chromium / native_client / nacl-gcc / f80d6b9ee7f94755c697ffb7194fb01dd0c537dd / . / mpfr-2.4.1 / mpfr.info

This is ../mpfr.info, produced by makeinfo version 4.12 from | |

../mpfr.texi. | |

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

Floating-Point Reliable Library, version 2.4.1. | |

Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, | |

2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 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 2.4.1. | |

Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, | |

2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 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. | |

* Contributors:: | |

* References:: | |

* GNU Free Documentation License:: | |

* Concept Index:: | |

* Function Index:: | |

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

MPFR Copying Conditions | |

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

This library is "free"; this means that everyone is free to use it and | |

free to redistribute it on a free basis. The library is not in the | |

public domain; it is copyrighted and there are restrictions on its | |

distribution, but these restrictions are designed to permit everything | |

that a good cooperating citizen would want to do. What is not allowed | |

is to try to prevent others from further sharing any version of this | |

library that they might get from you. | |

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

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

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

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

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

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

copies of the 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.LIB. | |

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 extend the class of floating-point numbers provided by the | |

GNU MP library by a precise semantics. The main differences with the | |

`mpf' class from GNU MP are: | |

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

not depend (or should not) on the machine word size | |

`mp_bits_per_limb' (32 or 64 on most machines); | |

* the precision in bits can be set exactly to any valid value for | |

each variable (including very small precision); | |

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

standard. | |

In particular, with a precision of 53 bits, MPFR should be able to | |

exactly reproduce all computations with double-precision machine | |

floating-point numbers (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 2.1 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. | |

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 | |

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

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 a standard | |

Unix `make' program, plus some other standard Unix utility | |

programs. | |

2. In the MPFR build directory, type `./configure' | |

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

your system. If you get error messages, you might check that you | |

use the same compiler and compile options as for GNU MP (see the | |

`INSTALL' file). | |

3. `make' | |

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

`libmpfr.a'. A dynamic library may be produced too (see | |

configure). | |

4. `make check' | |

This will make sure MPFR was built correctly. If you get error | |

messages, please report this to `mpfr@loria.fr'. (*Note Reporting | |

Bugs::, for information on what to include in useful bug reports.) | |

5. `make install' | |

This will copy the files `mpfr.h' and `mpf2mpfr.h' to the directory | |

`/usr/local/include', the file `libmpfr.a' to the directory | |

`/usr/local/lib', and the file `mpfr.info' to the directory | |

`/usr/local/share/info' (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 an info version of the manual, in `mpfr.info'. | |

* `mpfr.pdf' or `pdf' | |

Create a PDF version of the manual, in `mpfr.pdf'. | |

* `mpfr.dvi' or `dvi' | |

Create a DVI version of the manual, in `mpfr.dvi'. | |

* `mpfr.ps' or `ps' | |

Create a Postscript version of the manual, in `mpfr.ps'. | |

* `mpfr.html' or `html' | |

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

directory `mpfr.html'; if you want only one output HTML file, then | |

type `makeinfo --html --no-split mpfr.texi' instead. | |

* `clean' | |

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

configuration files. | |

* `distclean' | |

Delete all 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 `mpfr@loria.fr'. *Note Reporting Bugs::. | |

Some bug fixes are available on the MPFR 2.4.1 web page | |

`http://www.mpfr.org/mpfr-2.4.1/'. | |

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 2.4.1 web page `http://www.mpfr.org/mpfr-2.4.1/' and the | |

FAQ `http://www.mpfr.org/faq.html': perhaps this bug is already known, | |

in which case you may find there a workaround for it. Otherwise, please | |

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

and it is not to ask too much from you, to ask you to report the bugs | |

that you find. | |

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

report together. | |

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

reproduce the bug. Include instructions on how to run the test case. | |

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

the results printed are incorrect and in that case, in what way. | |

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

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

gcc, `gcc -v'. Also, include the output from `uname -a' and the MPFR | |

version (the GMP version may be useful too). | |

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

corrected version of the library; if the bug report is poor, we will | |

not do anything about it (aside of chiding you to send better bug | |

reports). | |

Send your bug report to: `mpfr@loria.fr'. | |

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

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

to the same address. | |

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

4 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'. | |

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. | |

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 | |

754-1985 standard, zero is signed, i.e. there are both +0 and -0; the | |

behavior is the same as in the IEEE 754-1985 standard and it is | |

generalized to the other functions supported by MPFR. | |

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

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

`mp_prec_t'. The precision can be any integer between `MPFR_PREC_MIN' | |

and `MPFR_PREC_MAX'. In the current implementation, `MPFR_PREC_MIN' is | |

equal to 2. | |

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 `mp_rnd_t'. | |

A "limb" means the part of a multi-precision number that fits in a | |

single word. (We chose this word because a limb of the human body is | |

analogous to a digit, only larger, and containing several digits.) | |

Normally a limb contains 32 or 64 bits. The C data type for a limb is | |

`mp_limb_t'. | |

4.3 Function Classes | |

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

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

1. Functions for floating-point arithmetic, with names beginning with | |

`mpfr_'. The associated type is `mpfr_t'. | |

4.4 MPFR Variable Conventions | |

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

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

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

assignment operator. | |

MPFR allows you to use the same variable for both input and output | |

in the same expression. For example, the main function for | |

floating-point multiplication, `mpfr_mul', can be used like this: | |

`mpfr_mul (x, x, x, rnd_mode)'. This computes the square of X with | |

rounding mode `rnd_mode' and puts the result back in X. | |

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

by calling one of the special initialization functions. When you're | |

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

functions for that purpose. | |

A variable should only be initialized once, or at least cleared out | |

between each initialization. After a variable has been initialized, it | |

may be assigned to any number of times. | |

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

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

it out after the loop has exited. | |

You 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. | |

4.5 Rounding Modes | |

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

The following four rounding modes are supported: | |

* `GMP_RNDN': round to nearest | |

* `GMP_RNDZ': round toward zero | |

* `GMP_RNDU': round toward plus infinity | |

* `GMP_RNDD': round toward minus infinity | |

The `round to nearest' mode works as in the IEEE 754-1985 standard: | |

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

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

significant bit set to zero. For example, the number 5/2, which is | |

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

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

"drift" phenomenon mentioned by Knuth in volume 2 of The Art of | |

Computer Programming (Section 4.2.2). | |

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

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

a rounding mode, and have a return value of type `int', called the | |

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

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 the target in the rounding to nearest mode, and less than 1 | |

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

of the least significant represented bit of the target 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 | |

`GMP_RNDU' rounding mode, the ternary value is usually positive, except | |

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

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

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

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

ternary value is guaranteed to be representable in an `int'. | |

Unless documented otherwise, functions returning a `1' (or any other | |

value specified in this manual) for special cases (like `acos(0)') | |

should return an overflow or an underflow if `1' is not representable | |

in the current exponent range. | |

4.6 Floating-Point Values on Special Numbers | |

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

This section specifies the floating-point values (of type `mpfr_t') | |

returned by MPFR 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 must be 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). In | |

the other cases, the sign must be specified in the description of the | |

MPFR function. Example: `mpfr_sin' on +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. | |

4.7 Exceptions | |

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

MPFR supports 5 exception types: | |

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

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

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

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

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

toward zero. | |

Note: This is not the single 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) in the current range, 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 exponent of the current | |

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

* NaN: A NaN exception occurs when the result of a function is a 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 in a conversion to an integer). | |

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. | |

4.8 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). | |

File: mpfr.info, Node: MPFR Interface, Next: Contributors, 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 integer functions. The function prefix for | |

floating-point operations is `mpfr_'. | |

There is one significant characteristic of floating-point numbers | |

that has motivated a difference between this function class and other | |

GNU MP function classes: the inherent inexactness of floating-point | |

arithmetic. The user has to specify the precision for each variable. | |

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

precision of the assigned variable; the cost of that computation should | |

not depend from the precision of variables used as input (on average). | |

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

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

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

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

be a smooth extension of the IEEE 754-1985 arithmetic. The results | |

obtained on one computer should not differ from the results obtained on | |

a computer with a different word size. | |

MPFR does not keep track of the accuracy of a computation. This is | |

left to the user or to a higher layer. As a consequence, if two | |

variables are used to store only a few significant bits, and their | |

product is stored in a variable with large precision, then MPFR will | |

still compute the result with full precision. | |

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, mp_prec_t PREC) | |

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

value to NaN. (Warning: the corresponding `mpf' functions | |

initialize to zero instead.) | |

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

be cleared, using `mpfr_clear', between initializations. To | |

change the precision of a variable which has already been | |

initialized, use `mpfr_set_prec'. The precision PREC must be an | |

integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX' (otherwise the | |

behavior is undefined). | |

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

Initialize all the `mpfr_t' variables of the given `va_list', set | |

their precision to 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. It 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 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. It 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: | |

{ | |

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 and set its value to NaN. | |

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

be cleared, using `mpfr_clear', between initializations. The | |

precision of X is the default precision, which 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 be the default precision and their value to NaN. | |

See `mpfr_init' for more details. The `va_list' is assumed to be | |

composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It | |

begins from X. It 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 (mp_prec_t PREC) | |

Set the default precision to be *exactly* PREC bits. The | |

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

significand. All subsequent calls to `mpfr_init' will use this | |

precision, but previously initialized variables are unaffected. | |

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

can be any integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. | |

-- Function: mp_prec_t mpfr_get_default_prec (void) | |

Return the default MPFR precision in bits. | |

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

{ | |

mpfr_t x, y; | |

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

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

... | |

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

mpfr_clear (x); | |

mpfr_clear (y); | |

mpfr_free_cache (); | |

} | |

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

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

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

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

numbers. | |

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

Reset the precision of 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. | |

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

Return the precision actually used for assignments of X, i.e. the | |

number of bits used to store its 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::). When using any functions using `intmax_t', | |

you must include `<stdint.h>' or `<inttypes.h>' before `mpfr.h', to | |

allow `mpfr.h' to define prototypes for these functions. | |

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

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

mp_rnd_t RND) | |

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

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

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

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

-- Function: int mpfr_set_ld (mpfr_t ROP, long double OP, mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

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

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

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_sj', `mpfr_set_uj', `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_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 (GCC version 4.2.0 is known to | |

support this data type, but only when configured with | |

`--enable-decimal-float' too). `mpfr_set_q' might not be able to | |

work if the numerator (or the denominator) can not be | |

representable 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_d', | |

`mpfr_set_ld' or `mpfr_set_decimal64'. Otherwise the | |

floating-point constant will be first converted into a | |

reduced-precision (e.g., 53-bit) binary number before MPFR can | |

work with it. | |

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

mp_exp_t E, mp_rnd_t RND) | |

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

E, mp_rnd_t RND) | |

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

E, mp_rnd_t RND) | |

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

E, mp_rnd_t RND) | |

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, | |

mp_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. This function returns 0 | |

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

number in base BASE; otherwise it returns -1, and ROP may have | |

changed. | |

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

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

Read a floating-point number from a string 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 36 (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 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. Case is ignored. 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; its value 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 it introduces a binary exponent: it | |

indicates a multiplication by a power of 2 (there is a difference | |

only for base 16). The value of an exponent is always written in | |

base 10. In base 2, the significand can start with `0b' or `0B', | |

and in base 16, it can start with `0x' or `0X'. | |

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 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'. | |

Special data (for infinities and NaN) can be `@inf@' or | |

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

`infinity', `inf', `nan' or `nan(n-char-sequence)', all case | |

insensitive. A `n-char-sequence' is a non-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. | |

