X-Git-Url: https://oss.titaniummirror.com/gitweb/?a=blobdiff_plain;f=mpfr%2Fmpfr.info;fp=mpfr%2Fmpfr.info;h=e1673794eba5dc2a2f8b5224bb855079c73f1c06;hb=6fed43773c9b0ce596dca5686f37ac3fc0fa11c0;hp=0000000000000000000000000000000000000000;hpb=27b11d56b743098deb193d510b337ba22dc52e5c;p=msp430-gcc.git diff --git a/mpfr/mpfr.info b/mpfr/mpfr.info new file mode 100644 index 00000000..e1673794 --- /dev/null +++ b/mpfr/mpfr.info @@ -0,0 +1,3570 @@ +This is ../mpfr.info, produced by makeinfo version 4.12 from +../mpfr.texi. + +This manual documents how to install and use the Multiple Precision +Floating-Point Reliable Library, version 2.4.1. + + Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, +2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software +Foundation, Inc. + + Permission is granted to copy, distribute and/or modify this +document under the terms of the GNU Free Documentation License, Version +1.2 or any later version published by the Free Software Foundation; +with no Invariant Sections, with no Front-Cover Texts, and with no +Back-Cover Texts. A copy of the license is included in *note GNU Free +Documentation License::. + +INFO-DIR-SECTION Software libraries +START-INFO-DIR-ENTRY +* mpfr: (mpfr). Multiple Precision Floating-Point Reliable Library. +END-INFO-DIR-ENTRY + + +File: mpfr.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir) + +GNU MPFR +******** + + This manual documents how to install and use the Multiple Precision +Floating-Point Reliable Library, version 2.4.1. + + Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, +2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009 Free Software +Foundation, Inc. + + Permission is granted to copy, distribute and/or modify this +document under the terms of the GNU Free Documentation License, Version +1.2 or any later version published by the Free Software Foundation; +with no Invariant Sections, with no Front-Cover Texts, and with no +Back-Cover Texts. A copy of the license is included in *note GNU Free +Documentation License::. + + +* Menu: + +* Copying:: MPFR Copying Conditions (LGPL). +* Introduction to MPFR:: Brief introduction to GNU MPFR. +* Installing MPFR:: How to configure and compile the MPFR library. +* Reporting Bugs:: How to usefully report bugs. +* MPFR Basics:: What every MPFR user should now. +* MPFR Interface:: MPFR functions and macros. +* Contributors:: +* References:: +* GNU Free Documentation License:: +* Concept Index:: +* Function Index:: + + +File: mpfr.info, Node: Copying, Next: Introduction to MPFR, Prev: Top, Up: Top + +MPFR Copying Conditions +*********************** + +This library is "free"; this means that everyone is free to use it and +free to redistribute it on a free basis. The library is not in the +public domain; it is copyrighted and there are restrictions on its +distribution, but these restrictions are designed to permit everything +that a good cooperating citizen would want to do. What is not allowed +is to try to prevent others from further sharing any version of this +library that they might get from you. + + Specifically, we want to make sure that you have the right to give +away copies of the library, that you receive source code or else can +get it if you want it, that you can change this library or use pieces +of it in new free programs, and that you know you can do these things. + + To make sure that everyone has such rights, we have to forbid you to +deprive anyone else of these rights. For example, if you distribute +copies of the GNU MPFR library, you must give the recipients all the +rights that you have. You must make sure that they, too, receive or +can get the source code. And you must tell them their rights. + + Also, for our own protection, we must make certain that everyone +finds out that there is no warranty for the GNU MPFR library. If it is +modified by someone else and passed on, we want their recipients to +know that what they have is not what we distributed, so that any +problems introduced by others will not reflect on our reputation. + + The precise conditions of the license for the GNU MPFR library are +found in the Lesser General Public License that accompanies the source +code. See the file COPYING.LIB. + + +File: mpfr.info, Node: Introduction to MPFR, Next: Installing MPFR, Prev: Copying, Up: Top + +1 Introduction to MPFR +********************** + +MPFR is a portable library written in C for arbitrary precision +arithmetic on floating-point numbers. It is based on the GNU MP library. +It aims to extend the class of floating-point numbers provided by the +GNU MP library by a precise semantics. The main differences with the +`mpf' class from GNU MP are: + + * the MPFR code is portable, i.e. the result of any operation does + not depend (or should not) on the machine word size + `mp_bits_per_limb' (32 or 64 on most machines); + + * the precision in bits can be set exactly to any valid value for + each variable (including very small precision); + + * MPFR provides the four rounding modes from the IEEE 754-1985 + standard. + + In particular, with a precision of 53 bits, MPFR should be able to +exactly reproduce all computations with double-precision machine +floating-point numbers (e.g., `double' type in C, with a C +implementation that rigorously follows Annex F of the ISO C99 standard +and `FP_CONTRACT' pragma set to `OFF') on the four arithmetic +operations and the square root, except the default exponent range is +much wider and subnormal numbers are not implemented (but can be +emulated). + + This version of MPFR is released under the GNU Lesser General Public +License, Version 2.1 or any later version. It is permitted to link +MPFR to most non-free programs, as long as when distributing them the +MPFR source code and a means to re-link with a modified MPFR library is +provided. + +1.1 How to Use This Manual +========================== + +Everyone should read *note MPFR Basics::. If you need to install the +library yourself, you need to read *note Installing MPFR::, too. + + The rest of the manual can be used for later reference, although it +is probably a good idea to glance through it. + + +File: mpfr.info, Node: Installing MPFR, Next: Reporting Bugs, Prev: Introduction to MPFR, Up: Top + +2 Installing MPFR +***************** + +2.1 How to Install +================== + +Here are the steps needed to install the library on Unix systems (more +details are provided in the `INSTALL' file): + + 1. To build MPFR, you first have to install GNU MP (version 4.1 or + higher) on your computer. You need a C compiler, preferably GCC, + but any reasonable compiler should work. And you need a standard + Unix `make' program, plus some other standard Unix utility + programs. + + 2. In the MPFR build directory, type `./configure' + + This will prepare the build and setup the options according to + your system. If you get error messages, you might check that you + use the same compiler and compile options as for GNU MP (see the + `INSTALL' file). + + 3. `make' + + This will compile MPFR, and create a library archive file + `libmpfr.a'. A dynamic library may be produced too (see + configure). + + 4. `make check' + + This will make sure MPFR was built correctly. If you get error + messages, please report this to `mpfr@loria.fr'. (*Note Reporting + Bugs::, for information on what to include in useful bug reports.) + + 5. `make install' + + This will copy the files `mpfr.h' and `mpf2mpfr.h' to the directory + `/usr/local/include', the file `libmpfr.a' to the directory + `/usr/local/lib', and the file `mpfr.info' to the directory + `/usr/local/share/info' (or if you passed the `--prefix' option to + `configure', using the prefix directory given as argument to + `--prefix' instead of `/usr/local'). + +2.2 Other `make' Targets +======================== + +There are some other useful make targets: + + * `mpfr.info' or `info' + + Create an info version of the manual, in `mpfr.info'. + + * `mpfr.pdf' or `pdf' + + Create a PDF version of the manual, in `mpfr.pdf'. + + * `mpfr.dvi' or `dvi' + + Create a DVI version of the manual, in `mpfr.dvi'. + + * `mpfr.ps' or `ps' + + Create a Postscript version of the manual, in `mpfr.ps'. + + * `mpfr.html' or `html' + + Create a HTML version of the manual, in several pages in the + directory `mpfr.html'; if you want only one output HTML file, then + type `makeinfo --html --no-split mpfr.texi' instead. + + * `clean' + + Delete all object files and archive files, but not the + configuration files. + + * `distclean' + + Delete all files not included in the distribution. + + * `uninstall' + + Delete all files copied by `make install'. + +2.3 Build Problems +================== + +In case of problem, please read the `INSTALL' file carefully before +reporting a bug, in particular section "In case of problem". Some +problems are due to bad configuration on the user side (not specific to +MPFR). Problems are also mentioned in the FAQ +`http://www.mpfr.org/faq.html'. + + Please report problems to `mpfr@loria.fr'. *Note Reporting Bugs::. +Some bug fixes are available on the MPFR 2.4.1 web page +`http://www.mpfr.org/mpfr-2.4.1/'. + +2.4 Getting the Latest Version of MPFR +====================================== + +The latest version of MPFR is available from +`ftp://ftp.gnu.org/gnu/mpfr/' or `http://www.mpfr.org/'. + + +File: mpfr.info, Node: Reporting Bugs, Next: MPFR Basics, Prev: Installing MPFR, Up: Top + +3 Reporting Bugs +**************** + +If you think you have found a bug in the MPFR library, first have a look +on the MPFR 2.4.1 web page `http://www.mpfr.org/mpfr-2.4.1/' and the +FAQ `http://www.mpfr.org/faq.html': perhaps this bug is already known, +in which case you may find there a workaround for it. Otherwise, please +investigate and report it. We have made this library available to you, +and it is not to ask too much from you, to ask you to report the bugs +that you find. + + There are a few things you should think about when you put your bug +report together. + + You have to send us a test case that makes it possible for us to +reproduce the bug. Include instructions on how to run the test case. + + You also have to explain what is wrong; if you get a crash, or if +the results printed are incorrect and in that case, in what way. + + Please include compiler version information in your bug report. This +can be extracted using `cc -V' on some machines, or, if you're using +gcc, `gcc -v'. Also, include the output from `uname -a' and the MPFR +version (the GMP version may be useful too). + + If your bug report is good, we will do our best to help you to get a +corrected version of the library; if the bug report is poor, we will +not do anything about it (aside of chiding you to send better bug +reports). + + Send your bug report to: `mpfr@loria.fr'. + + If you think something in this manual is unclear, or downright +incorrect, or if the language needs to be improved, please send a note +to the same address. + + +File: mpfr.info, Node: MPFR Basics, Next: MPFR Interface, Prev: Reporting Bugs, Up: Top + +4 MPFR Basics +************* + +4.1 Headers and Libraries +========================= + +All declarations needed to use MPFR are collected in the include file +`mpfr.h'. It is designed to work with both C and C++ compilers. You +should include that file in any program using the MPFR library: + + #include + + Note however that prototypes for MPFR functions with `FILE *' +parameters are provided only if `' is included too (before +`mpfr.h'). + + #include + #include + + Likewise `' (or `') is required for prototypes +with `va_list' parameters, such as `mpfr_vprintf'. + + You can avoid the use of MPFR macros encapsulating functions by +defining the `MPFR_USE_NO_MACRO' macro before `mpfr.h' is included. In +general this should not be necessary, but this can be useful when +debugging user code: with some macros, the compiler may emit spurious +warnings with some warning options, and macros can prevent some +prototype checking. + + All programs using MPFR must link against both `libmpfr' and +`libgmp' libraries. On a typical Unix-like system this can be done +with `-lmpfr -lgmp' (in that order), for example + + gcc myprogram.c -lmpfr -lgmp + + MPFR is built using Libtool and an application can use that to link +if desired, *note GNU Libtool: (libtool.info)Top. + + If MPFR has been installed to a non-standard location, then it may be +necessary to set up environment variables such as `C_INCLUDE_PATH' and +`LIBRARY_PATH', or use `-I' and `-L' compiler options, in order to +point to the right directories. For a shared library, it may also be +necessary to set up some sort of run-time library path (e.g., +`LD_LIBRARY_PATH') on some systems. Please read the `INSTALL' file for +additional information. + +4.2 Nomenclature and Types +========================== + +A "floating-point number" or "float" for short, is an arbitrary +precision significand (also called mantissa) with a limited precision +exponent. The C data type for such objects is `mpfr_t' (internally +defined as a one-element array of a structure, and `mpfr_ptr' is the C +data type representing a pointer to this structure). A floating-point +number can have three special values: Not-a-Number (NaN) or plus or +minus Infinity. NaN represents an uninitialized object, the result of +an invalid operation (like 0 divided by 0), or a value that cannot be +determined (like +Infinity minus +Infinity). Moreover, like in the IEEE +754-1985 standard, zero is signed, i.e. there are both +0 and -0; the +behavior is the same as in the IEEE 754-1985 standard and it is +generalized to the other functions supported by MPFR. + +The "precision" is the number of bits used to represent the significand +of a floating-point number; the corresponding C data type is +`mp_prec_t'. The precision can be any integer between `MPFR_PREC_MIN' +and `MPFR_PREC_MAX'. In the current implementation, `MPFR_PREC_MIN' is +equal to 2. + + Warning! MPFR needs to increase the precision internally, in order to +provide accurate results (and in particular, correct rounding). Do not +attempt to set the precision to any value near `MPFR_PREC_MAX', +otherwise MPFR will abort due to an assertion failure. Moreover, you +may reach some memory limit on your platform, in which case the program +may abort, crash or have undefined behavior (depending on your C +implementation). + +The "rounding mode" specifies the way to round the result of a +floating-point operation, in case the exact result can not be +represented exactly in the destination significand; the corresponding C +data type is `mp_rnd_t'. + +A "limb" means the part of a multi-precision number that fits in a +single word. (We chose this word because a limb of the human body is +analogous to a digit, only larger, and containing several digits.) +Normally a limb contains 32 or 64 bits. The C data type for a limb is +`mp_limb_t'. + +4.3 Function Classes +==================== + +There is only one class of functions in the MPFR library: + + 1. Functions for floating-point arithmetic, with names beginning with + `mpfr_'. The associated type is `mpfr_t'. + +4.4 MPFR Variable Conventions +============================= + +As a general rule, all MPFR functions expect output arguments before +input arguments. This notation is based on an analogy with the +assignment operator. + + MPFR allows you to use the same variable for both input and output +in the same expression. For example, the main function for +floating-point multiplication, `mpfr_mul', can be used like this: +`mpfr_mul (x, x, x, rnd_mode)'. This computes the square of X with +rounding mode `rnd_mode' and puts