of OP, rounded in the | |

direction RND. MPFR defines the dilogarithm function as the | |

integral of -log(1-t)/t from 0 to OP. | |

-- Function: int mpfr_gamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the value of the Gamma function on OP, rounded in the | |

direction RND. When OP is a negative integer, ROP is set to NaN. | |

-- Function: int mpfr_lngamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the value of the logarithm of the Gamma function on OP, | |

rounded in the direction RND. When −2K−1 <= OP <= −2K, K being a | |

non-negative integer, ROP is set to NaN. See also ‘mpfr_lgamma’. | |

-- Function: int mpfr_lgamma (mpfr_t ROP, int *SIGNP, mpfr_t OP, | |

mpfr_rnd_t RND) | |

Set ROP to the value of the logarithm of the absolute value of the | |

Gamma function on OP, rounded in the direction RND. The sign (1 or | |

−1) of Gamma(OP) is returned in the object pointed to by SIGNP. | |

When OP is an infinity or a non-positive integer, set ROP to +Inf. | |

When OP is NaN, −Inf or a negative integer, *SIGNP is undefined, | |

and when OP is ±0, *SIGNP is the sign of the zero. | |

-- Function: int mpfr_digamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the value of the Digamma (sometimes also called Psi) | |

function on OP, rounded in the direction RND. When OP is a | |

negative integer, set ROP to NaN. | |

-- Function: int mpfr_zeta (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_zeta_ui (mpfr_t ROP, unsigned long OP, mpfr_rnd_t | |

RND) | |

Set ROP to the value of the Riemann Zeta function on OP, rounded in | |

the direction RND. | |

-- Function: int mpfr_erf (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_erfc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the value of the error function on OP (resp. the | |

complementary error function on OP) rounded in the direction RND. | |

-- Function: int mpfr_j0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_j1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_jn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t | |

RND) | |

Set ROP to the value of the first kind Bessel function of order 0, | |

(resp. 1 and N) on OP, rounded in the direction RND. When OP is | |

NaN, ROP is always set to NaN. When OP is plus or minus Infinity, | |

ROP is set to +0. When OP is zero, and N is not zero, ROP is set | |

to +0 or −0 depending on the parity and sign of N, and the sign of | |

OP. | |

-- Function: int mpfr_y0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_y1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_yn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t | |

RND) | |

Set ROP to the value of the second kind Bessel function of order 0 | |

(resp. 1 and N) on OP, rounded in the direction RND. When OP is | |

NaN or negative, ROP is always set to NaN. When OP is +Inf, ROP is | |

set to +0. When OP is zero, ROP is set to +Inf or −Inf depending | |

on the parity and sign of N. | |

-- Function: int mpfr_fma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t | |

OP3, mpfr_rnd_t RND) | |

-- Function: int mpfr_fms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t | |

OP3, mpfr_rnd_t RND) | |

Set ROP to (OP1 times OP2) + OP3 (resp. (OP1 times OP2) - OP3) | |

rounded in the direction RND. | |

-- Function: int mpfr_agm (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, | |

