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MPFR

This manual documents how to install and use the Multiple Precision Floating-Point Reliable Library, version 2.3.2.

Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008 Free Software Foundation, Inc.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the license is included in GNU Free Documentation License.



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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 MPFR library, you must give the recipients all the rights that you have. You must make sure that they, too, receive or can get the source code. And you must tell them their rights.

Also, for our own protection, we must make certain that everyone finds out that there is no warranty for the MPFR library. If it is modified by someone else and passed on, we want their recipients to know that what they have is not what we distributed, so that any problems introduced by others will not reflect on our reputation.

The precise conditions of the license for the MPFR library are found in the Lesser General Public License that accompanies the source code. See the file COPYING.LIB.


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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:

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

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


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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’. (See 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:

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’. See Reporting Bugs. Some bug fixes are available on the MPFR 2.3.2 web page http://www.mpfr.org/mpfr-2.3.2/.

2.4 Getting the Latest Version of MPFR

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


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3 Reporting Bugs

If you think you have found a bug in the MPFR library, first have a look on the MPFR 2.3.2 web page http://www.mpfr.org/mpfr-2.3.2/ 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 won't do anything about it (aside of chiding you to send better bug reports).

Send your bug report to: ‘mpfr@loria.fr’.

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


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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>

You can avoid the use of MPFR macros encapsulating functions by defining the ‘MPFR_USE_NO_MACRO’ macro before mpfr.h is included. In general this should not be necessary, but this can be useful when debugging user code: with some macros, the compiler may emit spurious warnings with some warning options, and macros can prevent some prototype checking.

All programs using MPFR must link against both libmpfr and libgmp libraries. On a typical Unix-like system this can be done with ‘-lmpfr -lgmp’ (in that order), for example

     gcc myprogram.c -lmpfr -lgmp

MPFR is built using Libtool and an application can use that to link if desired, see GNU Libtool.

If MPFR has been installed to a non-standard location, then it may be necessary to set up environment variables such as ‘C_INCLUDE_PATH’ and ‘LIBRARY_PATH’, or use ‘-I’ and ‘-L’ compiler options, in order to point to the right directories. For a shared library, it may also be necessary to set up some sort of run-time library path (e.g., ‘LD_LIBRARY_PATH’) on some systems. Please read the INSTALL file for additional information.

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 don't 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:

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 (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 MPFR Interface. Example: mpfr_hypot on (NaN,0) gives NaN, but mpfr_hypot on (NaN,+Inf) gives +Inf (as specified in Special Functions), since for any finite input x, mpfr_hypot on (x,+Inf) gives +Inf.

4.7 Exceptions

MPFR supports 5 exception types:

MPFR has a global flag for each exception, which can be cleared, set or tested by functions described in Exception Related Functions.

Differences with the ISO C99 standard:

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).


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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.


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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_clear (mpfr_t x)

Free the space occupied by x. Make sure to call this function for all mpfr_t variables when you are done with them.

— Function: void mpfr_init (mpfr_t x)

Initialize 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.

— 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.


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5.2 Assignment Functions

These functions assign new values to already initialized floats (see Initialization Functions). When using any functions using intmax_t, you must include <stdint.h> or <inttypes.h> before mpfr.h, to allow mpfr.h to define prototypes for these functions.

— Function: int mpfr_set (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)
— Function: int mpfr_set_ui (mpfr_t rop, unsigned long int op, mp_rnd_t rnd)
— Function: int mpfr_set_si (mpfr_t rop, long int op, mp_rnd_t rnd)
— Function: int mpfr_set_uj (mpfr_t rop, uintmax_t op, mp_rnd_t rnd)
— Function: int mpfr_set_sj (mpfr_t rop, intmax_t op, mp_rnd_t rnd)
— Function: int mpfr_set_d (mpfr_t rop, double op, mp_rnd_t rnd)
— Function: int mpfr_set_ld (mpfr_t rop, long double op, mp_rnd_t rnd)
— Function: int mpfr_set_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 doesn't 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 whole 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. 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 = 36; 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.


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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.


Next: , Previous: 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 ±Inf, an infinity of the same sign or the result of ±1.0/0.0 is returned. If op is zero, these functions return a zero, trying to preserve its sign, if possible. The mpfr_get_decimal64 function is built only under some conditions: see the documentation of mpfr_set_decimal64.

— Function: 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 flag erange is set too. See also mpfr_fits_slong_p, mpfr_fits_ulong_p, mpfr_fits_intmax_p and mpfr_fits_uintmax_p.

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

Put the scaled 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: see section “Custom Allocation” in GNU MP.

— Function: int mpfr_fits_ulong_p (mpfr_t op, mp_rnd_t rnd)
— Function: int mpfr_fits_slong_p (mpfr_t op, mp_rnd_t rnd)
— Function: int mpfr_fits_uint_p (mpfr_t op, mp_rnd_t rnd)
— Function: int mpfr_fits_sint_p (mpfr_t op, mp_rnd_t rnd)
— Function: int mpfr_fits_ushort_p (mpfr_t op, mp_rnd_t rnd)
— Function: int mpfr_fits_sshort_p (mpfr_t op, mp_rnd_t rnd)
— Function: int mpfr_fits_intmax_p (mpfr_t op, mp_rnd_t rnd)
— Function: int mpfr_fits_uintmax_p (mpfr_t op, mp_rnd_t rnd)

Return non-zero if op would fit in the respective C data type, when rounded to an integer in the direction rnd.


Next: , Previous: 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_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)).

— Function: int mpfr_sub (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)
— Function: int mpfr_ui_sub (mpfr_t rop, unsigned long int op1, mpfr_t op2, mp_rnd_t rnd)
— Function: int mpfr_sub_ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mp_rnd_t rnd)
— Function: int mpfr_si_sub (mpfr_t rop, long int op1, mpfr_t op2, mp_rnd_t rnd)
— Function: int mpfr_sub_si (mpfr_t rop, mpfr_t op1, long int op2, mp_rnd_t rnd)
— Function: int mpfr_sub_z (mpfr_t rop, mpfr_t op1, mpz_t op2, mp_rnd_t rnd)
— Function: int mpfr_sub_q (mpfr_t rop, mpfr_t op1, mpq_t op2, mp_rnd_t rnd)

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)).

— Function: int mpfr_mul (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)
— Function: int mpfr_mul_ui (mpfr_t rop, mpfr_t op1, unsigned long int op2, mp_rnd_t rnd)
— Function: int mpfr_mul_si (mpfr_t rop, mpfr_t op1, long int op2, mp_rnd_t rnd)
— Function: int mpfr_mul_z (mpfr_t rop, mpfr_t op1, mpz_t op2, mp_rnd_t rnd)
— Function: int mpfr_mul_q (mpfr_t rop, mpfr_t op1, mpq_t op2, mp_rnd_t rnd)

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).

— 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_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).

— 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_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):

— 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.


Next: , Previous: 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.


Next: , Previous: 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):

— 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_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_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 ±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 order 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 order 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 e valu and non-zero ihenishey cannot g sqnstefiner> woutthe ordinn the directiue of the erras if the othse functionzero.


Next: Special Functions, Up: MPFR Interface

Input and Output Functions

Thr> isirectias descrshse functiote thy ofn noi"#Inr k fris an#In/on out st thm[n], ase functiote thon outpes ts an#In/on out st thmeroP cl being r fuof poinlue for, *x awrfinito ed , respectivise. 0. Whul beinsy4.3all these functi, if ym. Just lutside ted &stdio.h y &og2 — Functiose s_ int mpon _at sum ()
nd Outpen opxop, rounded in the direction rnd. Tbassris mvmina k fr2eed 36. Prn: ins nnopmpge _at sun to the sifiosecahegi, lue,d>Spmafuof po ( undefised by s (currneden e) a set trt migue of the firhegi,lt, andtrailendihe exponeed bassr10,ded in tfn no&lsock;eNNN...pts01"sam &rsock;valefopr poin.s): If op@...pts01"sam &rsock;owi fuay be d i cothrrue &lsock;e...pts01"sam &rsock;oasihe exponed(prelreaege. . Retuin ter numbue bytesrwrfitenNaN, ns inthe errocs (cuated, retu0ly).