the result back in X. + + Before you can assign to an MPFR variable, you need to initialize it +by calling one of the special initialization functions. When you're +done with a variable, you need to clear it out, using one of the +functions for that purpose. + + A variable should only be initialized once, or at least cleared out +between each initialization. After a variable has been initialized, it +may be assigned to any number of times. + + For efficiency reasons, avoid to initialize and clear out a variable +in loops. Instead, initialize it before entering the loop, and clear +it out after the loop has exited. + + You do not need to be concerned about allocating additional space +for MPFR variables, since any variable has a significand of fixed size. +Hence unless you change its precision, or clear and reinitialize it, a +floating-point variable will have the same allocated space during all +its life. + +4.5 Rounding Modes +================== + +The following four rounding modes are supported: + + * `GMP_RNDN': round to nearest + + * `GMP_RNDZ': round toward zero + + * `GMP_RNDU': round toward plus infinity + + * `GMP_RNDD': round toward minus infinity + + The `round to nearest' mode works as in the IEEE 754-1985 standard: +in case the number to be rounded lies exactly in the middle of two +representable numbers, it is rounded to the one with the least +significant bit set to zero. For example, the number 5/2, which is +represented by (10.1) in binary, is rounded to (10.0)=2 with a +precision of two bits, and not to (11.0)=3. This rule avoids the +"drift" phenomenon mentioned by Knuth in volume 2 of The Art of +Computer Programming (Section 4.2.2). + + Most MPFR functions take as first argument the destination variable, +as second and following arguments the input variables, as last argument +a rounding mode, and have a return value of type `int', called the +"ternary value". The value stored in the destination variable is +correctly rounded, i.e. MPFR behaves as if it computed the result with +an infinite precision, then rounded it to the precision of this +variable. The input variables are regarded as exact (in particular, +their precision does not affect the result). + + As a consequence, in case of a non-zero real rounded result, the +error on the result is less or equal to 1/2 ulp (unit in the last +place) of the target in the rounding to nearest mode, and less than 1 +ulp of the target in the directed rounding modes (a ulp is the weight +of the least significant represented bit of the target after rounding). + + Unless documented otherwise, functions returning an `int' return a +ternary value. If the ternary value is zero, it means that the value +stored in the destination variable is the exact result of the +corresponding mathematical function. If the ternary value is positive +(resp. negative), it means the value stored in the destination variable +is greater (resp. lower) than the exact result. For example with the +`GMP_RNDU' rounding mode, the ternary value is usually positive, except +when the result is exact, in which case it is zero. In the case of an +infinite result, it is considered as inexact when it was obtained by +overflow, and exact otherwise. A NaN result (Not-a-Number) always +corresponds to an exact return value. The opposite of a returned +ternary value is guaranteed to be representable in an `int'. + + Unless documented otherwise, functions returning a `1' (or any other +value specified in this manual) for special cases (like `acos(0)') +should return an overflow or an underflow if `1' is not representable +in the current exponent range. + +4.6 Floating-Point Values on Special Numbers +============================================ + +This section specifies the floating-point values (of type `mpfr_t') +returned by MPFR functions. For functions returning several values (like +`mpfr_sin_cos'), the rules apply to each result separately. + + Functions can have one or several input arguments. An input point is +a mapping from these input arguments to the set of the MPFR numbers. +When none of its components are NaN, an input point can also be seen as +a tuple in the extended real numbers (the set of the real numbers with +both infinities). + + When the input point is in the domain of the mathematical function, +the result is rounded as described in Section "Rounding Modes" (but see +below for the specification of the sign of an exact zero). Otherwise +the general rules from this section apply unless stated otherwise in +the description of the MPFR function (*note MPFR Interface::). + + When the input point is not in the domain of the mathematical +function but is in its closure in the extended real numbers and the +function can be extended by continuity, the result is the obtained +limit. Examples: `mpfr_hypot' on (+Inf,0) gives +Inf. But `mpfr_pow' +cannot be defined on (1,+Inf) using this rule, as one can find +sequences (X_N,Y_N) such that X_N goes to 1, Y_N goes to +Inf and X_N +to the Y_N goes to any positive value when N goes to the infinity. + + When the input point is in the closure of the domain of the +mathematical function and an input argument is +0 (resp. -0), one +considers the limit when the corresponding argument approaches 0 from +above (resp. below). If the limit is not defined (e.g., `mpfr_log' on +-0), the behavior must be specified in the description of the MPFR +function. + + When the result is equal to 0, its sign is determined by considering +the limit as if the input point were not in the domain: If one +approaches 0 from above (resp. below), the result is +0 (resp. -0). In +the other cases, the sign must be specified in the description of the +MPFR function. Example: `mpfr_sin' on +0 gives +0. + + When the input point is not in the closure of the domain of the +function, the result is NaN. Example: `mpfr_sqrt' on -17 gives NaN. + + When an input argument is NaN, the result is NaN, possibly except +when a partial function is constant on the finite floating-point +numbers; such a case is always explicitly specified in *note MPFR +Interface::. Example: `mpfr_hypot' on (NaN,0) gives NaN, but +`mpfr_hypot' on (NaN,+Inf) gives +Inf (as specified in *note Special +Functions::), since for any finite input X, `mpfr_hypot' on (X,+Inf) +gives +Inf. + +4.7 Exceptions +============== + +MPFR supports 5 exception types: + + * Underflow: An underflow occurs when the exact result of a function + is a non-zero real number and the result obtained after the + rounding, assuming an unbounded exponent range (for the rounding), + has an exponent smaller than the minimum exponent of the current + range. In the round-to-nearest mode, the halfway case is rounded + toward zero. + + Note: This is not the single definition of the underflow. MPFR + chooses to consider the underflow after rounding. The underflow + before rounding can also be defined. For instance, consider a + function that has the exact result 7 multiplied by two to the power + E-4, where E is the smallest exponent (for a significand between + 1/2 and 1) in the current range, with a 2-bit target precision and + rounding toward plus infinity. The exact result has the exponent + E-1. With the underflow before rounding, such a function call + would yield an underflow, as E-1 is outside the current exponent + range. However, MPFR first considers the rounded result assuming + an unbounded exponent range. The exact result cannot be + represented exactly in precision 2, and here, it is rounded to 0.5 + times 2 to E, which is representable in the current exponent + range. As a consequence, this will not yield an underflow in MPFR. + + * Overflow: An overflow occurs when the exact result of a function + is a non-zero real number and the result obtained after the + rounding, assuming an unbounded exponent range (for the rounding), + has an exponent larger than the maximum exponent of the current + range. In the round-to-nearest mode, the result is infinite. + + * NaN: A NaN exception occurs when the result of a function is a NaN. + + * Inexact: An inexact exception occurs when the result of a function + cannot be represented exactly and must be rounded. + + * Range error: A range exception occurs when a function that does + not return a MPFR number (such as comparisons and conversions to + an integer) has an invalid result (e.g. an argument is NaN in + `mpfr_cmp' or in a conversion to an integer). + + + MPFR has a global flag for each exception, which can be cleared, set +or tested by functions described in *note Exception Related Functions::. + + Differences with the ISO C99 standard: + + * In C, only quiet NaNs are specified, and a NaN propagation does not + raise an invalid exception. Unless explicitly stated otherwise, + MPFR sets the NaN flag whenever a NaN is generated, even when a + NaN is propagated (e.g. in NaN + NaN), as if all NaNs were + signaling. + + * An invalid exception in C corresponds to either a NaN exception or + a range error in MPFR. + + +4.8 Memory Handling +=================== + +MPFR functions may create caches, e.g. when computing constants such as +Pi, either because the user has called a function like `mpfr_const_pi' +directly or because such a function was called internally by the MPFR +library itself to compute some other function. + + At any time, the user can free the various caches with +`mpfr_free_cache'. It is strongly advised to do that before terminating +a thread, or before exiting when using tools like `valgrind' (to avoid +memory leaks being reported). + + MPFR internal data such as flags, the exponent range, the default +precision and rounding mode, and caches (i.e., data that are not +accessed via parameters) are either global (if MPFR has not been +compiled as thread safe) or per-thread (thread local storage). + + +File: mpfr.info, Node: MPFR Interface, Next: Contributors, Prev: MPFR Basics, Up: Top + +5 MPFR Interface +**************** + +The floating-point functions expect arguments of type `mpfr_t'. + + The MPFR floating-point functions have an interface that is similar +to the GNU MP integer functions. The function prefix for +floating-point operations is `mpfr_'. + + There is one significant characteristic of floating-point numbers +that has motivated a difference between this function class and other +GNU MP function classes: the inherent inexactness of floating-point +arithmetic. The user has to specify the precision for each variable. +A computation that assigns a variable will take place with the +precision of the assigned variable; the cost of that computation should +not depend from the precision of variables used as input (on average). + + The semantics of a calculation in MPFR is specified as follows: +Compute the requested operation exactly (with "infinite accuracy"), and +round the result to the precision of the destination variable, with the +given rounding mode. The MPFR floating-point functions are intended to +be a smooth extension of the IEEE 754-1985 arithmetic. The results +obtained on one computer should not differ from the results obtained on +a computer with a different word size. + + MPFR does not keep track of the accuracy of a computation. This is +left to the user or to a higher layer. As a consequence, if two +variables are used to store only a few significant bits, and their +product is stored in a variable with large precision, then MPFR will +still compute the result with full precision. + + The value of the standard C macro `errno' may be set to non-zero by +any MPFR function or macro, whether or not there is an error. + +* Menu: + +* Initialization Functions:: +* Assignment Functions:: +* Combined Initialization and Assignment Functions:: +* Conversion Functions:: +* Basic Arithmetic Functions:: +* Comparison Functions:: +* Special Functions:: +* Input and Output Functions:: +* Formatted Output Functions:: +* Integer Related Functions:: +* Rounding Related Functions:: +* Miscellaneous Functions:: +* Exception Related Functions:: +* Compatibility with MPF:: +* Custom Interface:: +* Internals:: + + +File: mpfr.info, Node: Initialization Functions, Next: Assignment Functions, Prev: MPFR Interface, Up: MPFR Interface + +5.1 Initialization Functions +============================ + +An `mpfr_t' object must be initialized before storing the first value in +it. The functions `mpfr_init' and `mpfr_init2' are used for that +purpose. + + -- Function: void mpfr_init2 (mpfr_t X, mp_prec_t PREC) + Initialize X, set its precision to be *exactly* PREC bits and its + value to NaN. (Warning: the corresponding `mpf' functions + initialize to zero instead.) + + Normally, a variable should be initialized once only or at least + be cleared, using `mpfr_clear', between initializations. To + change the precision of a variable which has already been + initialized, use `mpfr_set_prec'. The precision PREC must be an + integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX' (otherwise the + behavior is undefined). + + -- Function: void mpfr_inits2 (mp_prec_t PREC, mpfr_t X, ...) + Initialize all the `mpfr_t' variables of the given `va_list', set + their precision to be *exactly* PREC bits and their value to NaN. + See `mpfr_init2' for more details. The `va_list' is assumed to be + composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It + begins from X. It ends when it encounters a null pointer (whose + type must also be `mpfr_ptr'). + + -- Function: void mpfr_clear (mpfr_t X) + Free the space occupied by X. Make sure to call this function for + all `mpfr_t' variables when you are done with them. + + -- Function: void mpfr_clears (mpfr_t X, ...) + Free the space occupied by all the `mpfr_t' variables of the given + `va_list'. See `mpfr_clear' for more details. The `va_list' is + assumed to be composed only of type `mpfr_t' (or equivalently + `mpfr_ptr'). It begins from X. It ends when it encounters a null + pointer (whose type must also be `mpfr_ptr'). + + Here is an example of how to use multiple initialization functions: + + { + mpfr_t x, y, z, t; + mpfr_inits2 (256, x, y, z, t, (mpfr_ptr) 0); + ... + mpfr_clears (x, y, z, t, (mpfr_ptr) 0); + } + + -- Function: void mpfr_init (mpfr_t X) + Initialize X and set its value to NaN. + + Normally, a variable should be initialized once only or at least + be cleared, using `mpfr_clear', between initializations. The + precision of X is the default precision, which can be changed by a + call to `mpfr_set_default_prec'. + + Warning! In a given program, some other libraries might change the + default precision and not restore it. Thus it is safer to use + `mpfr_init2'. + + -- Function: void mpfr_inits (mpfr_t X, ...) + Initialize all the `mpfr_t' variables of the given `va_list', set + their precision to be the default precision and their value to NaN. + See `mpfr_init' for more details. The `va_list' is assumed to be + composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It + begins from X. It ends when it encounters a null pointer (whose + type must also be `mpfr_ptr'). + + Warning! In a given program, some other libraries might change the + default precision and not restore it. Thus it is safer to use + `mpfr_inits2'. + + -- Macro: MPFR_DECL_INIT (NAME, PREC) + This macro declares NAME as an automatic variable of type `mpfr_t', + initializes it and sets its precision to be *exactly* PREC bits + and its value to NaN. NAME must be a valid identifier. You must + use this macro in the declaration section. This macro is much + faster than using `mpfr_init2' but has some drawbacks: + + * You *must not* call `mpfr_clear' with variables created with + this macro (the storage is allocated