mpfr_rnd_t RND) | |

Set ROP to the arithmetic-geometric mean of OP1 and OP2, rounded in | |

the direction RND. The arithmetic-geometric mean is the common | |

limit of the sequences U_N and V_N, where U_0=OP1, V_0=OP2, U_(N+1) | |

is the arithmetic mean of U_N and V_N, and V_(N+1) is the geometric | |

mean of U_N and V_N. If any operand is negative, set ROP to NaN. | |

-- Function: int mpfr_hypot (mpfr_t ROP, mpfr_t X, mpfr_t Y, mpfr_rnd_t | |

RND) | |

Set ROP to the Euclidean norm of X and Y, i.e., the square root of | |

the sum of the squares of X and Y, rounded in the direction RND. | |

Special values are handled as described in Section F.9.4.3 of the | |

ISO C99 and IEEE 754-2008 standards: If X or Y is an infinity, then | |

+Inf is returned in ROP, even if the other number is NaN. | |

-- Function: int mpfr_ai (mpfr_t ROP, mpfr_t X, mpfr_rnd_t RND) | |

Set ROP to the value of the Airy function Ai on X, rounded in the | |

direction RND. When X is NaN, ROP is always set to NaN. When X is | |

+Inf or −Inf, ROP is +0. The current implementation is not | |

intended to be used with large arguments. It works with abs(X) | |

typically smaller than 500. For larger arguments, other methods | |

should be used and will be implemented in a future version. | |

-- Function: int mpfr_const_log2 (mpfr_t ROP, mpfr_rnd_t RND) | |

-- Function: int mpfr_const_pi (mpfr_t ROP, mpfr_rnd_t RND) | |

-- Function: int mpfr_const_euler (mpfr_t ROP, mpfr_rnd_t RND) | |

-- Function: int mpfr_const_catalan (mpfr_t ROP, mpfr_rnd_t RND) | |

Set ROP to the logarithm of 2, the value of Pi, of Euler’s constant | |

0.577…, of Catalan’s constant 0.915…, respectively, rounded in the | |

direction RND. These functions cache the computed values to avoid | |

other calculations if a lower or equal precision is requested. To | |

free these caches, use ‘mpfr_free_cache’. | |

-- Function: void mpfr_free_cache (void) | |

Free various caches used by MPFR internally, in particular the | |

caches used by the functions computing constants | |

(‘mpfr_const_log2’, ‘mpfr_const_pi’, ‘mpfr_const_euler’ and | |

‘mpfr_const_catalan’). You should call this function before | |

terminating a thread, even if you did not call these functions | |

directly (they could have been called internally). | |

-- Function: int mpfr_sum (mpfr_t ROP, mpfr_ptr const TAB[], unsigned | |

long int N, mpfr_rnd_t RND) | |

Set ROP to the sum of all elements of TAB, whose size is N, rounded | |

in the direction RND. Warning: for efficiency reasons, TAB is an | |

array of pointers to ‘mpfr_t’, not an array of ‘mpfr_t’. If the | |

returned ‘int’ value is zero, ROP is guaranteed to be the exact | |

sum; otherwise ROP might be smaller than, equal to, or larger than | |

the exact sum (in accordance to the rounding mode). However, | |

‘mpfr_sum’ does guarantee the result is correctly rounded. | |

File: mpfr.info, Node: Input and Output Functions, Next: Formatted Output Functions, Prev: Special Functions, Up: MPFR Interface | |

5.8 Input and Output Functions | |

============================== | |

This section describes functions that perform input from an input/output | |

stream, and functions that output to an input/output stream. Passing a | |

null pointer for a ‘stream’ to any of these functions will make them | |

read from ‘stdin’ and write to ‘stdout’, respectively. | |

When using any of these functions, you must include the ‘<stdio.h>’ | |

standard header before ‘mpfr.h’, to allow ‘mpfr.h’ to define prototypes | |

for these functions. | |

-- Function: size_t mpfr_out_str (FILE *STREAM, int BASE, size_t N, | |

mpfr_t OP, mpfr_rnd_t RND) | |

Output OP on stream STREAM, as a string of digits in base BASE, | |

rounded in the direction RND. The base may vary from 2 to 62. | |

Print N significant digits exactly, or if N is 0, enough digits so | |

that OP can be read back exactly (see ‘mpfr_get_str’). | |

In addition to the significant digits, a decimal point (defined by | |

the current locale) at the right of the first digit and a trailing | |

exponent in base 10, in the form ‘eNNN’, are printed. If BASE is | |

greater than 10, ‘@’ will be used instead of ‘e’ as exponent | |

delimiter. | |

Return the number of characters written, or if an error occurred, | |

return 0. | |

-- Function: size_t mpfr_inp_str (mpfr_t ROP, FILE *STREAM, int BASE, | |

mpfr_rnd_t RND) | |

Input a string in base BASE from stream STREAM, rounded in the | |

direction RND, and put the read float in ROP. | |

This function reads a word (defined as a sequence of characters | |

between whitespace) and parses it using ‘mpfr_set_str’. See the | |

documentation of ‘mpfr_strtofr’ for a detailed description of the | |

valid string formats. | |

Return the number of bytes read, or if an error occurred, return 0. | |

File: mpfr.info, Node: Formatted Output Functions, Next: Integer Related Functions, Prev: Input and Output Functions, Up: MPFR Interface | |