— Functiose s_ int mpan#_at sum (mpfr_t roFILE *st thm[nn: ibassinp, mp_rnd_t rnd)
s">Inpuist endied bassrIf opx, rounded in the direction rndrop . . . . Spmafud, for tng spfiplieassinsended bassr16in@sam {B}lue is tup --> . Spmafu;0 (i@:ly, inpredics to the powhm osed up --> . Tl this functi thrr aiwond ( undefisar aise sequewhm chsqncoint sttwe bon, icsperfave, anar casy, ul beise mpse _at sunmpstrtof sun does nnatoin):rup -. i
mpan#_raw (ar>mpfr@pmpon _raw}ive, anu set cal resulnr@p


Next: Input and Output Functions, Up: MPFR Interface

— Function: int mprr_eint (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)
— Function: int mpfeilint (mpfr_t rop, mpfr_trnd)
— Function: int mp x-ouler (mpfr_t rop, mpfr_trnd)
— Function: int mp>, roler (mpfr_t rop, mpfr_trnd)
— Function: int mptr; Fler (mpfr_t rop, mpfr_trnd)

Set rop en op, roundes ts an-intely).e mprde>int, rosar> to thesquaxact, sguma taban-inteded in the given rounding modnd mpfeilint, ros r> to thext ht mlower or equct, sguma taban-inteodnd mp x-ouletmp>, rolet and mptr; Flet value is zero when tal result he ex,For positiro why, it essgrea er thif thriglimlthe value on op, the returned value 0and when op ts an-inte ct, sguma taban in ropop ts an-inte te thr> is nct, sguma taban in ropopdid nts an-intelro.

Note thed mp>, roletmprde>int, roundes ts d, e an-intedn foe sifiosen)ne

Noee alte thnoong doubhe roundir> i ofn ned;tfn i coaque, 4.5 (100.1ded bliminar>) i, roundto bd mp>, roletn 2-by, ll precisi wough bd , ro(4.5)let a5 (101ded bliminar>) i, roundwo 6 (11s ed bliminar>n 2-by, ll precisly).

— Function: int mprde>pfeilint (mpfr_t rop, mpfr_t op, mp_rnd_t rnd) — Function: int mprde>p x-ouler (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)
— Function: int mprr_ep>, roler (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)
— Function: int mprr_eptr; Fler (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)

Set rop en op, roundes ts an-intely).e mprde>pfeilint, ros r> to thext ht mlower or equve intege.e mprde>p x-ouletmprde>p>, rolet and mprr_eptr; Fletrnd. The returned value in t interyned val cloci stapr> with the consider>, ro-to-an-intede> functiora nrounded in the saws mar and he ot ct mathematice> functlly).

— Function: int mpfraFler (mpfr_t rop, mpfr_t op, mp_rnd_t rnd)

Set rop to tfraFunctafuoargent of opop, rounded in the direction rndmprde>intrnd

— Function: int mpre domdeuler (mpfr_rop, mpfr_t x, mpfr_t y, mp_rnd_t rnd)
— Function: int mpre ockler (mpfr_roed l* qrop, mpfr_t x, mpfr_t y, mp_rnd_t rnd)

Set to tre domdeulue of tk, dsectiue spxy, roundes in hesquaxaan-inte (tt li>, roundes d, e)r>|, aet , roundPi according to the direction rnd): If op is zedy, hal is the siue spx. The retned value in t interyned valhe correspondir> en xAddirecternallid mpre ocklet is t a ohe sifiosecabi, i k fris tockqiferean in x minus ), r> with the siue spxy. Note thspx en y. these functioes aus="dfud, faddirev The argumpideufunctly).

— Function: int mpan-inteero_p (mpfr_t op)

Return non-zerof if op ts an-intely).

— Function: void mp, ntowando_p (mpfr_t x, mpfr_ op)
): If x or y is Nasethspxxyx with tal precisies of x ath th (currnhe exponeut ra) ed in the directit2 of y.

— Function: void mphextabovuo_p (mpfr_ op)
Equial gumpes id mp, ntowando_ty is plus infinde>.

— Function: void mphextbe a o_p (mpfr_ op)
Equial gumpes id mp, ntowando_ty r> minus infinde>.

— Function: int mpm_asin (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)

Set rop to tm_aimuithm of op1 and op2op1 and op2rop is set to Na): If op1 or op2 is Natd when rop is set in ter emetral vaNa): or op1 and op2rop is set ns −0.

— Function: int mpmaxsin (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)

Set rop to tmaximuithm of op1 and op2op1 and op2rop is set to Na): If op1 or op2 is Natd when rop is set in ter emetral vaNa): or op1 and op2rop is set +us;0.

— Function: int mpuoperombsin (mpfr_t rog2, mperly st_ttly strnd)
Gener stforusiformivudint ebomputoperomr x-Flned y ( us Inal ge 0 <= rop &l1les. Returar>,05fl is the exponeat is ned y ( h (currnhe exponeut raedyn w, whiin caen rop is set to lt, and n non-zeed value is returne. he secohe argumpr> isid functNaseeiy ( GMPsmanutiar>.

— Function: void mpoperomsin (mpfr_t rnd)
Gener stforusiformivudint ebomputoperomr x-Flned y ( us Inal ge 0 <= rop &l1lesTl this functiis bt, edicd; id mpuoperombsit.

— Function: void mpoperomlog2 (mpfr_t rop, se s_ ise srop, he _rnhe rnd)
Gener stforoperomr x-Flnum oxangs>Set , ann lies in tbliminact, sgumarectce). The exponeue of ter number yn y ( us Inal ns &minet en MPwal buil, ul bei&lsock;--er w-g2,-build...pts01"sam &rsock;lly).

— Functiop, he _rnnt mpfet_he rnp (mpfr_ op)
Get is the exponehm of xssum beiy thspx is n non-zeaccormin er numb>, and the sifiosendelucf whnded [1/2,1ace). Tbeha oonlue fis N an infinity on-zery is undefiar>.

— Function: int mpfrt_he rnp (mpfr_rop, he _rnh op)

Sis the exponehm of x ad, retu0 (d, even spx t, not n non-zeaccorminrer num); ot otherw, d, retund n non-zeed vaion. The sifiosendelu>ssumundes st ed [1/2,1ace>.

— Function: int mpfrgnbieint (mpfrt rnd)
R, retund n non-zeed vaerof if op whoct, sgumarectihal i, ise siby, is ace>.

— Function: int mpfetse sler (mpfr_t rop, mpfr_t on: is op, mp_rnd_t rnd)

St, the value in ropop, roundtowand t, the githe direction rnd n non-ze;corr.don-z)ead, evnd when op t is NaN.

— Function: int mpfrpyse sler (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)

St, the value in ropop, roundtowand t, the githe direction rndop2op1 or op2 t is ace). this functiis equial gumpes se mpse se si(sitropitop1mpfrgnbiei(sitop2itrnd.

" r f-re termiunhchsqncoinuesray",aup -. t, nonthe erre, anrobassyrbetoinaup -. Sus=rs.oup -.
— Functiotr conchsq *nnt mpfet_re versche (void)
. Retuin te">MPre vers,lar ai r f-re termiunhst endNaN.

&mdasMacro:nnt e">M_VERSIONche)M_fr_0VERSION-24-203">
&mdasMacro:nnt e">M_VERSION_MAJORche)M_fr_0VERSION_fr_0MAJOR-24-204">
&mdasMacro:nnt e">M_VERSION_MINORche)M_fr_0VERSION_fr_0MINOR-24-175">
&mdasMacro:nnt e">M_VERSION_PATCHLEVELche)M_fr_0VERSION_fr_0PATCHLEVEL-24-196">
&mdasMacro:nnt e">M_VERSION_STRINGche)M_fr_0VERSION_fr_0STRING-24-147">
<2M_VERSIONchtMPar aipct, o accputing constly).e M_VERSION_MAJORch2, M_VERSION_MINORchr ).e M_VERSION_PATCHLEVELchr, respectiv to tmajN,,vtererre, anatch ld, lhhm e">MPre vers,lar pct, o accputing constsly).e M_VERSION_STRINGcht wi meapunctafusuffi_robe d i tu, loprgumpe, anre-relen care versiolar aist endi's constanw, whiinn b the ceredar> to tre resuue se mpfet_re verscht t y ( hthrinlfilfput aoibrminres usmatchns): ): en (st cmp (ar>mpfet_re vers ()dee">M_VERSION_STRING))s): fpr pon (strin,,v">. Warn: hthrinlut aoibrminrdooes nnatch\n"); . re>o.

N: ObtaiWarnthe different endlur> t, nne accarilynonthe er,lar i eener l, aipcogramthe cicaller w_ a saocoue">MPre versurs can dyMPoibrminrre vers (if tl a d to to toibrminrre versputinystemace>.

&mdasMacro:ned lont e">M_VERSION_NUMler ()M_fr_0VERSION_fr_0NUM-24-198">
Cssgre ts an-inteded in the safn naneaches used <2M_VERSIONcht ( ( and H ralue onthxa comuue how io check in te">MPre vers oxahe cicaar> tns): ): #en (! undefi(e">M_VERSION) || (e">M_VERSIONr> &e">M_VERSION_NUM(2,1,0)))s): #the err"Wrd loe">MPre vers."s): #depef . re>o. p>
— Functiotr conchsq *nnt mpfet_natchesche (void)
R, retund r f-re termiunhst endotr taiWarnty ( udslue of tnatches appliinted in te">MPoibrminr(tr tlements in t. N:s): If tpcogramthal ve beee cicaller w_ mealdlowe">MPre vers oendel dyMPoibrminrre vers,ty ( udslue of tnatches appliintedhif thcou(ee cica-r> t)we">MPre vers os at, novaila tab(howd, e tend edfn naners su shout, nhd hamuwhius Inuaxaan eener lace>.


Next: , Up: MPFR Interface

— Function: void mpset_kefares_en round_ng mler ()

St, tkefaresven rounding mir> en rnd. Tkefaresven rounding mielues hesquaxaaninentlnde>.

— Functioar mp_rndnt mpfet_kefares_en round_ng mler (void)
Gp>St, tkefaresven rounding mNaN.

— Function: int mp, ep>, roler (mpfr_rop, , ep, ll pop2, mp_rnd_t rnd)
Rn ro of xrndyM_PREC_MINler and M_PREC_MAXog2): If y to thl precisies of x filcaller w_is zsar>Ot otherw, nd the sifiosendelu>, roundes al precisind y with the githe direct.PFRaff boao ca,ith tal precisies of xy
— Function: int mpr, ro_nl pr>r (mpfr_rop, mp_rnd_t rop, , ep, ll prnd)
[). this functiis oe abetmNaPlen cas, use mp, ep>, rolet
— Functiotr conchsq *nnt mpprde>p>ro_ng mler ()
. Retuin ti"#Inrst endo(GMP_RNDD, GMP_RNDU, GMP_RNDN, GMP_RNDZ) he correspondir> ty the rounding mien rndrnd ts anct idTen rounding mNaN.

— Functiop, he _rnnt mpfet_hm_asin (void)
— Functiop, he _rnnt mpfet_hmaxsin (void)
. Retuin tts (curr)th. Th tim2 raisintedhif h, and tor last he valhal is thn no(1 - epsilon)ar> tim2 raisintedhif aor last he exponar>.

— Function: int mpfrt_hm_asin () — Function: int mpfrt_hmaxsin ()

St, th aon-zeat otherwe 0. If tus=rncft ral is the exponeut raedyniat ot/hue isrressibilfiniio check te thmpcheck_ut rasun ie al ot/hue isrressibilfiniio check y oxa@m {em_a <= emax}. up -.

— Functiop, he _rnnt mpfet_hm_a_m_asin (void)
— Functiop, he _rnnt mpfet_hm_a_maxsin (void)
— Functiop, he _rnnt mpfet_hmax_m_asin (void)
— Functiop, he _rnnt mpfet_hmax_maxsin (void)
. Retuin tm_aimuitr> amaximuithm t, thmpse _hm_asir and mpfrt_hmaxsie cssgre td n porma tabpcogramted wrfi beise mpse _hmax(mpfet_hmax_max())che and mpse _hm_a(mpfet_hm_a_m_a())che
— Function: int mpfheck_ut rasur (mpfr_ron: itop2, mp_rnd_t rnd)
). this functid, ctimes xxxx or laower t if the exact e r> aon-zeen spx he exa(on befor ( inllaceIt eener st> ts is urf a oor_ meavurf a oufSis the exponehm of xxl result hr equr> to the exaexp, aipr positihe valen ty the roder>l result or laower t to the exaexp, aina negatihe valen ty the roder>l result iNoy oxaunve lings>Sse functi, If tal result he ceredar> to the exaexp, is nin ti"#Inrhe valof x, i.e. t interyned valat eropag stadro.

Nns): If x is an infinar> and x be differe k fron-ze;>, iNr>n ty the rodetal result is anhe exaan infin)Natd whif thvurf a ofladies se ce). this us="dfuaycas, use mpcheck_ut rasunMPse functi) afoinlquaxorarnty ( fladiote they could h ve beis sdumir> us Intafuhe comculatily).

— Function: int mpfrban nale sror (mpfr_ron: itop2, mp_rnd_t rnd)
). this functid, ros If xx hutconsty ( nrban nalrhe exponeut raedynijh mueropag st] is t interyned valnd xxrndxx >, rounded fixeding-po he arithmetes ts an-inte malticomuue twour> to tha lo by to tha lo by x on-z elu>, roundes twour> to tha lo by wihe roundiwo hesquaxdro. .e rndxx, and the return interyned valnd whns computiof x fk fred ): If tal resuey cannot ct, sgumunded in th (currnhe exponeut ra (dumir> atedo i. Note thunve lings>Sse functi, If tal result he ceredar> to the exaexp, is nin ti"#Inrhe valof x, i.e. t interyned valat eropag stadesTl thir aipctpreliminary interfly).

M:h3> s arv clasexa com">): {o. ar>mpfr_a, xb;o. n: ii;o. volcullethg douba, b;o. o. ar>mpset_kefares_pr p (53);o. ar>mpset_em_a (-1073);o. ar>mpset_emax (1024);o. o. ar>mpinfi (xa); ar>mpinfi (xb);o. o. b = 34.3; ar>mpset_k (xb, b, GMP_RNDN);o. a = 0x1.1235P-1021; ar>mpset_k (xa, a, GMP_RNDN);o. o. a /= b;o. n = ar>mp mpfrban nale sd(xa, i, GMP_RNDN);o. o. de>mpcvear (xa); ar>mpcvear (xb);o. } . re>o. 0. Warn: ll themalcuesoaong doubIEEE-754ahe arithmepr> wihe cocsven round ed in thrban nalrut raedw, whiis mcannot r ( insmud, fyouruldrdwereent.
— Function: void mpcvear_is urf a sin (void)
— Function: void mpcvear_hvurf a sin (void)
— Function: void mpcvear_nanfladsin (void) — Function: void mpcvear_anhefladsin (void) — Function: void mpcvear_eut rafladsin (void)
Cvear If tus urf a ,thvurf a , anct id, anhe exa>, aeut ra fladiar>.

— Function: void mpset_is urf a sin (void)
— Function: void mpset_hvurf a sin (void)
— Function: void mpset_nanfladsin (void)
— Function: void mpset_anhefladsin (void)
— Function: void mpset_eut rafladsin (void)

SIf tus urf a ,thvurf a , anct id, anhe exa>, aeut ra fladiar>.

— Function: void mpcvear_fladisin (void)
Cvear /sm globticeladio(us urf a ,thvurf a , anhe ex,Fanct id, eut raace>.

— Function: int mpus urf a ero_p (void)
— Function: int mphvurf a ero_p (void) — Function: int mpnanfladero_p (void) — Function: int mpanhefladero_p (void)
— Function: int mpeut rafladero_p (void)
. Retuin the correspondi(us urf a ,thvurf a , anct id, anhe ex, eut raa fladedw, whir> n non-zerof y ( fladar> is NaN.