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

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

Set the variable X to infinity or NaN (Not-a-Number) respectively. | |

In `mpfr_set_inf', X is set to plus infinity iff SIGN is | |

nonnegative. | |

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

Swap the values X and Y efficiently. Warning: the precisions are | |

exchanged too; in case the precisions are different, `mpfr_swap' | |

is thus not equivalent to three `mpfr_set' calls using a third | |

auxiliary variable. | |

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, mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

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

RND) | |

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

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

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

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, | |

mp_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: double mpfr_get_d (mpfr_t OP, mp_rnd_t RND) | |

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

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

Convert OP to a `double' (respectively `_Decimal64' or `long | |

double'), 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 Â±Inf, an infinity of the same sign or the | |

result of Â±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: double mpfr_get_d_2exp (long *EXP, mpfr_t OP, mp_rnd_t | |

RND) | |

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

mp_rnd_t RND) | |

Return D and set 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: long mpfr_get_si (mpfr_t OP, mp_rnd_t RND) | |

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

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

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

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, the result is undefined. If OP is too big for the | |

return type, it returns the maximum or the minimum of the | |

corresponding C type, depending on the direction of the overflow. | |

The _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: mp_exp_t mpfr_get_z_exp (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 multiplied by two exponent EXP. If the exponent is not | |

representable in the `mp_exp_t' type, the behavior is undefined. | |

-- Function: void mpfr_get_z (mpz_t ROP, mpfr_t OP, mp_rnd_t RND) | |

Convert OP to a `mpz_t', after rounding it with respect to RND. If | |

OP is NaN or Inf, the result is undefined. | |

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

Convert OP to a `mpf_t', after rounding it with respect to RND. | |

Return zero iff no error occurred, in particular a non-zero value | |

is returned if OP is NaN or Inf, which do not exist in `mpf'. | |

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

size_t N, mpfr_t OP, mp_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; in the latter case, N must be greater or equal | |

to 2. The base may vary from 2 to 36. | |

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

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

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

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

middle of two possible outputs, the one with an even last digit is | |

chosen (for an odd base, this may not correspond to an even | |

significand). | |

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 | |

depending on N and B only that satisfies the above property, i.e., | |

m = 1 + ceil(N*log(2)/log(B)), but in some very rare cases, it | |

might be m+1. | |

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. 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. | |

If the input number is an ordinary number, the exponent is written | |

through the pointer EXPPTR (the current minimal exponent for 0). | |

A pointer to the string is returned, unless there is an error, in | |

which case a null pointer is returned. | |

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

Free a string allocated by `mpfr_get_str' using the current | |

unallocation function (preliminary interface). The block is | |

assumed to be `strlen(STR)+1' bytes. For more information about | |

how it is done: *note Custom Allocation: (gmp.info)Custom | |

Allocation. | |

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

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

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

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

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

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

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

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

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

mp_rnd_t RND) | |

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

int OP2, mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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, | |

mp_rnd_t RND) | |

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

mpfr_t OP2, mp_rnd_t RND) | |

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

int OP2, mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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, | |

mp_rnd_t RND) | |

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

int OP2, mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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, mp_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, | |

mp_rnd_t RND) | |

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

mpfr_t OP2, mp_rnd_t RND) | |

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

int OP2, mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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, mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

Set ROP to the square root of OP rounded in the direction RND. | |

Return -0 if OP is -0 (to be consistent with the IEEE 754-1985 | |

standard). Set ROP to NaN if OP is negative. | |

-- Function: int mpfr_rec_sqrt (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

Set ROP to the reciprocal square root of OP rounded in the | |

direction RND. Return +Inf if OP is Â±0, and +0 if OP is +Inf. Set | |

ROP to NaN if OP is negative. | |

-- Function: int mpfr_cbrt (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

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

K, mp_rnd_t RND) | |

Set ROP to the cubic root (resp. the Kth root) of OP rounded in | |

the direction RND. An odd (resp. even) root of a negative number | |

(including -Inf) returns 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, | |

mp_rnd_t RND) | |

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

int OP2, mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

mp_rnd_t RND) | |

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

unsigned long int OP2, mp_rnd_t RND) | |

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

mpfr_t OP2, mp_rnd_t RND) | |

Set ROP to OP1 raised to OP2, rounded in the direction RND. | |

Special values are currently handled as described in the ISO C99 | |

standard for the `pow' function (note this may change in future | |

versions): | |

* `pow(Â±0, Y)' returns plus or minus infinity for Y a negative | |

odd integer. | |

* `pow(Â±0, Y)' returns plus infinity for Y negative and not an | |

odd integer. | |

* `pow(Â±0, Y)' returns plus or minus zero for Y a positive odd | |

integer. | |

* `pow(Â±0, Y)' returns plus zero for Y positive and not an odd | |

integer. | |

* `pow(-1, Â±Inf)' returns 1. | |

* `pow(+1, Y)' returns 1 for any Y, even a NaN. | |

* `pow(X, Â±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, mp_rnd_t RND) | |

Set ROP to -OP rounded in the direction RND. Just changes the | |

sign if ROP and OP are the same variable. | |

-- Function: int mpfr_abs (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

Set ROP to the absolute value of OP, rounded in the direction RND. | |

Just changes the sign if ROP and OP are the same variable. | |

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

mp_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, and +0 otherwise. | |

Returns NaN when OP1 or OP2 is NaN. | |

-- Function: int mpfr_mul_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long | |

int OP2, mp_rnd_t RND) | |

-- Function: int mpfr_mul_2si (mpfr_t ROP, mpfr_t OP1, long int OP2, | |

mp_rnd_t RND) | |

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, mp_rnd_t RND) | |

-- Function: int mpfr_div_2si (mpfr_t ROP, mpfr_t OP1, long int OP2, | |

mp_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, signed long int OP2) | |

-- Function: int mpfr_cmp_d (mpfr_t OP1, double OP2) | |

-- Function: int mpfr_cmp_ld (mpfr_t OP1, long double OP2) | |

-- Function: int mpfr_cmp_z (mpfr_t OP1, mpz_t OP2) | |

-- Function: int mpfr_cmp_q (mpfr_t OP1, mpq_t OP2) | |

-- Function: int mpfr_cmp_f (mpfr_t OP1, mpf_t OP2) | |

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, | |

mp_exp_t E) | |

-- Function: int mpfr_cmp_si_2exp (mpfr_t OP1, long int OP2, mp_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) | |

Return non-zero if OP is respectively NaN, an infinity, an ordinary | |

number (i.e. neither NaN nor an infinity) or 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. | |

-- Function: int mpfr_greater_p (mpfr_t OP1, mpfr_t OP2) | |

Return non-zero if OP1 > OP2, zero otherwise. | |

-- Function: int mpfr_greaterequal_p (mpfr_t OP1, mpfr_t OP2) | |

Return non-zero if OP1 >= OP2, zero otherwise. | |

-- Function: int mpfr_less_p (mpfr_t OP1, mpfr_t OP2) | |

Return non-zero if OP1 < OP2, zero otherwise. | |

-- Function: int mpfr_lessequal_p (mpfr_t OP1, mpfr_t OP2) | |

Return non-zero if OP1 <= OP2, zero otherwise. | |

-- Function: int mpfr_lessgreater_p (mpfr_t OP1, mpfr_t OP2) | |

Return non-zero if OP1 < OP2 or OP1 > OP2 (i.e. neither OP1, nor | |

OP2 is NaN, and OP1 <> OP2), zero otherwise (i.e. OP1 and/or OP2 | |

are NaN, or OP1 = OP2). | |

-- Function: int mpfr_equal_p (mpfr_t OP1, mpfr_t OP2) | |

Return non-zero if OP1 = OP2, zero otherwise (i.e. OP1 and/or OP2 | |

are 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, return 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, mp_rnd_t RND) | |

-- Function: int mpfr_log2 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_log10 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

Set ROP to the natural logarithm of OP, log2(OP) or log10(OP), | |

respectively, rounded in the direction RND. Return -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, mp_rnd_t RND) | |

-- Function: int mpfr_exp2 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_exp10 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

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, mp_rnd_t RND) | |

-- Function: int mpfr_sin (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_tan (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

Set ROP to the cosine of OP, sine of OP, tangent of OP, rounded in | |

the direction RND. | |

-- Function: int mpfr_sec (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_csc (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_cot (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

Set ROP to the secant of OP, cosecant of OP, cotangent of OP, | |

rounded in the direction RND. | |

-- Function: int mpfr_sin_cos (mpfr_t SOP, mpfr_t COP, mpfr_t OP, | |

mp_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. | |

-- Function: int mpfr_acos (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_asin (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_atan (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

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)'. | |

-- Function: int mpfr_atan2 (mpfr_t ROP, mpfr_t Y, mpfr_t X, mp_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)))'. 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 currently handled as described in the ISO C99 | |

standard for the `atan2' function (note this may change in future | |

versions): | |

* `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, mp_rnd_t RND) | |

-- Function: int mpfr_sinh (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_tanh (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

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, | |

mp_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. | |

-- Function: int mpfr_sech (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_csch (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_coth (mpfr_t ROP, mpfr_t OP, mp_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, mp_rnd_t RND) | |

-- Function: int mpfr_asinh (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_atanh (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

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, | |

mp_rnd_t RND) | |

Set ROP to the factorial of the `unsigned long int' OP, rounded in | |

the direction RND. | |

-- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mp_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, mp_rnd_t RND) | |

Set ROP to the exponential of OP minus one, rounded in the | |

direction RND. | |

-- Function: int mpfr_eint (mpfr_t Y, mpfr_t X, mp_rnd_t RND) | |

Set Y to the exponential integral of X, rounded in the direction | |

RND. For positive X, the exponential integral is the sum of | |

Euler's constant, of the logarithm of X, and of the sum for k from | |

1 to infinity of X to the power k, divided by k and factorial(k). | |

For negative X, the returned value is NaN. | |

-- Function: int mpfr_li2 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND_MODE) | |

Set ROP to real part of the dilogarithm of OP, rounded in the | |

direction RND_MODE. The dilogarithm function is defined here as | |

the integral of -log(1-t)/t from 0 to x. | |

-- Function: int mpfr_gamma (mpfr_t ROP, mpfr_t OP, mp_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, NaN is returned. | |

-- Function: int mpfr_lngamma (mpfr_t ROP, mpfr_t OP, mp_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 <= X <= -2K, K being a | |

non-negative integer, NaN is returned. See also `mpfr_lgamma'. | |

-- Function: int mpfr_lgamma (mpfr_t ROP, int *SIGNP, mpfr_t OP, | |

mp_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, +Inf is | |

returned. When OP is NaN, -Inf or a negative integer, *SIGNP is | |

undefined, and when OP is Â±0, *SIGNP is the sign of the zero. | |

-- Function: int mpfr_zeta (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_zeta_ui (mpfr_t ROP, unsigned long OP, mp_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, mp_rnd_t RND) | |

Set ROP to the value of the error function on OP, rounded in the | |

direction RND. | |

-- Function: int mpfr_erfc (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

Set ROP to the value of the complementary error function on OP, | |

rounded in the direction RND. | |

-- Function: int mpfr_j0 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_j1 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_jn (mpfr_t ROP, long N, mpfr_t OP, mp_rnd_t RND) | |

Set ROP to the value of the first kind Bessel function of order 0, | |

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 +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, mp_rnd_t RND) | |

-- Function: int mpfr_y1 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_yn (mpfr_t ROP, long N, mpfr_t OP, mp_rnd_t RND) | |