5.9 Formatted Output Functions | |

============================== | |

5.9.1 Requirements | |

------------------ | |

The class of ‘mpfr_printf’ functions provides formatted output in a | |

similar manner as the standard C ‘printf’. These functions are defined | |

only if your system supports ISO C variadic functions and the | |

corresponding argument access macros. | |

When using any of these functions, you must include the ‘<stdio.h>’ | |

standard header before ‘mpfr.h’, to allow ‘mpfr.h’ to define prototypes | |

for these functions. | |

5.9.2 Format String | |

------------------- | |

The format specification accepted by ‘mpfr_printf’ is an extension of | |

the ‘printf’ one. The conversion specification is of the form: | |

% [flags] [width] [.[precision]] [type] [rounding] conv | |

‘flags’, ‘width’, and ‘precision’ have the same meaning as for the | |

standard ‘printf’ (in particular, notice that the ‘precision’ is related | |

to the number of digits displayed in the base chosen by ‘conv’ and not | |

related to the internal precision of the ‘mpfr_t’ variable). | |

‘mpfr_printf’ accepts the same ‘type’ specifiers as GMP (except the | |

non-standard and deprecated ‘q’, use ‘ll’ instead), namely the length | |

modifiers defined in the C standard: | |

‘h’ ‘short’ | |

‘hh’ ‘char’ | |

‘j’ ‘intmax_t’ or ‘uintmax_t’ | |

‘l’ ‘long’ or ‘wchar_t’ | |

‘ll’ ‘long long’ | |

‘L’ ‘long double’ | |

‘t’ ‘ptrdiff_t’ | |

‘z’ ‘size_t’ | |

and the ‘type’ specifiers defined in GMP plus ‘R’ and ‘P’ specific to | |

MPFR (the second column in the table below shows the type of the | |

argument read in the argument list and the kind of ‘conv’ specifier to | |

use after the ‘type’ specifier): | |

‘F’ ‘mpf_t’, float conversions | |

‘Q’ ‘mpq_t’, integer conversions | |

‘M’ ‘mp_limb_t’, integer conversions | |

‘N’ ‘mp_limb_t’ array, integer conversions | |

‘Z’ ‘mpz_t’, integer conversions | |

‘P’ ‘mpfr_prec_t’, integer conversions | |

‘R’ ‘mpfr_t’, float conversions | |

The ‘type’ specifiers have the same restrictions as those mentioned | |

in the GMP documentation: *note (gmp.info)Formatted Output Strings::. | |

In particular, the ‘type’ specifiers (except ‘R’ and ‘P’) are supported | |

only if they are supported by ‘gmp_printf’ in your GMP build; this | |

implies that the standard specifiers, such as ‘t’, must _also_ be | |

supported by your C library if you want to use them. | |

The ‘rounding’ field is specific to ‘mpfr_t’ arguments and should not | |

be used with other types. | |

With conversion specification not involving ‘P’ and ‘R’ types, | |

‘mpfr_printf’ behaves exactly as ‘gmp_printf’. | |

The ‘P’ type specifies that a following ‘d’, ‘i’, ‘o’, ‘u’, ‘x’, or | |

‘X’ conversion specifier applies to a ‘mpfr_prec_t’ argument. It is | |

needed because the ‘mpfr_prec_t’ type does not necessarily correspond to | |

an ‘int’ or any fixed standard type. The ‘precision’ field specifies | |

the minimum number of digits to appear. The default ‘precision’ is 1. | |

For example: | |

mpfr_t x; | |

mpfr_prec_t p; | |

mpfr_init (x); | |

… | |

p = mpfr_get_prec (x); | |

mpfr_printf ("variable x with %Pu bits", p); | |

The ‘R’ type specifies that a following ‘a’, ‘A’, ‘b’, ‘e’, ‘E’, ‘f’, | |

‘F’, ‘g’, ‘G’, or ‘n’ conversion specifier applies to a ‘mpfr_t’ | |

argument. The ‘R’ type can be followed by a ‘rounding’ specifier | |

denoted by one of the following characters: | |

‘U’ round toward plus infinity | |

‘D’ round toward minus infinity | |

‘Y’ round away from zero | |

‘Z’ round toward zero | |

‘N’ round to nearest (with ties to even) | |

‘*’ rounding mode indicated by the | |

‘mpfr_rnd_t’ argument just before the | |

corresponding ‘mpfr_t’ variable. | |

The default rounding mode is rounding to nearest. The following | |

three examples are equivalent: | |

mpfr_t x; | |

mpfr_init (x); | |

… | |

mpfr_printf ("%.128Rf", x); | |

mpfr_printf ("%.128RNf", x); | |