A calh the giary interfsues apctpreliminde. y mt migcft raaryce cetibiveed fuufutu prversient.
&mdasMacro:nnt e">M_DECL_INITo_p ()M_fr_0DECL_fr_0INITo28-194">
Tl thmacro d>Slsqua > (mp>int, ase ssi, ial precisiet st he exivind y (Slsqaners sdirect.P Tl thmacro thmuwhifasrea er thul beise mpanin2sirYf ym. Jucanic/sm id mpcvearo_t wiee va tas cssgred r> with thmacro (. Thtoragfieslisedenstapa set tng-pornfth>Slsqaners senddeisedenstapro when tbrerf-ld, lhr> heistadace>.li>Yf yrs ccanicft raaen ir ll precisly).li>Yf ysu shout, icssgre ee va tas r> wihug tal precisir> with thmacroly).li>Yf reee cicarym. Jusuppormi&lsock;Nrs-Cs constPFRinente srs...pts01"sam &rsock;o(99 stand ed C++ >, ahe IS99ave, a&lsock;Tok whPastund...pts01"sam &rsock; (99 stand ed he IS89)Na): If yVe va ta-ldng w_outothemaresrays...pts01"sam &rsock;o(99 stand ed he IS99)Na &lsock;GCC 2.95.3...pts01"sam &rsock;osuppormsliseaen smudsgrfutsNa): yf ahe cicaayf repcogram r> wigcc ed c89ing mue, ar> wi&lsock;-pednstic...pts01"sam &rsock;anyf iis mwaumpes undef is tnd M_USE_EXTENSIONcht is tnd M_DECL_INITo_n
— Function: void mpaninsror (mpfr_ro...oid)
FRinente sliseaen use mp>inttintSIf ir ll precisiet st t, tkefaresvll precisi>, and ir ed valet to Nad. Sse mpaninsuntintssumundes st he cos=dTtt onue typ use mp>intmppt sun)mustgin> fk fref x wh typ um. Juie albe bd mppt sun p

— Function: void mpaninslog2 (mpfr_ro...oid)
FRinente sliseaen use mp>inttintSIf ir ll precisiet st he exiv nd y, and ir ed valet to Nad. Sse mpanin2suntintssumundes st he cos=dTtt onue typ use mp>intmppt sun)mustgin> fk fref x wh typ um. Juie albe bd mppt sun p

— Function: void mpcvearsror (mpfr_ro...oid)
Freeiy ( sperfiocs piplieyfoseaen use mp>inttintmpcvearo_ttintssumundes st he cos=dTtt onue typ use mp>intmppt sun)mustgin> fk fref x wh typ um. Juie albe bd mppt sun p

H ralue onthxa comuue how io usinmalticomuaninente aners al-functi:h3> s arv clasexa com">): {o. ar>mpfr_,r_t z, t;o. ar>mpinfis2 (256,r_,r_t z, t, (ar>mppt ) 0);o. ...o. de>mpcvears (_,r_t z, t, (ar>mppt ) 0);o. } . re>o.


">s"> Next: , Up: MPFR Interface

o288154">

wie">ons

A hthrinlfilfp2ar>m.h...pts01"sam end edcluounded in thent ebomershhm e">MPd, ce cetibilfinpr> wiin tGNU e" h3 cl e">ar>Afoinlin>ertarnty ( fol a arntywonlinfsuefrea ere bd &gmp.h>int s arv clasverbetimode#edcluouar> &ar>m.hde#edcluouar> &ar>2ar>m.hde. re>oand pcogramtwrfitetid, e">urs caneee cicallhe dirlnpr> wie">Rpr> wout and cft ralar>A fuopeqanerssues atd whi ofn nedpr> wiin tkefaresve">Rpen rounding mo w, whiinnnot ctsp>Sr> wind mpse _kefares_en round_ng mletpaninsun and panin2sunMPse functiianinente slet to : tend es us="dfues utocsvunaninente srned vas,dbom r> ilt miiveedce cetibie r> wind let
— Function: void mpset_, ep>a o_p (mpfr_rop, , ep, ll prnd)
R,sp>SIf tal precisies of xhe exivind y wind mpse _nl pr>tyssumundes b ti to tl (currntl aenstapmemoey sperfifr> or x.

— Function: int mpeqint (mpfrt op1, mpfr_t opunhe staped lo-pornp3oid)

Return non-zero of op1 and op2 wiin the sahe expone>, and the safirs>Set Retuon-zeat otherwe Tl t is functiis btndefiud, fns cetibilfinpr> wind let, a1.100000deree>, nrounda> be differe or_ mythe value in .

— Function: void mpoelbe dler (mpfr_t rop, mpfr_t op1, mpfr_t op2, mp_rnd_t rnd)
Cs come If tall sgatibe differfisttwe b op1 and op2ropop1op2op1rndrop
— Function: int mpmul_2he rnp (mpfr_t rop, mpfr_t opunhe staped lo-pornp op2, mp_rnd_t rnd)
— Function: int mp <_2he rnp (mpfr_t rop, mpfr_t opunhe staped lo-pornp op2, mp_rnd_t rnd)
mpmul_2uisir and mp <_2uisirr Ttt onkept d, fns cetibilfinpr> wie">ar>.


Next: ">Ce cetibilfinpr> wie">ons, Up: MPFR Interface

Sa saapplicanerssuusinaistack yo r tdleiy ( memoeyi>, and ir objicmece>Howd, e, If te">MPmemoeyidese siue t, nwelfusuistapd, fsuwhiaith ndceSalte t suwhiapplicanerssues apssmir> usine">M, ontauxilfminamemoeyiry interf hal ve becssgred: lle C. JomPFR Interf.iv Tl( fol a arntry interf tl a l is mir> usine">Mded iwon alwns): Eit otty y he dirlnpstoef If te">MdFPrer numah> isid mp>intEit otty y stoef If ir ownact, sgumarectiue adFPrer numaon If stack >, a's corufu_or a wtemporminrse mp>int tdyniat needfiar>. ul he exce nrbagarnty ( be d memoey:foseaen umemoeyistuff (tl aensarn, ustrdoarn, nrbagarn)iat keptted in tapplicaners.iv Eawhiis functiin tend ed interf r> ie ale complemunda> admacro d, effioifernactascti: d, fhxa comuse mpc. Jompinfi ( ,ap)chempc. Jompinfi) ( ,ap)cheMPse functiiis msti fuaninente sleemporminrFPrer num> ul bei99 stand ar>mpinfi. . SC. JomPAl aensactilGNU e")liv . No2:ie">MPse functiiis must r ( inchtapde functii(de>mpcs co_pi d, exa com)ead, ev>n ty y os at, n05flicitlynen cal. Yf yhd har> cn c se mp0oee_inchtint tdyf i nrbageiy ( memoeyirof ar>mpinfi, Ifrough GMPsC. JomPAl aensact,ntl aenstssi, imemoeyion If tapplicanerststackliv . No3:e). thid interf r> pctprelimindiv>
— Functiose s_ iid mpc. Jompfet_se sror ()
. Retuin tneedfihse sizn byst] iopstoef If the sifiosendue adFPrer numahm al precisind y p

— Function: void mpc. Jompinfio_p (v *ne sifioseninar , ep, ll prnd)
FRinente slithe sifiosendue al precisind y ympc. Jompfet_se s (nl p)cheint p

— Function: void mpc. Jompinfipse ror (mpfr_ron: iki, rop, he _rnhe rop, , ep, ll popar>v *ne sifiosenrnd)
P ofn noaonummyuaninente aners ue adse mp>int, ase y, ta:s): ie se M_NAN_KINDsirx is set to ;brli>ie se M_INF_KINDsirx is set in ti" infinahm se sise ie se M_ZERO_KINDsirx is set in ton-zeaf se sise ie se M_REGULAR_KINDsirx is set p;x with this functiey cannot ctse srnul be nd mpse _nl pr>tmpcvearo_t y wise mpc. Jompinfichey p

— Function: int mpf. Jompfet_ki, rnp (mpfr_ op)
. Retuin tl (currnki, and adse mp>intches used se mpc. Jompinfipse rotTd tbeha oonlhm t, this functid, _ mytse mp>int wise mpc. Jompinfipse rot.

— Functioar>v *int mpf. Jompfet_mnstissarnp (mpfr_ op)
R, retundof poinlet in the sifiosendes used adse mp>int w se mpc. Jompinfipse rotTd tbeha oonlhm t, this functid, _ mytse mp>int wise mpc. Jompinfipse rot.

— Functiop, he _rnnt mpf. Jompfet_he rnp (mpfr_ op)

Retuis the exponehm of xssum beiy thspx is n non-zeaccormin er numar>. Td, retuhe value fis N In infinity Zn-zery is respfiplieutidoeen't product_ my trap. >Td tbeha oonlhm t, this functid, _ mytse mp>int wise mpc. Jompinfipse rot.

— Function: void mpc. Jompmovsror (mpfr_roar>v *new_pr posrschp)
Idfn nie">MPte thnd the sifiosendhal movsd dumir> at nrbageico cact sendupdnst_i, i a wpr posrsdes id Howd, e If tapplicanersthal it movshnd the sifiosende, and tid mp>intTd tbeha oonlhm t, this functid, _ mytse mp>int wise mpc. Jompinfipse rot.


, Previous: , Up: MPFR Interface

Tl( fol a arnttyp sue, de functiiw ral domlyidese stapd, for ti complemeunctihm e">M, eutiis mst be "dfud, Sus=rstedo. >Howd, e nodupwand ns cetibilfinpue euaraumeedly)Yf yms mceundes edcluouar-i co.h...pts01"sam er> usinis m.iv Tl( se mp>intTl( se mppl pr>tM_PREC_MINlerTl( se mpse sletTl( se mphe rneShe sifioset limb. No non-zeed vas iues ap malticoipr 2^ntall sgatitalte t ng-poce Adis N is an infinar> aaton-zees aeddicmiunhed a respalthe value is the exponcro.
  • Formlly,and tid mpdrot. >Td ter numaoftlimba innusinysotr trolcalled <2mppl pr>tmppl pr>tNo nearnulareed vas is alwahd hard tngs>She sifiosetiby, ue is tngs> he sifiosettlimb is set 1. Wo when tal precisidoee t, nhe correspir> a whol ter numaoftlimba, is the accpbi, i>thnd t a oe, and in tkataees aon-z.iv ul mpadd_exp_ulp (ar>mpt @he {x}op2, mp_rnd@he {mp_})aup -. @he {x}v>n @he {x}v>> finist up -. n @he {x}v>> finist e, ana negat; up -. > is ncft rad. up -. . Td, retuhe valr> on-zhunvaccp meavurf a oocs rsneed w, whiinst r ( up -. mpadd_exp_ulp}his functibeha ee ve liaitr vlempctafuaddiunctNaup -. mphrb_exp_ulp (ar>mpt @he {x}op2, mp_rnd@he {mp_})aup -. @he {x}v>n @he {x}v>> finist e, anr posit,tadd expiup -. n @he {x}v>> finist e, ana negat; ot otherw, @he {x}v>> is ncft rad. up -. . Td, retuhe valr> on-zhunvaccp meis urf a oocs rsneed w, whiinst r ( up -. mphrb_exp_ulp}his functibeha ee ve liaitr vlempctafuhrbtr exnctNaup -. — Function: int mpfanp>, roler (mpfrbrop, he _rnhn,,v2, mp_rnd_t 1,v2, mp_rnd_t 2inar , ep, ll prnd)
    Assum beir>)x is approximaners ue ameisknownrer numaspx>op1 wihe errettngs>Stwour> to tha lo E(b)-nd x issmir> >, ro he cocsly If xy wiin tke direction >op2 a0 ot otherw (edcluo beiue fis >, ahnf)lesTl this functidoee t, nnodpfvii, i>rguplemsdro.

    Nnsen expiwauml it ie al uto tereiy ( he cocsv interyned valnd when round r>)xy>op2ttend wi fucheck en expiinn >, ro ng en y, ro ng hesquaxang en yx hesq fk franed valhe exivact, sguma tauus).py

  • — Functiong doubnt mpfet_k1int (mpfrt rnd)
    Cr vlr>Set xtbearnty ( kefaresve">Rpen rounding m (seehis functimpse _kefares_en round_ng mlet
    v de>mpse _str_biteryn(ar>mpt @he {x}optr conchsq *@he {s})aup -. to the value is tbiteryner numaanrst endo@he {s}edw, whihal it up -. poinnl r=dTup -. be malticoiplieyftd the sifiosen. up -. . The sifiosendldng w_ofo@he {s}lhal it st vaccper or equr> to thl precisiup -. -a-Nr num); up -. wiin the correspondind sneup -. v de>mpprr_epbiteryn(ar>mpt @he {ny fl})aup -.


    Next: MPFR Interface, Up: Topace

    Ve vers 1.2ouNovs numa2002>

    s arv clasdisplayod. Copyrt mig©a2000,2001,2002 FreeiSoftwerenFevstaunct,!Inc.o. 51nFrnsklitiSt,nFif wiFy oo, Bostct,!MA 02110-1301, USAo. o. Ere yexpir> po tettedar> copy >, ahent ebome verbetim copatso. hm t, thlice ceidocuplem, bom cft rarntrniat is ntl a d. . re>o. > ol typ =1 tar,=0PREAMBLEro. < Tl( purcos= hm t, thLice ceiil it ma liaitanu l, textbookedonlot ot de functafuaendes "dfudocuplem 0oee> fn>eed in thenstlue 0oeedom:ted >ssurenere yexpit hffdireve 0oeedomar> copy >, amedent ebome it, r> wionlwi wout nodpfvarntrn, eit ottns merspallnity nonns merspallnNad. heotanily,and thLice cei, sgr ee d, for tau wor e, anublis ottan al es fetecssdiuud, fIf ir work,iw, omucannotarntcs cidumed isrressible ue fmodifiosunctiiis"n eyfot otidro. Tl thLice ceiil anki, and &ldock;copyleft&rdock;anw, whiieassute thdumiv sgat workshhm tl( docuplem ch muteetsel ee be 0oeeeed in the sane ce. eIt ns complems to tGNU Gener l PublichLice ceedw, whir> a copyleft lice ceidese stapd, f0oeeesoftweredro. Weuhd hadese stapt, thLice ceiineaccinlet usiny, ue fmanu lspd, f0oee softwere,uaycas, u0oeeesoftweremceunsu0oeeedocuplemeunct: a 0oee pcogramtsu shouca sar> wimanu lspprovavarntin the sa0oeedomsPte thnd softwereidoee. eBut t, thLice ceiil is noimitedar> softwereimanu ls; itiinnnot us=d d, _ myttext equwork,i>, nrovaccpefuhrbjicm themeruor ], t ottyniat eublis onda> adprr_eplieook. W Td,ns menapt, thLice ce prr_cipallnid, _workshr> wh purcos= nd edcorufuers unlqudifferfdro.
  • APPLICABILITY AND DEFINITIONSro. Tl thLice ceiapplit] iop mytmanu ldonlot otuwork,iomdamytmsdium,lte t heonain> is ntice plerflieyftd tcopyrt migholcinlsavarntrniinnnot dent ebomedeis uriy ( tarms hm t, thLice ce. Suwhiai ntice graums a world-widu,i>oyalty-0oeeelice ceedunvemitedaomdduraunct,!r> usinis t workeis uriy ( heotiunctii tateda, whin. Tl( &ldock;Docuplem&rdock;anbe a , qudif] iop mytsuwhimanu ldonlwork. Amytms numaoftor tpublichr> a lice cef, e, ar> addisrsonda> &ldock;you&rdock;. Yf ya" apo in tlice ceii: yf copy, nodpfvdonldent ebome in tworkeomdan allququirarntpo tesvers is uricopyrt miglawdro. A &ldock;ModpfipliVe vers&rdock;aoftor tDocuplem ceassuenytworkeheonain beiy ( Docuplem , _ porm rs ue rn, eit ottnspipliverbetim,ionlwi w modifiosunctiie, /, fIrausl stapinr> anot otult ruagfdro. A &ldock;. heotany Sdirect&rdock;ar> is amundappledix , _ freon-themerus="sect oftor tDocuplem te thdu lspexclusgatlnpr> wiin tall sgctihipahm If nublis otcper au woriioftor tDocuplem totor tDocuplem'seavurn c hrbjicm (er r> >el stapthemers) e, aheonain> noth ndlte they coufn c he dirlnpr> wed in thavurn cuhrbjicm. (Thviouiftor tDocuplem nd ed ear,oaotextbookioftmateethemas, ai. heotany Sdirectiis mcann05flain mytmateethemas.) !>. Td,l sgctihipaey coubliaitaemeruof , txoracal heondirectir> wiin thrbjicm onlwi w >el stapthemersedonlof l, nlo ce merspal, p, oosop, cal, t ical onlpoliemaafupr posrsd>, nround is m.iv < Tl( &ldock;Inhe vaem Sdirects&rdock;aes acernaini. heotany Sdirectshr> wh oittas ereidese s sta,>asnotarntthos= hm Inhe vaem Sdirects,eed in t ntice in thsalwate thnd tDocuplem nd >eleasedeis uriy thLice ce. If a s="sectidoee t, nfithnd tabovsh undeim rs ue . heotany to why, il is tl a d it st dese s sta>asnInhe vaem. Tl( Docuplem cayaheonain on-z Inhe vaem Sdirects. If nd tDocuplem doee t, niplempfyuenytInhe vaem Sdirectshto when s aps at,ne.iv < Tl( &ldock;Cavur Texts&rdock;aes acernainisu r,op>ssagfiioftoextate thes alis>ta, >snFreon-Cavur Texts onlBack-Cavur Texts,eed in t ntice in thsalwate t nd tDocuplem nd >eleasedeis uriy thLice ce. AnFreon-Cavur Text cay ot ettngs>S5twords,ar> aatBack-Cavur Textiis mst ettngs>S25twordsdro. A &ldock;Trauscereem&rdock; copy oftor tDocuplem ceassuetmacwede-reana talcopy, ct, sgumunded anfn naner> wh respficanerstr> ivailissmir> If gener l public,lte t r> iuisassmud, f prvernty ( kocuplem nt at mid, wandlnpr> wieeneric oextasdiuotcper (d, fimagfiihe cos=dTtf pixels) eeneric pa-porpcogramcper (d, fdrawarns) a s widusyrnvailissm drawarnasdiuot, e, and t r> iuisassmud, fi"#Inrr> Iextafn nantotcper ue foutothemarIrausl ssrsdes athe veinphm fn nan> iuisassmud, fi"#In r> Iextafn nantotc. Ancopy is"n omdampot otherw Trauscereemlfilf fn naner> wh markupedonlabsferfioftmarkupedhasnotes osra radir> Ifwant onldeney ragfihrbsuqulem codifiosunctieyfrthrind es is nTrauscereemce Antimagfnfn nanees is nTrauscereemuiftus=d d, _ mythrbsonstaticameuno oftoext. Ancopy nd t r> is n&ldock;Trauscereem&rdock; ysotn caln&ldock;Opaqul&rdock;.ro. Exa comipefuhrisassmud, nan> d, _Trauscereemucopats edcluouaflain ascii...pts0lwi wout narkupedTexinfofi"#Inrd, nan, LaTeXfi"#In d, nan, SGMLdonlXMLdbearnta publicsyrnvailissm DTD, e, a99 stand-heod, nondind comusacronym>HTML, Pgs>ScriptdonlPDFddese stapd, fhuman codifiosunct. Exa comi oftorauscereemuimagfnfn nans edcluouaPNG, XCFde, JPG. Opaqulnfn nans edcluouapcop veierynfn nanwate thinnnot reanda, aediu=dTtt onbyapcop veierynwordapco accorsedSGMLdon XMLdd, _w, whiis tsacronym>DTDie, /, fpco accerntyoolsues t, ngener lsyrnvailissm, e, and tmacwede-gener u=dTsacronym>HTML, Pgs>ScriptdonlPDFdproductlieyf a s wordapco accors d, out#Inrpurcos=sTtt o.iv < Tl( &ldock;Titta Pagf&rdock; ceass, d, _ dprr_eplieook,and toittaeeagf i, elf, plviPsuwhiiol a arnteagfida> ar tneedfihyo rolc,ilegibiv,and tmatotati t, thLice ceiququirt] iop pplareed in toittaeeagf. F, _workshed fn nanwaw, whidout, uhd ha myttittaeeagf aiPsuwh, &ldock;Titta Pagf&rdock; ceass y ( taxtanear If tngs>Spcotereporhpplaraerfioftin twork'setitta, p dievarntin tstginnund ofPte affdy oftor toext.ro. A s="secti&ldock;Entittad XYZ&rdock; ceass is amundhrbunfi oftor tDocuplem r> wh titta eit otti ial preusyrXYZdonlceonain> XYZdibeearlem thsiiol a arn oextate thIrausl st> XYZdibeanot otult ruagfd (H ralXYZd99 st> d, _a srespfic s="secti amu mlempctpliee a ,fsuwhia> &ldock;Acknowtadgmplems&rdock;a &ldock;Dediosuncti&rdock;an&ldock;Endorsmplems&rdock;adonl&ldock;H txory&rdock;.) !>ol&ldock;P sgr etor tTitta&rdock;ahm suwhiais="sectind whyf iiodpfvdor tDocuplem ceassund t rtiqumain> i s="secti&ldock;Entittad XYZ&rdock; accaccorgir> Ifis btndeiunctliv < Tl( Docuplem cayaedcluouaWosra inpDenelaimind nextato in t ntice w, wh tatewate thnd thLice ceiapplit] iopor tDocuplem. Tl(suaWosra in Denelaimind es acs cidumed it st edcluoundeyfrtdifferfiin tend Lice ceedbom tt onasd>, nrosldenelaimorgiwosra iit]:a mytot ot d coiosunctite thnd suaWosra inpDenelaimind cayahd haasner>v sendhal t, hffdirion If tceasund ofPte thLice ce.ro.