at the point of + declaration and deallocated when the brace-level is exited). + + * You *cannot* change their precision. + + * You *should not* create variables with huge precision with + this macro. + + * Your compiler must support `Non-Constant Initializers' + (standard in C++ and ISO C99) and `Token Pasting' (standard + in ISO C89). If PREC is not a constant expression, your + compiler must support `variable-length automatic arrays' + (standard in ISO C99). `GCC 2.95.3' and above supports all + these features. If you compile your program with gcc in c89 + mode and with `-pedantic', you may want to define the + `MPFR_USE_EXTENSION' macro to avoid warnings due to the + `MPFR_DECL_INIT' implementation. + + -- Function: void mpfr_set_default_prec (mp_prec_t PREC) + Set the default precision to be *exactly* PREC bits. The + precision of a variable means the number of bits used to store its + significand. All subsequent calls to `mpfr_init' will use this + precision, but previously initialized variables are unaffected. + This default precision is set to 53 bits initially. The precision + can be any integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. + + -- Function: mp_prec_t mpfr_get_default_prec (void) + Return the default MPFR precision in bits. + + Here is an example on how to initialize floating-point variables: + + { + mpfr_t x, y; + mpfr_init (x); /* use default precision */ + mpfr_init2 (y, 256); /* precision _exactly_ 256 bits */ + ... + /* When the program is about to exit, do ... */ + mpfr_clear (x); + mpfr_clear (y); + mpfr_free_cache (); + } + + The following functions are useful for changing the precision during +a calculation. A typical use would be for adjusting the precision +gradually in iterative algorithms like Newton-Raphson, making the +computation precision closely match the actual accurate part of the +numbers. + + -- Function: void mpfr_set_prec (mpfr_t X, mp_prec_t PREC) + Reset the precision of X to be *exactly* PREC bits, and set its + value to NaN. The previous value stored in X is lost. It is + equivalent to a call to `mpfr_clear(x)' followed by a call to + `mpfr_init2(x, prec)', but more efficient as no allocation is done + in case the current allocated space for the significand of X is + enough. The precision PREC can be any integer between + `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. + + In case you want to keep the previous value stored in X, use + `mpfr_prec_round' instead. + + -- Function: mp_prec_t mpfr_get_prec (mpfr_t X) + Return the precision actually used for assignments of X, i.e. the + number of bits used to store its significand. + + +File: mpfr.info, Node: Assignment Functions, Next: Combined Initialization and Assignment Functions, Prev: Initialization Functions, Up: MPFR Interface + +5.2 Assignment Functions +======================== + +These functions assign new values to already initialized floats (*note +Initialization Functions::). When using any functions using `intmax_t', +you must include `' or `' before `mpfr.h', to +allow `mpfr.h' to define prototypes for these functions. + + -- Function: int mpfr_set (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_set_ui (mpfr_t ROP, unsigned long int OP, + mp_rnd_t RND) + -- Function: int mpfr_set_si (mpfr_t ROP, long int OP, mp_rnd_t RND) + -- Function: int mpfr_set_uj (mpfr_t ROP, uintmax_t OP, mp_rnd_t RND) + -- Function: int mpfr_set_sj (mpfr_t ROP, intmax_t OP, mp_rnd_t RND) + -- Function: int mpfr_set_d (mpfr_t ROP, double OP, mp_rnd_t RND) + -- Function: int mpfr_set_ld (mpfr_t ROP, long double OP, mp_rnd_t RND) + -- Function: int mpfr_set_decimal64 (mpfr_t ROP, _Decimal64 OP, + mp_rnd_t RND) + -- Function: int mpfr_set_z (mpfr_t ROP, mpz_t OP, mp_rnd_t RND) + -- Function: int mpfr_set_q (mpfr_t ROP, mpq_t OP, mp_rnd_t RND) + -- Function: int mpfr_set_f (mpfr_t ROP, mpf_t OP, mp_rnd_t RND) + Set the value of ROP from OP, rounded toward the given direction + RND. Note that the input 0 is converted to +0 by `mpfr_set_ui', + `mpfr_set_si', `mpfr_set_sj', `mpfr_set_uj', `mpfr_set_z', + `mpfr_set_q' and `mpfr_set_f', regardless of the rounding mode. + If the system does not support the IEEE-754 standard, `mpfr_set_d', + `mpfr_set_ld' and `mpfr_set_decimal64' might not preserve the + signed zeros. The `mpfr_set_decimal64' function is built only + with the configure option `--enable-decimal-float', which also + requires `--with-gmp-build', and when the compiler or system + provides the `_Decimal64' data type (GCC version 4.2.0 is known to + support this data type, but only when configured with + `--enable-decimal-float' too). `mpfr_set_q' might not be able to + work if the numerator (or the denominator) can not be + representable as a `mpfr_t'. + + Note: If you want to store a floating-point constant to a `mpfr_t', + you should use `mpfr_set_str' (or one of the MPFR constant + functions, such as `mpfr_const_pi' for Pi) instead of `mpfr_set_d', + `mpfr_set_ld' or `mpfr_set_decimal64'. Otherwise the + floating-point constant will be first converted into a + reduced-precision (e.g., 53-bit) binary number before MPFR can + work with it. + + -- Function: int mpfr_set_ui_2exp (mpfr_t ROP, unsigned long int OP, + mp_exp_t E, mp_rnd_t RND) + -- Function: int mpfr_set_si_2exp (mpfr_t ROP, long int OP, mp_exp_t + E, mp_rnd_t RND) + -- Function: int mpfr_set_uj_2exp (mpfr_t ROP, uintmax_t OP, intmax_t + E, mp_rnd_t RND) + -- Function: int mpfr_set_sj_2exp (mpfr_t ROP, intmax_t OP, intmax_t + E, mp_rnd_t RND) + Set the value of ROP from OP multiplied by two to the power E, + rounded toward the given direction RND. Note that the input 0 is + converted to +0. + + -- Function: int mpfr_set_str (mpfr_t ROP, const char *S, int BASE, + mp_rnd_t RND) + Set ROP to the value of the string S in base BASE, rounded in the + direction RND. See the documentation of `mpfr_strtofr' for a + detailed description of the valid string formats. Contrary to + `mpfr_strtofr', `mpfr_set_str' requires the _whole_ string to + represent a valid floating-point number. This function returns 0 + if the entire string up to the final null character is a valid + number in base BASE; otherwise it returns -1, and ROP may have + changed. + + -- Function: int mpfr_strtofr (mpfr_t ROP, const char *NPTR, char + **ENDPTR, int BASE, mp_rnd_t RND) + Read a floating-point number from a string NPTR in base BASE, + rounded in the direction RND; BASE must be either 0 (to detect the + base, as described below) or a number from 2 to 36 (otherwise the + behavior is undefined). If NPTR starts with valid data, the result + is stored in ROP and `*ENDPTR' points to the character just after + the valid data (if ENDPTR is not a null pointer); otherwise ROP is + set to zero and the value of NPTR is stored in the location + referenced by ENDPTR (if ENDPTR is not a null pointer). The usual + ternary value is returned. + + Parsing follows the standard C `strtod' function with some + extensions. Case is ignored. After optional leading whitespace, + one has a subject sequence consisting of an optional sign (`+' or + `-'), and either numeric data or special data. The subject + sequence is defined as the longest initial subsequence of the + input string, starting with the first non-whitespace character, + that is of the expected form. + + The form of numeric data is a non-empty sequence of significand + digits with an optional decimal point, and an optional exponent + consisting of an exponent prefix followed by an optional sign and + a non-empty sequence of decimal digits. A significand digit is + either a decimal digit or a Latin letter (62 possible characters), + with `a' = 10, `b' = 11, ..., `z' = 35; its value must be strictly + less than the base. The decimal point can be either the one + defined by the current locale or the period (the first one is + accepted for consistency with the C standard and the practice, the + second one is accepted to allow the programmer to provide MPFR + numbers from strings in a way that does not depend on the current + locale). The exponent prefix can be `e' or `E' for bases up to + 10, or `@' in any base; it indicates a multiplication by a power + of the base. In bases 2 and 16, the exponent prefix can also be + `p' or `P', in which case it introduces a binary exponent: it + indicates a multiplication by a power of 2 (there is a difference + only for base 16). The value of an exponent is always written in + base 10. In base 2, the significand can start with `0b' or `0B', + and in base 16, it can start with `0x' or `0X'. + + If the argument BASE is 0, then the base is automatically detected + as follows. If the significand starts with `0b' or `0B', base 2 is + assumed. If the significand starts with `0x' or `0X', base 16 is + assumed. Otherwise base 10 is assumed. + + Note: The exponent must contain at least a digit. Otherwise the + possible exponent prefix and sign are not part of the number + (which ends with the significand). Similarly, if `0b', `0B', `0x' + or `0X' is not followed by a binary/hexadecimal digit, then the + subject sequence stops at the character `0'. + + Special data (for infinities and NaN) can be `@inf@' or + `@nan@(n-char-sequence)', and if BASE <= 16, it can also be + `infinity', `inf', `nan' or `nan(n-char-sequence)', all case + insensitive. A `n-char-sequence' is a non-empty string containing + only digits, Latin letters and the underscore (0, 1, 2, ..., 9, a, + b, ..., z, A, B, ..., Z, _). Note: one has an optional sign for + all data, even NaN. + + + -- Function: void mpfr_set_inf (mpfr_t X, int SIGN) + -- Function: void mpfr_set_nan (mpfr_t X) + Set the variable X to infinity or NaN (Not-a-Number) respectively. + In `mpfr_set_inf', X is set to plus infinity iff SIGN is + nonnegative. + + -- Function: void mpfr_swap (mpfr_t X, mpfr_t Y) + Swap the values X and Y efficiently. Warning: the precisions are + exchanged too; in case the precisions are different, `mpfr_swap' + is thus not equivalent to three `mpfr_set' calls using a third + auxiliary variable. + + +File: mpfr.info, Node: Combined Initialization and Assignment Functions, Next: Conversion Functions, Prev: Assignment Functions, Up: MPFR Interface + +5.3 Combined Initialization and Assignment Functions +==================================================== + + -- Macro: int mpfr_init_set (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Macro: int mpfr_init_set_ui (mpfr_t ROP, unsigned long int OP, + mp_rnd_t RND) + -- Macro: int mpfr_init_set_si (mpfr_t ROP, signed long int OP, + mp_rnd_t RND) + -- Macro: int mpfr_init_set_d (mpfr_t ROP, double OP, mp_rnd_t RND) + -- Macro: int mpfr_init_set_ld (mpfr_t ROP, long double OP, mp_rnd_t + RND) + -- Macro: int mpfr_init_set_z (mpfr_t ROP, mpz_t OP, mp_rnd_t RND) + -- Macro: int mpfr_init_set_q (mpfr_t ROP, mpq_t OP, mp_rnd_t RND) + -- Macro: int mpfr_init_set_f (mpfr_t ROP, mpf_t OP, mp_rnd_t RND) + Initialize ROP and set its value from OP, rounded in the direction + RND. The precision of ROP will be taken from the active default + precision, as set by `mpfr_set_default_prec'. + + -- Function: int mpfr_init_set_str (mpfr_t X, const char *S, int BASE, + mp_rnd_t RND) + Initialize X and set its value from the string S in base BASE, + rounded in the direction RND. See `mpfr_set_str'. + + +File: mpfr.info, Node: Conversion Functions, Next: Basic Arithmetic Functions, Prev: Combined Initialization and Assignment Functions, Up: MPFR Interface + +5.4 Conversion Functions +======================== + + -- Function: double mpfr_get_d (mpfr_t OP, mp_rnd_t RND) + -- Function: long double mpfr_get_ld (mpfr_t OP, mp_rnd_t RND) + -- Function: _Decimal64 mpfr_get_decimal64 (mpfr_t OP, mp_rnd_t RND) + Convert OP to a `double' (respectively `_Decimal64' or `long + double'), using the rounding mode RND. If OP is NaN, some fixed + NaN (either quiet or signaling) or the result of 0.0/0.0 is + returned. If OP is ±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 _erange_ flag is set too. See also `mpfr_fits_slong_p', + `mpfr_fits_ulong_p', `mpfr_fits_intmax_p' and + `mpfr_fits_uintmax_p'. + + -- Function: mp_exp_t mpfr_get_z_exp (mpz_t ROP, mpfr_t OP) + Put the scaled significand of OP (regarded as an integer, with the + precision of OP) into ROP, and return the exponent EXP (which may + be outside the current exponent range) such that OP exactly equals + ROP multiplied by two exponent EXP. If the exponent is not + representable in the `mp_exp_t' type, the behavior is undefined. + + -- Function: void mpfr_get_z (mpz_t ROP, mpfr_t OP, mp_rnd_t RND) + Convert OP to a `mpz_t', after rounding it with respect to RND. If + OP is NaN or Inf, the result is undefined. + + -- Function: int mpfr_get_f (mpf_t ROP, mpfr_t OP, mp_rnd_t RND) + Convert OP to a `mpf_t', after rounding it with respect to RND. + Return zero iff no error occurred, in particular a non-zero value + is returned if OP is NaN or Inf, which do not exist in `mpf'. + + -- Function: char * mpfr_get_str (char *STR, mp_exp_t *EXPPTR, int B, + size_t N, mpfr_t OP, mp_rnd_t RND) + Convert OP to a string of digits in base B, with rounding in the + direction RND, where N is either zero (see below) or the number of + significant digits; in the latter case, N must be greater or equal + to 2. The base may vary from 2 to 36. + + The generated string is a fraction, with an implicit radix point + immediately to the left of the first digit. For example, the + number -3.1416 would be returned as "-31416" in the string and 1 + written at EXPPTR. If RND is to nearest, and OP is exactly in the + middle of two possible outputs, the one with an even last digit is + chosen (for an odd base, this may not correspond to an even + significand). + + If N is zero, the number of digits of the significand is chosen + large enough so that re-reading the printed value with the same + precision, assuming both output and input use rounding to nearest, + will recover the original value of OP. More precisely, in most + cases, the chosen precision of STR is the minimal precision + depending on N and B only that satisfies the above property, i.e., + m = 1 + ceil(N*log(2)/log(B)), but in some very rare cases, it + might be m+1. + + If STR is a null pointer, space for the significand is allocated + using the current allocation function, and a pointer to the string + is returned. To free the returned string, you must use + `mpfr_free_str'. + + If STR is not a null pointer, it should point to a block of storage + large enough for the significand, i.e., at least `max(N + 2, 7)'. + The extra two bytes are for a possible minus sign, and for the + terminating null character. + + If the input number is an ordinary number, the exponent is written + through the pointer EXPPTR (the current minimal exponent for 0). + + A pointer to the string is returned, unless there is an error, in + which case a null pointer is returned. + + -- Function: void mpfr_free_str (char *STR) + Free a string allocated by `mpfr_get_str' using the current + unallocation function (preliminary interface). The block is + assumed to be `strlen(STR)+1' bytes. For more information about + how it is done: *note Custom Allocation: (gmp.info)Custom + Allocation. + + -- Function: int mpfr_fits_ulong_p (mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_fits_slong_p (mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_fits_uint_p (mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_fits_sint_p (mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_fits_ushort_p (mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_fits_sshort_p (mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_fits_intmax_p (mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_fits_uintmax_p (mpfr_t OP, mp_rnd_t RND) + Return non-zero if OP would fit in the respective C data type, when + rounded to an integer in the direction RND. + + +File: mpfr.info, Node: Basic Arithmetic Functions, Next: Comparison Functions, Prev: Conversion Functions, Up: MPFR Interface + +5.5 Basic Arithmetic Functions +============================== + + -- Function: int mpfr_add (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_add_ui (mpfr_t ROP, mpfr_t OP1, unsigned long + int OP2, mp_rnd_t RND) + -- Function: int mpfr_add_si (mpfr_t ROP, mpfr_t OP1, long int OP2, + mp_rnd_t RND) + -- Function: int mpfr_add_d (mpfr_t ROP, mpfr_t OP1, double OP2, + mp_rnd_t RND) + -- Function: int mpfr_add_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_add_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, + mp_rnd_t RND) + Set ROP to OP1 + OP2 rounded in the direction RND. For types + having no signed zero, it is considered unsigned (i.e. (+0) + 0 = + (+0) and (-0) + 0 = (-0)). The `mpfr_add_d' function assumes that + the radix of the `double' type is a power of 2, with a precision + at most that declared by the C implementation (macro + `IEEE_DBL_MANT_DIG', and if not defined 53 bits). + + -- Function: int mpfr_sub (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_ui_sub (mpfr_t ROP, unsigned long int OP1, + mpfr_t OP2, mp_rnd_t RND) + -- Function: int mpfr_sub_ui (mpfr_t ROP, mpfr_t OP1, unsigned long + int OP2, mp_rnd_t RND) + -- Function: int mpfr_si_sub (mpfr_t ROP, long int OP1, mpfr_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_sub_si (mpfr_t ROP, mpfr_t OP1, long int OP2, + mp_rnd_t RND) + -- Function: int mpfr_d_sub (mpfr_t ROP, double OP1, mpfr_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_sub_d (mpfr_t ROP, mpfr_t OP1, double OP2, + mp_rnd_t RND) + -- Function: int mpfr_sub_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_sub_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, + mp_rnd_t RND) + Set ROP to OP1 - OP2 rounded in the direction RND. For types + having no signed zero, it is considered unsigned (i.e. (+0) - 0 = + (+0), (-0) - 0 = (-0), 0 - (+0) = (-0) and 0 - (-0) = (+0)). The + same restrictions than for `mpfr_add_d' apply to `mpfr_d_sub' and + `mpfr_sub_d'. + + -- Function: int mpfr_mul (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_mul_ui (mpfr_t ROP, mpfr_t OP1, unsigned long + int OP2, mp_rnd_t RND) + -- Function: int mpfr_mul_si (mpfr_t ROP, mpfr_t OP1, long int OP2, + mp_rnd_t RND) + -- Function: int mpfr_mul_d (mpfr_t ROP, mpfr_t OP1, double OP2, + mp_rnd_t RND) + -- Function: int mpfr_mul_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_mul_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, + mp_rnd_t RND) + Set ROP to OP1 times OP2 rounded in the direction RND. When a + result is zero, its sign is the product of the signs of the + operands (for types having no signed zero, it is considered + positive). The same restrictions than for `mpfr_add_d' apply to + `mpfr_mul_d'. + + -- Function: int mpfr_sqr (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the square of OP rounded in the direction RND. + + -- Function: int mpfr_div (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_ui_div (mpfr_t ROP, unsigned long int OP1, + mpfr_t OP2, mp_rnd_t RND) + -- Function: int mpfr_div_ui (mpfr_t ROP, mpfr_t OP1, unsigned long + int OP2, mp_rnd_t RND) + -- Function: int mpfr_si_div (mpfr_t ROP, long int OP1, mpfr_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_div_si (mpfr_t ROP, mpfr_t OP1, long int OP2, + mp_rnd_t RND) + -- Function: int mpfr_d_div (mpfr_t ROP, double OP1, mpfr_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_div_d (mpfr_t ROP, mpfr_t OP1, double OP2, + mp_rnd_t RND) + -- Function: int mpfr_div_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_div_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, + mp_rnd_t RND) + Set ROP to OP1/OP2 rounded in the direction RND. When a result is + zero, its sign is the product of the signs of the operands (for + types having no signed zero, it is considered positive). The same + restrictions than for `mpfr_add_d' apply to `mpfr_d_div' and + `mpfr_div_d'. + + -- Function: int mpfr_sqrt (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_sqrt_ui (mpfr_t ROP, unsigned long int OP, + mp_rnd_t RND) + Set ROP to the square root of OP rounded in the direction RND. + Return -0 if OP is -0 (to be consistent with the IEEE 754-1985 + standard). Set ROP to NaN if OP is negative. + + -- Function: int mpfr_rec_sqrt (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the reciprocal square root of OP rounded in the + direction RND. Return +Inf if OP is ±0, and +0 if OP is +Inf. Set + ROP to NaN if OP is negative. + + -- Function: int mpfr_cbrt (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_root (mpfr_t ROP, mpfr_t OP, unsigned long int + K, mp_rnd_t RND) + Set ROP to the cubic root (resp. the Kth root) of OP rounded in + the direction RND. An odd (resp. even) root of a negative number + (including -Inf) returns a negative number (resp. NaN). The Kth + root of -0 is defined to be -0, whatever the parity of K. + + -- Function: int mpfr_pow (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_pow_ui (mpfr_t ROP, mpfr_t OP1, unsigned long + int OP2, mp_rnd_t RND) + -- Function: int mpfr_pow_si (mpfr_t ROP, mpfr_t OP1, long int OP2, + mp_rnd_t RND) + -- Function: int mpfr_pow_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, + mp_rnd_t RND) + -- Function: int mpfr_ui_pow_ui (mpfr_t ROP, unsigned long int OP1, + unsigned long int OP2, mp_rnd_t RND) + -- Function: int mpfr_ui_pow (mpfr_t ROP, unsigned long int OP1, + mpfr_t OP2, mp_rnd_t RND) + Set ROP to OP1 raised to OP2, rounded in the direction RND. + Special values are currently handled as described in the ISO C99 + standard for the `pow' function (note this may change in future + versions): + * `pow(±0, Y)' returns plus or minus infinity for Y a negative + odd integer. + + * `pow(±0, Y)' returns plus infinity for Y negative and not an + odd integer. + + * `pow(±0, Y)' returns plus or minus zero for Y a positive odd + integer. + + * `pow(±0, Y)' returns plus zero for Y positive and not an odd + integer. + + * `pow(-1, ±Inf)' returns 1. + + * `pow(+1, Y)' returns 1 for any Y, even a NaN. + + * `pow(X, ±0)' returns 1 for any X, even a NaN. + + * `pow(X, Y)' returns NaN for finite negative X and finite + non-integer Y. + + * `pow(X, -Inf)' returns plus infinity for 0 < abs(x) < 1, and + plus zero for abs(x) > 1. + + * `pow(X, +Inf)' returns plus zero for 0 < abs(x) < 1, and plus + infinity for abs(x) > 1. + + * `pow(-Inf, Y)' returns minus zero for Y a negative odd + integer. + + * `pow(-Inf, Y)' returns plus zero for Y negative and not an + odd integer. + + * `pow(-Inf, Y)' returns minus infinity for Y a positive odd + integer. + + * `pow(-Inf, Y)' returns plus infinity for Y positive and not + an odd integer. + + * `pow(+Inf, Y)' returns plus zero for Y negative, and plus + infinity for Y positive. + + -- Function: int mpfr_neg (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to -OP rounded in the direction RND. Just changes the + sign if ROP and OP are the same variable. + + -- Function: int mpfr_abs (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the absolute value of OP, rounded in the direction RND. + Just changes the sign if ROP and OP are the same variable. + + -- Function: int mpfr_dim (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, + mp_rnd_t RND) + Set ROP to the positive difference of OP1 and OP2, i.e., OP1 - OP2 + rounded in the direction RND if OP1 > OP2, and +0 otherwise. + Returns NaN when OP1 or OP2 is NaN. + + -- Function: int mpfr_mul_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long + int OP2, mp_rnd_t RND) + -- Function: int mpfr_mul_2si (mpfr_t ROP, mpfr_t OP1, long int OP2, + mp_rnd_t RND) + Set ROP to OP1 times 2 raised to OP2 rounded in the direction RND. + Just increases the exponent by OP2 when ROP and OP1 are identical. + + -- Function: int mpfr_div_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long + int OP2, mp_rnd_t RND) + -- Function: int mpfr_div_2si (mpfr_t ROP, mpfr_t OP1, long int OP2, + mp_rnd_t RND) + Set ROP to OP1 divided by 2 raised to OP2 rounded in the direction + RND. Just decreases the exponent by OP2 when ROP and OP1 are + identical. + + +File: mpfr.info, Node: Comparison Functions, Next: Special Functions, Prev: Basic Arithmetic Functions, Up: MPFR Interface + +5.6 Comparison Functions +======================== + + -- Function: int mpfr_cmp (mpfr_t OP1, mpfr_t OP2) + -- Function: int mpfr_cmp_ui (mpfr_t OP1, unsigned long int OP2) + -- Function: int mpfr_cmp_si (mpfr_t OP1, signed long int OP2) + -- Function: int mpfr_cmp_d (mpfr_t OP1, double OP2) + -- Function: int mpfr_cmp_ld (mpfr_t OP1, long double OP2) + -- Function: int mpfr_cmp_z (mpfr_t OP1, mpz_t OP2) + -- Function: int mpfr_cmp_q (mpfr_t OP1, mpq_t OP2) + -- Function: int mpfr_cmp_f (mpfr_t OP1, mpf_t OP2) + Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero + if OP1 = OP2, and a negative value if OP1 < OP2. Both OP1 and OP2 + are considered to their full own precision, which may differ. If + one of the operands is NaN, set the _erange_ flag and return zero. + + Note: These functions may be useful to distinguish the three + possible cases. If you need to distinguish two cases only, it is + recommended to use the predicate functions (e.g., `mpfr_equal_p' + for the equality) described below; they behave like the IEEE-754 + comparisons, in particular when one or both arguments are NaN. But + only floating-point numbers can be compared (you may need to do a + conversion first). + + -- Function: int mpfr_cmp_ui_2exp (mpfr_t OP1, unsigned long int OP2, + mp_exp_t E) + -- Function: int mpfr_cmp_si_2exp (mpfr_t OP1, long int OP2, mp_exp_t + E) + Compare OP1 and OP2 multiplied by two to the power E. Similar as + above. + + -- Function: int mpfr_cmpabs (mpfr_t OP1, mpfr_t OP2) + Compare |OP1| and |OP2|. Return a positive value if |OP1| > + |OP2|, zero if |OP1| = |OP2|, and a negative value if |OP1| < + |OP2|. If one of the operands is NaN, set the _erange_ flag and + return zero. + + -- Function: int mpfr_nan_p (mpfr_t OP) + -- Function: int mpfr_inf_p (mpfr_t OP) + -- Function: int mpfr_number_p (mpfr_t OP) + -- Function: int mpfr_zero_p (mpfr_t OP) + Return non-zero if OP is respectively NaN, an infinity, an ordinary + number (i.e. neither NaN nor an infinity) or zero. Return zero + otherwise. + + -- Macro: int mpfr_sgn (mpfr_t OP) + Return a positive value if OP > 0, zero if OP = 0, and a negative + value if OP < 0. If the operand is NaN, set the _erange_ flag and + return zero. + + -- Function: int mpfr_greater_p (mpfr_t OP1, mpfr_t OP2) + Return non-zero if OP1 > OP2, zero otherwise. + + -- Function: int mpfr_greaterequal_p (mpfr_t OP1, mpfr_t OP2) + Return non-zero if OP1 >= OP2, zero otherwise. + + -- Function: int mpfr_less_p (mpfr_t OP1, mpfr_t OP2) + Return non-zero if OP1 < OP2, zero otherwise. + + -- Function: int mpfr_lessequal_p (mpfr_t OP1, mpfr_t OP2) + Return non-zero if OP1 <= OP2, zero otherwise. + + -- Function: int mpfr_lessgreater_p (mpfr_t OP1, mpfr_t OP2) + Return non-zero if OP1 < OP2 or OP1 > OP2 (i.e. neither OP1, nor + OP2 is NaN, and OP1 <> OP2), zero otherwise (i.e. OP1 and/or OP2 + are NaN, or OP1 = OP2). + + -- Function: int mpfr_equal_p (mpfr_t OP1, mpfr_t OP2) + Return non-zero if OP1 = OP2, zero otherwise (i.e. OP1 and/or OP2 + are NaN, or OP1 <> OP2). + + -- Function: int mpfr_unordered_p (mpfr_t OP1, mpfr_t OP2) + Return non-zero if OP1 or OP2 is a NaN (i.e. they cannot be + compared), zero otherwise. + + +File: mpfr.info, Node: Special Functions, Next: Input and Output Functions, Prev: Comparison Functions, Up: MPFR Interface + +5.7 Special Functions +===================== + +All those functions, except explicitly stated, return zero for an exact +return value, a positive value for a return value larger than the exact +result, and a negative value otherwise. + + Important note: in some domains, computing special functions (either +with correct or incorrect rounding) is expensive, even for small +precision, for example the trigonometric and Bessel functions for large +argument. + + -- Function: int mpfr_log (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_log2 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_log10 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the natural logarithm of OP, log2(OP) or log10(OP), + respectively, rounded in the direction RND. Return -Inf if OP is + -0 (i.e. the sign of the zero has no influence on the result). + + -- Function: int mpfr_exp (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_exp2 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_exp10 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the exponential of OP, to 2 power of OP or to 10 power + of OP, respectively, rounded in the direction RND. + + -- Function: int mpfr_cos (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_sin (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_tan (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the cosine of OP, sine of OP, tangent of OP, rounded in + the direction RND. + + -- Function: int mpfr_sec (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_csc (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_cot (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the secant of OP, cosecant of OP, cotangent of OP, + rounded in the direction RND. + + -- Function: int mpfr_sin_cos (mpfr_t SOP, mpfr_t COP, mpfr_t OP, + mp_rnd_t RND) + Set simultaneously SOP to the sine of OP and + COP to the cosine of OP, rounded in the direction RND with the + corresponding precisions of SOP and COP, which must be different + variables. Return 0 iff both results are exact. + + -- Function: int mpfr_acos (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_asin (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_atan (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the arc-cosine, arc-sine or arc-tangent of OP, rounded + in the direction RND. Note that since `acos(-1)' returns the + floating-point number closest to Pi according to the given + rounding mode, this number might not be in the output range 0 <= + ROP < \pi of the arc-cosine function; still, the result lies in + the image of the output range by the rounding function. The same + holds for `asin(-1)', `asin(1)', `atan(-Inf)', `atan(+Inf)'. + + -- Function: int mpfr_atan2 (mpfr_t ROP, mpfr_t Y, mpfr_t X, mp_rnd_t + RND) + Set ROP to the arc-tangent2 of Y and X, rounded in the direction + RND: if `x > 0', `atan2(y, x) = atan (y/x)'; if `x < 0', `atan2(y, + x) = sign(y)*(Pi - atan (abs(y/x)))'. As for `atan', in case the + exact mathematical result is +Pi or -Pi, its rounded result might + be outside the function output range. + + `atan2(y, 0)' does not raise any floating-point exception. + Special values are currently handled as described in the ISO C99 + standard for the `atan2' function (note this may change in future + versions): + * `atan2(+0, -0)' returns +Pi. + + * `atan2(-0, -0)' returns -Pi. + + * `atan2(+0, +0)' returns +0. + + * `atan2(-0, +0)' returns -0. + + * `atan2(+0, x)' returns +Pi for x < 0. + + * `atan2(-0, x)' returns -Pi for x < 0. + + * `atan2(+0, x)' returns +0 for x > 0. + + * `atan2(-0, x)' returns -0 for x > 0. + + * `atan2(y, 0)' returns -Pi/2 for y < 0. + + * `atan2(y, 0)' returns +Pi/2 for y > 0. + + * `atan2(+Inf, -Inf)' returns +3*Pi/4. + + * `atan2(-Inf, -Inf)' returns -3*Pi/4. + + * `atan2(+Inf, +Inf)' returns +Pi/4. + + * `atan2(-Inf, +Inf)' returns -Pi/4. + + * `atan2(+Inf, x)' returns +Pi/2 for finite x. + + * `atan2(-Inf, x)' returns -Pi/2 for finite x. + + * `atan2(y, -Inf)' returns +Pi for finite y > 0. + + * `atan2(y, -Inf)' returns -Pi for finite y < 0. + + * `atan2(y, +Inf)' returns +0 for finite y > 0. + + * `atan2(y, +Inf)' returns -0 for finite y < 0. + + -- Function: int mpfr_cosh (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_sinh (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_tanh (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the hyperbolic cosine, sine or tangent of OP, rounded + in the direction RND. + + -- Function: int mpfr_sinh_cosh (mpfr_t SOP, mpfr_t COP, mpfr_t OP, + mp_rnd_t RND) + Set simultaneously SOP to the hyperbolic sine of OP and + COP to the hyperbolic cosine of OP, rounded in the + direction RND with the corresponding precision of SOP and COP + which must be different variables. Return 0 iff both results are + exact. + + -- Function: int mpfr_sech (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_csch (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_coth (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the hyperbolic secant of OP, cosecant of OP, cotangent + of OP, rounded in the direction RND. + + -- Function: int mpfr_acosh (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_asinh (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_atanh (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the inverse hyperbolic cosine, sine or tangent of OP, + rounded in the direction RND. + + -- Function: int mpfr_fac_ui (mpfr_t ROP, unsigned long int OP, + mp_rnd_t RND) + Set ROP to the factorial of the `unsigned long int' OP, rounded in + the direction RND. + + -- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the logarithm of one plus OP, rounded in the direction + RND. + + -- Function: int mpfr_expm1 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the exponential of OP minus one, rounded in the + direction RND. + + -- Function: int mpfr_eint (mpfr_t Y, mpfr_t X, mp_rnd_t RND) + Set Y to the exponential integral of X, rounded in the direction + RND. For positive X, the exponential integral is the sum of + Euler's constant, of the logarithm of X, and of the sum for k from + 1 to infinity of X to the power k, divided by k and factorial(k). + For negative X, the returned value is NaN. + + -- Function: int mpfr_li2 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND_MODE) + Set ROP to real part of the dilogarithm of OP, rounded in the + direction RND_MODE. The dilogarithm function is defined here as + the integral of -log(1-t)/t from 0 to x. + + -- Function: int mpfr_gamma (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the value of the Gamma function on OP, rounded in the + direction RND. When OP is a negative integer, NaN is returned. + + -- Function: int mpfr_lngamma (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the value of the logarithm of the Gamma function on OP, + rounded in the direction RND. When -2K-1 <= X <= -2K, K being a + non-negative integer, NaN is returned. See also `mpfr_lgamma'. + + -- Function: int mpfr_lgamma (mpfr_t ROP, int *SIGNP, mpfr_t OP, + mp_rnd_t RND) + Set ROP to the value of the logarithm of the absolute value of the + Gamma function on OP, rounded in the direction RND. The sign (1 or + -1) of Gamma(OP) is returned in the object pointed to by SIGNP. + When OP is an infinity or a non-positive integer, +Inf is + returned. When OP is NaN, -Inf or a negative integer, *SIGNP is + undefined, and when OP is ±0, *SIGNP is the sign of the zero. + + -- Function: int mpfr_zeta (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_zeta_ui (mpfr_t ROP, unsigned long OP, mp_rnd_t + RND) + Set ROP to the value of the Riemann Zeta function on OP, rounded + in the direction RND. + + -- Function: int mpfr_erf (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the value of the error function on OP, rounded in the + direction RND. + + -- Function: int mpfr_erfc (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the value of the complementary error function on OP, + rounded in the direction RND. + + -- Function: int mpfr_j0 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_j1 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_jn (mpfr_t ROP, long N, mpfr_t OP, mp_rnd_t RND) + Set ROP to the value of the first kind Bessel function of order 0, + 1 and N on OP, rounded in the direction RND. When OP is NaN, ROP + is always set to NaN. When OP is plus or minus Infinity, ROP is + set to +0. When OP is zero, and N is not zero, ROP is +0 or -0 + depending on the parity and sign of N, and the sign of OP. + + -- Function: int mpfr_y0 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_y1 (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_yn (mpfr_t ROP, long N, mpfr_t OP, mp_rnd_t RND) + Set ROP to the value of the second kind Bessel function of order + 0, 1 and N on OP, rounded in the direction RND. When OP is NaN or + negative, ROP is always set to NaN. When OP is +Inf, ROP is +0. + When OP is zero, ROP is +Inf or -Inf depending on the parity and + sign of N. + + -- Function: int mpfr_fma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t + OP3, mp_rnd_t RND) + Set ROP to (OP1 times OP2) + OP3, rounded in the direction RND. + + -- Function: int mpfr_fms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t + OP3, mp_rnd_t RND) + Set ROP to (OP1 times OP2) - OP3, rounded in the direction RND. + + -- Function: int mpfr_agm (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, + mp_rnd_t RND) + Set ROP to the arithmetic-geometric mean of OP1 and OP2, rounded + in the direction RND. The arithmetic-geometric mean is the common + limit of the sequences u[n] and v[n], where u[0]=OP1, v[0]=OP2, + u[n+1] is the arithmetic mean of u[n] and v[n], and v[n+1] is the + geometric mean of u[n] and v[n]. If any operand is negative, the + return value is NaN. + + -- Function: int mpfr_hypot (mpfr_t ROP, mpfr_t X, mpfr_t Y, mp_rnd_t + RND) + Set ROP to the Euclidean norm of X and Y, i.e. the square root of + the sum of the squares of X and Y, rounded in the direction RND. + Special values are currently handled as described in Section + F.9.4.3 of the ISO C99 standard, for the `hypot' function (note + this may change in future versions): If X or Y is an infinity, + then plus infinity is returned in ROP, even if the other number is + NaN. + + -- Function: int mpfr_const_log2 (mpfr_t ROP, mp_rnd_t RND) + -- Function: int mpfr_const_pi (mpfr_t ROP, mp_rnd_t RND) + -- Function: int mpfr_const_euler (mpfr_t ROP, mp_rnd_t RND) + -- Function: int mpfr_const_catalan (mpfr_t ROP, mp_rnd_t RND) + Set ROP to the logarithm of 2, the value of Pi, of Euler's + constant 0.577..., of Catalan's constant 0.915..., respectively, + rounded in the direction RND. These functions cache the computed + values to avoid other calculations if a lower or equal precision + is requested. To free these caches, use `mpfr_free_cache'. + + -- Function: void mpfr_free_cache (void) + Free various caches used by MPFR internally, in particular the + caches used by the functions computing constants (currently + `mpfr_const_log2', `mpfr_const_pi', `mpfr_const_euler' and + `mpfr_const_catalan'). You should call this function before + terminating a thread, even if you did not call these functions + directly (they could have been called internally). + + -- Function: int mpfr_sum (mpfr_t ROP, mpfr_ptr const TAB[], unsigned + long N, mp_rnd_t RND) + Set RET to the sum of all elements of TAB whose size is N, rounded + in the direction RND. Warning, TAB is a table of pointers to + mpfr_t, not a table of mpfr_t (preliminary interface). The returned + `int' value is zero when the computed value is the exact value, + and non-zero when this cannot be guaranteed, without giving the + direction of the error as the other functions do. + + +File: mpfr.info, Node: Input and Output Functions, Next: Formatted Output Functions, Prev: Special Functions, Up: MPFR Interface + +5.8 Input and Output Functions +============================== + +This section describes functions that perform input from an input/output +stream, and functions that output to an input/output stream. Passing a +null pointer for a `stream' to any of these functions will make them +read from `stdin' and write to `stdout', respectively. + + When using any of these functions, you must include the `' +standard header before `mpfr.h', to allow `mpfr.h' to define prototypes +for these functions. + + -- Function: size_t mpfr_out_str (FILE *STREAM, int BASE, size_t N, + mpfr_t OP, mp_rnd_t RND) + Output OP on stream STREAM, as a string of digits in base BASE, + rounded in the direction RND. The base may vary from 2 to 36. + Print N significant digits exactly, or if N is 0, enough digits so + that OP can be read back exactly (see `mpfr_get_str'). + + In addition to the significant digits, a decimal point (defined by + the current locale) at the right of the first digit and a trailing + exponent in base 10, in the form `eNNN', are printed. If BASE is + greater than 10, `@' will be used instead of `e' as exponent + delimiter. + + Return the number of bytes written, or if an error occurred, + return 0. + + -- Function: size_t mpfr_inp_str (mpfr_t ROP, FILE *STREAM, int BASE, + mp_rnd_t RND) + Input a string in base BASE from stream STREAM, rounded in the + direction RND, and put the read float in ROP. + + This function reads a word (defined as a sequence of characters + between whitespace) and parses it using `mpfr_set_str' (it may + change). See the documentation of `mpfr_strtofr' for a detailed + description of the valid string formats. + + Return the number of bytes read, or if an error occurred, return 0. + + +File: mpfr.info, Node: Formatted Output Functions, Next: Integer Related Functions, Prev: Input and Output Functions, Up: MPFR Interface + +5.9 Formatted Output Functions +============================== + +5.9.1 Requirements +------------------ + +The class of `mpfr_printf' functions provides formatted output in a +similar manner as the standard C `printf'. These functions are defined +only if your system supports ISO C variadic functions and the +corresponding argument access macros. + + When using any of these functions, you must include the `' +standard header before `mpfr.h', to allow `mpfr.h' to define prototypes +for these functions. + +5.9.2 Format String +------------------- + +The format specification accepted by `mpfr_printf' is an extension of +the `printf' one. The conversion specification is of the form: + % [flags] [width] [.[precision]] [type] [rounding] conv + `flags', `width', and `precision' have the same meaning as for the +standard C function `printf' (in particular, notice that the precision +is related to the number of digits displayed in the base chosen by +`conv' and not related to the internal precision of the `mpfr_t' +variable). `mpfr_printf' accepts the same `type' specifiers as `gmp' +(except the non-standard and deprecated `q', use `ll' instead), plus +`R' and `P': + + `h' `short' + `hh' `char' + `j' `intmax_t' or `uintmax_t' + `l' `long' or `wchar_t' + `ll' `long long' + `L' `long double' + `t' `ptrdiff_t' + `z' `size_t' + `F' `mpf_t', float conversions + `Q' `mpq_t', integer conversions + `M' `mp_limb_t', integer conversions + `N' `mp_limb_t' array, integer conversions + `Z' `mpz_t', integer conversions + `R' `mpfr_t' input, float conversions + `P' `mpfr_prec_t' input, integer conversions + + The `type' specifiers have the same restrictions as those mentioned +in the GMP documentation: *note Formatted Output Strings: +(gmp.info)Formatted Output Strings. More precisely, except for `R' and +`P' (which are defined by MPFR), the `type' specifiers are supported +only if they are supported by `gmp_printf' in your GMP build; this +implies that the standard specifiers, such as `t', must _also_ be +supported by your C library if you want to use them. + + The `rounding' specifier is specific to `mpfr_t' parameter and shall +not be used with other types. `mpfr_printf' accepts the same conversion +specifier character `conv' as `gmp_printf' plus `b'. + + The `P' type outputs the precision of an `mpfr_t' variable. It is +needed because the `mpfr_prec_t' type does not necessarily correspond +to an `unsigned int' or any fixed standard type. For example: + mpfr_t x; + mpfr_prec_t p; + mpfr_init (x); + ... + p = mpfr_get_prec (x); + mpfr_printf ("variable x with %Pu bits", p); + + The `R' type is used for a `mpfr_t' output and can be followed by a +rounding specifier denoted by one of the following characters: + + `U' round toward plus infinity + `D' round toward minus infinity + `Z' round toward zero + `N' round to nearest + `*' rounding mode (as a `mpfr_rnd_t') + indicated by the argument just before + the corresponding `mpfr_t' variable. + + If the precision field is not empty, the `mpfr_t' number is rounded +to the given precision in the direction specified by the rounding mode. +If the precision field is empty (as in `%.Rf'), the number is displayed +with enough digits so that it can be read back exactly (assuming +rounding to nearest, see `mpfr_get_str'). If no rounding is specified, +the `mpfr_t' argument is rounded to nearest. The following three +examples are equivalent: + mpfr_t x; + mpfr_init (x); + ... + mpfr_printf ("%.128Rf", x); + mpfr_printf ("%.128RNf", x); + mpfr_printf ("%.128R*f", GMP_RNDN, x); + + `mpfr_printf' also adds a new conversion specifier `b' which +displays the `mpfr_t' parameter in binary, the behavior is undefined +with other parameter type. The `conv' specifiers allowed with `mpfr_t' +parameter are: + + `a' `A' hex float, C99 style + `b' binary output + `e' `E' scientific format float + `f' fixed point float + `g' `G' fixed or scientific float + + In case of non-decimal output, only the significand is written in the +specified base, the exponent is always displayed in decimal. Special +values are always displayed as `nan', `-inf', and `inf' for `a', `b', +`e', `f', and `g' specifiers and `NAN', `-INF', and `INF' for `A', `E', +`F', and `G' specifiers. In binary output, the precision is silently +increased up to 2 if it equals 1. + +5.9.3 Functions +--------------- + + -- Function: int mpfr_fprintf (FILE *STREAM, const char *TEMPLATE, ...) + -- Function: int mpfr_vfprintf (FILE *STREAM, const char *TEMPLATE, + va_list AP) + Print to the stream STREAM the optional arguments under the + control of the template string TEMPLATE. + + Return the number of characters written or a negative value if an + error occurred. If the number of characters which ought to be + written appears to exceed the maximum limit for an `int', nothing + is written in the stream, the function returns -1, sets the + _erange_ flag, and (in POSIX system only) `errno' is set to + `EOVERFLOW'. + + -- Function: int mpfr_printf (const char *TEMPLATE, ...) + -- Function: int mpfr_vprintf (const char *TEMPLATE, va_list AP) + Print to STDOUT the optional arguments under the control of the + template string TEMPLATE. + + Return the number of characters written or a negative value if an + error occurred. If the number of characters which ought to be + written appears to exceed the maximum limit for an `int', nothing + is written in `stdout', the function returns -1, sets the _erange_ + flag, and (in POSIX system only) `errno' is set to `EOVERFLOW'. + + -- Function: int mpfr_sprintf (char *BUF, const char *TEMPLATE, ...) + -- Function: int mpfr_vsprintf (char *BUF, const char *TEMPLATE, + va_list AP) + Form a null-terminated