Set ROP to the value of the second kind Bessel function of order | |

0, 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 +0. | |

When OP is zero, ROP is +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, mp_rnd_t RND) | |

Set ROP to (OP1 times OP2) + OP3, rounded in the direction RND. | |

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

OP3, mp_rnd_t RND) | |

Set ROP to (OP1 times OP2) - OP3, rounded in the direction RND. | |

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

mp_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, the | |

return value is NaN. | |

-- Function: int mpfr_hypot (mpfr_t ROP, mpfr_t X, mpfr_t Y, mp_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 currently handled as described in Section | |

F.9.4.3 of the ISO C99 standard, for the `hypot' function (note | |

this may change in future versions): If X or Y is an infinity, | |

then plus infinity is returned in ROP, even if the other number is | |

NaN. | |

-- Function: int mpfr_const_log2 (mpfr_t ROP, mp_rnd_t RND) | |

-- Function: int mpfr_const_pi (mpfr_t ROP, mp_rnd_t RND) | |

-- Function: int mpfr_const_euler (mpfr_t ROP, mp_rnd_t RND) | |

-- Function: int mpfr_const_catalan (mpfr_t ROP, mp_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 (currently | |

`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 N, mp_rnd_t RND) | |

Set RET to the sum of all elements of TAB whose size is N, rounded | |

in the direction RND. Warning, TAB is a table of pointers to | |

mpfr_t, not a table of mpfr_t (preliminary interface). The returned | |

`int' value is zero when the computed value is the exact value, | |

and non-zero when this cannot be guaranteed, without giving the | |

direction of the error as the other functions do. | |

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, mp_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 36. | |

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 bytes written, or if an error occurred, | |

return 0. | |

-- Function: size_t mpfr_inp_str (mpfr_t ROP, FILE *STREAM, int BASE, | |

mp_rnd_t RND) | |

Input a string in base 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' (it may | |

change). See the documentation of `mpfr_strtofr' for a detailed | |

description of the valid string formats. | |

Return the number of bytes read, or if an error occurred, return 0. | |

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 C function `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), plus | |

`R' and `P': | |

`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' | |

`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 | |

`R' `mpfr_t' input, float conversions | |

`P' `mpfr_prec_t' input, integer conversions | |

The `type' specifiers have the same restrictions as those mentioned | |

in the GMP documentation: *note Formatted Output Strings: | |

(gmp.info)Formatted Output Strings. More precisely, except for `R' and | |

`P' (which are defined by MPFR), the `type' specifiers 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' specifier is specific to `mpfr_t' parameter and shall | |

not be used with other types. `mpfr_printf' accepts the same conversion | |

specifier character `conv' as `gmp_printf' plus `b'. | |

The `P' type outputs the precision of an `mpfr_t' variable. It is | |

needed because the `mpfr_prec_t' type does not necessarily correspond | |

to an `unsigned int' or any fixed standard type. 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 is used for a `mpfr_t' output and can be followed by a | |

rounding specifier denoted by one of the following characters: | |

`U' round toward plus infinity | |

`D' round toward minus infinity | |

`Z' round toward zero | |

`N' round to nearest | |

`*' rounding mode (as a `mpfr_rnd_t') | |

indicated by the argument just before | |

the corresponding `mpfr_t' variable. | |

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 field is empty (as in `%.Rf'), the number is displayed | |

with enough digits so that it can be read back exactly (assuming | |

rounding to nearest, see `mpfr_get_str'). If no rounding is specified, | |

the `mpfr_t' argument is rounded 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", GMP_RNDN, x); | |

`mpfr_printf' also adds a new conversion specifier `b' which | |

displays the `mpfr_t' parameter in binary, the behavior is undefined | |

with other parameter type. The `conv' specifiers allowed with `mpfr_t' | |

parameter are: | |

`a' `A' hex float, C99 style | |

`b' binary output | |

`e' `E' scientific format float | |

`f' fixed point float | |

`g' `G' fixed or scientific float | |

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. In binary output, the precision is silently | |

increased up to 2 if it equals 1. | |

5.9.3 Functions | |

--------------- | |

-- 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. 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, the function returns -1, sets the | |

_erange_ flag, and (in POSIX system only) `errno' is set to | |

`EOVERFLOW'. | |

-- 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. If the number of characters which ought to be | |

written appears to exceed the maximum limit for an `int', nothing | |

is written in `stdout', the function returns -1, sets the _erange_ | |

flag, and (in POSIX system only) `errno' is set to `EOVERFLOW'. | |

-- 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 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. If the number of characters which ought to be | |

written appears to exceed the maximum limit for an `int', nothing | |

is written in BUF, the function returns -1, sets the _erange_ | |

flag, and (in POSIX system only) `errno' is set to `EOVERFLOW'. | |

-- 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 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. If the number of | |

characters produced by the optional arguments under the control of | |

the template string TEMPLATE appears to exceed the maximum limit | |

for an `int', nothing is written in BUF, the function returns -1, | |

sets the _erange_ flag, and (in POSIX system only) `errno' is set | |

to `EOVERFLOW'. | |

-- 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. If the number of characters produced by the | |

optional arguments under the control of the template string | |

TEMPLATE appears to exceed the maximum limit for an `int', STR is | |

a null pointer, the function returns -1, sets the _erange_ flag, | |

and (in POSIX system only) `errno' is set to `EOVERFLOW'. | |

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, mp_rnd_t RND) | |

-- Function: int mpfr_ceil (mpfr_t ROP, mpfr_t OP) | |

-- Function: int mpfr_floor (mpfr_t ROP, mpfr_t OP) | |

-- Function: int mpfr_round (mpfr_t ROP, mpfr_t OP) | |

-- Function: int mpfr_trunc (mpfr_t ROP, mpfr_t OP) | |

Set ROP to OP rounded to an integer. `mpfr_rint' rounds to the | |

nearest representable integer in the given rounding mode, | |

`mpfr_ceil' rounds to the next higher or equal representable | |

integer, `mpfr_floor' to the next lower or equal representable | |

integer, `mpfr_round' to the nearest representable integer, | |

rounding halfway cases away from zero, and `mpfr_trunc' to the | |

next representable integer 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. | |

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, 4.5 (100.1 in binary) is rounded by | |

`mpfr_round' to 4 (100 in binary) in 2-bit precision, though | |

`round(4.5)' is equal to 5 and 5 (101 in binary) is rounded to 6 | |

(110 in binary) in 2-bit precision. | |

-- Function: int mpfr_rint_ceil (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_rint_floor (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_rint_round (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

-- Function: int mpfr_rint_trunc (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) | |

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). | |

-- Function: int mpfr_frac (mpfr_t ROP, mpfr_t OP, mp_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, | |

mp_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. | |

-- Function: int mpfr_fmod (mpfr_t R, mpfr_t X, mpfr_t Y, mp_rnd_t RND) | |

-- Function: int mpfr_remainder (mpfr_t R, mpfr_t X, mpfr_t Y, | |

mp_rnd_t RND) | |

-- Function: int mpfr_remquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y, | |

mp_rnd_t RND) | |

Set R to the value of x - n y, 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 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. `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 (mp_rnd_t RND) | |

Set the default rounding mode to RND. The default rounding mode | |

is to nearest initially. | |

-- Function: mp_rnd_t mpfr_get_default_rounding_mode (void) | |

Get the default rounding mode. | |

-- Function: int mpfr_prec_round (mpfr_t X, mp_prec_t PREC, mp_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. | |

-- Function: int mpfr_round_prec (mpfr_t X, mp_rnd_t RND, mp_prec_t | |

PREC) | |

[This function is obsolete. Please use `mpfr_prec_round' instead.] | |

-- Function: int mpfr_can_round (mpfr_t B, mp_exp_t ERR, mp_rnd_t | |

RND1, mp_rnd_t RND2, mp_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. | |

Note: if one wants to also determine the correct ternary value | |

when rounding B to precision PREC, a useful trick is the following: if (mpfr_can_round (b, err, rnd1, GMP_RNDZ, prec + (rnd2 == GMP_RNDN))) | |

... | |

Indeed, if RND2 is `GMP_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: const char * mpfr_print_rnd_mode (mp_rnd_t RND) | |

Return the input string (GMP_RNDD, GMP_RNDU, GMP_RNDN, GMP_RNDZ) | |

corresponding to the rounding mode 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. 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, if | |

there is one (the infinite values are seen as the smallest and | |

largest floating-point numbers). If the result is zero, it keeps | |

the same sign. No underflow or overflow is generated. | |

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

Equivalent to `mpfr_nexttoward' where Y is plus infinity. | |

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

Equivalent to `mpfr_nexttoward' where Y is minus infinity. | |

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

mp_rnd_t RND) | |

Set ROP to the minimum of OP1 and OP2. If OP1 and OP2 are both | |

NaN, then ROP is set to NaN. If OP1 or OP2 is NaN, then ROP is set | |

to the numeric value. If OP1 and OP2 are zeros of different signs, | |

then ROP is set to -0. | |

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

mp_rnd_t RND) | |

Set ROP to the maximum of OP1 and OP2. If OP1 and OP2 are both | |

NaN, then ROP is set to NaN. If OP1 or OP2 is NaN, then ROP is set | |

to the numeric value. If OP1 and OP2 are zeros of different signs, | |

then ROP is set to +0. | |

-- Function: int mpfr_urandomb (mpfr_t ROP, gmp_randstate_t STATE) | |

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

ROP < 1. More precisely, the number can be seen as a float with a | |

random non-normalized significand and exponent 0, which is then | |

normalized (thus if E denotes the exponent after normalization, | |

then the least -E significant bits of the significand are always | |

0). Return 0, unless the exponent is not in the current exponent | |

range, in which case ROP is set to NaN and a non-zero value is | |

returned (this should never happen in practice, except in very | |

specific cases). The second argument is a `gmp_randstate_t' | |

structure which should be created using the GMP `gmp_randinit' | |

function, see the GMP manual. | |

-- Function: void mpfr_random (mpfr_t ROP) | |

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

ROP < 1. | |

This function is deprecated and will be suppressed in the next | |

release; `mpfr_urandomb' should be used instead. | |

-- Function: void mpfr_random2 (mpfr_t ROP, mp_size_t SIZE, mp_exp_t | |

EXP) | |

Generate a random float of at most SIZE limbs, with long strings of | |

zeros and ones in the binary representation. The exponent of the | |

number is in the interval -EXP to EXP. This function is useful for | |

testing functions and algorithms, since this kind of random | |

numbers have proven to be more likely to trigger corner-case bugs. | |

Negative random numbers are generated when SIZE is negative. Put | |

+0 in ROP when size if zero. The internal state of the default | |

pseudorandom number generator is modified by a call to this | |

function (the same one as GMP if MPFR was built using | |

`--with-gmp-build'). | |

This function is deprecated and will be suppressed in the next | |

release. | |

-- Function: mp_exp_t mpfr_get_exp (mpfr_t X) | |

Get the exponent of X, assuming that X is a non-zero ordinary | |

number and the significand is chosen in [1/2,1). The behavior for | |

NaN, infinity or zero is undefined. | |

-- Function: int mpfr_set_exp (mpfr_t X, mp_exp_t E) | |

Set the exponent of X if E is in the current exponent range, and | |

return 0 (even if X is not a non-zero ordinary number); otherwise, | |

return a non-zero value. The significand is assumed to be in | |

[1/2,1). | |

-- Function: int mpfr_signbit (mpfr_t OP) | |

Return a non-zero value iff OP has its sign bit set (i.e. if it is | |

negative, -0, or a NaN whose representation has its sign bit set). | |

-- Function: int mpfr_setsign (mpfr_t ROP, mpfr_t OP, int S, mp_rnd_t | |

RND) | |

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

RND, then set (resp. clear) its sign bit if S is non-zero (resp. | |

zero), even when OP is a NaN. | |

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

mp_rnd_t RND) | |

Set the value of ROP from OP1, rounded toward the given direction | |

RND, then set its sign bit to that of OP2 (even when OP1 or OP2 is | |

a NaN). This function is equivalent to `mpfr_setsign (ROP, OP1, | |

mpfr_signbit (OP2), RND)'. | |

-- Function: const char * mpfr_get_version (void) | |

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

-- Macro: MPFR_VERSION | |

-- Macro: MPFR_VERSION_MAJOR | |

-- Macro: MPFR_VERSION_MINOR | |

-- Macro: MPFR_VERSION_PATCHLEVEL | |

-- Macro: MPFR_VERSION_STRING | |

`MPFR_VERSION' is the version of MPFR as a preprocessing constant. | |

`MPFR_VERSION_MAJOR', `MPFR_VERSION_MINOR' and | |

`MPFR_VERSION_PATCHLEVEL' are respectively the major, minor and | |

patch level of MPFR version, as preprocessing constants. | |

`MPFR_VERSION_STRING' is the version (with an optional suffix, used | |

in development and pre-release versions) as a string constant, | |

which can be compared to the result of `mpfr_get_version' to check | |

at run time the header file and library used match: | |

if (strcmp (mpfr_get_version (), MPFR_VERSION_STRING)) | |

fprintf (stderr, "Warning: header and library do not match\n"); | |

Note: Obtaining different strings is not necessarily an error, as | |

in general, a program compiled with some old MPFR version can be | |

dynamically linked with a newer MPFR library version (if allowed | |

by the library versioning system). | |

-- Macro: long MPFR_VERSION_NUM (MAJOR, MINOR, PATCHLEVEL) | |

Create an integer in the same format as used by `MPFR_VERSION' | |

from the given MAJOR, MINOR and PATCHLEVEL. Here is an example of | |

how to check the MPFR version at compile time: | |

#if (!defined(MPFR_VERSION) || (MPFR_VERSION<MPFR_VERSION_NUM(2,1,0))) | |

# error "Wrong MPFR version." | |

#endif | |

-- Function: const char * mpfr_get_patches (void) | |

Return a null-terminated string containing the ids of the patches | |

applied to the MPFR library (contents of the `PATCHES' file), | |

separated by spaces. Note: If the program has been compiled with | |

an older MPFR version and is dynamically linked with a new MPFR | |

library version, the ids of the patches applied to the old | |

(compile-time) MPFR version are not available (however this | |

information should not have much interest in general). | |

File: mpfr.info, Node: Exception Related Functions, Next: Compatibility with MPF, Prev: Miscellaneous Functions, Up: MPFR Interface | |

5.13 Exception Related Functions | |

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

-- Function: mp_exp_t mpfr_get_emin (void) | |

-- Function: mp_exp_t mpfr_get_emax (void) | |

Return the (current) smallest and largest exponents allowed for a | |

floating-point variable. The smallest positive value of a | |

floating-point variable is one half times 2 raised to the smallest | |

exponent and the largest value has the form (1 - epsilon) times 2 | |

raised to the largest exponent. | |

-- Function: int mpfr_set_emin (mp_exp_t EXP) | |

-- Function: int mpfr_set_emax (mp_exp_t EXP) | |

Set the smallest and largest exponents allowed for a | |

floating-point variable. Return a non-zero value when EXP is not | |

in the range accepted by the implementation (in that case the | |

smallest or largest exponent is not changed), and zero otherwise. | |

If the user changes the exponent range, it is her/his | |

responsibility to check that all current floating-point variables | |

are in the new allowed range (for example using | |

`mpfr_check_range'), otherwise the subsequent behavior will be | |

undefined, in the sense of the ISO C standard. | |

-- Function: mp_exp_t mpfr_get_emin_min (void) | |

-- Function: mp_exp_t mpfr_get_emin_max (void) | |

-- Function: mp_exp_t mpfr_get_emax_min (void) | |

-- Function: mp_exp_t mpfr_get_emax_max (void) | |

Return the minimum and maximum of the smallest and largest | |

exponents allowed for `mpfr_set_emin' and `mpfr_set_emax'. These | |

values are implementation dependent; it is possible to create a non | |

portable program by writing `mpfr_set_emax(mpfr_get_emax_max())' | |

and `mpfr_set_emin(mpfr_get_emin_min())' since the values of the | |

smallest and largest exponents become implementation dependent. | |

-- Function: int mpfr_check_range (mpfr_t X, int T, mp_rnd_t RND) | |

This function forces X to be in the current range of acceptable | |

values, T being the current ternary value: negative if X is | |

smaller than the exact value, positive if X is larger than the | |

exact value and zero if X is exact (before the call). It generates | |

an underflow or an overflow if the exponent of X is outside the | |

current allowed range; the value of T may be used to avoid a | |

double rounding. This function returns zero if the rounded result | |

is equal to the exact one, a positive value if the rounded result | |

is larger than the exact one, a negative value if the rounded | |

result is smaller than the exact one. Note that unlike most | |

functions, the result is compared to the exact one, not the input | |

value X, i.e. the ternary value is propagated. | |

Note: If X is an infinity and T is different from zero (i.e., if | |

the rounded result is an inexact infinity), then the overflow flag | |

is set. This is useful because `mpfr_check_range' is typically | |

called (at least in MPFR functions) after restoring the flags that | |

could have been set due to internal computations. | |

-- Function: int mpfr_subnormalize (mpfr_t X, int T, mp_rnd_t RND) | |

This function rounds X emulating subnormal number arithmetic: if X | |

is outside the subnormal exponent range, it just propagates the | |

ternary value T; otherwise, it rounds X to precision | |

`EXP(x)-emin+1' according to rounding mode RND and previous | |

ternary value T, avoiding double rounding problems. More | |

precisely in the subnormal domain, denoting by E the value of | |

`emin', X is rounded in fixed-point arithmetic to an integer | |

multiple of two to the power E-1; as a consequence, 1.5 multiplied | |

by two to the power E-1 when T is zero is rounded to two to the | |

power E with rounding to nearest. | |

`PREC(x)' is not modified by this function. RND and T must be the | |

used rounding mode for computing X and the returned ternary value | |

when computing X. The subnormal exponent range is from `emin' to | |

`emin+PREC(x)-1'. If the result cannot be represented in the | |

current exponent range (due to a too small `emax'), the behavior | |

is undefined. Note that unlike most functions, the result is | |

compared to the exact one, not the input value X, i.e. the ternary | |

value is propagated. This is a preliminary interface. | |

This is an example of how to emulate double IEEE-754 arithmetic | |

using MPFR: | |

{ | |

mpfr_t xa, xb; | |

int i; | |

volatile double a, b; | |

mpfr_set_default_prec (53); | |

mpfr_set_emin (-1073); | |

mpfr_set_emax (1024); | |

mpfr_init (xa); mpfr_init (xb); | |

b = 34.3; mpfr_set_d (xb, b, GMP_RNDN); | |

a = 0x1.1235P-1021; mpfr_set_d (xa, a, GMP_RNDN); | |

a /= b; | |

i = mpfr_div (xa, xa, xb, GMP_RNDN); | |

i = mpfr_subnormalize (xa, i, GMP_RNDN); | |

mpfr_clear (xa); mpfr_clear (xb); | |

} | |

Warning: this emulates a double IEEE-754 arithmetic with correct | |

rounding in the subnormal range, which may not be the case for your | |

hardware. | |

-- Function: void mpfr_clear_underflow (void) | |

-- Function: void mpfr_clear_overflow (void) | |

-- Function: void mpfr_clear_nanflag (void) | |

-- Function: void mpfr_clear_inexflag (void) | |

-- Function: void mpfr_clear_erangeflag (void) | |

Clear the underflow, overflow, invalid, inexact and _erange_ flags. | |

-- Function: void mpfr_set_underflow (void) | |

-- Function: void mpfr_set_overflow (void) | |

-- Function: void mpfr_set_nanflag (void) | |

-- Function: void mpfr_set_inexflag (void) | |

-- Function: void mpfr_set_erangeflag (void) | |

Set the underflow, overflow, invalid, inexact and _erange_ flags. | |

-- Function: void mpfr_clear_flags (void) | |

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

_erange_). | |

-- Function: int mpfr_underflow_p (void) | |

-- Function: int mpfr_overflow_p (void) | |

-- Function: int mpfr_nanflag_p (void) | |

-- Function: int mpfr_inexflag_p (void) | |

-- Function: int mpfr_erangeflag_p (void) | |

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

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

File: mpfr.info, Node: Compatibility with MPF, Next: Custom Interface, Prev: Exception Related Functions, Up: MPFR Interface | |

5.14 Compatibility With MPF | |

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

A header file `mpf2mpfr.h' is included in the distribution of MPFR for | |

compatibility with the GNU MP class MPF. After inserting the following | |

two lines after the `#include <gmp.h>' line, | |

#include <mpfr.h> | |

#include <mpf2mpfr.h> | |

any program written for MPF can be compiled directly with MPFR without | |

any changes. All operations are then performed with the default MPFR | |

rounding mode, which can be reset with `mpfr_set_default_rounding_mode'. | |

Warning: the `mpf_init' and `mpf_init2' functions initialize to | |

zero, whereas the corresponding MPFR functions initialize to NaN: this | |

is useful to detect uninitialized values, but is slightly incompatible | |

with `mpf'. | |

-- Function: void mpfr_set_prec_raw (mpfr_t X, mp_prec_t PREC) | |

Reset the precision of X to be *exactly* PREC bits. The only | |

difference with `mpfr_set_prec' is that PREC is assumed to be | |

small enough so that the significand fits into the current | |

allocated memory space for X. Otherwise the behavior is undefined. | |

-- Function: int mpfr_eq (mpfr_t OP1, mpfr_t OP2, unsigned long int | |

OP3) | |

Return non-zero if OP1 and OP2 are both non-zero ordinary numbers | |

with the same exponent and the same first OP3 bits, both zero, or | |

both infinities of the same sign. Return zero otherwise. This | |

function is defined for compatibility with `mpf'. Do not use it if | |

you want to know whether two numbers are close to each other; for | |

instance, 1.011111 and 1.100000 are currently regarded as | |

different for any value of OP3 larger than 1 (but this may change | |

in the next release). | |

-- Function: void mpfr_reldiff (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, | |

mp_rnd_t RND) | |

Compute the relative difference between OP1 and OP2 and store the | |

result in ROP. This function does not guarantee the correct | |

rounding on the relative difference; it just computes | |

|OP1-OP2|/OP1, using the rounding mode RND for all operations and | |

the precision of ROP. | |

-- Function: int mpfr_mul_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long | |

int OP2, mp_rnd_t RND) | |

-- Function: int mpfr_div_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long | |

int OP2, mp_rnd_t RND) | |

See `mpfr_mul_2ui' and `mpfr_div_2ui'. These functions are only | |

kept for compatibility with MPF. | |

File: mpfr.info, Node: Custom Interface, Next: Internals, Prev: Compatibility with MPF, Up: MPFR Interface | |

5.15 Custom Interface | |

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

Some applications use a stack to handle the memory and their objects. | |

However, the MPFR memory design is not well suited for such a thing. So | |

that such applications are able to use MPFR, an auxiliary memory | |

interface has been created: the Custom Interface. | |

The following interface allows them to use MPFR in two ways: | |

* Either they directly store the MPFR FP number as a `mpfr_t' on the | |

stack. | |

* Either they store their own representation of a FP number on the | |

stack and construct a new temporary `mpfr_t' each time it is | |

needed. | |

Nothing has to be done to destroy the FP numbers except garbaging | |

the used memory: all the memory stuff (allocating, destroying, | |

garbaging) is kept to the application. | |

Each function in this interface is also implemented as a macro for | |

efficiency reasons: for example `mpfr_custom_init (s, p)' uses the | |

macro, while `(mpfr_custom_init) (s, p)' uses the function. | |

Note 1: MPFR functions may still initialize temporary FP numbers | |

using standard mpfr_init. See Custom Allocation (GNU MP). | |

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

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

`mpfr_free_cache' each time you garbage the memory iff mpfr_init, | |

through GMP Custom Allocation, allocates its memory on the application | |

stack. | |

Note 3: This interface is preliminary. | |

-- Function: size_t mpfr_custom_get_size (mp_prec_t PREC) | |

Return the needed size in bytes to store the significand of a FP | |

number of precision PREC. | |

-- Function: void mpfr_custom_init (void *SIGNIFICAND, mp_prec_t PREC) | |

Initialize a significand of precision PREC. SIGNIFICAND must be | |

an area of `mpfr_custom_get_size (prec)' bytes at least and be | |

suitably aligned for an array of `mp_limb_t'. | |

-- Function: void mpfr_custom_init_set (mpfr_t X, int KIND, mp_exp_t | |

EXP, mp_prec_t PREC, void *SIGNIFICAND) | |

Perform a dummy initialization of a `mpfr_t' and set it to: | |

* if `ABS(kind) == MPFR_NAN_KIND', X is set to NaN; | |

* if `ABS(kind) == MPFR_INF_KIND', X is set to the infinity of | |

sign `sign(kind)'; | |

* if `ABS(kind) == MPFR_ZERO_KIND', X is set to the zero of | |

sign `sign(kind)'; | |

* if `ABS(kind) == MPFR_REGULAR_KIND', X is set to a regular | |

number: `x = sign(kind)*significand*2^exp' | |

In all cases, it uses SIGNIFICAND directly for further computing | |

involving X. It will not allocate anything. A FP number | |

initialized with this function cannot be resized using | |

`mpfr_set_prec', or cleared using `mpfr_clear'! SIGNIFICAND must | |

have been initialized with `mpfr_custom_init' using the same | |

precision PREC. | |

-- Function: int mpfr_custom_get_kind (mpfr_t X) | |

Return the current kind of a `mpfr_t' as used by | |

`mpfr_custom_init_set'. The behavior of this function for any | |

`mpfr_t' not initialized with `mpfr_custom_init_set' is undefined. | |

-- Function: void * mpfr_custom_get_mantissa (mpfr_t X) | |

Return a pointer to the significand used by a `mpfr_t' initialized | |

with `mpfr_custom_init_set'. The behavior of this function for | |

any `mpfr_t' not initialized with `mpfr_custom_init_set' is | |

undefined. | |

-- Function: mp_exp_t mpfr_custom_get_exp (mpfr_t X) | |

Return the exponent of X, assuming that X is a non-zero ordinary | |

number. The return value for NaN, Infinity or Zero is unspecified | |

but does not produce any trap. The behavior of this function for | |

any `mpfr_t' not initialized with `mpfr_custom_init_set' is | |

undefined. | |

-- Function: void mpfr_custom_move (mpfr_t X, void *NEW_POSITION) | |

Inform MPFR that the significand has moved due to a garbage collect | |

and update its new position to `new_position'. However the | |

application has to move the significand and the `mpfr_t' itself. | |

The behavior of this function for any `mpfr_t' not initialized | |

with `mpfr_custom_init_set' is undefined. | |

See the test suite for examples. | |

File: mpfr.info, Node: Internals, Prev: Custom Interface, Up: MPFR Interface | |

5.16 Internals | |

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

The following types and functions were mainly designed for the | |

implementation of MPFR, but may be useful for users too. However no | |

upward compatibility is guaranteed. You may need to include | |

`mpfr-impl.h' to use them. | |

The `mpfr_t' type consists of four fields. | |

* The `_mpfr_prec' field is used to store the precision of the | |

variable (in bits); this is not less than `MPFR_PREC_MIN'. | |

* The `_mpfr_sign' field is used to store the sign of the variable. | |

* The `_mpfr_exp' field stores the exponent. An exponent of 0 means | |

a radix point just above the most significant limb. Non-zero | |

values n are a multiplier 2^n relative to that point. A NaN, an | |

infinity and a zero are indicated by a special value of the | |

exponent. | |

* Finally, the `_mpfr_d' is a pointer to the limbs, least | |

significant limbs stored first. The number of limbs in use is | |

controlled by `_mpfr_prec', namely | |

ceil(`_mpfr_prec'/`mp_bits_per_limb'). Non-singular values always | |

have the most significant bit of the most significant limb set to | |

1. When the precision does not correspond to a whole number of | |

limbs, the excess bits at the low end of the data are zero. | |

File: mpfr.info, Node: Contributors, Next: References, Prev: MPFR Interface, Up: Top | |

Contributors | |

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

The main developers of MPFR are Guillaume Hanrot, Vincent LefÃ¨vre, | |

Patrick PÃ©lissier, Philippe ThÃ©veny and Paul Zimmermann. | |

Sylvie Boldo from ENS-Lyon, France, contributed the functions | |

`mpfr_agm' and `mpfr_log'. Emmanuel Jeandel, from ENS-Lyon too, | |

contributed the generic hypergeometric code, as well as the `mpfr_exp3', | |

a first implementation of the sine and cosine, and improved versions of | |

`mpfr_const_log2' and `mpfr_const_pi'. Mathieu Dutour contributed the | |

functions `mpfr_atan' and `mpfr_asin', and a previous version of | |

`mpfr_gamma'; David Daney contributed the hyperbolic and inverse | |

hyperbolic functions, the base-2 exponential, and the factorial | |

function. Fabrice Rouillier contributed the original version of | |

`mul_ui.c', the `gmp_op.c' file, and helped to the Microsoft Windows | |

porting. Jean-Luc RÃ©my contributed the `mpfr_zeta' code. Ludovic | |

Meunier helped in the design of the `mpfr_erf' code. Damien StehlÃ© | |

contributed the `mpfr_get_ld_2exp' function. | |

We would like to thank Jean-Michel Muller and Joris van der Hoeven | |

for very fruitful discussions at the beginning of that project, | |

TorbjÃ¶rn Granlund and Kevin Ryde for their help about design issues, | |

and Nathalie Revol for her careful reading of a previous version of | |

this documentation. Kevin Ryde did a tremendous job for the | |

portability of MPFR in 2002-2004. | |

The development of the MPFR library would not have been possible | |

without the continuous support of INRIA, and of the LORIA (Nancy, | |

France) and LIP (Lyon, France) laboratories. In particular the main | |

authors were or are members of the PolKA, Spaces, Cacao project-teams | |

at LORIA and of the Arenaire project-team at LIP. This project was | |

started during the Fiable (reliable in French) action supported by | |

INRIA, and continued during the AOC action. The development of MPFR | |

was also supported by a grant (202F0659 00 MPN 121) from the Conseil | |

RÃ©gional de Lorraine in 2002, and from INRIA by an "associate engineer" | |

grant (2003-2005) and an "opÃ©ration de dÃ©veloppement logiciel" grant | |

(2007-2009). | |

File: mpfr.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top | |

References | |

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

* Laurent Fousse, Guillaume Hanrot, Vincent LefÃ¨vre, Patrick | |

PÃ©lissier and Paul Zimmermann, "MPFR: A Multiple-Precision Binary | |

Floating-Point Library With Correct Rounding", ACM Transactions on | |

Mathematical Software, volume 33, issue 2, article 13, 15 pages, | |

2007, `http://doi.acm.org/10.1145/1236463.1236468'. | |

* TorbjÃ¶rn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic | |

Library", version 4.2.2, 2007, `http://gmplib.org'. | |

* IEEE standard for binary floating-point arithmetic, Technical | |

Report ANSI-IEEE Standard 754-1985, New York, 1985. Approved | |

March 21, 1985: IEEE Standards Board; approved July 26, 1985: | |

American National Standards Institute, 18 pages. | |

* Donald E. Knuth, "The Art of Computer Programming", vol 2, | |

"Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981. | |

* Jean-Michel Muller, "Elementary Functions, Algorithms and | |

Implementation", Birkhauser, Boston, 2nd edition, 2006. | |

File: mpfr.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top | |

Appendix A GNU Free Documentation License | |

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

Version 1.2, November 2002 | |

Copyright (C) 2000,2001,2002 Free Software Foundation, Inc. | |

51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA | |

Everyone is permitted to copy and distribute verbatim copies | |

of this license document, but changing it is not allowed. | |

0. PREAMBLE | |

The purpose of this License is to make a manual, textbook, or other | |

functional and useful document "free" in the sense of freedom: to | |

assure everyone the effective freedom to copy and redistribute it, | |

with or without modifying it, either commercially or | |

noncommercially. Secondarily, this License preserves for the | |

author and publisher a way to get credit for their work, while not | |

being considered responsible for modifications made by others. | |

This License is a kind of "copyleft", which means that derivative | |

works of the document must themselves be free in the same sense. | |

It complements the GNU General Public License, which is a copyleft | |

license designed for free software. | |

We have designed this License in order to use it for manuals for | |

free software, because free software needs free documentation: a | |

free program should come with manuals providing the same freedoms | |

that the software does. But this License is not limited to | |

software manuals; it can be used for any textual work, regardless | |

of subject matter or whether it is published as a printed book. | |

We recommend this License principally for works whose purpose is | |

instruction or reference. | |

1. APPLICABILITY AND DEFINITIONS | |

This License applies to any manual or other work, in any medium, | |

that contains a notice placed by the copyright holder saying it | |

can be distributed under the terms of this License. Such a notice | |

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A "Modified Version" of the Document means any work containing the | |

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A "Secondary Section" is a named appendix or a front-matter section | |

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following text that translates XYZ in another language. (Here XYZ | |

stands for a specific section name mentioned below, such as | |

"Acknowledgements", "Dedications", "Endorsements", or "History".) | |

To "Preserve the Title" of such a section when you modify the | |

Document means that it remains a section "Entitled XYZ" according | |

to this definition. | |

The Document may include Warranty Disclaimers next to the notice | |

which states that this License applies to the Document. These | |

Warranty Disclaimers are considered to be included by reference in | |

this License, but only as regards disclaiming warranties: any other | |

implication that these Warranty Disclaimers may have is void and | |

has no effect on the meaning of this License. | |

2. VERBATIM COPYING | |

You may copy and distribute the Document in any medium, either | |

commercially or noncommercially, provided that this License, the | |

copyright notices, and the license notice saying this License | |

applies to the Document are reproduced in all copies, and that you | |

add no other conditions whatsoever to those of this License. You | |

may not use technical measures to obstruct or control the reading | |

or further copying of the copies you make or distribute. However, | |

you may accept compensation in exchange for copies. If you | |

distribute a large enough number of copies you must also follow | |

the conditions in section 3. | |

You may also lend copies, under the same conditions stated above, | |

and you may publicly display copies. | |

3. COPYING IN QUANTITY | |

If you publish printed copies (or copies in media that commonly | |

have printed covers) of the Document, numbering more than 100, and | |

the Document's license notice requires Cover Texts, you must | |

enclose the copies in covers that carry, clearly and legibly, all | |

these Cover Texts: Front-Cover Texts on the front cover, and | |

Back-Cover Texts on the back cover. Both covers must also clearly | |

and legibly identify you as the publisher of these copies. The | |

front cover must present the full title with all words of the | |

title equally prominent and visible. You may add other material | |

on the covers in addition. Copying with changes limited to the | |

covers, as long as they preserve the title of the Document and | |

satisfy these conditions, can be treated as verbatim copying in | |

other respects. | |

If the required texts for either cover are too voluminous to fit | |

legibly, you should put the first ones listed (as many as fit | |

reasonably) on the actual cover, and continue the rest onto | |

adjacent pages. | |

If you publish or distribute Opaque copies of the Document | |

numbering more than 100, you must either include a | |

machine-readable Transparent copy along with each Opaque copy, or | |

state in or with each Opaque copy a computer-network location from | |

which the general network-using public has access to download | |

using public-standard network protocols a complete Transparent | |

copy of the Document, free of added material. If you use the | |

latter option, you must take reasonably prudent steps, when you | |

begin distribution of Opaque copies in quantity, to ensure that | |

this Transparent copy will remain thus accessible at the stated | |

location until at least one year after the last time you | |

distribute an Opaque copy (directly or through your agents or | |

retailers) of that edition to the public. | |

It is requested, but not required, that you contact the authors of | |

the Document well before redistributing any large number of | |

copies, to give them a chance to provide you with an updated | |

version of the Document. | |

4. MODIFICATIONS | |

You may copy and distribute a Modified Version of the Document | |

under the conditions of sections 2 and 3 above, provided that you | |

release the Modified Version under precisely this License, with | |

the Modified Version filling the role of the Document, thus | |

licensing distribution and modification of the Modified Version to | |

whoever possesses a copy of it. In addition, you must do these | |

things in the Modified Version: | |

A. Use in the Title Page (and on the covers, if any) a title | |

distinct from that of the Document, and from those of | |

previous versions (which should, if there were any, be listed | |

in the History section of the Document). You may use the | |

same title as a previous version if the original publisher of | |

that version gives permission. | |

B. List on the Title Page, as authors, one or more persons or | |

entities responsible for authorship of the modifications in | |

the Modified Version, together with at least five of the | |

principal authors of the Document (all of its principal | |

authors, if it has fewer than five), unless they release you | |

from this requirement. | |

C. State on the Title page the name of the publisher of the | |

Modified Version, as the publisher. | |

D. Preserve all the copyright notices of the Document. | |

E. Add an appropriate copyright notice for your modifications | |

adjacent to the other copyright notices. | |

F. Include, immediately after the copyright notices, a license | |

notice giving the public permission to use the Modified | |

Version under the terms of this License, in the form shown in | |

the Addendum below. | |

G. Preserve in that license notice the full lists of Invariant | |

Sections and required Cover Texts given in the Document's | |

license notice. | |

H. Include an unaltered copy of this License. | |

I. Preserve the section Entitled "History", Preserve its Title, | |

and add to it an item stating at least the title, year, new | |

authors, and publisher of the Modified Version as given on | |

the Title Page. If there is no section Entitled "History" in | |

the Document, create one stating the title, year, authors, | |

and publisher of the Document as given on its Title Page, | |

then add an item describing the Modified Version as stated in | |

the previous sentence. | |

J. Preserve the network location, if any, given in the Document | |

for public access to a Transparent copy of the Document, and | |

likewise the network locations given in the Document for | |

previous versions it was based on. These may be placed in | |

the "History" section. You may omit a network location for a | |

work that was published at least four years before the | |

Document itself, or if the original publisher of the version | |

it refers to gives permission. | |

K. For any section Entitled "Acknowledgements" or "Dedications", | |

Preserve the Title of the section, and preserve in the | |

section all the substance and tone of each of the contributor | |

acknowledgements and/or dedications given therein. | |

L. Preserve all the Invariant Sections of the Document, | |

unaltered in their text and in their titles. Section numbers | |

or the equivalent are not considered part of the section | |

titles. | |

M. Delete any section Entitled "Endorsements". Such a section | |

may not be included in the Modified Version. | |

N. Do not retitle any existing section to be Entitled | |

"Endorsements" or to conflict in title with any Invariant | |

Section. | |

O. Preserve any Warranty Disclaimers. | |

If the Modified Version includes new front-matter sections or | |

appendices that qualify as Secondary Sections and contain no | |

material copied from the Document, you may at your option | |

designate some or all of these sections as invariant. To do this, | |

add their titles to the list of Invariant Sections in the Modified | |

Version's license notice. These titles must be distinct from any | |

other section titles. | |

You may add a section Entitled "Endorsements", provided it contains | |

nothing but endorsements of your Modified Version by various | |

parties--for example, statements of peer review or that the text | |

has been approved by an organization as the authoritative | |

definition of a standard. | |

You may add a passage of up to five words as a Front-Cover Text, | |

and a passage of up to 25 words as a Back-Cover Text, to the end | |

of the list of Cover Texts in the Modified Version. Only one | |

passage of Front-Cover Text and one of Back-Cover Text may be | |

added by (or through arrangements made by) any one entity. If the | |

Document already includes a cover text for the same cover, | |

previously added by you or by arrangement made by the same entity | |

you are acting on behalf of, you may not add another; but you may | |

replace the old one, on explicit permission from the previous | |

publisher that added the old one. | |

The author(s) and publisher(s) of the Document do not by this | |

License give permission to use their names for publicity for or to | |

assert or imply endorsement of any Modified Version. | |

5. COMBINING DOCUMENTS | |

You may combine the Document with other documents released under | |

this License, under the terms defined in section 4 above for | |

modified versions, provided that you include in the combination | |

all of the Invariant Sections of all of the original documents, | |

unmodified, and list them all as Invariant Sections of your | |

combined work in its license notice, and that you preserve all | |

their Warranty Disclaimers. | |

The combined work need only contain one copy of this License, and | |

multiple identical Invariant Sections may be replaced with a single | |

copy. If there are multiple Invariant Sections with the same name | |

but different contents, make the title of each such section unique | |

by adding at the end of it, in parentheses, the name of the | |

original author or publisher of that section if known, or else a | |

unique number. Make the same adjustment to the section titles in | |

the list of Invariant Sections in the license notice of the | |

combined work. | |

In the combination, you must combine any sections Entitled | |

"History" in the various original documents, forming one section | |

Entitled "History"; likewise combine any sections Entitled | |

"Acknowledgements", and any sections Entitled "Dedications". You | |

must delete all sections Entitled "Endorsements." | |

6. COLLECTIONS OF DOCUMENTS | |

You may make a collection consisting of the Document and other | |

documents released under this License, and replace the individual | |

copies of this License in the various documents with a single copy | |

that is included in the collection, provided that you follow the | |

rules of this License for verbatim copying of each of the | |

documents in all other respects. | |

You may extract a single document from such a collection, and | |

distribute it individually under this License, provided you insert | |

a copy of this License into the extracted document, and follow | |

this License in all other respects regarding verbatim copying of | |

that document. | |

7. AGGREGATION WITH INDEPENDENT WORKS | |

A compilation of the Document or its derivatives with other | |

separate and independent documents or works, in or on a volume of | |

a storage or distribution medium, is called an "aggregate" if the | |

copyright resulting from the compilation is not used to limit the | |

legal rights of the compilation's users beyond what the individual | |

works permit. When the Document is included in an aggregate, this | |

License does not apply to the other works in the aggregate which | |

are not themselves derivative works of the Document. | |

If the Cover Text requirement of section 3 is applicable to these | |

copies of the Document, then if the Document is less than one half | |

of the entire aggregate, the Document's Cover Texts may be placed | |

on covers that bracket the Document within the aggregate, or the | |

electronic equivalent of covers if the Document is in electronic | |

form. Otherwise they must appear on printed covers that bracket | |

the whole aggregate. | |

8. TRANSLATION | |

Translation is considered a kind of modification, so you may | |

distribute translations of the Document under the terms of section | |

4. Replacing Invariant Sections with translations requires special | |

permission from their copyright holders, but you may include | |

translations of some or all Invariant Sections in addition to the | |

original versions of these Invariant Sections. You may include a | |

translation of this License, and all the license notices in the | |

Document, and any Warranty Disclaimers, provided that you also | |

include the original English version of this License and the | |

original versions of those notices and disclaimers. In case of a | |

disagreement between the translation and the original version of | |

this License or a notice or disclaimer, the original version will | |

prevail. | |

If a section in the Document is Entitled "Acknowledgements", | |

"Dedications", or "History", the requirement (section 4) to | |

Preserve its Title (section 1) will typically require changing the | |

actual title. | |

9. TERMINATION | |

You may not copy, modify, sublicense, or distribute the Document | |

except as expressly provided for under this License. Any other | |

attempt to copy, modify, sublicense or distribute the Document is | |

void, and will automatically terminate your rights under this | |

License. However, parties who have received copies, or rights, | |

from you under this License will not have their licenses | |

terminated so long as such parties remain in full compliance. | |

10. FUTURE REVISIONS OF THIS LICENSE | |

The Free Software Foundation may publish new, revised versions of | |

the GNU Free Documentation License from time to time. Such new | |

versions will be similar in spirit to the present version, but may | |

differ in detail to address new problems or concerns. See | |

`http://www.gnu.org/copyleft/'. | |

Each version of the License is given a distinguishing version | |

number. If the Document specifies that a particular numbered | |

version of this License "or any later version" applies to it, you | |

have the option of following the terms and conditions either of | |

that specified version or of any later version that has been | |

published (not as a draft) by the Free Software Foundation. If | |

the Document does not specify a version number of this License, | |

you may choose any version ever published (not as a draft) by the | |

Free Software Foundation. | |

A.1 ADDENDUM: How to use this License for your documents | |

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

To use this License in a document you have written, include a copy of | |

the License in the document and put the following copyright and license | |

notices just after the title page: | |

Copyright (C) YEAR YOUR NAME. | |

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

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

or any later version published by the Free Software Foundation; | |

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

Texts. A copy of the license is included in the section entitled ``GNU | |

Free Documentation License''. | |

If you have Invariant Sections, Front-Cover Texts and Back-Cover | |

Texts, replace the "with...Texts." line with this: | |

with the Invariant Sections being LIST THEIR TITLES, with | |

the Front-Cover Texts being LIST, and with the Back-Cover Texts | |

being LIST. | |

If you have Invariant Sections without Cover Texts, or some other | |

combination of the three, merge those two alternatives to suit the | |

situation. | |

If your document contains nontrivial examples of program code, we | |

recommend releasing these examples in parallel under your choice of | |

free software license, such as the GNU General Public License, to | |

permit their use in free software. | |

File: mpfr.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top | |

Concept Index | |

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

�[index�] | |

* Menu: | |

* Accuracy: MPFR Interface. (line 28) | |

* Arithmetic functions: Basic Arithmetic Functions. | |

(line 3) | |

* Assignment functions: Assignment Functions. (line 3) | |

* Basic arithmetic functions: Basic Arithmetic Functions. | |

(line 3) | |

* Combined initialization and assignment functions: Combined Initialization and Assignment Functions. | |

(line 3) | |

* Comparison functions: Comparison Functions. (line 3) | |

* Compatibility with MPF: Compatibility with MPF. | |

(line 3) | |

* Conditions for copying MPFR: Copying. (line 6) | |

* Conversion functions: Conversion Functions. (line 3) | |

* Copying conditions: Copying. (line 6) | |

* Custom interface: Custom Interface. (line 3) | |

* Exception related functions: Exception Related Functions. | |

(line 3) | |

* FDL, GNU Free Documentation License: GNU Free Documentation License. | |

(line 6) | |

* Float arithmetic functions: Basic Arithmetic Functions. | |

(line 3) | |

* Float comparisons functions: Comparison Functions. (line 3) | |

* Float functions: MPFR Interface. (line 6) | |

* Float input and output functions: Input and Output Functions. | |

(line 3) | |

* Float output functions: Formatted Output Functions. | |

(line 3) | |

* Floating-point functions: MPFR Interface. (line 6) | |

* Floating-point number: MPFR Basics. (line 52) | |

* GNU Free Documentation License: GNU Free Documentation License. | |

(line 6) | |

* I/O functions <1>: Formatted Output Functions. | |

(line 3) | |

* I/O functions: Input and Output Functions. | |

(line 3) | |

* Initialization functions: Initialization Functions. | |

(line 3) | |

* Input functions: Input and Output Functions. | |

(line 3) | |

* Installation: Installing MPFR. (line 6) | |

* Integer related functions: Integer Related Functions. | |

(line 3) | |

* Internals: Internals. (line 3) | |

* libmpfr: MPFR Basics. (line 32) | |

* Libraries: MPFR Basics. (line 32) | |

* Libtool: MPFR Basics. (line 38) | |

* Limb: MPFR Basics. (line 84) | |

* Linking: MPFR Basics. (line 32) | |

* Miscellaneous float functions: Miscellaneous Functions. | |

(line 3) | |

* mpfr.h: MPFR Basics. (line 9) | |

* Output functions <1>: Formatted Output Functions. | |

(line 3) | |

* Output functions: Input and Output Functions. | |

(line 3) | |

* Precision <1>: MPFR Interface. (line 20) | |

* Precision: MPFR Basics. (line 65) | |

* Reporting bugs: Reporting Bugs. (line 6) | |

* Rounding mode related functions: Rounding Related Functions. | |

(line 3) | |

* Rounding Modes: MPFR Basics. (line 79) | |

* Special functions: Special Functions. (line 3) | |

* stdarg.h: MPFR Basics. (line 22) | |

* stdio.h: MPFR Basics. (line 15) | |

File: mpfr.info, Node: Function Index, Prev: Concept Index, Up: Top | |

Function and Type Index | |

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

�[index�] | |

* Menu: | |

* mp_prec_t: MPFR Basics. (line 65) | |

* mp_rnd_t: MPFR Basics. (line 79) | |

* mpfr_abs: Basic Arithmetic Functions. | |

(line 177) | |

* mpfr_acos: Special Functions. (line 48) | |

* mpfr_acosh: Special Functions. (line 131) | |

* mpfr_add: Basic Arithmetic Functions. | |

(line 8) | |

* mpfr_add_d: Basic Arithmetic Functions. | |

(line 14) | |

* mpfr_add_q: Basic Arithmetic Functions. | |

(line 18) | |

* mpfr_add_si: Basic Arithmetic Functions. | |

(line 12) | |

* mpfr_add_ui: Basic Arithmetic Functions. | |

(line 10) | |

* mpfr_add_z: Basic Arithmetic Functions. | |

(line 16) | |

* mpfr_agm: Special Functions. (line 221) | |

* mpfr_asin: Special Functions. (line 49) | |

* mpfr_asinh: Special Functions. (line 132) | |

* mpfr_asprintf: Formatted Output Functions. | |

(line 171) | |

* mpfr_atan: Special Functions. (line 50) | |

* mpfr_atan2: Special Functions. (line 60) | |

* mpfr_atanh: Special Functions. (line 133) | |

* mpfr_can_round: Rounding Related Functions. | |

(line 29) | |

* mpfr_cbrt: Basic Arithmetic Functions. | |

(line 107) | |

* mpfr_ceil: Integer Related Functions. | |

(line 8) | |

* mpfr_check_range: Exception Related Functions. | |

(line 38) | |

* mpfr_clear: Initialization Functions. | |

(line 31) | |

* mpfr_clear_erangeflag: Exception Related Functions. | |

(line 111) | |

* mpfr_clear_flags: Exception Related Functions. | |

(line 121) | |

* mpfr_clear_inexflag: Exception Related Functions. | |

(line 110) | |

* mpfr_clear_nanflag: Exception Related Functions. | |

(line 109) | |

* mpfr_clear_overflow: Exception Related Functions. | |

(line 108) | |

* mpfr_clear_underflow: Exception Related Functions. | |

(line 107) | |

* mpfr_clears: Initialization Functions. | |

(line 35) | |

* mpfr_cmp: Comparison Functions. | |

(line 7) | |

* mpfr_cmp_d: Comparison Functions. | |

(line 10) | |

* mpfr_cmp_f: Comparison Functions. | |

(line 14) | |

* mpfr_cmp_ld: Comparison Functions. | |

(line 11) | |

* mpfr_cmp_q: Comparison Functions. | |

(line 13) | |

* mpfr_cmp_si: Comparison Functions. | |

(line 9) | |

* mpfr_cmp_si_2exp: Comparison Functions. | |

(line 31) | |

* mpfr_cmp_ui: Comparison Functions. | |

(line 8) | |

* mpfr_cmp_ui_2exp: Comparison Functions. | |

(line 29) | |

* mpfr_cmp_z: Comparison Functions. | |

(line 12) | |

* mpfr_cmpabs: Comparison Functions. | |

(line 35) | |

* mpfr_const_catalan: Special Functions. (line 242) | |

* mpfr_const_euler: Special Functions. (line 241) | |

* mpfr_const_log2: Special Functions. (line 239) | |

* mpfr_const_pi: Special Functions. (line 240) | |

* mpfr_copysign: Miscellaneous Functions. | |

(line 93) | |

* mpfr_cos: Special Functions. (line 29) | |

* mpfr_cosh: Special Functions. (line 111) | |

* mpfr_cot: Special Functions. (line 37) | |

* mpfr_coth: Special Functions. (line 127) | |

* mpfr_csc: Special Functions. (line 36) | |

* mpfr_csch: Special Functions. (line 126) | |

* mpfr_custom_get_exp: Custom Interface. (line 78) | |

* mpfr_custom_get_kind: Custom Interface. (line 67) | |

* mpfr_custom_get_mantissa: Custom Interface. (line 72) | |

* mpfr_custom_get_size: Custom Interface. (line 38) | |

* mpfr_custom_init: Custom Interface. (line 42) | |

* mpfr_custom_init_set: Custom Interface. (line 48) | |

* mpfr_custom_move: Custom Interface. (line 85) | |

* mpfr_d_div: Basic Arithmetic Functions. | |

(line 82) | |

* mpfr_d_sub: Basic Arithmetic Functions. | |

(line 37) | |

* MPFR_DECL_INIT: Initialization Functions. | |

(line 75) | |

* mpfr_dim: Basic Arithmetic Functions. | |

(line 182) | |

* mpfr_div: Basic Arithmetic Functions. | |

(line 72) | |

* mpfr_div_2exp: Compatibility with MPF. | |

(line 49) | |

* mpfr_div_2si: Basic Arithmetic Functions. | |

(line 197) | |

* mpfr_div_2ui: Basic Arithmetic Functions. | |

(line 195) | |

* mpfr_div_d: Basic Arithmetic Functions. | |

(line 84) | |

* mpfr_div_q: Basic Arithmetic Functions. | |

(line 88) | |

* mpfr_div_si: Basic Arithmetic Functions. | |

(line 80) | |

* mpfr_div_ui: Basic Arithmetic Functions. | |

(line 76) | |

* mpfr_div_z: Basic Arithmetic Functions. | |

(line 86) | |

* mpfr_eint: Special Functions. (line 150) | |

* mpfr_eq: Compatibility with MPF. | |

(line 28) | |

* mpfr_equal_p: Comparison Functions. | |

(line 71) | |

* mpfr_erangeflag_p: Exception Related Functions. | |

(line 129) | |

* mpfr_erf: Special Functions. (line 186) | |

* mpfr_erfc: Special Functions. (line 190) | |

* mpfr_exp: Special Functions. (line 23) | |

* mpfr_exp10: Special Functions. (line 25) | |

* mpfr_exp2: Special Functions. (line 24) | |

* mpfr_expm1: Special Functions. (line 146) | |

* mpfr_fac_ui: Special Functions. (line 138) | |

* mpfr_fits_intmax_p: Conversion Functions. | |

(line 113) | |

* mpfr_fits_sint_p: Conversion Functions. | |

(line 110) | |

* mpfr_fits_slong_p: Conversion Functions. | |

(line 108) | |

* mpfr_fits_sshort_p: Conversion Functions. | |

(line 112) | |

* mpfr_fits_uint_p: Conversion Functions. | |

(line 109) | |

* mpfr_fits_uintmax_p: Conversion Functions. | |

(line 114) | |

* mpfr_fits_ulong_p: Conversion Functions. | |

(line 107) | |

* mpfr_fits_ushort_p: Conversion Functions. | |

(line 111) | |

* mpfr_floor: Integer Related Functions. | |

(line 9) | |

* mpfr_fma: Special Functions. (line 213) | |

* mpfr_fmod: Integer Related Functions. | |

(line 63) | |

* mpfr_fms: Special Functions. (line 217) | |

* mpfr_fprintf: Formatted Output Functions. | |

(line 117) | |

* mpfr_frac: Integer Related Functions. | |

(line 48) | |

* mpfr_free_cache: Special Functions. (line 249) | |

* mpfr_free_str: Conversion Functions. | |

(line 100) | |

* mpfr_gamma: Special Functions. (line 162) | |

* mpfr_get_d: Conversion Functions. | |

(line 7) | |

* mpfr_get_d_2exp: Conversion Functions. | |

(line 20) | |

* mpfr_get_decimal64: Conversion Functions. | |

(line 9) | |

* mpfr_get_default_prec: Initialization Functions. | |

(line 109) | |

* mpfr_get_default_rounding_mode: Rounding Related Functions. | |

(line 11) | |

* mpfr_get_emax: Exception Related Functions. | |

(line 8) | |

* mpfr_get_emax_max: Exception Related Functions. | |

(line 30) | |

* mpfr_get_emax_min: Exception Related Functions. | |

(line 29) | |

* mpfr_get_emin: Exception Related Functions. | |

(line 7) | |

* mpfr_get_emin_max: Exception Related Functions. | |

(line 28) | |

* mpfr_get_emin_min: Exception Related Functions. | |

(line 27) | |

* mpfr_get_exp: Miscellaneous Functions. | |

(line 71) | |

* mpfr_get_f: Conversion Functions. | |

(line 55) | |

* mpfr_get_ld: Conversion Functions. | |

(line 8) | |

* mpfr_get_ld_2exp: Conversion Functions. | |

(line 22) | |

* mpfr_get_patches: Miscellaneous Functions. | |

(line 130) | |

* mpfr_get_prec: Initialization Functions. | |

(line 143) | |