  • VERBATIM COPYINGiv < Yf yms mcopy >, ahent ebome nd tDocuplem nmdamytmsdium,leit ot ns merspallnity nonns merspalln,pprovavedate thnd thLice ce,and copyrt mig ntices, e, and tlice cei ntice savarntnd thLice ceiapplit] iopor tDocuplemderee>,productliinhCOPYING IN QUANTITYiv < ): yf anublis dprr_eplicopats (, fnspats ed dsdialte theymmtt onhd h prr_eplicovers) oftor tDocuplem,ner numundingef If t 100, e, and Docuplem'selice cei ntice ququirt] Cavur Texts,eyf ii. Juferl whoid copats ed coversate thinrrn,pcvearly >, alegibiv,aiseaen smuCavur Texts:nFreon-Cavur Texts on If tfreon cover, e, aBack-Cavur Textsuus)If tbnck cover. eBf bocoversai. Juie alcvearly >, alegibivniplempfy yf ies to thublis ottoftor sefnspatsNaaTf tfreon coverai. Ju, sgum is tisseaeitta er w_olltwords oftor toitta er eqlySpcotereporh, prve tad Yf yms madd et otumatotati on If tcoversaenuaddiunctce Copyund er w_cft ralnoimitedar> If tcovers,>asned loes to yi, sgr e or toitta oftor tDocuplem e, a9aunsfvdor ssaheotiuncti,iinnnot tssgred >snverbetim copvarntrmpot ot isrricmeciv < ): in talquirtdtoext> d, _eit ottnsveraes atoo vofupineviPr> fit legibiv,ayf isu shou#Inrrhsafirs>Sonalnois>ta (al mamytas fit ctasctassy)ion If tact equcover, e, aheoninus in tals>Soniop djasion eagfidro. < ): yf anublis donldent ebome Opaqulncopats oftor tDocuplem er numund ngef If t 100, yf ii. Jufit ottidcluouaetmacwede-reana talTrauscereem copy >ed lor> wihawhiOpaqulncopyadonl tateiineacor> wihawhiOpaqulncopy auns comer-networke aensactifk frw, whiis tgener l network-ul be publichhasnp" accies ownload ul beipublic-99 stand networkepcotocols auns cbetm_Trauscereemucopy oftor tDocuplem,u0oeeeofPaddtaptheotatiar>): yf aust r ( laemeruopunct,!yf ii. Jutskedctasctassyepcupleml tepso w, whyf istginthent ebomershhm Opaqulncopats intr entity,!r> e cure te thnd thTrauscereemucopy wi fuqumainhndusnp" accossmi thnd thtated aensactiuntil oxalen nexpiyear efrea ere ln nr> tdyf ihent ebome an Opaqulncopy (he dirlnp, fIfroughayf reagfnts onlretailers) oftorat ediunctur> to thublicdro. < )m nd >equls>ta,dbom cs nalquirtd,and t yf iceonacthnd tau woriioftor Docuplem welfubeforee>,hent ebome loemytor la er numaoftcopats,!r> he g is m a chaerfies alovave yf ier w_ondupdnstdPre vers ue or tDocuplem.ro.
  • MODIFICATIONSro. Yf yms mcopy >, ahent ebome a ModpfipliVe vers oftor tDocuplem is ur in theotiunctiihm sdirectsh2 >, a3aabovs, provavedate thyf i>elease in tModpfipliVe vers is urial preusyrnd thLice ce,ar> wiin tModpfipl Ve vers fi firntin trota oftor tDocuplem,hndusnlice cirnthent ebomers r> acodifiosunctihm If teodpfipliVe vers es r> d, e ee psrso> a copy ue rn. Inh ol typ =A tar,=1Useeed in tTitta Pagf (a, ann If tcoversouiftemy)oaotitta hentidct fk frllai oftor tDocuplem, e, afk frllos= hm , previPre verss (w, whisu shoouiftor reew ralemy, st vis>ta ed in tH txoryus="sect oftor tDocuplem)d Yf yms must r ( he satitta a> adpr previPre vers iftor toragormlPhublis ottoftorai re vers he gstpo tesvers.ro. > li>Lisrion If tTitta Pagf,>asnau woriedoxpie fmoefopeqsctiih _entitit] isrressible fer au worihipahm If fmodifiosunctiied in teodpfipl Ve vers,!r>g t otter w_oxalen nfeve oftor tprr_cipaltau woriioftor Docuplem ( fuofii, iprr_cipaltau wori,!en itlhasnfewaower t feve), unvaccpto yi>elease!yf ifk frllnd >equirmplet.ro. > li>Stateion If tTitta pageiy ( amu oftor tpublis ottoftor eodpfipliVe vers,ies to thublis ot.ro. > li>P sgr etoseaen ucopyrt mig ntices ue or tDocuplem.ro. .
  • Add is approprrateicopyrt mig ntice fer yf remodifiosuncti djasionur> to tot ottnspyrt mig ntices.ro. .
  • Idcluou,!emdsdiatusyrnfrea ere copyrt mig ntices, eelice cei ntice he irntin tpublichpo tesverser> usinis tModpfipliVe vers is urior tarms hm t, thLice ce,eed in tfn nisu wneed in tAdde, umiee a .ro. > li>P sgr eted in thlice cei ntice is tissealisments Inhe vaem Sdirects >, amequirtdtCavur Textsuhe gived in tDocuplem'selice cei ntice.ro. .
  • Idcluou ameisaltumed copy oftor thLice ce.ro. > li>P sgr etr ( hdirect Entittad &ldock;H txory&rdock;, P sgr etetsuTitta,ar> aadd iopit is at m htate loexalen nor toitta,iyear,i a wau wori,!h, publis ottoftor tModpfipliVe vers asuhe givon If tTitta PagfNaa): or reer> is hdirect Entittad &ldock;H txory&rdock;ved in tDocuplem,ecssgredoxp htate loor toitta,iyear,iau wori,!h, tpublis ottoftor tDocuplem es he givon etsuTitta Pagf,>t, whadd is at m desc ebe loor tModpfipl Ve vers aii tatedaiwhen tal previPsgumunce.ro. .
  • P sgr etr ( networke aensactouiftemy,uhe gived in tDocuplem d, publichp" accies a_Trauscereemucopy oftor tDocuplem,u>, alikeherw r ( networke aensactsuhe gived in tDocuplem fer , previPre verss im wal bas=dTtt. Tl(suais mst plerflied in t&ldock;H txory&rdock;vhdirectly)Yf yms momiu_or atworke aensactif, _ dworkein thwal eublis ondat len nff reyears beforeend tDocuplem n, elf,ih _iftor toragorml publis ottoftor tre vers rtiqudif] iophe gstpo tesvers.ro. > li>F, _ mythdirect Entittad &ldock;Acknowtadgmplems&rdock;donl&ldock;Dediosuncti&rdock;anP sgr e or tTitta oftor thdirect,!h, tp sgr eted in( hdirect oseaen hrbsonsce >, atoxpief eawhiafPte acont ebomorhp"knowtadgmplemsie, /, dediosunctiuhe givt, whin.ro. > li>P sgr etoseaen uInhe vaem Sdirects oftor tDocuplem, isaltumed ed in(ir taxtasendenuin(ir tittas. S="secti r num> , for tequivallemdereet, nhe cidumed ear,ooftor thdirect tittas.ro. > li>Debetm_ mythdirect Entittad &ldock;Endorsmplems&rdock;. Suwhiais="sect ms mcs nst edcluounded in teodpfipliVe vers.ro. > li>Do cs naltitta amytexentidgthdirect tt st Entittad &ldock;Endorsmplems&rdock;don r> conflicm nmdeitta er w_onytInhe vaem S="sect.ro. > li>P sgr etonytWosra inpDenelaimind.o. > /ol>iv < ): in tModpfipliVe vers edcluou i a wfreon-themerus="sectcper appledicewate thr eqpfyuesi. heotany Sdirectshe, aheonain is matotati nspiplifk frlle Docuplem,uyf iis mat yf reopunct dese s stf a s or_ f oftor sefs="sectcpal anhe vaem. Toudodoris,tadd in(ir tittasir> If lisrios Inhe vaem Sdirectsded in teodpfipliVe vers'selice cei ntice.a Tl(wh oittaslch must hentidct fk framytot otthdirect tittas.ro. < Yf yms midd ithdirect Entittad &ldock;Endorsmplems&rdock;, provavedaitlceonain> noth ndlbom endorsmplemsios yf reModpfipliVe vers bythe vevi ear,it]"> &mdad, fhxa com,i tateplemsios pee f pra w, for thnd ttaxtahal ve beapprovsd eyfan organe aners es to tau worit sgatibundeim rs ue a 99 stand.ro. < Yf yms midd itp>ssagf ue upPr> fivetwords a> adFreon-Cavur Text,ar> aa p>ssagf ue upPr> 25twords a> adBack-Cavur Text,ur> to te, and in tlisr hm Csvur Textsued in teodpfipliVe vers. Ot onoxpip>ssagf ue Freon-Cavur Text a, annf ue Back-Cavur Textiis mst edoundeyf(on rfroughaosra raplemsiis"n ey)ramytone_entity. If nd tDocuplem alreany edcluou iatnsveraIextafn r ( he sacover, , previly edoundeyfyf ion eyfasra raplemiis"n eyfr ( he saentity yf iar tactund on behalf ue, yf iis mis ntdd isat ot;ubom yf iis mreplerf or tol annfedoxn05flicit po tesversefk frlle , previPpublis ottte thedded is tol annfliv < Tl( au wor(s) e, apublis ot(s) oftor tDocuplemidout, ubyrnd thLice ce he ghpo tesverser> usinis ir amu> d, _publicity d, _er r> >sslr>Sot d coy endorsmplem ue amyteodpfipliVe vers.ro.
  • COMBINING DOCUMENTSro. Yf yms mcombereiy ( Docuplem w> wiot ottkocuplemd >eleasedeis uriy t Lice ceedis uriy ( tarms btndefiuis sdirect 4tabovshue fmodifipl re verss, provavedate thyf iidcluouain If tcomberarect oseaoftor Inhe vaem Sdirects oftoseaoftor toragormlPkocuplemdedismodifipl,!h, lisriis m alfues Inhe vaem Sdirects oftyo reeemberedtworkeomdn, lice cei ntice, e, and t yf ip sgr etoseaen ir Wosra inpDenelaimind.ov < Tl( eemberedtworkeceundtt onheonain oxpiiopy oftor thLice ce,!h, malticomuiplempaafuInhe vaem Sdirects is mst replerfdier w_orcirnl copy. If nd s aps amalticomuInhe vaem Sdirects r> wiin the sa amu bom be diffetlceonlemdediskedor toittaief eawhisuwhisdirect uniqulnby aanythdirects Entittad &ldock;Dediosuncti&rdock;d Yf ym. Judebetm_ c hdirects Entittad &ldock;Endorsmplems.&rdock; o.
  • COLLECTIONS OF DOCUMENTSro. Yf yms mma liaico cactect cs cismund ofPte aDocuplem e, aot ottkocuplemd >eleasedeis uriy thLice ce, >, ameplerf or teddivav equcopats oftor t Lice ce in If the vevitkocuplemd er w_orcirnl ncopy nd t r> edcluounded in the cactect, provavedate thyf iiol a iy ( rules hm t, thLice ce d, verbetim copvarntef eawhiafPte akocuplemd inh, ahent ebome fi indivav eq onis uriy thLice ce, provavedayf iidslr>Saiiopy oftor t Lice ce inr> to textr exsd docuplem, e, afol a iy thLice ceiine f ot ot isrricmed>, nroundnverbetim copvarntoftorai docuplem.ro.
  • AGGREGATION WITH INDEPENDENT WORKSro. Auns cilsunctihm If tDocuplem , _etsudumiv sgats w> wiot ottseearate sendendepledlemidocuplemd , _works,iineacoct o vofupl ofPattxoragf ur hent ebomershmsdium,lysotn calnati&ldock;agg>, ntf&rdock; ifPte acopyrt mi israltibeiuk frlle ns cilsunctir> is nus=d iopoimit If tl, nl rt mis afPte acomcilsunct'sSus=rstbeyespiwr thnd teddivav equworkshpo tetar Wo when tDocuplem nd edcluounded is agg>, ntf,and thLice ceidoee t, appsyrn> to tot ottworkshed to tagg>, ntfrw, whiereet, nteetsel ee dumiv sgat workshhm tl( Docuplem.ro. < ): in tCavur Texti>equirmpletihm sdirect 3tr> ipplicassmir> If s copats oftor tDocuplem,hnd ev>n ty tDocuplem nd omiswer t oxpihalf ue to te,tis aagg>, ntf,and tDocuplem'seCsvur Textsuis mst plerflion coversate thbr ekethnd tDocuplem r> wed in aagg>, ntf,an r ( ecactreoictequivallemdoftcoversaeftor tDocuplem nd ed ecactreoictd, nar Ot otherw to yii. Juipplarect prr_eplicoversate thbr ekethnd twhol agg>, ntfdro.
  • TRANSLATIONov < Trausl ssrsdysotr cidumed anki, and modifiosunct,i o yf iis hent ebome nrausl ssrss oftor tDocuplem is uriy ( tarms hm sdirect 4ar RepleribeiInhe vaem Sdirects r> wiirausl ssrss ququirt] respal po tesversefk frlleittnspyrt migholcins,ubom yf iis medcluou nrausl ssrss oft a s or_ f Inhe vaem Sdirectsded addiunctdr> If oragormlPre verss oftor smuInhe vaem Sdirectsd Yf yms midcluouae nrausl ssrs oftor thLice ce,!h, toseaen ulice cei nticeshed to Docuplem,u>, aonytWosra inpDenelaimind, provavedate thyf iie aledcluou nr toragormlPErnlis dre vers ue or thLice ceiand is toragormlPre verss afPteocei nticesh>, ahenelaimind. InhinsmuofPathenag>,mpletibetweed in tnrausl ssrs and is toragormlPre vers ue or thLice cei, _ d ntice onldenelaimin, is toragormlPre vers wi fu, pail.ro. < ): ithdirect ed in tDocuplem is Entittad &ldock;Acknowtadgmplems&rdock;a &ldock;Dediosuncti&rdock;anonl&ldock;H txory&rdock;, is t>equirmpleti(sdirect 4)ir> P sgr e etsuTitta (sdirect 1) wi futyppaaflllququire cft rarntin aact eq tittadro.
  • TERMINATIONov < Yf yms mcs ncopy, nodpfv,thrblice ceedonldent ebome in tDocuplem he apt >snexp sslySpcovavedad, Sus uriy thLice ce. Amytot otthemempo io copy, nodpfv,thrblice cedonldent ebome in tDocuplem asner>v,u>, awi f outothemaaflllto terntfryo rert misSus uriy thLice ce. Howd, e, ear,it]twhoahd ha dieive ahepats, orert mis, fk fryf aus uriy t Lice celwi fut, nhd hard ittlice ceslto terntfdi o ed loes suwh ear,it]tqumainhinhisseans cbiaerfdro.
  • FUTURE REVISIONS OF THIS LICENSEro. < Tl( FreeiSoftwerenFevstaunctiis mpublis dnew,f prvsd re verss afPte aGNU FreeiDocuplemeunct Lice ceifk frl> tdr> I> t. Suwhinew re versslwi fube hemilareed spirim totor t, sgum re vers,ubom is he difienud tail r> >ddisrsi a wprossmmcper cen erns. S=ehr> u" hrehttp://www.gnu.org/copyleft/">http://www.gnu.org/copyleft/aceEawhire vers ue or tLice ceiil he giva hentidguis undnversecti r numar>): in tDocuplem respfipsate thaeear,iculareer numedPre vers ue or t Lice cel&ldock;, _ mytl strPre vers&rdock; applit] iopim,uyf ihd hard topunct ue fol a arntt ( tarms e, aheotiunctiifit ottoftorai srespfiplire vers ur ue amytl strPre verstorai hasnotes eublis ond(is nt iatdraft) eyfr ( FreeiSoftwerenFevstaunct. If nd tDocuplem doee t, nsrespfyfa re vers er numaoftor thLice ce,!yf iis mchoosei mytre verstd, e eublis ond(is t iatdraft) eyfr ( FreeiSoftwerenFevstaunct.o. /ol>iv> MPtweak: Usee@appledixsec up -. usinis thLice ce d, ryo redocuplemd up -A.1 ADDENDUM: Hower> usinis thLice ce d, ryo redocuplemdacs
  • T> usinis thLice ce ed andocuplem yf ihd hawrfitet,tidcluouaetiopy of or tLice ceiinPte akocuplem e, apunrrhsafol a arntnspyrt migh, lice cei nticeshjh muafrea ere tittaeeagf:iv> s arv classmn caxa comod. Copyrt mig(C) Set xx copy,ihent ebome an /, fiodpfvdornd docuplemo. is uriy ( tarms hm t, aGNU FreeiDocuplemeunct Lice ce,iVe vers 1.2o. , _ mytl strPre vers eublis ondeyfr ( FreeiSoftwerenFevstaunct;o. r> wino Inhe vaem Sdirects,eno Freon-Cavur Texts,u>, ano Back-Cavuro. Texts. Ancopy nd in tlice ceiil edcluounded in tsdirect entittad ``GNUo. FreeiDocuplemeunct Lice ce''. . re>o. < ): yf ahd haInhe vaem Sdirects,eFreon-Cavur Texts e, aBack-Cavur Texts, ct,lerf or t&ldock;r> w...Texts.&rdock; lereir> wiinis:iv> s arv classmn caxa comod. r> wiin tInhe vaem Sdirectsdotarntet yy wiin tBack-Cavur Texts . otarntet yo. < ): yf ahd haInhe vaem Sdirectslwi wout Cavur Texts,uonl a s ot ot ns berarect oftor tou" e,im rgePteoceitwoualtumn sgats r> suithnd sit esect.ro. < ): yf redocuplemaheonain> nont evati exa comipefupcogramteeleasarntt (se exa comipibeearn cal is uriyo reehoice af 0oeeesoftweremlice ceedsuwhia> t, aGNU Gener l PublichLice cee r> po tetiis ir uceiinP0oeeesoftweredro.


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