mpfr_printf ("%.128R*f", MPFR_RNDN, x); | |

Note that the rounding away from zero mode is specified with ‘Y’ | |

because ISO C reserves the ‘A’ specifier for hexadecimal output (see | |

below). | |

The output ‘conv’ specifiers allowed with ‘mpfr_t’ parameter are: | |

‘a’ ‘A’ hex float, C99 style | |

‘b’ binary output | |

‘e’ ‘E’ scientific format float | |

‘f’ ‘F’ fixed point float | |

‘g’ ‘G’ fixed or scientific float | |

The conversion specifier ‘b’ which displays the argument in binary is | |

specific to ‘mpfr_t’ arguments and should not be used with other types. | |

Other conversion specifiers have the same meaning as for a ‘double’ | |

argument. | |

In case of non-decimal output, only the significand is written in the | |

specified base, the exponent is always displayed in decimal. Special | |

values are always displayed as ‘nan’, ‘-inf’, and ‘inf’ for ‘a’, ‘b’, | |

‘e’, ‘f’, and ‘g’ specifiers and ‘NAN’, ‘-INF’, and ‘INF’ for ‘A’, ‘E’, | |

‘F’, and ‘G’ specifiers. | |

If the ‘precision’ field is not empty, the ‘mpfr_t’ number is rounded | |

to the given precision in the direction specified by the rounding mode. | |

If the precision is zero with rounding to nearest mode and one of the | |

following ‘conv’ specifiers: ‘a’, ‘A’, ‘b’, ‘e’, ‘E’, tie case is | |

rounded to even when it lies between two consecutive values at the | |

wanted precision which have the same exponent, otherwise, it is rounded | |

away from zero. For instance, 85 is displayed as "8e+1" and 95 is | |

displayed as "1e+2" with the format specification ‘"%.0RNe"’. This also | |

applies when the ‘g’ (resp. ‘G’) conversion specifier uses the ‘e’ | |

(resp. ‘E’) style. If the precision is set to a value greater than the | |

maximum value for an ‘int’, it will be silently reduced down to | |

‘INT_MAX’. | |

If the ‘precision’ field is empty (as in ‘%Re’ or ‘%.RE’) with ‘conv’ | |

specifier ‘e’ and ‘E’, the number is displayed with enough digits so | |

that it can be read back exactly, assuming that the input and output | |

variables have the same precision and that the input and output rounding | |

modes are both rounding to nearest (as for ‘mpfr_get_str’). The default | |

precision for an empty ‘precision’ field with ‘conv’ specifiers ‘f’, | |

‘F’, ‘g’, and ‘G’ is 6. | |

5.9.3 Functions | |

--------------- | |

For all the following functions, if the number of characters which ought | |

to be written appears to exceed the maximum limit for an ‘int’, nothing | |

is written in the stream (resp. to ‘stdout’, to BUF, to STR), the | |

function returns −1, sets the _erange_ flag, and (in POSIX system only) | |

‘errno’ is set to ‘EOVERFLOW’. | |

-- Function: int mpfr_fprintf (FILE *STREAM, const char *TEMPLATE, …) | |

-- Function: int mpfr_vfprintf (FILE *STREAM, const char *TEMPLATE, | |

va_list AP) | |

Print to the stream STREAM the optional arguments under the control | |

of the template string TEMPLATE. Return the number of characters | |

written or a negative value if an error occurred. | |

-- Function: int mpfr_printf (const char *TEMPLATE, …) | |

-- Function: int mpfr_vprintf (const char *TEMPLATE, va_list AP) | |

Print to ‘stdout’ the optional arguments under the control of the | |

template string TEMPLATE. Return the number of characters written | |

or a negative value if an error occurred. | |

-- Function: int mpfr_sprintf (char *BUF, const char *TEMPLATE, …) | |

-- Function: int mpfr_vsprintf (char *BUF, const char *TEMPLATE, | |

va_list AP) | |

Form a null-terminated string corresponding to the optional | |

arguments under the control of the template string TEMPLATE, and | |

print it in BUF. No overlap is permitted between BUF and the other | |

arguments. Return the number of characters written in the array | |

BUF _not counting_ the terminating null character or a negative | |

value if an error occurred. | |

-- Function: int mpfr_snprintf (char *BUF, size_t N, const char | |

*TEMPLATE, …) | |

-- Function: int mpfr_vsnprintf (char *BUF, size_t N, const char | |

*TEMPLATE, va_list AP) | |

Form a null-terminated string corresponding to the optional | |

arguments under the control of the template string TEMPLATE, and | |

print it in BUF. If N is zero, nothing is written and BUF may be a | |

null pointer, otherwise, the N−1 first characters are written in | |

BUF and the N-th is a null character. Return the number of | |

characters that would have been written had N be sufficiently | |

large, _not counting_ the terminating null character, or a negative | |

value if an error occurred. | |

-- Function: int mpfr_asprintf (char **STR, const char *TEMPLATE, …) | |

-- Function: int mpfr_vasprintf (char **STR, const char *TEMPLATE, | |

va_list AP) | |

Write their output as a null terminated string in a block of memory | |

allocated using the current allocation function. A pointer to the | |

block is stored in STR. The block of memory must be freed using | |

‘mpfr_free_str’. The return value is the number of characters | |

written in the string, excluding the null-terminator, or a negative | |

value if an error occurred. | |

File: mpfr.info, Node: Integer Related Functions, Next: Rounding Related Functions, Prev: Formatted Output Functions, Up: MPFR Interface | |

5.10 Integer and Remainder Related Functions | |

============================================ | |

-- Function: int mpfr_rint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_ceil (mpfr_t ROP, mpfr_t OP) | |

-- Function: int mpfr_floor (mpfr_t ROP, mpfr_t OP) | |

-- Function: int mpfr_round (mpfr_t ROP, mpfr_t OP) | |

-- Function: int mpfr_trunc (mpfr_t ROP, mpfr_t OP) | |

Set ROP to OP rounded to an integer. ‘mpfr_rint’ rounds to the | |

nearest representable integer in the given direction RND, | |

‘mpfr_ceil’ rounds to the next higher or equal representable | |

integer, ‘mpfr_floor’ to the next lower or equal representable | |

integer, ‘mpfr_round’ to the nearest representable integer, | |

rounding halfway cases away from zero (as in the roundTiesToAway | |

mode of IEEE 754-2008), and ‘mpfr_trunc’ to the next representable | |

integer toward zero. | |

The returned value is zero when the result is exact, positive when | |

it is greater than the original value of OP, and negative when it | |

is smaller. More precisely, the returned value is 0 when OP is an | |

integer representable in ROP, 1 or −1 when OP is an integer that is | |

not representable in ROP, 2 or −2 when OP is not an integer. | |

When OP is NaN, the NaN flag is set as usual. In the other cases, | |

the inexact flag is set when ROP differs from OP, following the ISO | |

C99 rule for the ‘rint’ function. If you want the behavior to be | |

more like IEEE 754 / ISO TS 18661-1, i.e., the usual behavior where | |

the round-to-integer function is regarded as any other mathematical | |

function, you should use one the ‘mpfr_rint_*’ functions instead | |

(however it is not possible to round to nearest with the even | |

rounding rule yet). | |

Note that ‘mpfr_round’ is different from ‘mpfr_rint’ called with | |

the rounding to nearest mode (where halfway cases are rounded to an | |

even integer or significand). Note also that no double rounding is | |

performed; for instance, 10.5 (1010.1 in binary) is rounded by | |

‘mpfr_rint’ with rounding to nearest to 12 (1100 in binary) in | |

2-bit precision, because the two enclosing numbers representable on | |

two bits are 8 and 12, and the closest is 12. (If one first | |

rounded to an integer, one would round 10.5 to 10 with even | |

rounding, and then 10 would be rounded to 8 again with even | |

rounding.) | |

-- Function: int mpfr_rint_ceil (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

-- Function: int mpfr_rint_floor (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t | |

RND) | |

-- Function: int mpfr_rint_round (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t | |

RND) | |

-- Function: int mpfr_rint_trunc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t | |

RND) | |

Set ROP to OP rounded to an integer. ‘mpfr_rint_ceil’ rounds to | |

the next higher or equal integer, ‘mpfr_rint_floor’ to the next | |

lower or equal integer, ‘mpfr_rint_round’ to the nearest integer, | |

rounding halfway cases away from zero, and ‘mpfr_rint_trunc’ to the | |

next integer toward zero. If the result is not representable, it | |

is rounded in the direction RND. The returned value is the ternary | |

value associated with the considered round-to-integer function | |

(regarded in the same way as any other mathematical function). | |

Contrary to ‘mpfr_rint’, those functions do perform a double | |

rounding: first OP is rounded to the nearest integer in the | |

direction given by the function name, then this nearest integer (if | |

not representable) is rounded in the given direction RND. Thus | |

these round-to-integer functions behave more like the other | |

mathematical functions, i.e., the returned result is the correct | |

rounding of the exact result of the function in the real numbers. | |

For example, ‘mpfr_rint_round’ with rounding to nearest and a | |

precision of two bits rounds 6.5 to 7 (halfway cases away from | |

zero), then 7 is rounded to 8 by the round-even rule, despite the | |

fact that 6 is also representable on two bits, and is closer to 6.5 | |

than 8. | |

-- Function: int mpfr_frac (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND) | |

Set ROP to the fractional part of OP, having the same sign as OP, | |

rounded in the direction RND (unlike in ‘mpfr_rint’, RND affects | |

only how the exact fractional part is rounded, not how the | |

fractional part is generated). | |

-- Function: int mpfr_modf (mpfr_t IOP, mpfr_t FOP, mpfr_t OP, | |

mpfr_rnd_t RND) | |

Set simultaneously IOP to the integral part of OP and FOP to the | |

fractional part of OP, rounded in the direction RND with the | |

corresponding precision of IOP and FOP (equivalent to | |

‘mpfr_trunc(IOP, OP, RND)’ and ‘mpfr_frac(FOP, OP, RND)’). The | |

variables IOP and FOP must be different. Return 0 iff both results | |

are exact (see ‘mpfr_sin_cos’ for a more detailed description of | |

the return value). | |

-- Function: int mpfr_fmod (mpfr_t R, mpfr_t X, mpfr_t Y, mpfr_rnd_t | |

RND) | |

-- Function: int mpfr_remainder (mpfr_t R, mpfr_t X, mpfr_t Y, | |

mpfr_rnd_t RND) | |

-- Function: int mpfr_remquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y, | |

mpfr_rnd_t RND) | |

Set R to the value of X - NY, rounded according to the direction | |

RND, where N is the integer quotient of X divided by Y, defined as | |

follows: N is rounded toward zero for ‘mpfr_fmod’, and to the | |

nearest integer (ties rounded to even) for ‘mpfr_remainder’ and | |

‘mpfr_remquo’. | |

Special values are handled as described in Section F.9.7.1 of the | |

ISO C99 standard: If X is infinite or Y is zero, R is NaN. If Y is | |

infinite and X is finite, R is X rounded to the precision of R. If | |

R is zero, it has the sign of X. The return value is the ternary | |

value corresponding to R. | |

Additionally, ‘mpfr_remquo’ stores the low significant bits from | |

the quotient N in *Q (more precisely the number of bits in a ‘long’ | |

minus one), with the sign of X divided by Y (except if those low | |

bits are all zero, in which case zero is returned). Note that X | |

may be so large in magnitude relative to Y that an exact | |

representation of the quotient is not practical. The | |

‘mpfr_remainder’ and ‘mpfr_remquo’ functions are useful for | |

additive argument reduction. | |

-- Function: int mpfr_integer_p (mpfr_t OP) | |

Return non-zero iff OP is an integer. | |

File: mpfr.info, Node: Rounding Related Functions, Next: Miscellaneous Functions, Prev: Integer Related Functions, Up: MPFR Interface | |

5.11 Rounding Related Functions | |

=============================== | |

-- Function: void mpfr_set_default_rounding_mode (mpfr_rnd_t RND) | |

Set the default rounding mode to RND. The default rounding mode is | |

to nearest initially. | |

-- Function: mpfr_rnd_t mpfr_get_default_rounding_mode (void) | |

Get the default rounding mode. | |

-- Function: int mpfr_prec_round (mpfr_t X, mpfr_prec_t PREC, | |

mpfr_rnd_t RND) | |

Round X according to RND with precision PREC, which must be an | |

integer between ‘MPFR_PREC_MIN’ and ‘MPFR_PREC_MAX’ (otherwise the | |

behavior is undefined). If PREC is greater or equal to the | |

precision of X, then new space is allocated for the significand, | |

and it is filled with zeros. Otherwise, the significand is rounded | |

to precision PREC with the given direction. In both cases, the | |

precision of X is changed to PREC. | |

Here is an example of how to use ‘mpfr_prec_round’ to implement | |

Newton’s algorithm to compute the inverse of A, assuming X is | |

already an approximation to N bits: | |

mpfr_set_prec (t, 2 * n); | |

mpfr_set (t, a, MPFR_RNDN); /* round a to 2n bits */ | |

mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to 2n bits */ | |

mpfr_ui_sub (t, 1, t, MPFR_RNDN); /* high n bits cancel with 1 */ | |

mpfr_prec_round (t, n, MPFR_RNDN); /* t is correct to n bits */ | |

mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to n bits */ | |

mpfr_prec_round (x, 2 * n, MPFR_RNDN); /* exact */ | |

mpfr_add (x, x, t, MPFR_RNDN); /* x is correct to 2n bits */ | |

Warning! You must not use this function if X was initialized with | |

‘MPFR_DECL_INIT’ or with ‘mpfr_custom_init_set’ (*note Custom | |

Interface::). | |

-- Function: int mpfr_can_round (mpfr_t B, mpfr_exp_t ERR, mpfr_rnd_t | |

RND1, mpfr_rnd_t RND2, mpfr_prec_t PREC) | |

Assuming B is an approximation of an unknown number X in the | |

direction RND1 with error at most two to the power E(b)-ERR where | |

E(b) is the exponent of B, return a non-zero value if one is able | |

to round correctly X to precision PREC with the direction RND2, and | |

0 otherwise (including for NaN and Inf). This function *does not | |

modify* its arguments. | |

If RND1 is ‘MPFR_RNDN’, then the sign of the error is unknown, but | |

its absolute value is the same, so that the possible range is twice | |

as large as with a directed rounding for RND1. | |

Note: if one wants to also determine the correct *note ternary | |

value:: when rounding B to precision PREC with rounding mode RND, a | |

useful trick is the following: | |

if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ, | |

prec + (rnd == MPFR_RNDN))) | |

... | |

Indeed, if RND is ‘MPFR_RNDN’, this will check if one can round to | |

PREC+1 bits with a directed rounding: if so, one can surely round | |

to nearest to PREC bits, and in addition one can determine the | |

correct ternary value, which would not be the case when B is near | |

from a value exactly representable on PREC bits. | |

-- Function: mpfr_prec_t mpfr_min_prec (mpfr_t X) | |

Return the minimal number of bits required to store the significand | |

of X, and 0 for special values, including 0. (Warning: the | |

returned value can be less than ‘MPFR_PREC_MIN’.) | |

The function name is subject to change. | |

-- Function: const char * mpfr_print_rnd_mode (mpfr_rnd_t RND) | |

Return a string ("MPFR_RNDD", "MPFR_RNDU", "MPFR_RNDN", | |

"MPFR_RNDZ", "MPFR_RNDA") corresponding to the rounding mode RND, | |

or a null pointer if RND is an invalid rounding mode. | |

File: mpfr.info, Node: Miscellaneous Functions, Next: Exception Related Functions, Prev: Rounding Related Functions, Up: MPFR Interface | |

5.12 Miscellaneous Functions | |

============================ | |

-- Function: void mpfr_nexttoward (mpfr_t X, mpfr_t Y) | |

If X or Y is NaN, set X to NaN. If X and Y are equal, X is | |

unchanged. Otherwise, if X is different from Y, replace X by the | |

next floating-point number (with the precision of X and the current | |

exponent range) in the direction of Y (the infinite values are seen | |

as the smallest and largest floating-point numbers). If the result | |

is |