string in BUF. No overlap is permitted + between BUF and the other arguments. + + Return the number of characters written in the array BUF not + counting the terminating null character or a negative value if an + error occurred. If the number of characters which ought to be + written appears to exceed the maximum limit for an `int', nothing + is written in BUF, the function returns -1, sets the _erange_ + flag, and (in POSIX system only) `errno' is set to `EOVERFLOW'. + + -- Function: int mpfr_snprintf (char *BUF, size_t N, const char + *TEMPLATE, ...) + -- Function: int mpfr_vsnprintf (char *BUF, size_t N, const char + *TEMPLATE, va_list AP) + Form a null-terminated string in BUF. If N is zero, nothing is + written and BUF may be a null pointer, otherwise, the `n-1' first + characters are written in BUF and the N-th is a null character. + + Return the number of characters that would have been written had N + be sufficiently large, not counting the terminating null character + or a negative value if an error occurred. If the number of + characters produced by the optional arguments under the control of + the template string TEMPLATE appears to exceed the maximum limit + for an `int', nothing is written in BUF, the function returns -1, + sets the _erange_ flag, and (in POSIX system only) `errno' is set + to `EOVERFLOW'. + + -- Function: int mpfr_asprintf (char **STR, const char *TEMPLATE, ...) + -- Function: int mpfr_vasprintf (char **STR, const char *TEMPLATE, + va_list AP) + Write their output as a null terminated string in a block of + memory allocated using the current allocation function. A pointer + to the block is stored in STR. The block of memory must be freed + using `mpfr_free_str'. + + The return value is the number of characters written in the + string, excluding the null-terminator or a negative value if an + error occurred. If the number of characters produced by the + optional arguments under the control of the template string + TEMPLATE appears to exceed the maximum limit for an `int', STR is + a null pointer, the function returns -1, sets the _erange_ flag, + and (in POSIX system only) `errno' is set to `EOVERFLOW'. + + +File: mpfr.info, Node: Integer Related Functions, Next: Rounding Related Functions, Prev: Formatted Output Functions, Up: MPFR Interface + +5.10 Integer and Remainder Related Functions +============================================ + + -- Function: int mpfr_rint (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_ceil (mpfr_t ROP, mpfr_t OP) + -- Function: int mpfr_floor (mpfr_t ROP, mpfr_t OP) + -- Function: int mpfr_round (mpfr_t ROP, mpfr_t OP) + -- Function: int mpfr_trunc (mpfr_t ROP, mpfr_t OP) + Set ROP to OP rounded to an integer. `mpfr_rint' rounds to the + nearest representable integer in the given rounding mode, + `mpfr_ceil' rounds to the next higher or equal representable + integer, `mpfr_floor' to the next lower or equal representable + integer, `mpfr_round' to the nearest representable integer, + rounding halfway cases away from zero, and `mpfr_trunc' to the + next representable integer toward zero. + + The returned value is zero when the result is exact, positive when + it is greater than the original value of OP, and negative when it + is smaller. More precisely, the returned value is 0 when OP is an + integer representable in ROP, 1 or -1 when OP is an integer that + is not representable in ROP, 2 or -2 when OP is not an integer. + + Note that `mpfr_round' is different from `mpfr_rint' called with + the rounding to nearest mode (where halfway cases are rounded to + an even integer or significand). Note also that no double rounding + is performed; for instance, 4.5 (100.1 in binary) is rounded by + `mpfr_round' to 4 (100 in binary) in 2-bit precision, though + `round(4.5)' is equal to 5 and 5 (101 in binary) is rounded to 6 + (110 in binary) in 2-bit precision. + + -- Function: int mpfr_rint_ceil (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_rint_floor (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_rint_round (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + -- Function: int mpfr_rint_trunc (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to OP rounded to an integer. `mpfr_rint_ceil' rounds to + the next higher or equal integer, `mpfr_rint_floor' to the next + lower or equal integer, `mpfr_rint_round' to the nearest integer, + rounding halfway cases away from zero, and `mpfr_rint_trunc' to + the next integer toward zero. If the result is not representable, + it is rounded in the direction RND. The returned value is the + ternary value associated with the considered round-to-integer + function (regarded in the same way as any other mathematical + function). + + -- Function: int mpfr_frac (mpfr_t ROP, mpfr_t OP, mp_rnd_t RND) + Set ROP to the fractional part of OP, having the same sign as OP, + rounded in the direction RND (unlike in `mpfr_rint', RND affects + only how the exact fractional part is rounded, not how the + fractional part is generated). + + -- Function: int mpfr_modf (mpfr_t IOP, mpfr_t FOP, mpfr_t OP, + mp_rnd_t RND) + Set simultaneously IOP to the integral part of OP and FOP to the + fractional part of OP, rounded in the direction RND with the + corresponding precision of IOP and FOP (equivalent to + `mpfr_trunc(IOP, OP, RND)' and `mpfr_frac(FOP, OP, RND)'). The + variables IOP and FOP must be different. Return 0 iff both results + are exact. + + -- Function: int mpfr_fmod (mpfr_t R, mpfr_t X, mpfr_t Y, mp_rnd_t RND) + -- Function: int mpfr_remainder (mpfr_t R, mpfr_t X, mpfr_t Y, + mp_rnd_t RND) + -- Function: int mpfr_remquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y, + mp_rnd_t RND) + Set R to the value of x - n y, rounded according to the direction + RND, where n is the integer quotient of X divided by Y, defined as + follows: n is rounded toward zero for `mpfr_fmod', and to the + nearest integer (ties rounded to even) for `mpfr_remainder' and + `mpfr_remquo'. + + Special values are handled as described in Section F.9.7.1 of the + ISO C99 standard: If X is infinite or Y is zero, R is NaN. If Y + is infinite and X is finite, R is X rounded to the precision of R. + If R is zero, it has the sign of X. The return value is the + ternary value corresponding to R. + + Additionally, `mpfr_remquo' stores the low significant bits from + the quotient in *Q (more precisely the number of bits in a `long' + minus one), with the sign of X divided by Y (except if those low + bits are all zero, in which case zero is returned). Note that X + may be so large in magnitude relative to Y that an exact + representation of the quotient is not practical. `mpfr_remainder' + and `mpfr_remquo' functions are useful for additive argument + reduction. + + -- Function: int mpfr_integer_p (mpfr_t OP) + Return non-zero iff OP is an integer. + + +File: mpfr.info, Node: Rounding Related Functions, Next: Miscellaneous Functions, Prev: Integer Related Functions, Up: MPFR Interface + +5.11 Rounding Related Functions +=============================== + + -- Function: void mpfr_set_default_rounding_mode (mp_rnd_t RND) + Set the default rounding mode to RND. The default rounding mode + is to nearest initially. + + -- Function: mp_rnd_t mpfr_get_default_rounding_mode (void) + Get the default rounding mode. + + -- Function: int mpfr_prec_round (mpfr_t X, mp_prec_t PREC, mp_rnd_t + RND) + Round X according to RND with precision PREC, which must be an + integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX' (otherwise the + behavior is undefined). If PREC is greater or equal to the + precision of X, then new space is allocated for the significand, + and it is filled with zeros. Otherwise, the significand is + rounded to precision PREC with the given direction. In both cases, + the precision of X is changed to PREC. + + -- Function: int mpfr_round_prec (mpfr_t X, mp_rnd_t RND, mp_prec_t + PREC) + [This function is obsolete. Please use `mpfr_prec_round' instead.] + + -- Function: int mpfr_can_round (mpfr_t B, mp_exp_t ERR, mp_rnd_t + RND1, mp_rnd_t RND2, mp_prec_t PREC) + Assuming B is an approximation of an unknown number X in the + direction RND1 with error at most two to the power E(b)-ERR where + E(b) is the exponent of B, return a non-zero value if one is able + to round correctly X to precision PREC with the direction RND2, + and 0 otherwise (including for NaN and Inf). This function *does + not modify* its arguments. + + Note: if one wants to also determine the correct ternary value + when rounding B to precision PREC, a useful trick is the following: if (mpfr_can_round (b, err, rnd1, GMP_RNDZ, prec + (rnd2 == GMP_RNDN))) + ... + Indeed, if RND2 is `GMP_RNDN', this will check if one can round + to PREC+1 bits with a directed rounding: if so, one can surely + round to nearest to PREC bits, and in addition one can determine + the correct ternary value, which would not be the case when B is + near from a value exactly representable on PREC bits. + + -- Function: const char * mpfr_print_rnd_mode (mp_rnd_t RND) + Return the input string (GMP_RNDD, GMP_RNDU, GMP_RNDN, GMP_RNDZ) + corresponding to the rounding mode RND or a null pointer if RND is + an invalid rounding mode. + + +File: mpfr.info, Node: Miscellaneous Functions, Next: Exception Related Functions, Prev: Rounding Related Functions, Up: MPFR Interface + +5.12 Miscellaneous Functions +============================ + + -- Function: void mpfr_nexttoward (mpfr_t X, mpfr_t Y) + If X or Y is NaN, set X to NaN. Otherwise, if X is different from + Y, replace X by the next floating-point number (with the precision + of X and the current exponent range) in the direction of Y, if + there is one (the infinite values are seen as the smallest and + largest floating-point numbers). If the result is zero, it keeps + the same sign. No underflow or overflow is generated. + + -- Function: void mpfr_nextabove (mpfr_t X) + Equivalent to `mpfr_nexttoward' where Y is plus infinity. + + -- Function: void mpfr_nextbelow (mpfr_t X) + Equivalent to `mpfr_nexttoward' where Y is minus infinity. + + -- Function: int mpfr_min (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, + mp_rnd_t RND) + Set ROP to the minimum of OP1 and OP2. If OP1 and OP2 are both + NaN, then ROP is set to NaN. If OP1 or OP2 is NaN, then ROP is set + to the numeric value. If OP1 and OP2 are zeros of different signs, + then ROP is set to -0. + + -- Function: int mpfr_max (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, + mp_rnd_t RND) + Set ROP to the maximum of OP1 and OP2. If OP1 and OP2 are both + NaN, then ROP is set to NaN. If OP1 or OP2 is NaN, then ROP is set + to the numeric value. If OP1 and OP2 are zeros of different signs, + then ROP is set to +0. + + -- Function: int mpfr_urandomb (mpfr_t ROP, gmp_randstate_t STATE) + Generate a uniformly distributed random float in the interval 0 <= + ROP < 1. More precisely, the number can be seen as a float with a + random non-normalized significand and exponent 0, which is then + normalized (thus if E denotes the exponent after normalization, + then the least -E significant bits of the significand are always + 0). Return 0, unless the exponent is not in the current exponent + range, in which case ROP is set to NaN and a non-zero value is + returned (this should never happen in practice, except in very + specific cases). The second argument is a `gmp_randstate_t' + structure which should be created using the GMP `gmp_randinit' + function, see the GMP manual. + + -- Function: void mpfr_random (mpfr_t ROP) + Generate a uniformly distributed random float in the interval 0 <= + ROP < 1. + + This function is deprecated and will be suppressed in the next + release; `mpfr_urandomb' should be used instead. + + -- Function: void mpfr_random2 (mpfr_t ROP, mp_size_t SIZE, mp_exp_t + EXP) + Generate a random float of at most SIZE limbs, with long strings of + zeros and ones in the binary representation. The exponent of the + number is in the interval -EXP to EXP. This function is useful for + testing functions and algorithms, since this kind of random + numbers have proven to be more likely to trigger corner-case bugs. + Negative random numbers are generated when SIZE is negative. Put + +0 in ROP when size if zero. The internal state of the default + pseudorandom number generator is modified by a call to this + function (the same one as GMP if MPFR was built using + `--with-gmp-build'). + + This function is deprecated and will be suppressed in the next + release. + + -- Function: mp_exp_t mpfr_get_exp (mpfr_t X) + Get the exponent of X, assuming that X is a non-zero ordinary + number and the significand is chosen in [1/2,1). The behavior for + NaN, infinity or zero is undefined. + + -- Function: int mpfr_set_exp (mpfr_t X, mp_exp_t E) + Set the exponent of X if E is in the current exponent range, and + return 0 (even if X is not a non-zero ordinary number); otherwise, + return a non-zero value. The significand is assumed to be in + [1/2,1). + + -- Function: int mpfr_signbit (mpfr_t OP) + Return a non-zero value iff OP has its sign bit set (i.e. if it is + negative, -0, or a NaN whose representation has its sign bit set). + + -- Function: int mpfr_setsign (mpfr_t ROP, mpfr_t OP, int S, mp_rnd_t + RND) + Set the value of ROP from OP, rounded toward the given direction + RND, then set (resp. clear) its sign bit if S is non-zero (resp. + zero), even when OP is a NaN. + + -- Function: int mpfr_copysign (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, + mp_rnd_t RND) + Set the value of ROP from OP1, rounded toward the given direction + RND, then set its sign bit to that of OP2 (even when OP1 or OP2 is + a NaN). This function is equivalent to `mpfr_setsign (ROP, OP1, + mpfr_signbit (OP2), RND)'. + + -- Function: const char * mpfr_get_version (void) + Return the MPFR version, as a null-terminated string. + + -- Macro: MPFR_VERSION + -- Macro: MPFR_VERSION_MAJOR + -- Macro: MPFR_VERSION_MINOR + -- Macro: MPFR_VERSION_PATCHLEVEL + -- Macro: MPFR_VERSION_STRING + `MPFR_VERSION' is the version of MPFR as a preprocessing constant. + `MPFR_VERSION_MAJOR', `MPFR_VERSION_MINOR' and + `MPFR_VERSION_PATCHLEVEL' are respectively the major, minor and + patch level of MPFR version, as preprocessing constants. + `MPFR_VERSION_STRING' is the version (with an optional suffix, used + in development and pre-release versions) as a string constant, + which can be compared to the result of `mpfr_get_version' to check + at run time the header file and library used match: + if (strcmp (mpfr_get_version (), MPFR_VERSION_STRING)) + fprintf (stderr, "Warning: header and library do not match\n"); + Note: Obtaining different strings is not necessarily an error, as + in general, a program compiled with some old MPFR version can be + dynamically linked with a newer MPFR library version (if allowed + by the library versioning system). + + -- Macro: long MPFR_VERSION_NUM (MAJOR, MINOR, PATCHLEVEL) + Create an integer in the same format as used by `MPFR_VERSION' + from the given MAJOR, MINOR and PATCHLEVEL. Here is an example of + how to check the MPFR version at compile time: + #if (!defined(MPFR_VERSION) || (MPFR_VERSION' line, +#include +#include + any program written for MPF can be compiled directly with MPFR without +any changes. All operations are then performed with the default MPFR +rounding mode, which can be reset with `mpfr_set_default_rounding_mode'. + + Warning: the `mpf_init' and `mpf_init2' functions initialize to +zero, whereas the corresponding MPFR functions initialize to NaN: this +is useful to detect uninitialized values, but is slightly incompatible +with `mpf'. + + -- Function: void mpfr_set_prec_raw (mpfr_t X, mp_prec_t PREC) + Reset the precision of X to be *exactly* PREC bits. The only + difference with `mpfr_set_prec' is that PREC is assumed to be + small enough so that the significand fits into the current + allocated memory space for X. Otherwise the behavior is undefined. + + -- Function: int mpfr_eq (mpfr_t OP1, mpfr_t OP2, unsigned long int + OP3) + Return non-zero if OP1 and OP2 are both non-zero ordinary numbers + with the same exponent and the same first OP3 bits, both zero, or + both infinities of the same sign. Return zero otherwise. This + function is defined for compatibility with `mpf'. Do not use it if + you want to know whether two numbers are close to each other; for + instance, 1.011111 and 1.100000 are currently regarded as + different for any value of OP3 larger than 1 (but this may change + in the next release). + + -- Function: void mpfr_reldiff (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, + mp_rnd_t RND) + Compute the relative difference between OP1 and OP2 and store the + result in ROP. This function does not guarantee the correct + rounding on the relative difference; it just computes + |OP1-OP2|/OP1, using the rounding mode RND for all operations and + the precision of ROP. + + -- Function: int mpfr_mul_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long + int OP2, mp_rnd_t RND) + -- Function: int mpfr_div_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long + int OP2, mp_rnd_t RND) + See `mpfr_mul_2ui' and `mpfr_div_2ui'. These functions are only + kept for compatibility with MPF. + + +File: mpfr.info, Node: Custom Interface, Next: Internals, Prev: Compatibility with MPF, Up: MPFR Interface + +5.15 Custom Interface +===================== + +Some applications use a stack to handle the memory and their objects. +However, the MPFR memory design is not well suited for such a thing. So +that such applications are able to use MPFR, an auxiliary memory +interface has been created: the Custom Interface. + + The following interface allows them to use MPFR in two ways: + * Either they directly store the MPFR FP number as a `mpfr_t' on the + stack. + + * Either they store their own representation of a FP number on the + stack and construct a new temporary `mpfr_t' each time it is + needed. + Nothing has to be done to destroy the FP numbers except garbaging +the used memory: all the memory stuff (allocating, destroying, +garbaging) is kept to the application. + + Each function in this interface is also implemented as a macro for +efficiency reasons: for example `mpfr_custom_init (s, p)' uses the +macro, while `(mpfr_custom_init) (s, p)' uses the function. + + Note 1: MPFR functions may still initialize temporary FP numbers +using standard mpfr_init. See Custom Allocation (GNU MP). + + Note 2: MPFR functions may use the cached functions (mpfr_const_pi +for example), even if they are not explicitly called. You have to call +`mpfr_free_cache' each time you garbage the memory iff mpfr_init, +through GMP Custom Allocation, allocates its memory on the application +stack. + + Note 3: This interface is preliminary. + + -- Function: size_t mpfr_custom_get_size (mp_prec_t PREC) + Return the needed size in bytes to store the significand of a FP + number of precision PREC. + + -- Function: void mpfr_custom_init (void *SIGNIFICAND, mp_prec_t PREC) + Initialize a significand of precision PREC. SIGNIFICAND must be + an area of `mpfr_custom_get_size (prec)' bytes at least and be + suitably aligned for an array of `mp_limb_t'. + + -- Function: void mpfr_custom_init_set (mpfr_t X, int KIND, mp_exp_t + EXP, mp_prec_t PREC, void *SIGNIFICAND) + Perform a dummy initialization of a `mpfr_t' and set it to: + * if `ABS(kind) == MPFR_NAN_KIND', X is set to NaN; + + * if `ABS(kind) == MPFR_INF_KIND', X is set to the infinity of + sign `sign(kind)'; + + * if `ABS(kind) == MPFR_ZERO_KIND', X is set to the zero of + sign `sign(kind)'; + + * if `ABS(kind) == MPFR_REGULAR_KIND', X is set to a regular + number: `x = sign(kind)*significand*2^exp' + In all cases, it uses SIGNIFICAND directly for further computing + involving X. It will not allocate anything. A FP number + initialized with this function cannot be resized using + `mpfr_set_prec', or cleared using `mpfr_clear'! SIGNIFICAND must + have been initialized with `mpfr_custom_init' using the same + precision PREC. + + -- Function: int mpfr_custom_get_kind (mpfr_t X) + Return the current kind of a `mpfr_t' as used by + `mpfr_custom_init_set'. The behavior of this function for any + `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined. + + -- Function: void * mpfr_custom_get_mantissa (mpfr_t X) + Return a pointer to the significand used by a `mpfr_t' initialized + with `mpfr_custom_init_set'. The behavior of this function for + any `mpfr_t' not initialized with `mpfr_custom_init_set' is + undefined. + + -- Function: mp_exp_t mpfr_custom_get_exp (mpfr_t X) + Return the exponent of X, assuming that X is a non-zero ordinary + number. The return value for NaN, Infinity or Zero is unspecified + but does not produce any trap. The behavior of this function for + any `mpfr_t' not initialized with `mpfr_custom_init_set' is + undefined. + + -- Function: void mpfr_custom_move (mpfr_t X, void *NEW_POSITION) + Inform MPFR that the significand has moved due to a garbage collect + and update its new position to `new_position'. However the + application has to move the significand and the `mpfr_t' itself. + The behavior of this function for any `mpfr_t' not initialized + with `mpfr_custom_init_set' is undefined. + + See the test suite for examples. + + +File: mpfr.info, Node: Internals, Prev: Custom Interface, Up: MPFR Interface + +5.16 Internals +============== + +The following types and functions were mainly designed for the +implementation of MPFR, but may be useful for users too. However no +upward compatibility is guaranteed. You may need to include +`mpfr-impl.h' to use them. + + The `mpfr_t' type consists of four fields. + + * The `_mpfr_prec' field is used to store the precision of the + variable (in bits); this is not less than `MPFR_PREC_MIN'. + + * The `_mpfr_sign' field is used to store the sign of the variable. + + * The `_mpfr_exp' field stores the exponent. An exponent of 0 means + a radix point just above the most significant limb. Non-zero + values n are a multiplier 2^n relative to that point. A NaN, an + infinity and a zero are indicated by a special value of the + exponent. + + * Finally, the `_mpfr_d' is a pointer to the limbs, least + significant limbs stored first. The number of limbs in use is + controlled by `_mpfr_prec', namely + ceil(`_mpfr_prec'/`mp_bits_per_limb'). Non-singular values always + have the most significant bit of the most significant limb set to + 1. When the precision does not correspond to a whole number of + limbs, the excess bits at the low end of the data are zero. + + + +File: mpfr.info, Node: Contributors, Next: References, Prev: MPFR Interface, Up: Top + +Contributors +************ + +The main developers of MPFR are Guillaume Hanrot, Vincent Lefèvre, +Patrick Pélissier, Philippe Théveny and Paul Zimmermann. + + Sylvie Boldo from ENS-Lyon, France, contributed the functions +`mpfr_agm' and `mpfr_log'. Emmanuel Jeandel, from ENS-Lyon too, +contributed the generic hypergeometric code, as well as the `mpfr_exp3', +a first implementation of the sine and cosine, and improved versions of +`mpfr_const_log2' and `mpfr_const_pi'. Mathieu Dutour contributed the +functions `mpfr_atan' and `mpfr_asin', and a previous version of +`mpfr_gamma'; David Daney contributed the hyperbolic and inverse +hyperbolic functions, the base-2 exponential, and the factorial +function. Fabrice Rouillier contributed the original version of +`mul_ui.c', the `gmp_op.c' file, and helped to the Microsoft Windows +porting. Jean-Luc Rémy contributed the `mpfr_zeta' code. Ludovic +Meunier helped in the design of the `mpfr_erf' code. Damien Stehlé +contributed the `mpfr_get_ld_2exp' function. + + We would like to thank Jean-Michel Muller and Joris van der Hoeven +for very fruitful discussions at the beginning of that project, +Torbjörn Granlund and Kevin Ryde for their help about design issues, +and Nathalie Revol for her careful reading of a previous version of +this documentation. Kevin Ryde did a tremendous job for the +portability of MPFR in 2002-2004. + + The development of the MPFR library would not have been possible +without the continuous support of INRIA, and of the LORIA (Nancy, +France) and LIP (Lyon, France) laboratories. In particular the main +authors were or are members of the PolKA, Spaces, Cacao project-teams +at LORIA and of the Arenaire project-team at LIP. This project was +started during the Fiable (reliable in French) action supported by +INRIA, and continued during the AOC action. The development of MPFR +was also supported by a grant (202F0659 00 MPN 121) from the Conseil +Régional de Lorraine in 2002, and from INRIA by an "associate engineer" +grant (2003-2005) and an "opération de développement logiciel" grant +(2007-2009). + + +File: mpfr.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top + +References +********** + + * Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick + Pélissier and Paul Zimmermann, "MPFR: A Multiple-Precision Binary + Floating-Point Library With Correct Rounding", ACM Transactions on + Mathematical Software, volume 33, issue 2, article 13, 15 pages, + 2007, `http://doi.acm.org/10.1145/1236463.1236468'. + + * Torbjörn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic + Library", version 4.2.2, 2007, `http://gmplib.org'. + + * IEEE standard for binary floating-point arithmetic, Technical + Report ANSI-IEEE Standard 754-1985, New York, 1985. Approved + March 21, 1985: IEEE Standards Board; approved July 26, 1985: + American National Standards Institute, 18 pages. + + * Donald E. Knuth, "The Art of Computer Programming", vol 2, + "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981. + + * Jean-Michel Muller, "Elementary Functions, Algorithms and + Implementation", Birkhauser, Boston, 2nd edition, 2006. + + + +File: mpfr.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top + +Appendix A GNU Free Documentation License +***************************************** + + Version 1.2, November 2002 + + Copyright (C) 2000,2001,2002 Free Software Foundation, Inc. + 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA + + Everyone is permitted to copy and distribute verbatim copies + of this license document, but changing it is not allowed. + + 0. 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FUTURE REVISIONS OF THIS LICENSE + + The Free Software Foundation may publish new, revised versions of + the GNU Free Documentation License from time to time. Such new + versions will be similar in spirit to the present version, but may + differ in detail to address new problems or concerns. See + `http://www.gnu.org/copyleft/'. + + Each version of the License is given a distinguishing version + number. If the Document specifies that a particular numbered + version of this License "or any later version" applies to it, you + have the option of following the terms and conditions either of + that specified version or of any later version that has been + published (not as a draft) by the Free Software Foundation. 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A copy of the license is included in the section entitled ``GNU + Free Documentation License''. + + If you have Invariant Sections, Front-Cover Texts and Back-Cover +Texts, replace the "with...Texts." line with this: + + with the Invariant Sections being LIST THEIR TITLES, with + the Front-Cover Texts being LIST, and with the Back-Cover Texts + being LIST. + + If you have Invariant Sections without Cover Texts, or some other +combination of the three, merge those two alternatives to suit the +situation. + + If your document contains nontrivial examples of program code, we +recommend releasing these examples in parallel under your choice of +free software license, such as the GNU General Public License, to +permit their use in free software. + + +File: mpfr.info, Node: Concept Index, Next: Function Index, Prev: GNU Free Documentation License, Up: Top + +Concept Index +************* + +[index] +* Menu: + +* Accuracy: MPFR Interface. (line 28) +* Arithmetic functions: Basic Arithmetic Functions. + (line 3) +* Assignment functions: Assignment Functions. (line 3) +* Basic arithmetic functions: Basic Arithmetic Functions. + (line 3) +* Combined initialization and assignment functions: Combined Initialization and Assignment Functions. + (line 3) +* Comparison functions: Comparison Functions. (line 3) +* Compatibility with MPF: Compatibility with MPF. + (line 3) +* Conditions for copying MPFR: Copying. (line 6) +* Conversion functions: Conversion Functions. (line 3) +* Copying conditions: Copying. (line 6) +* Custom interface: Custom Interface. (line 3) +* Exception related functions: Exception Related Functions. + (line 3) +* FDL, GNU Free Documentation License: GNU Free Documentation License. + (line 6) +* Float arithmetic functions: Basic Arithmetic Functions. + (line 3) +* Float comparisons functions: Comparison Functions. (line 3) +* Float functions: MPFR Interface. (line 6) +* Float input and output functions: Input and Output Functions. + (line 3) +* Float output functions: Formatted Output Functions. + (line 3) +* Floating-point functions: MPFR Interface. (line 6) +* Floating-point number: MPFR Basics. (line 52) +* GNU Free Documentation License: GNU Free Documentation License. + (line 6) +* I/O functions <1>: Formatted Output Functions. + (line 3) +* I/O functions: Input and Output Functions. + (line 3) +* Initialization functions: Initialization Functions. + (line 3) +* Input functions: Input and Output Functions. + (line 3) +* Installation: Installing MPFR. (line 6) +* Integer related functions: Integer Related Functions. + (line 3) +* Internals: Internals. (line 3) +* libmpfr: MPFR Basics. (line 32) +* Libraries: MPFR Basics. (line 32) +* Libtool: MPFR Basics. (line 38) +* Limb: MPFR Basics. (line 84) +* Linking: MPFR Basics. (line 32) +* Miscellaneous float functions: Miscellaneous Functions. + (line 3) +* mpfr.h: MPFR Basics. (line 9) +* Output functions <1>: Formatted Output Functions. + (line 3) +* Output functions: Input and Output Functions. + (line 3) +* Precision <1>: MPFR Interface. (line 20) +* Precision: MPFR Basics. (line 65) +* Reporting bugs: Reporting Bugs. (line 6) +* Rounding mode related functions: Rounding Related Functions. + (line 3) +* Rounding Modes: MPFR Basics. (line 79) +* Special functions: Special Functions. (line 3) +* stdarg.h: MPFR Basics. (line 22) +* stdio.h: MPFR Basics. (line 15) + + +File: mpfr.info, Node: Function Index, Prev: Concept Index, Up: Top + +Function and Type Index +*********************** + +[index] +* Menu: + +* mp_prec_t: MPFR Basics. (line 65) +* mp_rnd_t: MPFR Basics. (line 79) +* mpfr_abs: Basic Arithmetic Functions. + (line 177) +* mpfr_acos: Special Functions. (line 48) +* mpfr_acosh: Special Functions. (line 131) +* mpfr_add: Basic Arithmetic Functions. + (line 8) +* mpfr_add_d: Basic Arithmetic Functions. + (line 14) +* mpfr_add_q: Basic Arithmetic Functions. + (line 18) +* mpfr_add_si: Basic Arithmetic Functions. + (line 12) +* mpfr_add_ui: Basic Arithmetic Functions. + (line 10) +* mpfr_add_z: Basic Arithmetic Functions. + (line 16) +* mpfr_agm: Special Functions. (line 221) +* mpfr_asin: Special Functions. (line 49) +* mpfr_asinh: Special Functions. (line 132) +* mpfr_asprintf: Formatted