* mpfr_get_si: Conversion Functions. | |

(line 31) | |

* mpfr_get_sj: Conversion Functions. | |

(line 33) | |

* mpfr_get_str: Conversion Functions. | |

(line 61) | |

* mpfr_get_ui: Conversion Functions. | |

(line 32) | |

* mpfr_get_uj: Conversion Functions. | |

(line 34) | |

* mpfr_get_version: Miscellaneous Functions. | |

(line 99) | |

* mpfr_get_z: Conversion Functions. | |

(line 51) | |

* mpfr_get_z_exp: Conversion Functions. | |

(line 44) | |

* mpfr_greater_p: Comparison Functions. | |

(line 54) | |

* mpfr_greaterequal_p: Comparison Functions. | |

(line 57) | |

* mpfr_hypot: Special Functions. (line 230) | |

* mpfr_inexflag_p: Exception Related Functions. | |

(line 128) | |

* mpfr_inf_p: Comparison Functions. | |

(line 42) | |

* mpfr_init: Initialization Functions. | |

(line 51) | |

* mpfr_init2: Initialization Functions. | |

(line 11) | |

* mpfr_init_set: Combined Initialization and Assignment Functions. | |

(line 7) | |

* mpfr_init_set_d: Combined Initialization and Assignment Functions. | |

(line 12) | |

* mpfr_init_set_f: Combined Initialization and Assignment Functions. | |

(line 17) | |

* mpfr_init_set_ld: Combined Initialization and Assignment Functions. | |

(line 14) | |

* mpfr_init_set_q: Combined Initialization and Assignment Functions. | |

(line 16) | |

* mpfr_init_set_si: Combined Initialization and Assignment Functions. | |

(line 11) | |

* mpfr_init_set_str: Combined Initialization and Assignment Functions. | |

(line 23) | |

* mpfr_init_set_ui: Combined Initialization and Assignment Functions. | |

(line 9) | |

* mpfr_init_set_z: Combined Initialization and Assignment Functions. | |

(line 15) | |

* mpfr_inits: Initialization Functions. | |

(line 63) | |

* mpfr_inits2: Initialization Functions. | |

(line 23) | |

* mpfr_inp_str: Input and Output Functions. | |

(line 33) | |

* mpfr_integer_p: Integer Related Functions. | |

(line 89) | |

* mpfr_j0: Special Functions. (line 194) | |

* mpfr_j1: Special Functions. (line 195) | |

* mpfr_jn: Special Functions. (line 196) | |

* mpfr_less_p: Comparison Functions. | |

(line 60) | |

* mpfr_lessequal_p: Comparison Functions. | |

(line 63) | |

* mpfr_lessgreater_p: Comparison Functions. | |

(line 66) | |

* mpfr_lgamma: Special Functions. (line 172) | |

* mpfr_li2: Special Functions. (line 157) | |

* mpfr_lngamma: Special Functions. (line 166) | |

* mpfr_log: Special Functions. (line 16) | |

* mpfr_log10: Special Functions. (line 18) | |

* mpfr_log1p: Special Functions. (line 142) | |

* mpfr_log2: Special Functions. (line 17) | |

* mpfr_max: Miscellaneous Functions. | |

(line 29) | |

* mpfr_min: Miscellaneous Functions. | |

(line 22) | |

* mpfr_modf: Integer Related Functions. | |

(line 55) | |

* mpfr_mul: Basic Arithmetic Functions. | |

(line 51) | |

* mpfr_mul_2exp: Compatibility with MPF. | |

(line 47) | |

* mpfr_mul_2si: Basic Arithmetic Functions. | |

(line 190) | |

* mpfr_mul_2ui: Basic Arithmetic Functions. | |

(line 188) | |

* mpfr_mul_d: Basic Arithmetic Functions. | |

(line 57) | |

* mpfr_mul_q: Basic Arithmetic Functions. | |

(line 61) | |

* mpfr_mul_si: Basic Arithmetic Functions. | |

(line 55) | |

* mpfr_mul_ui: Basic Arithmetic Functions. | |

(line 53) | |

* mpfr_mul_z: Basic Arithmetic Functions. | |

(line 59) | |

* mpfr_nan_p: Comparison Functions. | |

(line 41) | |

* mpfr_nanflag_p: Exception Related Functions. | |

(line 127) | |

* mpfr_neg: Basic Arithmetic Functions. | |

(line 173) | |

* mpfr_nextabove: Miscellaneous Functions. | |

(line 15) | |

* mpfr_nextbelow: Miscellaneous Functions. | |

(line 18) | |

* mpfr_nexttoward: Miscellaneous Functions. | |

(line 7) | |

* mpfr_number_p: Comparison Functions. | |

(line 43) | |

* mpfr_out_str: Input and Output Functions. | |

(line 17) | |

* mpfr_overflow_p: Exception Related Functions. | |

(line 126) | |

* mpfr_pow: Basic Arithmetic Functions. | |

(line 116) | |

* mpfr_pow_si: Basic Arithmetic Functions. | |

(line 120) | |

* mpfr_pow_ui: Basic Arithmetic Functions. | |

(line 118) | |

* mpfr_pow_z: Basic Arithmetic Functions. | |

(line 122) | |

* mpfr_prec_round: Rounding Related Functions. | |

(line 15) | |

* mpfr_print_rnd_mode: Rounding Related Functions. | |

(line 46) | |

* mpfr_printf: Formatted Output Functions. | |

(line 130) | |

* mpfr_random: Miscellaneous Functions. | |

(line 48) | |

* mpfr_random2: Miscellaneous Functions. | |

(line 56) | |

* mpfr_rec_sqrt: Basic Arithmetic Functions. | |

(line 102) | |

* mpfr_reldiff: Compatibility with MPF. | |

(line 39) | |

* mpfr_remainder: Integer Related Functions. | |

(line 65) | |

* mpfr_remquo: Integer Related Functions. | |

(line 67) | |

* mpfr_rint: Integer Related Functions. | |

(line 7) | |

* mpfr_rint_ceil: Integer Related Functions. | |

(line 34) | |

* mpfr_rint_floor: Integer Related Functions. | |

(line 35) | |

* mpfr_rint_round: Integer Related Functions. | |

(line 36) | |

* mpfr_rint_trunc: Integer Related Functions. | |

(line 37) | |

* mpfr_root: Basic Arithmetic Functions. | |

(line 109) | |

* mpfr_round: Integer Related Functions. | |

(line 10) | |

* mpfr_round_prec: Rounding Related Functions. | |

(line 25) | |

* mpfr_sec: Special Functions. (line 35) | |

* mpfr_sech: Special Functions. (line 125) | |

* mpfr_set: Assignment Functions. | |

(line 12) | |

* mpfr_set_d: Assignment Functions. | |

(line 18) | |

* mpfr_set_decimal64: Assignment Functions. | |

(line 21) | |

* mpfr_set_default_prec: Initialization Functions. | |

(line 101) | |

* mpfr_set_default_rounding_mode: Rounding Related Functions. | |

(line 7) | |

* mpfr_set_emax: Exception Related Functions. | |

(line 16) | |

* mpfr_set_emin: Exception Related Functions. | |

(line 15) | |

* mpfr_set_erangeflag: Exception Related Functions. | |

(line 118) | |

* mpfr_set_exp: Miscellaneous Functions. | |

(line 76) | |

* mpfr_set_f: Assignment Functions. | |

(line 24) | |

* mpfr_set_inexflag: Exception Related Functions. | |

(line 117) | |

* mpfr_set_inf: Assignment Functions. | |

(line 131) | |

* mpfr_set_ld: Assignment Functions. | |

(line 19) | |

* mpfr_set_nan: Assignment Functions. | |

(line 132) | |

* mpfr_set_nanflag: Exception Related Functions. | |

(line 116) | |

* mpfr_set_overflow: Exception Related Functions. | |

(line 115) | |

* mpfr_set_prec: Initialization Functions. | |

(line 131) | |

* mpfr_set_prec_raw: Compatibility with MPF. | |

(line 21) | |

* mpfr_set_q: Assignment Functions. | |

(line 23) | |

* mpfr_set_si: Assignment Functions. | |

(line 15) | |

* mpfr_set_si_2exp: Assignment Functions. | |

(line 51) | |

* mpfr_set_sj: Assignment Functions. | |

(line 17) | |

* mpfr_set_sj_2exp: Assignment Functions. | |

(line 55) | |

* mpfr_set_str: Assignment Functions. | |

(line 61) | |

* mpfr_set_ui: Assignment Functions. | |

(line 14) | |

* mpfr_set_ui_2exp: Assignment Functions. | |

(line 49) | |

* mpfr_set_uj: Assignment Functions. | |

(line 16) | |

* mpfr_set_uj_2exp: Assignment Functions. | |

(line 53) | |

* mpfr_set_underflow: Exception Related Functions. | |

(line 114) | |

* mpfr_set_z: Assignment Functions. | |

(line 22) | |

* mpfr_setsign: Miscellaneous Functions. | |

(line 87) | |

* mpfr_sgn: Comparison Functions. | |

(line 49) | |

* mpfr_si_div: Basic Arithmetic Functions. | |

(line 78) | |

* mpfr_si_sub: Basic Arithmetic Functions. | |

(line 33) | |

* mpfr_signbit: Miscellaneous Functions. | |

(line 82) | |

* mpfr_sin: Special Functions. (line 30) | |

* mpfr_sin_cos: Special Functions. (line 42) | |

* mpfr_sinh: Special Functions. (line 112) | |

* mpfr_sinh_cosh: Special Functions. (line 118) | |

* mpfr_snprintf: Formatted Output Functions. | |

(line 155) | |

* mpfr_sprintf: Formatted Output Functions. | |

(line 141) | |

* mpfr_sqr: Basic Arithmetic Functions. | |

(line 68) | |

* mpfr_sqrt: Basic Arithmetic Functions. | |

(line 95) | |

* mpfr_sqrt_ui: Basic Arithmetic Functions. | |

(line 97) | |

* mpfr_strtofr: Assignment Functions. | |

(line 72) | |

* mpfr_sub: Basic Arithmetic Functions. | |

(line 27) | |

* mpfr_sub_d: Basic Arithmetic Functions. | |

(line 39) | |

* mpfr_sub_q: Basic Arithmetic Functions. | |

(line 43) | |

* mpfr_sub_si: Basic Arithmetic Functions. | |

(line 35) | |

* mpfr_sub_ui: Basic Arithmetic Functions. | |

(line 31) | |

* mpfr_sub_z: Basic Arithmetic Functions. | |

(line 41) | |

* mpfr_subnormalize: Exception Related Functions. | |

(line 58) | |

* mpfr_sum: Special Functions. (line 258) | |

* mpfr_swap: Assignment Functions. | |

(line 137) | |

* mpfr_t: MPFR Basics. (line 52) | |

* mpfr_tan: Special Functions. (line 31) | |

* mpfr_tanh: Special Functions. (line 113) | |

* mpfr_trunc: Integer Related Functions. | |

(line 11) | |

* mpfr_ui_div: Basic Arithmetic Functions. | |

(line 74) | |

* mpfr_ui_pow: Basic Arithmetic Functions. | |

(line 126) | |

* mpfr_ui_pow_ui: Basic Arithmetic Functions. | |

(line 124) | |

* mpfr_ui_sub: Basic Arithmetic Functions. | |

(line 29) | |

* mpfr_underflow_p: Exception Related Functions. | |

(line 125) | |

* mpfr_unordered_p: Comparison Functions. | |

(line 75) | |

* mpfr_urandomb: Miscellaneous Functions. | |

(line 35) | |

* mpfr_vasprintf: Formatted Output Functions. | |

(line 173) | |

* MPFR_VERSION: Miscellaneous Functions. | |