Output Functions. + (line 171) +* mpfr_atan: Special Functions. (line 50) +* mpfr_atan2: Special Functions. (line 60) +* mpfr_atanh: Special Functions. (line 133) +* mpfr_can_round: Rounding Related Functions. + (line 29) +* mpfr_cbrt: Basic Arithmetic Functions. + (line 107) +* mpfr_ceil: Integer Related Functions. + (line 8) +* mpfr_check_range: Exception Related Functions. + (line 38) +* mpfr_clear: Initialization Functions. + (line 31) +* mpfr_clear_erangeflag: Exception Related Functions. + (line 111) +* mpfr_clear_flags: Exception Related Functions. + (line 121) +* mpfr_clear_inexflag: Exception Related Functions. + (line 110) +* mpfr_clear_nanflag: Exception Related Functions. + (line 109) +* mpfr_clear_overflow: Exception Related Functions. + (line 108) +* mpfr_clear_underflow: Exception Related Functions. + (line 107) +* mpfr_clears: Initialization Functions. + (line 35) +* mpfr_cmp: Comparison Functions. + (line 7) +* mpfr_cmp_d: Comparison Functions. + (line 10) +* mpfr_cmp_f: Comparison Functions. + (line 14) +* mpfr_cmp_ld: Comparison Functions. + (line 11) +* mpfr_cmp_q: Comparison Functions. + (line 13) +* mpfr_cmp_si: Comparison Functions. + (line 9) +* mpfr_cmp_si_2exp: Comparison Functions. + (line 31) +* mpfr_cmp_ui: Comparison Functions. + (line 8) +* mpfr_cmp_ui_2exp: Comparison Functions. + (line 29) +* mpfr_cmp_z: Comparison Functions. + (line 12) +* mpfr_cmpabs: Comparison Functions. + (line 35) +* mpfr_const_catalan: Special Functions. (line 242) +* mpfr_const_euler: Special Functions. (line 241) +* mpfr_const_log2: Special Functions. (line 239) +* mpfr_const_pi: Special Functions. (line 240) +* mpfr_copysign: Miscellaneous Functions. + (line 93) +* mpfr_cos: Special Functions. (line 29) +* mpfr_cosh: Special Functions. (line 111) +* mpfr_cot: Special Functions. (line 37) +* mpfr_coth: Special Functions. (line 127) +* mpfr_csc: Special Functions. (line 36) +* mpfr_csch: Special Functions. (line 126) +* mpfr_custom_get_exp: Custom Interface. (line 78) +* mpfr_custom_get_kind: Custom Interface. (line 67) +* mpfr_custom_get_mantissa: Custom Interface. (line 72) +* mpfr_custom_get_size: Custom Interface. (line 38) +* mpfr_custom_init: Custom Interface. (line 42) +* mpfr_custom_init_set: Custom Interface. (line 48) +* mpfr_custom_move: Custom Interface. (line 85) +* mpfr_d_div: Basic Arithmetic Functions. + (line 82) +* mpfr_d_sub: Basic Arithmetic Functions. + (line 37) +* MPFR_DECL_INIT: Initialization Functions. + (line 75) +* mpfr_dim: Basic Arithmetic Functions. + (line 182) +* mpfr_div: Basic Arithmetic Functions. + (line 72) +* mpfr_div_2exp: Compatibility with MPF. + (line 49) +* mpfr_div_2si: Basic Arithmetic Functions. + (line 197) +* mpfr_div_2ui: Basic Arithmetic Functions. + (line 195) +* mpfr_div_d: Basic Arithmetic Functions. + (line 84) +* mpfr_div_q: Basic Arithmetic Functions. + (line 88) +* mpfr_div_si: Basic Arithmetic Functions. + (line 80) +* mpfr_div_ui: Basic Arithmetic Functions. + (line 76) +* mpfr_div_z: Basic Arithmetic Functions. + (line 86) +* mpfr_eint: Special Functions. (line 150) +* mpfr_eq: Compatibility with MPF. + (line 28) +* mpfr_equal_p: Comparison Functions. + (line 71) +* mpfr_erangeflag_p: Exception Related Functions. + (line 129) +* mpfr_erf: Special Functions. (line 186) +* mpfr_erfc: Special Functions. (line 190) +* mpfr_exp: Special Functions. (line 23) +* mpfr_exp10: Special Functions. (line 25) +* mpfr_exp2: Special Functions. (line 24) +* mpfr_expm1: Special Functions. (line 146) +* mpfr_fac_ui: Special Functions. (line 138) +* mpfr_fits_intmax_p: Conversion Functions. + (line 113) +* mpfr_fits_sint_p: Conversion Functions. + (line 110) +* mpfr_fits_slong_p: Conversion Functions. + (line 108) +* mpfr_fits_sshort_p: Conversion Functions. + (line 112) +* mpfr_fits_uint_p: Conversion Functions. + (line 109) +* mpfr_fits_uintmax_p: Conversion Functions. + (line 114) +* mpfr_fits_ulong_p: Conversion Functions. + (line 107) +* mpfr_fits_ushort_p: Conversion Functions. + (line 111) +* mpfr_floor: Integer Related Functions. + (line 9) +* mpfr_fma: Special Functions. (line 213) +* mpfr_fmod: Integer Related Functions. + (line 63) +* mpfr_fms: Special Functions. (line 217) +* mpfr_fprintf: Formatted Output Functions. + (line 117) +* mpfr_frac: Integer Related Functions. + (line 48) +* mpfr_free_cache: Special Functions. (line 249) +* mpfr_free_str: Conversion Functions. + (line 100) +* mpfr_gamma: Special Functions. (line 162) +* mpfr_get_d: Conversion Functions. + (line 7) +* mpfr_get_d_2exp: Conversion Functions. + (line 20) +* mpfr_get_decimal64: Conversion Functions. + (line 9) +* mpfr_get_default_prec: Initialization Functions. + (line 109) +* mpfr_get_default_rounding_mode: Rounding Related Functions. + (line 11) +* mpfr_get_emax: Exception Related Functions. + (line 8) +* mpfr_get_emax_max: Exception Related Functions. + (line 30) +* mpfr_get_emax_min: Exception Related Functions. + (line 29) +* mpfr_get_emin: Exception Related Functions. + (line 7) +* mpfr_get_emin_max: Exception Related Functions. + (line 28) +* mpfr_get_emin_min: Exception Related Functions. + (line 27) +* mpfr_get_exp: Miscellaneous Functions. + (line 71) +* mpfr_get_f: Conversion Functions. + (line 55) +* mpfr_get_ld: Conversion Functions. + (line 8) +* mpfr_get_ld_2exp: Conversion Functions. + (line 22) +* mpfr_get_patches: Miscellaneous Functions. + (line 130) +* mpfr_get_prec: Initialization Functions. + (line 143) +* mpfr_get_si: Conversion Functions. + (line 31) +* mpfr_get_sj: Conversion Functions. + (line 33) +* mpfr_get_str: Conversion Functions. + (line 61) +* mpfr_get_ui: Conversion Functions. + (line 32) +* mpfr_get_uj: Conversion Functions. + (line 34) +* mpfr_get_version: Miscellaneous Functions. + (line 99) +* mpfr_get_z: Conversion Functions. + (line 51) +* mpfr_get_z_exp: Conversion Functions. + (line 44) +* mpfr_greater_p: Comparison Functions. + (line 54) +* mpfr_greaterequal_p: Comparison Functions. + (line 57) +* mpfr_hypot: Special Functions. (line 230) +* mpfr_inexflag_p: Exception Related Functions. + (line 128) +* mpfr_inf_p: Comparison Functions. + (line 42) +* mpfr_init: Initialization Functions. + (line 51) +* mpfr_init2: Initialization Functions. + (line 11) +* mpfr_init_set: Combined Initialization and Assignment Functions. + (line 7) +* mpfr_init_set_d: Combined Initialization and Assignment Functions. + (line 12) +* mpfr_init_set_f: Combined Initialization and Assignment Functions. + (line 17) +* mpfr_init_set_ld: Combined Initialization and Assignment Functions. + (line 14) +* mpfr_init_set_q: Combined Initialization and Assignment Functions. + (line 16) +* mpfr_init_set_si: Combined Initialization and Assignment Functions. + (line 11) +* mpfr_init_set_str: Combined Initialization and Assignment Functions. + (line 23) +* mpfr_init_set_ui: Combined Initialization and Assignment Functions. + (line 9) +* mpfr_init_set_z: Combined Initialization and Assignment Functions. + (line 15) +* mpfr_inits: Initialization Functions. + (line 63) +* mpfr_inits2: Initialization Functions. + (line 23) +* mpfr_inp_str: Input and Output Functions. + (line 33) +* mpfr_integer_p: Integer Related Functions. + (line 89) +* mpfr_j0: Special Functions. (line 194) +* mpfr_j1: Special Functions. (line 195) +* mpfr_jn: Special Functions. (line 196) +* mpfr_less_p: Comparison Functions. + (line 60) +* mpfr_lessequal_p: Comparison Functions. + (line 63) +* mpfr_lessgreater_p: Comparison Functions. + (line 66) +* mpfr_lgamma: Special Functions. (line 172) +* mpfr_li2: Special Functions. (line 157) +* mpfr_lngamma: Special Functions. (line 166) +* mpfr_log: Special Functions. (line 16) +* mpfr_log10: Special Functions. (line 18) +* mpfr_log1p: Special Functions. (line 142) +* mpfr_log2: Special Functions. (line 17) +* mpfr_max: Miscellaneous Functions. + (line 29) +* mpfr_min: Miscellaneous Functions. + (line 22) +* mpfr_modf: Integer Related Functions. + (line 55) +* mpfr_mul: Basic Arithmetic Functions. + (line 51) +* mpfr_mul_2exp: Compatibility with MPF. + (line 47) +* mpfr_mul_2si: Basic Arithmetic Functions. + (line 190) +* mpfr_mul_2ui: Basic Arithmetic Functions. + (line 188) +* mpfr_mul_d: Basic Arithmetic Functions. + (line 57) +* mpfr_mul_q: Basic Arithmetic Functions. + (line 61) +* mpfr_mul_si: Basic Arithmetic Functions. + (line 55) +* mpfr_mul_ui: Basic Arithmetic Functions. + (line 53) +* mpfr_mul_z: Basic Arithmetic Functions. + (line 59) +* mpfr_nan_p: Comparison Functions. + (line 41) +* mpfr_nanflag_p: Exception Related Functions. + (line 127) +* mpfr_neg: Basic Arithmetic Functions. + (line 173) +* mpfr_nextabove: Miscellaneous Functions. + (line 15) +* mpfr_nextbelow: Miscellaneous Functions. + (line 18) +* mpfr_nexttoward: Miscellaneous Functions. + (line 7) +* mpfr_number_p: Comparison Functions. + (line 43) +* mpfr_out_str: Input and Output Functions. + (line 17) +* mpfr_overflow_p: Exception Related Functions. + (line 126) +* mpfr_pow: Basic Arithmetic Functions. + (line 116) +* mpfr_pow_si: Basic Arithmetic Functions. + (line 120) +* mpfr_pow_ui: Basic Arithmetic Functions. + (line 118) +* mpfr_pow_z: Basic Arithmetic Functions. + (line 122) +* mpfr_prec_round: Rounding Related Functions. + (line 15) +* mpfr_print_rnd_mode: Rounding Related Functions. + (line 46) +* mpfr_printf: Formatted Output Functions. + (line 130) +* mpfr_random: Miscellaneous Functions. + (line 48) +* mpfr_random2: Miscellaneous Functions. + (line 56) +* mpfr_rec_sqrt: Basic Arithmetic Functions. + (line 102) +* mpfr_reldiff: Compatibility with MPF. + (line 39) +* mpfr_remainder: Integer Related Functions. + (line 65) +* mpfr_remquo: Integer Related Functions. + (line 67) +* mpfr_rint: Integer Related Functions. + (line 7) +* mpfr_rint_ceil: Integer Related Functions. + (line 34) +* mpfr_rint_floor: Integer Related Functions. + (line 35) +* mpfr_rint_round: Integer Related Functions. + (line 36) +* mpfr_rint_trunc: Integer Related Functions. + (line 37) +* mpfr_root: Basic Arithmetic Functions. + (line 109) +* mpfr_round: Integer Related Functions. + (line 10) +* mpfr_round_prec: Rounding Related Functions. + (line 25) +* mpfr_sec: Special Functions. (line 35) +* mpfr_sech: Special Functions. (line 125) +* mpfr_set: Assignment Functions. + (line 12) +* mpfr_set_d: Assignment Functions. + (line 18) +* mpfr_set_decimal64: Assignment Functions. + (line 21) +* mpfr_set_default_prec: Initialization Functions. + (line 101) +* mpfr_set_default_rounding_mode: Rounding Related Functions. + (line 7) +* mpfr_set_emax: Exception Related Functions. + (line 16) +* mpfr_set_emin: Exception Related Functions. + (line 15) +* mpfr_set_erangeflag: Exception Related Functions. + (line 118) +* mpfr_set_exp: Miscellaneous Functions. + (line 76) +* mpfr_set_f: Assignment Functions. + (line 24) +* mpfr_set_inexflag: Exception Related Functions. + (line 117) +* mpfr_set_inf: Assignment Functions. + (line 131) +* mpfr_set_ld: Assignment Functions. + (line 19) +* mpfr_set_nan: Assignment Functions. + (line 132) +* mpfr_set_nanflag: Exception Related Functions. + (line 116) +* mpfr_set_overflow: Exception Related Functions. + (line 115) +* mpfr_set_prec: Initialization Functions. + (line 131) +* mpfr_set_prec_raw: Compatibility with MPF. + (line 21) +* mpfr_set_q: Assignment Functions. + (line 23) +* mpfr_set_si: Assignment Functions. + (line 15) +* mpfr_set_si_2exp: Assignment Functions. + (line 51) +* mpfr_set_sj: Assignment Functions. + (line 17) +* mpfr_set_sj_2exp: Assignment Functions. + (line 55) +* mpfr_set_str: Assignment Functions. + (line 61) +* mpfr_set_ui: Assignment Functions. + (line 14) +* mpfr_set_ui_2exp: Assignment Functions. + (line 49) +* mpfr_set_uj: Assignment Functions. + (line 16) +* mpfr_set_uj_2exp: Assignment Functions. + (line 53) +* mpfr_set_underflow: Exception Related Functions. + (line 114) +* mpfr_set_z: Assignment Functions. + (line 22) +* mpfr_setsign: Miscellaneous Functions. + (line 87) +* mpfr_sgn: Comparison Functions. + (line 49) +* mpfr_si_div: Basic Arithmetic Functions. + (line 78) +* mpfr_si_sub: Basic Arithmetic Functions. + (line 33) +* mpfr_signbit: Miscellaneous Functions. + (line 82) +* mpfr_sin: Special Functions. (line 30) +* mpfr_sin_cos: Special Functions. (line 42) +* mpfr_sinh: Special Functions. (line 112) +* mpfr_sinh_cosh: Special Functions. (line 118) +* mpfr_snprintf: Formatted Output Functions. + (line 155) +* mpfr_sprintf: Formatted Output Functions. + (line 141) +* mpfr_sqr: Basic Arithmetic Functions. + (line 68) +* mpfr_sqrt: Basic Arithmetic Functions. + (line 95) +* mpfr_sqrt_ui: Basic Arithmetic Functions. + (line 97) +* mpfr_strtofr: Assignment Functions. + (line 72) +* mpfr_sub: Basic Arithmetic Functions. + (line 27) +* mpfr_sub_d: Basic Arithmetic Functions. + (line 39) +* mpfr_sub_q: Basic Arithmetic Functions. + (line 43) +* mpfr_sub_si: Basic Arithmetic Functions. + (line 35) +* mpfr_sub_ui: Basic Arithmetic Functions. + (line 31) +* mpfr_sub_z: Basic Arithmetic Functions. + (line 41) +* mpfr_subnormalize: Exception Related Functions. + (line 58) +* mpfr_sum: Special Functions. (line 258) +* mpfr_swap: Assignment Functions. + (line 137) +* mpfr_t: MPFR Basics. (line 52) +* mpfr_tan: Special Functions. (line 31) +* mpfr_tanh: Special Functions. (line 113) +* mpfr_trunc: Integer Related Functions. + (line 11) +* mpfr_ui_div: Basic Arithmetic Functions. + (line 74) +* mpfr_ui_pow: Basic Arithmetic Functions. + (line 126) +* mpfr_ui_pow_ui: Basic Arithmetic Functions. + (line 124) +* mpfr_ui_sub: Basic Arithmetic Functions. + (line 29) +* mpfr_underflow_p: Exception Related Functions. + (line 125) +* mpfr_unordered_p: Comparison Functions. + (line 75) +* mpfr_urandomb: Miscellaneous Functions. + (line 35) +* mpfr_vasprintf: Formatted Output Functions. + (line 173) +* MPFR_VERSION: Miscellaneous Functions. + (line 102) +* MPFR_VERSION_MAJOR: Miscellaneous Functions. + (line 103) +* MPFR_VERSION_MINOR: Miscellaneous Functions. + (line 104) +* MPFR_VERSION_NUM: Miscellaneous Functions. + (line 122) +* MPFR_VERSION_PATCHLEVEL: Miscellaneous Functions. + (line 105) +* MPFR_VERSION_STRING: Miscellaneous Functions. + (line 106) +* mpfr_vfprintf: Formatted Output Functions. + (line 119) +* mpfr_vprintf: Formatted Output Functions. + (line 131) +* mpfr_vsnprintf: Formatted Output Functions. + (line 157) +* mpfr_vsprintf: Formatted Output Functions. + (line 143) +* mpfr_y0: Special Functions. (line 203) +* mpfr_y1: Special Functions. (line 204) +* mpfr_yn: Special Functions. (line 205) +* mpfr_zero_p: Comparison Functions. + (line 44) +* mpfr_zeta: Special Functions. (line 180) +* mpfr_zeta_ui: Special Functions. (line 182) + + + +Tag Table: +Node: Top874 +Node: Copying2113 +Node: Introduction to MPFR3843 +Node: Installing MPFR5755 +Node: Reporting Bugs8997 +Node: MPFR Basics10613 +Node: MPFR Interface25127 +Node: Initialization Functions27351 +Node: Assignment Functions33919 +Node: Combined Initialization and Assignment Functions41669 +Node: Conversion Functions42951 +Node: Basic Arithmetic Functions49151 +Node: Comparison Functions58008 +Node: Special Functions61430 +Node: Input and Output Functions73827 +Node: Formatted Output Functions75757 +Node: Integer Related Functions84168 +Node: Rounding Related Functions89004 +Node: Miscellaneous Functions91474 +Node: Exception Related Functions98260 +Node: Compatibility with MPF104378 +Node: Custom Interface106870 +Node: Internals111053 +Node: Contributors112376 +Node: References114541 +Node: GNU Free Documentation License115655 +Node: Concept Index138098 +Node: Function Index142890 + +End Tag Table + + +Local Variables: +coding: iso-8859-1 +End: