1 This is mpfr.info, produced by makeinfo version 4.13 from mpfr.texi.
3 This manual documents how to install and use the Multiple Precision
4 Floating-Point Reliable Library, version 3.1.0.
6 Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
7 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free
8 Software Foundation, Inc.
10 Permission is granted to copy, distribute and/or modify this
11 document under the terms of the GNU Free Documentation License, Version
12 1.2 or any later version published by the Free Software Foundation;
13 with no Invariant Sections, with no Front-Cover Texts, and with no
14 Back-Cover Texts. A copy of the license is included in *note GNU Free
15 Documentation License::.
17 INFO-DIR-SECTION Software libraries
19 * mpfr: (mpfr). Multiple Precision Floating-Point Reliable Library.
23 File: mpfr.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir)
28 This manual documents how to install and use the Multiple Precision
29 Floating-Point Reliable Library, version 3.1.0.
31 Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
32 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011 Free
33 Software Foundation, Inc.
35 Permission is granted to copy, distribute and/or modify this
36 document under the terms of the GNU Free Documentation License, Version
37 1.2 or any later version published by the Free Software Foundation;
38 with no Invariant Sections, with no Front-Cover Texts, and with no
39 Back-Cover Texts. A copy of the license is included in *note GNU Free
40 Documentation License::.
45 * Copying:: MPFR Copying Conditions (LGPL).
46 * Introduction to MPFR:: Brief introduction to GNU MPFR.
47 * Installing MPFR:: How to configure and compile the MPFR library.
48 * Reporting Bugs:: How to usefully report bugs.
49 * MPFR Basics:: What every MPFR user should now.
50 * MPFR Interface:: MPFR functions and macros.
51 * API Compatibility:: API compatibility with previous MPFR versions.
54 * GNU Free Documentation License::
56 * Function and Type Index::
59 File: mpfr.info, Node: Copying, Next: Introduction to MPFR, Prev: Top, Up: Top
61 MPFR Copying Conditions
62 ***********************
64 The GNU MPFR library (or MPFR for short) is "free"; this means that
65 everyone is free to use it and free to redistribute it on a free basis.
66 The library is not in the public domain; it is copyrighted and there
67 are restrictions on its distribution, but these restrictions are
68 designed to permit everything that a good cooperating citizen would
69 want to do. What is not allowed is to try to prevent others from
70 further sharing any version of this library that they might get from
73 Specifically, we want to make sure that you have the right to give
74 away copies of the library, that you receive source code or else can
75 get it if you want it, that you can change this library or use pieces
76 of it in new free programs, and that you know you can do these things.
78 To make sure that everyone has such rights, we have to forbid you to
79 deprive anyone else of these rights. For example, if you distribute
80 copies of the GNU MPFR library, you must give the recipients all the
81 rights that you have. You must make sure that they, too, receive or
82 can get the source code. And you must tell them their rights.
84 Also, for our own protection, we must make certain that everyone
85 finds out that there is no warranty for the GNU MPFR library. If it is
86 modified by someone else and passed on, we want their recipients to
87 know that what they have is not what we distributed, so that any
88 problems introduced by others will not reflect on our reputation.
90 The precise conditions of the license for the GNU MPFR library are
91 found in the Lesser General Public License that accompanies the source
92 code. See the file COPYING.LESSER.
95 File: mpfr.info, Node: Introduction to MPFR, Next: Installing MPFR, Prev: Copying, Up: Top
97 1 Introduction to MPFR
98 **********************
100 MPFR is a portable library written in C for arbitrary precision
101 arithmetic on floating-point numbers. It is based on the GNU MP library.
102 It aims to provide a class of floating-point numbers with precise
103 semantics. The main characteristics of MPFR, which make it differ from
104 most arbitrary precision floating-point software tools, are:
106 * the MPFR code is portable, i.e., the result of any operation does
107 not depend on the machine word size `mp_bits_per_limb' (64 on most
110 * the precision in bits can be set _exactly_ to any valid value for
111 each variable (including very small precision);
113 * MPFR provides the four rounding modes from the IEEE 754-1985
114 standard, plus away-from-zero, as well as for basic operations as
115 for other mathematical functions.
117 In particular, with a precision of 53 bits, MPFR is able to exactly
118 reproduce all computations with double-precision machine floating-point
119 numbers (e.g., `double' type in C, with a C implementation that
120 rigorously follows Annex F of the ISO C99 standard and `FP_CONTRACT'
121 pragma set to `OFF') on the four arithmetic operations and the square
122 root, except the default exponent range is much wider and subnormal
123 numbers are not implemented (but can be emulated).
125 This version of MPFR is released under the GNU Lesser General Public
126 License, version 3 or any later version. It is permitted to link MPFR
127 to most non-free programs, as long as when distributing them the MPFR
128 source code and a means to re-link with a modified MPFR library is
131 1.1 How to Use This Manual
132 ==========================
134 Everyone should read *note MPFR Basics::. If you need to install the
135 library yourself, you need to read *note Installing MPFR::, too. To
136 use the library you will need to refer to *note MPFR Interface::.
138 The rest of the manual can be used for later reference, although it
139 is probably a good idea to glance through it.
142 File: mpfr.info, Node: Installing MPFR, Next: Reporting Bugs, Prev: Introduction to MPFR, Up: Top
147 The MPFR library is already installed on some GNU/Linux distributions,
148 but the development files necessary to the compilation such as `mpfr.h'
149 are not always present. To check that MPFR is fully installed on your
150 computer, you can check the presence of the file `mpfr.h' in
151 `/usr/include', or try to compile a small program having `#include
152 <mpfr.h>' (since `mpfr.h' may be installed somewhere else). For
153 instance, you can try to compile:
159 printf ("MPFR library: %-12s\nMPFR header: %s (based on %d.%d.%d)\n",
160 mpfr_get_version (), MPFR_VERSION_STRING, MPFR_VERSION_MAJOR,
161 MPFR_VERSION_MINOR, MPFR_VERSION_PATCHLEVEL);
167 cc -o version version.c -lmpfr -lgmp
169 and if you get errors whose first line looks like
171 version.c:2:19: error: mpfr.h: No such file or directory
173 then MPFR is probably not installed. Running this program will give you
176 If MPFR is not installed on your computer, or if you want to install
177 a different version, please follow the steps below.
182 Here are the steps needed to install the library on Unix systems (more
183 details are provided in the `INSTALL' file):
185 1. To build MPFR, you first have to install GNU MP (version 4.1 or
186 higher) on your computer. You need a C compiler, preferably GCC,
187 but any reasonable compiler should work. And you need the
188 standard Unix `make' command, plus some other standard Unix
191 Then, in the MPFR build directory, type the following commands.
195 This will prepare the build and setup the options according to
196 your system. You can give options to specify the install
197 directories (instead of the default `/usr/local'), threading
198 support, and so on. See the `INSTALL' file and/or the output of
199 `./configure --help' for more information, in particular if you
204 This will compile MPFR, and create a library archive file
205 `libmpfr.a'. On most platforms, a dynamic library will be
210 This will make sure MPFR was built correctly. If you get error
211 messages, please report this to the MPFR mailing-list
212 `mpfr@loria.fr'. (*Note Reporting Bugs::, for information on what
213 to include in useful bug reports.)
217 This will copy the files `mpfr.h' and `mpf2mpfr.h' to the directory
218 `/usr/local/include', the library files (`libmpfr.a' and possibly
219 others) to the directory `/usr/local/lib', the file `mpfr.info' to
220 the directory `/usr/local/share/info', and some other documentation
221 files to the directory `/usr/local/share/doc/mpfr' (or if you
222 passed the `--prefix' option to `configure', using the prefix
223 directory given as argument to `--prefix' instead of `/usr/local').
225 2.2 Other `make' Targets
226 ========================
228 There are some other useful make targets:
230 * `mpfr.info' or `info'
232 Create or update an info version of the manual, in `mpfr.info'.
234 This file is already provided in the MPFR archives.
236 * `mpfr.pdf' or `pdf'
238 Create a PDF version of the manual, in `mpfr.pdf'.
240 * `mpfr.dvi' or `dvi'
242 Create a DVI version of the manual, in `mpfr.dvi'.
246 Create a Postscript version of the manual, in `mpfr.ps'.
248 * `mpfr.html' or `html'
250 Create a HTML version of the manual, in several pages in the
251 directory `doc/mpfr.html'; if you want only one output HTML file,
252 then type `makeinfo --html --no-split mpfr.texi' from the `doc'
257 Delete all object files and archive files, but not the
262 Delete all generated files not included in the distribution.
266 Delete all files copied by `make install'.
271 In case of problem, please read the `INSTALL' file carefully before
272 reporting a bug, in particular section "In case of problem". Some
273 problems are due to bad configuration on the user side (not specific to
274 MPFR). Problems are also mentioned in the FAQ
275 `http://www.mpfr.org/faq.html'.
277 Please report problems to the MPFR mailing-list `mpfr@loria.fr'.
278 *Note Reporting Bugs::. Some bug fixes are available on the MPFR 3.1.0
279 web page `http://www.mpfr.org/mpfr-3.1.0/'.
281 2.4 Getting the Latest Version of MPFR
282 ======================================
284 The latest version of MPFR is available from
285 `ftp://ftp.gnu.org/gnu/mpfr/' or `http://www.mpfr.org/'.
288 File: mpfr.info, Node: Reporting Bugs, Next: MPFR Basics, Prev: Installing MPFR, Up: Top
293 If you think you have found a bug in the MPFR library, first have a look
294 on the MPFR 3.1.0 web page `http://www.mpfr.org/mpfr-3.1.0/' and the
295 FAQ `http://www.mpfr.org/faq.html': perhaps this bug is already known,
296 in which case you may find there a workaround for it. You might also
297 look in the archives of the MPFR mailing-list:
298 `http://websympa.loria.fr/wwsympa/arc/mpfr'. Otherwise, please
299 investigate and report it. We have made this library available to you,
300 and it is not to ask too much from you, to ask you to report the bugs
303 There are a few things you should think about when you put your bug
306 You have to send us a test case that makes it possible for us to
307 reproduce the bug, i.e., a small self-content program, using no other
308 library than MPFR. Include instructions on how to run the test case.
310 You also have to explain what is wrong; if you get a crash, or if
311 the results you get are incorrect and in that case, in what way.
313 Please include compiler version information in your bug report. This
314 can be extracted using `cc -V' on some machines, or, if you're using
315 GCC, `gcc -v'. Also, include the output from `uname -a' and the MPFR
316 version (the GMP version may be useful too). If you get a failure
317 while running `make' or `make check', please include the `config.log'
318 file in your bug report.
320 If your bug report is good, we will do our best to help you to get a
321 corrected version of the library; if the bug report is poor, we will
322 not do anything about it (aside of chiding you to send better bug
325 Send your bug report to the MPFR mailing-list `mpfr@loria.fr'.
327 If you think something in this manual is unclear, or downright
328 incorrect, or if the language needs to be improved, please send a note
332 File: mpfr.info, Node: MPFR Basics, Next: MPFR Interface, Prev: Reporting Bugs, Up: Top
339 * Headers and Libraries::
340 * Nomenclature and Types::
341 * MPFR Variable Conventions::
343 * Floating-Point Values on Special Numbers::
348 File: mpfr.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: MPFR Basics, Up: MPFR Basics
350 4.1 Headers and Libraries
351 =========================
353 All declarations needed to use MPFR are collected in the include file
354 `mpfr.h'. It is designed to work with both C and C++ compilers. You
355 should include that file in any program using the MPFR library:
359 Note however that prototypes for MPFR functions with `FILE *'
360 parameters are provided only if `<stdio.h>' is included too (before
366 Likewise `<stdarg.h>' (or `<varargs.h>') is required for prototypes
367 with `va_list' parameters, such as `mpfr_vprintf'.
369 And for any functions using `intmax_t', you must include
370 `<stdint.h>' or `<inttypes.h>' before `mpfr.h', to allow `mpfr.h' to
371 define prototypes for these functions. Moreover, users of C++ compilers
372 under some platforms may need to define `MPFR_USE_INTMAX_T' (and should
373 do it for portability) before `mpfr.h' has been included; of course, it
374 is possible to do that on the command line, e.g., with
375 `-DMPFR_USE_INTMAX_T'.
377 Note: If `mpfr.h' and/or `gmp.h' (used by `mpfr.h') are included
378 several times (possibly from another header file), `<stdio.h>' and/or
379 `<stdarg.h>' (or `<varargs.h>') should be included *before the first
380 inclusion* of `mpfr.h' or `gmp.h'. Alternatively, you can define
381 `MPFR_USE_FILE' (for MPFR I/O functions) and/or `MPFR_USE_VA_LIST' (for
382 MPFR functions with `va_list' parameters) anywhere before the last
383 inclusion of `mpfr.h'. As a consequence, if your file is a public
384 header that includes `mpfr.h', you need to use the latter method.
386 When calling a MPFR macro, it is not allowed to have previously
387 defined a macro with the same name as some keywords (currently `do',
388 `while' and `sizeof').
390 You can avoid the use of MPFR macros encapsulating functions by
391 defining the `MPFR_USE_NO_MACRO' macro before `mpfr.h' is included. In
392 general this should not be necessary, but this can be useful when
393 debugging user code: with some macros, the compiler may emit spurious
394 warnings with some warning options, and macros can prevent some
397 All programs using MPFR must link against both `libmpfr' and
398 `libgmp' libraries. On a typical Unix-like system this can be done
399 with `-lmpfr -lgmp' (in that order), for example:
401 gcc myprogram.c -lmpfr -lgmp
403 MPFR is built using Libtool and an application can use that to link
404 if desired, *note GNU Libtool: (libtool.info)Top.
406 If MPFR has been installed to a non-standard location, then it may be
407 necessary to set up environment variables such as `C_INCLUDE_PATH' and
408 `LIBRARY_PATH', or use `-I' and `-L' compiler options, in order to
409 point to the right directories. For a shared library, it may also be
410 necessary to set up some sort of run-time library path (e.g.,
411 `LD_LIBRARY_PATH') on some systems. Please read the `INSTALL' file for
412 additional information.
415 File: mpfr.info, Node: Nomenclature and Types, Next: MPFR Variable Conventions, Prev: Headers and Libraries, Up: MPFR Basics
417 4.2 Nomenclature and Types
418 ==========================
420 A "floating-point number", or "float" for short, is an arbitrary
421 precision significand (also called mantissa) with a limited precision
422 exponent. The C data type for such objects is `mpfr_t' (internally
423 defined as a one-element array of a structure, and `mpfr_ptr' is the C
424 data type representing a pointer to this structure). A floating-point
425 number can have three special values: Not-a-Number (NaN) or plus or
426 minus Infinity. NaN represents an uninitialized object, the result of
427 an invalid operation (like 0 divided by 0), or a value that cannot be
428 determined (like +Infinity minus +Infinity). Moreover, like in the IEEE
429 754 standard, zero is signed, i.e., there are both +0 and -0; the
430 behavior is the same as in the IEEE 754 standard and it is generalized
431 to the other functions supported by MPFR. Unless documented otherwise,
432 the sign bit of a NaN is unspecified.
434 The "precision" is the number of bits used to represent the significand
435 of a floating-point number; the corresponding C data type is
436 `mpfr_prec_t'. The precision can be any integer between
437 `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. In the current implementation,
438 `MPFR_PREC_MIN' is equal to 2.
440 Warning! MPFR needs to increase the precision internally, in order to
441 provide accurate results (and in particular, correct rounding). Do not
442 attempt to set the precision to any value near `MPFR_PREC_MAX',
443 otherwise MPFR will abort due to an assertion failure. Moreover, you
444 may reach some memory limit on your platform, in which case the program
445 may abort, crash or have undefined behavior (depending on your C
448 The "rounding mode" specifies the way to round the result of a
449 floating-point operation, in case the exact result can not be
450 represented exactly in the destination significand; the corresponding C
451 data type is `mpfr_rnd_t'.
454 File: mpfr.info, Node: MPFR Variable Conventions, Next: Rounding Modes, Prev: Nomenclature and Types, Up: MPFR Basics
456 4.3 MPFR Variable Conventions
457 =============================
459 Before you can assign to an MPFR variable, you need to initialize it by
460 calling one of the special initialization functions. When you're done
461 with a variable, you need to clear it out, using one of the functions
462 for that purpose. A variable should only be initialized once, or at
463 least cleared out between each initialization. After a variable has
464 been initialized, it may be assigned to any number of times. For
465 efficiency reasons, avoid to initialize and clear out a variable in
466 loops. Instead, initialize it before entering the loop, and clear it
467 out after the loop has exited. You do not need to be concerned about
468 allocating additional space for MPFR variables, since any variable has
469 a significand of fixed size. Hence unless you change its precision, or
470 clear and reinitialize it, a floating-point variable will have the same
471 allocated space during all its life.
473 As a general rule, all MPFR functions expect output arguments before
474 input arguments. This notation is based on an analogy with the
475 assignment operator. MPFR allows you to use the same variable for both
476 input and output in the same expression. For example, the main
477 function for floating-point multiplication, `mpfr_mul', can be used
478 like this: `mpfr_mul (x, x, x, rnd)'. This computes the square of X
479 with rounding mode `rnd' and puts the result back in X.
482 File: mpfr.info, Node: Rounding Modes, Next: Floating-Point Values on Special Numbers, Prev: MPFR Variable Conventions, Up: MPFR Basics
487 The following five rounding modes are supported:
489 * `MPFR_RNDN': round to nearest (roundTiesToEven in IEEE 754-2008),
491 * `MPFR_RNDZ': round toward zero (roundTowardZero in IEEE 754-2008),
493 * `MPFR_RNDU': round toward plus infinity (roundTowardPositive in
496 * `MPFR_RNDD': round toward minus infinity (roundTowardNegative in
499 * `MPFR_RNDA': round away from zero.
501 The `round to nearest' mode works as in the IEEE 754 standard: in
502 case the number to be rounded lies exactly in the middle of two
503 representable numbers, it is rounded to the one with the least
504 significant bit set to zero. For example, the number 2.5, which is
505 represented by (10.1) in binary, is rounded to (10.0)=2 with a
506 precision of two bits, and not to (11.0)=3. This rule avoids the
507 "drift" phenomenon mentioned by Knuth in volume 2 of The Art of
508 Computer Programming (Section 4.2.2).
510 Most MPFR functions take as first argument the destination variable,
511 as second and following arguments the input variables, as last argument
512 a rounding mode, and have a return value of type `int', called the
513 "ternary value". The value stored in the destination variable is
514 correctly rounded, i.e., MPFR behaves as if it computed the result with
515 an infinite precision, then rounded it to the precision of this
516 variable. The input variables are regarded as exact (in particular,
517 their precision does not affect the result).
519 As a consequence, in case of a non-zero real rounded result, the
520 error on the result is less or equal to 1/2 ulp (unit in the last
521 place) of that result in the rounding to nearest mode, and less than 1
522 ulp of that result in the directed rounding modes (a ulp is the weight
523 of the least significant represented bit of the result after rounding).
525 Unless documented otherwise, functions returning an `int' return a
526 ternary value. If the ternary value is zero, it means that the value
527 stored in the destination variable is the exact result of the
528 corresponding mathematical function. If the ternary value is positive
529 (resp. negative), it means the value stored in the destination variable
530 is greater (resp. lower) than the exact result. For example with the
531 `MPFR_RNDU' rounding mode, the ternary value is usually positive,
532 except when the result is exact, in which case it is zero. In the case
533 of an infinite result, it is considered as inexact when it was obtained
534 by overflow, and exact otherwise. A NaN result (Not-a-Number) always
535 corresponds to an exact return value. The opposite of a returned
536 ternary value is guaranteed to be representable in an `int'.
538 Unless documented otherwise, functions returning as result the value
539 `1' (or any other value specified in this manual) for special cases
540 (like `acos(0)') yield an overflow or an underflow if that value is not
541 representable in the current exponent range.
544 File: mpfr.info, Node: Floating-Point Values on Special Numbers, Next: Exceptions, Prev: Rounding Modes, Up: MPFR Basics
546 4.5 Floating-Point Values on Special Numbers
547 ============================================
549 This section specifies the floating-point values (of type `mpfr_t')
550 returned by MPFR functions (where by "returned" we mean here the
551 modified value of the destination object, which should not be mixed
552 with the ternary return value of type `int' of those functions). For
553 functions returning several values (like `mpfr_sin_cos'), the rules
554 apply to each result separately.
556 Functions can have one or several input arguments. An input point is
557 a mapping from these input arguments to the set of the MPFR numbers.
558 When none of its components are NaN, an input point can also be seen as
559 a tuple in the extended real numbers (the set of the real numbers with
562 When the input point is in the domain of the mathematical function,
563 the result is rounded as described in Section "Rounding Modes" (but see
564 below for the specification of the sign of an exact zero). Otherwise
565 the general rules from this section apply unless stated otherwise in
566 the description of the MPFR function (*note MPFR Interface::).
568 When the input point is not in the domain of the mathematical
569 function but is in its closure in the extended real numbers and the
570 function can be extended by continuity, the result is the obtained
571 limit. Examples: `mpfr_hypot' on (+Inf,0) gives +Inf. But `mpfr_pow'
572 cannot be defined on (1,+Inf) using this rule, as one can find
573 sequences (X_N,Y_N) such that X_N goes to 1, Y_N goes to +Inf and X_N
574 to the Y_N goes to any positive value when N goes to the infinity.
576 When the input point is in the closure of the domain of the
577 mathematical function and an input argument is +0 (resp. -0), one
578 considers the limit when the corresponding argument approaches 0 from
579 above (resp. below). If the limit is not defined (e.g., `mpfr_log' on
580 -0), the behavior is specified in the description of the MPFR function.
582 When the result is equal to 0, its sign is determined by considering
583 the limit as if the input point were not in the domain: If one
584 approaches 0 from above (resp. below), the result is +0 (resp. -0); for
585 example, `mpfr_sin' on +0 gives +0. In the other cases, the sign is
586 specified in the description of the MPFR function; for example
587 `mpfr_max' on -0 and +0 gives +0.
589 When the input point is not in the closure of the domain of the
590 function, the result is NaN. Example: `mpfr_sqrt' on -17 gives NaN.
592 When an input argument is NaN, the result is NaN, possibly except
593 when a partial function is constant on the finite floating-point
594 numbers; such a case is always explicitly specified in *note MPFR
595 Interface::. Example: `mpfr_hypot' on (NaN,0) gives NaN, but
596 `mpfr_hypot' on (NaN,+Inf) gives +Inf (as specified in *note Special
597 Functions::), since for any finite input X, `mpfr_hypot' on (X,+Inf)
601 File: mpfr.info, Node: Exceptions, Next: Memory Handling, Prev: Floating-Point Values on Special Numbers, Up: MPFR Basics
606 MPFR supports 6 exception types:
608 * Underflow: An underflow occurs when the exact result of a function
609 is a non-zero real number and the result obtained after the
610 rounding, assuming an unbounded exponent range (for the rounding),
611 has an exponent smaller than the minimum value of the current
612 exponent range. (In the round-to-nearest mode, the halfway case is
613 rounded toward zero.)
615 Note: This is not the single possible definition of the underflow.
616 MPFR chooses to consider the underflow _after_ rounding. The
617 underflow before rounding can also be defined. For instance,
618 consider a function that has the exact result 7 multiplied by two
619 to the power E-4, where E is the smallest exponent (for a
620 significand between 1/2 and 1), with a 2-bit target precision and
621 rounding toward plus infinity. The exact result has the exponent
622 E-1. With the underflow before rounding, such a function call
623 would yield an underflow, as E-1 is outside the current exponent
624 range. However, MPFR first considers the rounded result assuming
625 an unbounded exponent range. The exact result cannot be
626 represented exactly in precision 2, and here, it is rounded to 0.5
627 times 2 to E, which is representable in the current exponent
628 range. As a consequence, this will not yield an underflow in MPFR.
630 * Overflow: An overflow occurs when the exact result of a function
631 is a non-zero real number and the result obtained after the
632 rounding, assuming an unbounded exponent range (for the rounding),
633 has an exponent larger than the maximum value of the current
634 exponent range. In the round-to-nearest mode, the result is
635 infinite. Note: unlike the underflow case, there is only one
636 possible definition of overflow here.
638 * Divide-by-zero: An exact infinite result is obtained from finite
641 * NaN: A NaN exception occurs when the result of a function is NaN.
643 * Inexact: An inexact exception occurs when the result of a function
644 cannot be represented exactly and must be rounded.
646 * Range error: A range exception occurs when a function that does
647 not return a MPFR number (such as comparisons and conversions to
648 an integer) has an invalid result (e.g., an argument is NaN in
649 `mpfr_cmp', or a conversion to an integer cannot be represented in
653 MPFR has a global flag for each exception, which can be cleared, set
654 or tested by functions described in *note Exception Related Functions::.
656 Differences with the ISO C99 standard:
658 * In C, only quiet NaNs are specified, and a NaN propagation does not
659 raise an invalid exception. Unless explicitly stated otherwise,
660 MPFR sets the NaN flag whenever a NaN is generated, even when a
661 NaN is propagated (e.g., in NaN + NaN), as if all NaNs were
664 * An invalid exception in C corresponds to either a NaN exception or
665 a range error in MPFR.
669 File: mpfr.info, Node: Memory Handling, Prev: Exceptions, Up: MPFR Basics
674 MPFR functions may create caches, e.g., when computing constants such
675 as Pi, either because the user has called a function like
676 `mpfr_const_pi' directly or because such a function was called
677 internally by the MPFR library itself to compute some other function.
679 At any time, the user can free the various caches with
680 `mpfr_free_cache'. It is strongly advised to do that before terminating
681 a thread, or before exiting when using tools like `valgrind' (to avoid
682 memory leaks being reported).
684 MPFR internal data such as flags, the exponent range, the default
685 precision and rounding mode, and caches (i.e., data that are not
686 accessed via parameters) are either global (if MPFR has not been
687 compiled as thread safe) or per-thread (thread local storage, TLS).
688 The initial values of TLS data after a thread is created entirely
689 depend on the compiler and thread implementation (MPFR simply does a
690 conventional variable initialization, the variables being declared with
691 an implementation-defined TLS specifier).
694 File: mpfr.info, Node: MPFR Interface, Next: API Compatibility, Prev: MPFR Basics, Up: Top
699 The floating-point functions expect arguments of type `mpfr_t'.
701 The MPFR floating-point functions have an interface that is similar
702 to the GNU MP functions. The function prefix for floating-point
703 operations is `mpfr_'.
705 The user has to specify the precision of each variable. A
706 computation that assigns a variable will take place with the precision
707 of the assigned variable; the cost of that computation should not
708 depend on the precision of variables used as input (on average).
710 The semantics of a calculation in MPFR is specified as follows:
711 Compute the requested operation exactly (with "infinite accuracy"), and
712 round the result to the precision of the destination variable, with the
713 given rounding mode. The MPFR floating-point functions are intended to
714 be a smooth extension of the IEEE 754 arithmetic. The results obtained
715 on a given computer are identical to those obtained on a computer with
716 a different word size, or with a different compiler or operating system.
718 MPFR _does not keep track_ of the accuracy of a computation. This is
719 left to the user or to a higher layer (for example the MPFI library for
720 interval arithmetic). As a consequence, if two variables are used to
721 store only a few significant bits, and their product is stored in a
722 variable with large precision, then MPFR will still compute the result
725 The value of the standard C macro `errno' may be set to non-zero by
726 any MPFR function or macro, whether or not there is an error.
730 * Initialization Functions::
731 * Assignment Functions::
732 * Combined Initialization and Assignment Functions::
733 * Conversion Functions::
734 * Basic Arithmetic Functions::
735 * Comparison Functions::
736 * Special Functions::
737 * Input and Output Functions::
738 * Formatted Output Functions::
739 * Integer Related Functions::
740 * Rounding Related Functions::
741 * Miscellaneous Functions::
742 * Exception Related Functions::
743 * Compatibility with MPF::
748 File: mpfr.info, Node: Initialization Functions, Next: Assignment Functions, Prev: MPFR Interface, Up: MPFR Interface
750 5.1 Initialization Functions
751 ============================
753 An `mpfr_t' object must be initialized before storing the first value in
754 it. The functions `mpfr_init' and `mpfr_init2' are used for that
757 -- Function: void mpfr_init2 (mpfr_t X, mpfr_prec_t PREC)
758 Initialize X, set its precision to be *exactly* PREC bits and its
759 value to NaN. (Warning: the corresponding MPF function initializes
762 Normally, a variable should be initialized once only or at least
763 be cleared, using `mpfr_clear', between initializations. To
764 change the precision of a variable which has already been
765 initialized, use `mpfr_set_prec'. The precision PREC must be an
766 integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX' (otherwise the
767 behavior is undefined).
769 -- Function: void mpfr_inits2 (mpfr_prec_t PREC, mpfr_t X, ...)
770 Initialize all the `mpfr_t' variables of the given variable
771 argument `va_list', set their precision to be *exactly* PREC bits
772 and their value to NaN. See `mpfr_init2' for more details. The
773 `va_list' is assumed to be composed only of type `mpfr_t' (or
774 equivalently `mpfr_ptr'). It begins from X, and ends when it
775 encounters a null pointer (whose type must also be `mpfr_ptr').
777 -- Function: void mpfr_clear (mpfr_t X)
778 Free the space occupied by the significand of X. Make sure to
779 call this function for all `mpfr_t' variables when you are done
782 -- Function: void mpfr_clears (mpfr_t X, ...)
783 Free the space occupied by all the `mpfr_t' variables of the given
784 `va_list'. See `mpfr_clear' for more details. The `va_list' is
785 assumed to be composed only of type `mpfr_t' (or equivalently
786 `mpfr_ptr'). It begins from X, and ends when it encounters a null
787 pointer (whose type must also be `mpfr_ptr').
789 Here is an example of how to use multiple initialization functions
790 (since `NULL' is not necessarily defined in this context, we use
791 `(mpfr_ptr) 0' instead, but `(mpfr_ptr) NULL' is also correct).
795 mpfr_inits2 (256, x, y, z, t, (mpfr_ptr) 0);
797 mpfr_clears (x, y, z, t, (mpfr_ptr) 0);
800 -- Function: void mpfr_init (mpfr_t X)
801 Initialize X, set its precision to the default precision, and set
802 its value to NaN. The default precision can be changed by a call
803 to `mpfr_set_default_prec'.
805 Warning! In a given program, some other libraries might change the
806 default precision and not restore it. Thus it is safer to use
809 -- Function: void mpfr_inits (mpfr_t X, ...)
810 Initialize all the `mpfr_t' variables of the given `va_list', set
811 their precision to the default precision and their value to NaN.
812 See `mpfr_init' for more details. The `va_list' is assumed to be
813 composed only of type `mpfr_t' (or equivalently `mpfr_ptr'). It
814 begins from X, and ends when it encounters a null pointer (whose
815 type must also be `mpfr_ptr').
817 Warning! In a given program, some other libraries might change the
818 default precision and not restore it. Thus it is safer to use
821 -- Macro: MPFR_DECL_INIT (NAME, PREC)
822 This macro declares NAME as an automatic variable of type `mpfr_t',
823 initializes it and sets its precision to be *exactly* PREC bits
824 and its value to NaN. NAME must be a valid identifier. You must
825 use this macro in the declaration section. This macro is much
826 faster than using `mpfr_init2' but has some drawbacks:
828 * You *must not* call `mpfr_clear' with variables created with
829 this macro (the storage is allocated at the point of
830 declaration and deallocated when the brace-level is exited).
832 * You *cannot* change their precision.
834 * You *should not* create variables with huge precision with
837 * Your compiler must support `Non-Constant Initializers'
838 (standard in C++ and ISO C99) and `Token Pasting' (standard
839 in ISO C89). If PREC is not a constant expression, your
840 compiler must support `variable-length automatic arrays'
841 (standard in ISO C99). GCC 2.95.3 and above supports all
842 these features. If you compile your program with GCC in C89
843 mode and with `-pedantic', you may want to define the
844 `MPFR_USE_EXTENSION' macro to avoid warnings due to the
845 `MPFR_DECL_INIT' implementation.
847 -- Function: void mpfr_set_default_prec (mpfr_prec_t PREC)
848 Set the default precision to be *exactly* PREC bits, where PREC
849 can be any integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX'.
850 The precision of a variable means the number of bits used to store
851 its significand. All subsequent calls to `mpfr_init' or
852 `mpfr_inits' will use this precision, but previously initialized
853 variables are unaffected. The default precision is set to 53 bits
856 Note: when MPFR is built with the `--enable-thread-safe' configure
857 option, the default precision is local to each thread. *Note
858 Memory Handling::, for more information.
860 -- Function: mpfr_prec_t mpfr_get_default_prec (void)
861 Return the current default MPFR precision in bits. See the
862 documentation of `mpfr_set_default_prec'.
864 Here is an example on how to initialize floating-point variables:
868 mpfr_init (x); /* use default precision */
869 mpfr_init2 (y, 256); /* precision _exactly_ 256 bits */
871 /* When the program is about to exit, do ... */
874 mpfr_free_cache (); /* free the cache for constants like pi */
877 The following functions are useful for changing the precision during
878 a calculation. A typical use would be for adjusting the precision
879 gradually in iterative algorithms like Newton-Raphson, making the
880 computation precision closely match the actual accurate part of the
883 -- Function: void mpfr_set_prec (mpfr_t X, mpfr_prec_t PREC)
884 Reset the precision of X to be *exactly* PREC bits, and set its
885 value to NaN. The previous value stored in X is lost. It is
886 equivalent to a call to `mpfr_clear(x)' followed by a call to
887 `mpfr_init2(x, prec)', but more efficient as no allocation is done
888 in case the current allocated space for the significand of X is
889 enough. The precision PREC can be any integer between
890 `MPFR_PREC_MIN' and `MPFR_PREC_MAX'. In case you want to keep the
891 previous value stored in X, use `mpfr_prec_round' instead.
893 -- Function: mpfr_prec_t mpfr_get_prec (mpfr_t X)
894 Return the precision of X, i.e., the number of bits used to store
898 File: mpfr.info, Node: Assignment Functions, Next: Combined Initialization and Assignment Functions, Prev: Initialization Functions, Up: MPFR Interface
900 5.2 Assignment Functions
901 ========================
903 These functions assign new values to already initialized floats (*note
904 Initialization Functions::).
906 -- Function: int mpfr_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
907 -- Function: int mpfr_set_ui (mpfr_t ROP, unsigned long int OP,
909 -- Function: int mpfr_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t RND)
910 -- Function: int mpfr_set_uj (mpfr_t ROP, uintmax_t OP, mpfr_rnd_t RND)
911 -- Function: int mpfr_set_sj (mpfr_t ROP, intmax_t OP, mpfr_rnd_t RND)
912 -- Function: int mpfr_set_flt (mpfr_t ROP, float OP, mpfr_rnd_t RND)
913 -- Function: int mpfr_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND)
914 -- Function: int mpfr_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t
916 -- Function: int mpfr_set_decimal64 (mpfr_t ROP, _Decimal64 OP,
918 -- Function: int mpfr_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND)
919 -- Function: int mpfr_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND)
920 -- Function: int mpfr_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND)
921 Set the value of ROP from OP, rounded toward the given direction
922 RND. Note that the input 0 is converted to +0 by `mpfr_set_ui',
923 `mpfr_set_si', `mpfr_set_uj', `mpfr_set_sj', `mpfr_set_z',
924 `mpfr_set_q' and `mpfr_set_f', regardless of the rounding mode.
925 If the system does not support the IEEE 754 standard,
926 `mpfr_set_flt', `mpfr_set_d', `mpfr_set_ld' and
927 `mpfr_set_decimal64' might not preserve the signed zeros. The
928 `mpfr_set_decimal64' function is built only with the configure
929 option `--enable-decimal-float', which also requires
930 `--with-gmp-build', and when the compiler or system provides the
931 `_Decimal64' data type (recent versions of GCC support this data
932 type). `mpfr_set_q' might fail if the numerator (or the
933 denominator) can not be represented as a `mpfr_t'.
935 Note: If you want to store a floating-point constant to a `mpfr_t',
936 you should use `mpfr_set_str' (or one of the MPFR constant
937 functions, such as `mpfr_const_pi' for Pi) instead of
938 `mpfr_set_flt', `mpfr_set_d', `mpfr_set_ld' or
939 `mpfr_set_decimal64'. Otherwise the floating-point constant will
940 be first converted into a reduced-precision (e.g., 53-bit) binary
941 (or decimal, for `mpfr_set_decimal64') number before MPFR can work
944 -- Function: int mpfr_set_ui_2exp (mpfr_t ROP, unsigned long int OP,
945 mpfr_exp_t E, mpfr_rnd_t RND)
946 -- Function: int mpfr_set_si_2exp (mpfr_t ROP, long int OP, mpfr_exp_t
948 -- Function: int mpfr_set_uj_2exp (mpfr_t ROP, uintmax_t OP, intmax_t
950 -- Function: int mpfr_set_sj_2exp (mpfr_t ROP, intmax_t OP, intmax_t
952 -- Function: int mpfr_set_z_2exp (mpfr_t ROP, mpz_t OP, mpfr_exp_t E,
954 Set the value of ROP from OP multiplied by two to the power E,
955 rounded toward the given direction RND. Note that the input 0 is
958 -- Function: int mpfr_set_str (mpfr_t ROP, const char *S, int BASE,
960 Set ROP to the value of the string S in base BASE, rounded in the
961 direction RND. See the documentation of `mpfr_strtofr' for a
962 detailed description of the valid string formats. Contrary to
963 `mpfr_strtofr', `mpfr_set_str' requires the _whole_ string to
964 represent a valid floating-point number.
966 The meaning of the return value differs from other MPFR functions:
967 it is 0 if the entire string up to the final null character is a
968 valid number in base BASE; otherwise it is -1, and ROP may have
969 changed (users interested in the *note ternary value:: should use
970 `mpfr_strtofr' instead).
972 Note: it is preferable to use `mpfr_set_str' if one wants to
973 distinguish between an infinite ROP value coming from an infinite
974 S or from an overflow.
976 -- Function: int mpfr_strtofr (mpfr_t ROP, const char *NPTR, char
977 **ENDPTR, int BASE, mpfr_rnd_t RND)
978 Read a floating-point number from a string NPTR in base BASE,
979 rounded in the direction RND; BASE must be either 0 (to detect the
980 base, as described below) or a number from 2 to 62 (otherwise the
981 behavior is undefined). If NPTR starts with valid data, the result
982 is stored in ROP and `*ENDPTR' points to the character just after
983 the valid data (if ENDPTR is not a null pointer); otherwise ROP is
984 set to zero (for consistency with `strtod') and the value of NPTR
985 is stored in the location referenced by ENDPTR (if ENDPTR is not a
986 null pointer). The usual ternary value is returned.
988 Parsing follows the standard C `strtod' function with some
989 extensions. After optional leading whitespace, one has a subject
990 sequence consisting of an optional sign (`+' or `-'), and either
991 numeric data or special data. The subject sequence is defined as
992 the longest initial subsequence of the input string, starting with
993 the first non-whitespace character, that is of the expected form.
995 The form of numeric data is a non-empty sequence of significand
996 digits with an optional decimal point, and an optional exponent
997 consisting of an exponent prefix followed by an optional sign and
998 a non-empty sequence of decimal digits. A significand digit is
999 either a decimal digit or a Latin letter (62 possible characters),
1000 with `A' = 10, `B' = 11, ..., `Z' = 35; case is ignored in bases
1001 less or equal to 36, in bases larger than 36, `a' = 36, `b' = 37,
1002 ..., `z' = 61. The value of a significand digit must be strictly
1003 less than the base. The decimal point can be either the one
1004 defined by the current locale or the period (the first one is
1005 accepted for consistency with the C standard and the practice, the
1006 second one is accepted to allow the programmer to provide MPFR
1007 numbers from strings in a way that does not depend on the current
1008 locale). The exponent prefix can be `e' or `E' for bases up to
1009 10, or `@' in any base; it indicates a multiplication by a power
1010 of the base. In bases 2 and 16, the exponent prefix can also be
1011 `p' or `P', in which case the exponent, called _binary exponent_,
1012 indicates a multiplication by a power of 2 instead of the base
1013 (there is a difference only for base 16); in base 16 for example
1014 `1p2' represents 4 whereas `1@2' represents 256. The value of an
1015 exponent is always written in base 10.
1017 If the argument BASE is 0, then the base is automatically detected
1018 as follows. If the significand starts with `0b' or `0B', base 2 is
1019 assumed. If the significand starts with `0x' or `0X', base 16 is
1020 assumed. Otherwise base 10 is assumed.
1022 Note: The exponent (if present) must contain at least a digit.
1023 Otherwise the possible exponent prefix and sign are not part of
1024 the number (which ends with the significand). Similarly, if `0b',
1025 `0B', `0x' or `0X' is not followed by a binary/hexadecimal digit,
1026 then the subject sequence stops at the character `0', thus 0 is
1029 Special data (for infinities and NaN) can be `@inf@' or
1030 `@nan@(n-char-sequence-opt)', and if BASE <= 16, it can also be
1031 `infinity', `inf', `nan' or `nan(n-char-sequence-opt)', all case
1032 insensitive. A `n-char-sequence-opt' is a possibly empty string
1033 containing only digits, Latin letters and the underscore (0, 1, 2,
1034 ..., 9, a, b, ..., z, A, B, ..., Z, _). Note: one has an optional
1035 sign for all data, even NaN. For example,
1036 `-@nAn@(This_Is_Not_17)' is a valid representation for NaN in base
1040 -- Function: void mpfr_set_nan (mpfr_t X)
1041 -- Function: void mpfr_set_inf (mpfr_t X, int SIGN)
1042 -- Function: void mpfr_set_zero (mpfr_t X, int SIGN)
1043 Set the variable X to NaN (Not-a-Number), infinity or zero
1044 respectively. In `mpfr_set_inf' or `mpfr_set_zero', X is set to
1045 plus infinity or plus zero iff SIGN is nonnegative; in
1046 `mpfr_set_nan', the sign bit of the result is unspecified.
1048 -- Function: void mpfr_swap (mpfr_t X, mpfr_t Y)
1049 Swap the values X and Y efficiently. Warning: the precisions are
1050 exchanged too; in case the precisions are different, `mpfr_swap'
1051 is thus not equivalent to three `mpfr_set' calls using a third
1055 File: mpfr.info, Node: Combined Initialization and Assignment Functions, Next: Conversion Functions, Prev: Assignment Functions, Up: MPFR Interface
1057 5.3 Combined Initialization and Assignment Functions
1058 ====================================================
1060 -- Macro: int mpfr_init_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1061 -- Macro: int mpfr_init_set_ui (mpfr_t ROP, unsigned long int OP,
1063 -- Macro: int mpfr_init_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t
1065 -- Macro: int mpfr_init_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND)
1066 -- Macro: int mpfr_init_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t
1068 -- Macro: int mpfr_init_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND)
1069 -- Macro: int mpfr_init_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND)
1070 -- Macro: int mpfr_init_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND)
1071 Initialize ROP and set its value from OP, rounded in the direction
1072 RND. The precision of ROP will be taken from the active default
1073 precision, as set by `mpfr_set_default_prec'.
1075 -- Function: int mpfr_init_set_str (mpfr_t X, const char *S, int BASE,
1077 Initialize X and set its value from the string S in base BASE,
1078 rounded in the direction RND. See `mpfr_set_str'.
1081 File: mpfr.info, Node: Conversion Functions, Next: Basic Arithmetic Functions, Prev: Combined Initialization and Assignment Functions, Up: MPFR Interface
1083 5.4 Conversion Functions
1084 ========================
1086 -- Function: float mpfr_get_flt (mpfr_t OP, mpfr_rnd_t RND)
1087 -- Function: double mpfr_get_d (mpfr_t OP, mpfr_rnd_t RND)
1088 -- Function: long double mpfr_get_ld (mpfr_t OP, mpfr_rnd_t RND)
1089 -- Function: _Decimal64 mpfr_get_decimal64 (mpfr_t OP, mpfr_rnd_t RND)
1090 Convert OP to a `float' (respectively `double', `long double' or
1091 `_Decimal64'), using the rounding mode RND. If OP is NaN, some
1092 fixed NaN (either quiet or signaling) or the result of 0.0/0.0 is
1093 returned. If OP is ±Inf, an infinity of the same sign or the
1094 result of ±1.0/0.0 is returned. If OP is zero, these functions
1095 return a zero, trying to preserve its sign, if possible. The
1096 `mpfr_get_decimal64' function is built only under some conditions:
1097 see the documentation of `mpfr_set_decimal64'.
1099 -- Function: long mpfr_get_si (mpfr_t OP, mpfr_rnd_t RND)
1100 -- Function: unsigned long mpfr_get_ui (mpfr_t OP, mpfr_rnd_t RND)
1101 -- Function: intmax_t mpfr_get_sj (mpfr_t OP, mpfr_rnd_t RND)
1102 -- Function: uintmax_t mpfr_get_uj (mpfr_t OP, mpfr_rnd_t RND)
1103 Convert OP to a `long', an `unsigned long', an `intmax_t' or an
1104 `uintmax_t' (respectively) after rounding it with respect to RND.
1105 If OP is NaN, 0 is returned and the _erange_ flag is set. If OP
1106 is too big for the return type, the function returns the maximum
1107 or the minimum of the corresponding C type, depending on the
1108 direction of the overflow; the _erange_ flag is set too. See also
1109 `mpfr_fits_slong_p', `mpfr_fits_ulong_p', `mpfr_fits_intmax_p' and
1110 `mpfr_fits_uintmax_p'.
1112 -- Function: double mpfr_get_d_2exp (long *EXP, mpfr_t OP, mpfr_rnd_t
1114 -- Function: long double mpfr_get_ld_2exp (long *EXP, mpfr_t OP,
1116 Return D and set EXP (formally, the value pointed to by EXP) such
1117 that 0.5<=abs(D)<1 and D times 2 raised to EXP equals OP rounded
1118 to double (resp. long double) precision, using the given rounding
1119 mode. If OP is zero, then a zero of the same sign (or an unsigned
1120 zero, if the implementation does not have signed zeros) is
1121 returned, and EXP is set to 0. If OP is NaN or an infinity, then
1122 the corresponding double precision (resp. long-double precision)
1123 value is returned, and EXP is undefined.
1125 -- Function: int mpfr_frexp (mpfr_exp_t *EXP, mpfr_t Y, mpfr_t X,
1127 Set EXP (formally, the value pointed to by EXP) and Y such that
1128 0.5<=abs(Y)<1 and Y times 2 raised to EXP equals X rounded to the
1129 precision of Y, using the given rounding mode. If X is zero, then
1130 Y is set to a zero of the same sign and EXP is set to 0. If X is
1131 NaN or an infinity, then Y is set to the same value and EXP is
1134 -- Function: mpfr_exp_t mpfr_get_z_2exp (mpz_t ROP, mpfr_t OP)
1135 Put the scaled significand of OP (regarded as an integer, with the
1136 precision of OP) into ROP, and return the exponent EXP (which may
1137 be outside the current exponent range) such that OP exactly equals
1138 ROP times 2 raised to the power EXP. If OP is zero, the minimal
1139 exponent `emin' is returned. If OP is NaN or an infinity, the
1140 _erange_ flag is set, ROP is set to 0, and the the minimal
1141 exponent `emin' is returned. The returned exponent may be less
1142 than the minimal exponent `emin' of MPFR numbers in the current
1143 exponent range; in case the exponent is not representable in the
1144 `mpfr_exp_t' type, the _erange_ flag is set and the minimal value
1145 of the `mpfr_exp_t' type is returned.
1147 -- Function: int mpfr_get_z (mpz_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1148 Convert OP to a `mpz_t', after rounding it with respect to RND. If
1149 OP is NaN or an infinity, the _erange_ flag is set, ROP is set to
1150 0, and 0 is returned.
1152 -- Function: int mpfr_get_f (mpf_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1153 Convert OP to a `mpf_t', after rounding it with respect to RND.
1154 The _erange_ flag is set if OP is NaN or an infinity, which do not
1155 exist in MPF. If OP is NaN, then ROP is undefined. If OP is an
1156 +Inf (resp. -Inf), then ROP is set to the maximum (resp. minimum)
1157 value in the precision of the MPF number; if a future MPF version
1158 supports infinities, this behavior will be considered incorrect
1159 and will change (portable programs should assume that ROP is set
1160 either to this finite number or to an infinite number). Note that
1161 since MPFR currently has the same exponent type as MPF (but not
1162 with the same radix), the range of values is much larger in MPF
1163 than in MPFR, so that an overflow or underflow is not possible.
1165 -- Function: char * mpfr_get_str (char *STR, mpfr_exp_t *EXPPTR, int
1166 B, size_t N, mpfr_t OP, mpfr_rnd_t RND)
1167 Convert OP to a string of digits in base B, with rounding in the
1168 direction RND, where N is either zero (see below) or the number of
1169 significant digits output in the string; in the latter case, N
1170 must be greater or equal to 2. The base may vary from 2 to 62. If
1171 the input number is an ordinary number, the exponent is written
1172 through the pointer EXPPTR (for input 0, the current minimal
1173 exponent is written).
1175 The generated string is a fraction, with an implicit radix point
1176 immediately to the left of the first digit. For example, the
1177 number -3.1416 would be returned as "-31416" in the string and 1
1178 written at EXPPTR. If RND is to nearest, and OP is exactly in the
1179 middle of two consecutive possible outputs, the one with an even
1180 significand is chosen, where both significands are considered with
1181 the exponent of OP. Note that for an odd base, this may not
1182 correspond to an even last digit: for example with 2 digits in
1183 base 7, (14) and a half is rounded to (15) which is 12 in decimal,
1184 (16) and a half is rounded to (20) which is 14 in decimal, and
1185 (26) and a half is rounded to (26) which is 20 in decimal.
1187 If N is zero, the number of digits of the significand is chosen
1188 large enough so that re-reading the printed value with the same
1189 precision, assuming both output and input use rounding to nearest,
1190 will recover the original value of OP. More precisely, in most
1191 cases, the chosen precision of STR is the minimal precision m
1192 depending only on P = PREC(OP) and B that satisfies the above
1193 property, i.e., m = 1 + ceil(P*log(2)/log(B)), with P replaced by
1194 P-1 if B is a power of 2, but in some very rare cases, it might be
1195 m+1 (the smallest case for bases up to 62 is when P equals
1196 186564318007 for bases 7 and 49).
1198 If STR is a null pointer, space for the significand is allocated
1199 using the current allocation function, and a pointer to the string
1200 is returned. To free the returned string, you must use
1203 If STR is not a null pointer, it should point to a block of storage
1204 large enough for the significand, i.e., at least `max(N + 2, 7)'.
1205 The extra two bytes are for a possible minus sign, and for the
1206 terminating null character, and the value 7 accounts for `-@Inf@'
1207 plus the terminating null character.
1209 A pointer to the string is returned, unless there is an error, in
1210 which case a null pointer is returned.
1212 -- Function: void mpfr_free_str (char *STR)
1213 Free a string allocated by `mpfr_get_str' using the current
1214 unallocation function. The block is assumed to be `strlen(STR)+1'
1215 bytes. For more information about how it is done: *note Custom
1216 Allocation: (gmp.info)Custom Allocation.
1218 -- Function: int mpfr_fits_ulong_p (mpfr_t OP, mpfr_rnd_t RND)
1219 -- Function: int mpfr_fits_slong_p (mpfr_t OP, mpfr_rnd_t RND)
1220 -- Function: int mpfr_fits_uint_p (mpfr_t OP, mpfr_rnd_t RND)
1221 -- Function: int mpfr_fits_sint_p (mpfr_t OP, mpfr_rnd_t RND)
1222 -- Function: int mpfr_fits_ushort_p (mpfr_t OP, mpfr_rnd_t RND)
1223 -- Function: int mpfr_fits_sshort_p (mpfr_t OP, mpfr_rnd_t RND)
1224 -- Function: int mpfr_fits_uintmax_p (mpfr_t OP, mpfr_rnd_t RND)
1225 -- Function: int mpfr_fits_intmax_p (mpfr_t OP, mpfr_rnd_t RND)
1226 Return non-zero if OP would fit in the respective C data type,
1227 respectively `unsigned long', `long', `unsigned int', `int',
1228 `unsigned short', `short', `uintmax_t', `intmax_t', when rounded
1229 to an integer in the direction RND.
1232 File: mpfr.info, Node: Basic Arithmetic Functions, Next: Comparison Functions, Prev: Conversion Functions, Up: MPFR Interface
1234 5.5 Basic Arithmetic Functions
1235 ==============================
1237 -- Function: int mpfr_add (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1239 -- Function: int mpfr_add_ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1240 int OP2, mpfr_rnd_t RND)
1241 -- Function: int mpfr_add_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1243 -- Function: int mpfr_add_d (mpfr_t ROP, mpfr_t OP1, double OP2,
1245 -- Function: int mpfr_add_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
1247 -- Function: int mpfr_add_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2,
1249 Set ROP to OP1 + OP2 rounded in the direction RND. For types
1250 having no signed zero, it is considered unsigned (i.e., (+0) + 0 =
1251 (+0) and (-0) + 0 = (-0)). The `mpfr_add_d' function assumes that
1252 the radix of the `double' type is a power of 2, with a precision
1253 at most that declared by the C implementation (macro
1254 `IEEE_DBL_MANT_DIG', and if not defined 53 bits).
1256 -- Function: int mpfr_sub (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1258 -- Function: int mpfr_ui_sub (mpfr_t ROP, unsigned long int OP1,
1259 mpfr_t OP2, mpfr_rnd_t RND)
1260 -- Function: int mpfr_sub_ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1261 int OP2, mpfr_rnd_t RND)
1262 -- Function: int mpfr_si_sub (mpfr_t ROP, long int OP1, mpfr_t OP2,
1264 -- Function: int mpfr_sub_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1266 -- Function: int mpfr_d_sub (mpfr_t ROP, double OP1, mpfr_t OP2,
1268 -- Function: int mpfr_sub_d (mpfr_t ROP, mpfr_t OP1, double OP2,
1270 -- Function: int mpfr_z_sub (mpfr_t ROP, mpz_t OP1, mpfr_t OP2,
1272 -- Function: int mpfr_sub_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
1274 -- Function: int mpfr_sub_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2,
1276 Set ROP to OP1 - OP2 rounded in the direction RND. For types
1277 having no signed zero, it is considered unsigned (i.e., (+0) - 0 =
1278 (+0), (-0) - 0 = (-0), 0 - (+0) = (-0) and 0 - (-0) = (+0)). The
1279 same restrictions than for `mpfr_add_d' apply to `mpfr_d_sub' and
1282 -- Function: int mpfr_mul (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1284 -- Function: int mpfr_mul_ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1285 int OP2, mpfr_rnd_t RND)
1286 -- Function: int mpfr_mul_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1288 -- Function: int mpfr_mul_d (mpfr_t ROP, mpfr_t OP1, double OP2,
1290 -- Function: int mpfr_mul_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
1292 -- Function: int mpfr_mul_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2,
1294 Set ROP to OP1 times OP2 rounded in the direction RND. When a
1295 result is zero, its sign is the product of the signs of the
1296 operands (for types having no signed zero, it is considered
1297 positive). The same restrictions than for `mpfr_add_d' apply to
1300 -- Function: int mpfr_sqr (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1301 Set ROP to the square of OP rounded in the direction RND.
1303 -- Function: int mpfr_div (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1305 -- Function: int mpfr_ui_div (mpfr_t ROP, unsigned long int OP1,
1306 mpfr_t OP2, mpfr_rnd_t RND)
1307 -- Function: int mpfr_div_ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1308 int OP2, mpfr_rnd_t RND)
1309 -- Function: int mpfr_si_div (mpfr_t ROP, long int OP1, mpfr_t OP2,
1311 -- Function: int mpfr_div_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1313 -- Function: int mpfr_d_div (mpfr_t ROP, double OP1, mpfr_t OP2,
1315 -- Function: int mpfr_div_d (mpfr_t ROP, mpfr_t OP1, double OP2,
1317 -- Function: int mpfr_div_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
1319 -- Function: int mpfr_div_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2,
1321 Set ROP to OP1/OP2 rounded in the direction RND. When a result is
1322 zero, its sign is the product of the signs of the operands (for
1323 types having no signed zero, it is considered positive). The same
1324 restrictions than for `mpfr_add_d' apply to `mpfr_d_div' and
1327 -- Function: int mpfr_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1328 -- Function: int mpfr_sqrt_ui (mpfr_t ROP, unsigned long int OP,
1330 Set ROP to the square root of OP rounded in the direction RND (set
1331 ROP to -0 if OP is -0, to be consistent with the IEEE 754
1332 standard). Set ROP to NaN if OP is negative.
1334 -- Function: int mpfr_rec_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1335 Set ROP to the reciprocal square root of OP rounded in the
1336 direction RND. Set ROP to +Inf if OP is ±0, +0 if OP is +Inf, and
1337 NaN if OP is negative.
1339 -- Function: int mpfr_cbrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1340 -- Function: int mpfr_root (mpfr_t ROP, mpfr_t OP, unsigned long int
1342 Set ROP to the cubic root (resp. the Kth root) of OP rounded in
1343 the direction RND. For K odd (resp. even) and OP negative
1344 (including -Inf), set ROP to a negative number (resp. NaN). The
1345 Kth root of -0 is defined to be -0, whatever the parity of K.
1347 -- Function: int mpfr_pow (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1349 -- Function: int mpfr_pow_ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1350 int OP2, mpfr_rnd_t RND)
1351 -- Function: int mpfr_pow_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1353 -- Function: int mpfr_pow_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
1355 -- Function: int mpfr_ui_pow_ui (mpfr_t ROP, unsigned long int OP1,
1356 unsigned long int OP2, mpfr_rnd_t RND)
1357 -- Function: int mpfr_ui_pow (mpfr_t ROP, unsigned long int OP1,
1358 mpfr_t OP2, mpfr_rnd_t RND)
1359 Set ROP to OP1 raised to OP2, rounded in the direction RND.
1360 Special values are handled as described in the ISO C99 and IEEE
1361 754-2008 standards for the `pow' function:
1362 * `pow(±0, Y)' returns plus or minus infinity for Y a negative
1365 * `pow(±0, Y)' returns plus infinity for Y negative and not an
1368 * `pow(±0, Y)' returns plus or minus zero for Y a positive odd
1371 * `pow(±0, Y)' returns plus zero for Y positive and not an odd
1374 * `pow(-1, ±Inf)' returns 1.
1376 * `pow(+1, Y)' returns 1 for any Y, even a NaN.
1378 * `pow(X, ±0)' returns 1 for any X, even a NaN.
1380 * `pow(X, Y)' returns NaN for finite negative X and finite
1383 * `pow(X, -Inf)' returns plus infinity for 0 < abs(x) < 1, and
1384 plus zero for abs(x) > 1.
1386 * `pow(X, +Inf)' returns plus zero for 0 < abs(x) < 1, and plus
1387 infinity for abs(x) > 1.
1389 * `pow(-Inf, Y)' returns minus zero for Y a negative odd
1392 * `pow(-Inf, Y)' returns plus zero for Y negative and not an
1395 * `pow(-Inf, Y)' returns minus infinity for Y a positive odd
1398 * `pow(-Inf, Y)' returns plus infinity for Y positive and not
1401 * `pow(+Inf, Y)' returns plus zero for Y negative, and plus
1402 infinity for Y positive.
1404 -- Function: int mpfr_neg (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1405 -- Function: int mpfr_abs (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1406 Set ROP to -OP and the absolute value of OP respectively, rounded
1407 in the direction RND. Just changes or adjusts the sign if ROP and
1408 OP are the same variable, otherwise a rounding might occur if the
1409 precision of ROP is less than that of OP.
1411 -- Function: int mpfr_dim (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1413 Set ROP to the positive difference of OP1 and OP2, i.e., OP1 - OP2
1414 rounded in the direction RND if OP1 > OP2, +0 if OP1 <= OP2, and
1415 NaN if OP1 or OP2 is NaN.
1417 -- Function: int mpfr_mul_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1418 int OP2, mpfr_rnd_t RND)
1419 -- Function: int mpfr_mul_2si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1421 Set ROP to OP1 times 2 raised to OP2 rounded in the direction RND.
1422 Just increases the exponent by OP2 when ROP and OP1 are identical.
1424 -- Function: int mpfr_div_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1425 int OP2, mpfr_rnd_t RND)
1426 -- Function: int mpfr_div_2si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1428 Set ROP to OP1 divided by 2 raised to OP2 rounded in the direction
1429 RND. Just decreases the exponent by OP2 when ROP and OP1 are
1433 File: mpfr.info, Node: Comparison Functions, Next: Special Functions, Prev: Basic Arithmetic Functions, Up: MPFR Interface
1435 5.6 Comparison Functions
1436 ========================
1438 -- Function: int mpfr_cmp (mpfr_t OP1, mpfr_t OP2)
1439 -- Function: int mpfr_cmp_ui (mpfr_t OP1, unsigned long int OP2)
1440 -- Function: int mpfr_cmp_si (mpfr_t OP1, long int OP2)
1441 -- Function: int mpfr_cmp_d (mpfr_t OP1, double OP2)
1442 -- Function: int mpfr_cmp_ld (mpfr_t OP1, long double OP2)
1443 -- Function: int mpfr_cmp_z (mpfr_t OP1, mpz_t OP2)
1444 -- Function: int mpfr_cmp_q (mpfr_t OP1, mpq_t OP2)
1445 -- Function: int mpfr_cmp_f (mpfr_t OP1, mpf_t OP2)
1446 Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
1447 if OP1 = OP2, and a negative value if OP1 < OP2. Both OP1 and OP2
1448 are considered to their full own precision, which may differ. If
1449 one of the operands is NaN, set the _erange_ flag and return zero.
1451 Note: These functions may be useful to distinguish the three
1452 possible cases. If you need to distinguish two cases only, it is
1453 recommended to use the predicate functions (e.g., `mpfr_equal_p'
1454 for the equality) described below; they behave like the IEEE 754
1455 comparisons, in particular when one or both arguments are NaN. But
1456 only floating-point numbers can be compared (you may need to do a
1459 -- Function: int mpfr_cmp_ui_2exp (mpfr_t OP1, unsigned long int OP2,
1461 -- Function: int mpfr_cmp_si_2exp (mpfr_t OP1, long int OP2,
1463 Compare OP1 and OP2 multiplied by two to the power E. Similar as
1466 -- Function: int mpfr_cmpabs (mpfr_t OP1, mpfr_t OP2)
1467 Compare |OP1| and |OP2|. Return a positive value if |OP1| >
1468 |OP2|, zero if |OP1| = |OP2|, and a negative value if |OP1| <
1469 |OP2|. If one of the operands is NaN, set the _erange_ flag and
1472 -- Function: int mpfr_nan_p (mpfr_t OP)
1473 -- Function: int mpfr_inf_p (mpfr_t OP)
1474 -- Function: int mpfr_number_p (mpfr_t OP)
1475 -- Function: int mpfr_zero_p (mpfr_t OP)
1476 -- Function: int mpfr_regular_p (mpfr_t OP)
1477 Return non-zero if OP is respectively NaN, an infinity, an ordinary
1478 number (i.e., neither NaN nor an infinity), zero, or a regular
1479 number (i.e., neither NaN, nor an infinity nor zero). Return zero
1482 -- Macro: int mpfr_sgn (mpfr_t OP)
1483 Return a positive value if OP > 0, zero if OP = 0, and a negative
1484 value if OP < 0. If the operand is NaN, set the _erange_ flag and
1485 return zero. This is equivalent to `mpfr_cmp_ui (op, 0)', but
1488 -- Function: int mpfr_greater_p (mpfr_t OP1, mpfr_t OP2)
1489 -- Function: int mpfr_greaterequal_p (mpfr_t OP1, mpfr_t OP2)
1490 -- Function: int mpfr_less_p (mpfr_t OP1, mpfr_t OP2)
1491 -- Function: int mpfr_lessequal_p (mpfr_t OP1, mpfr_t OP2)
1492 -- Function: int mpfr_equal_p (mpfr_t OP1, mpfr_t OP2)
1493 Return non-zero if OP1 > OP2, OP1 >= OP2, OP1 < OP2, OP1 <= OP2,
1494 OP1 = OP2 respectively, and zero otherwise. Those functions
1495 return zero whenever OP1 and/or OP2 is NaN.
1497 -- Function: int mpfr_lessgreater_p (mpfr_t OP1, mpfr_t OP2)
1498 Return non-zero if OP1 < OP2 or OP1 > OP2 (i.e., neither OP1, nor
1499 OP2 is NaN, and OP1 <> OP2), zero otherwise (i.e., OP1 and/or OP2
1500 is NaN, or OP1 = OP2).
1502 -- Function: int mpfr_unordered_p (mpfr_t OP1, mpfr_t OP2)
1503 Return non-zero if OP1 or OP2 is a NaN (i.e., they cannot be
1504 compared), zero otherwise.
1507 File: mpfr.info, Node: Special Functions, Next: Input and Output Functions, Prev: Comparison Functions, Up: MPFR Interface
1509 5.7 Special Functions
1510 =====================
1512 All those functions, except explicitly stated (for example
1513 `mpfr_sin_cos'), return a *note ternary value::, i.e., zero for an
1514 exact return value, a positive value for a return value larger than the
1515 exact result, and a negative value otherwise.
1517 Important note: in some domains, computing special functions (either
1518 with correct or incorrect rounding) is expensive, even for small
1519 precision, for example the trigonometric and Bessel functions for large
1522 -- Function: int mpfr_log (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1523 -- Function: int mpfr_log2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1524 -- Function: int mpfr_log10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1525 Set ROP to the natural logarithm of OP, log2(OP) or log10(OP),
1526 respectively, rounded in the direction RND. Set ROP to -Inf if OP
1527 is -0 (i.e., the sign of the zero has no influence on the result).
1529 -- Function: int mpfr_exp (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1530 -- Function: int mpfr_exp2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1531 -- Function: int mpfr_exp10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1532 Set ROP to the exponential of OP, to 2 power of OP or to 10 power
1533 of OP, respectively, rounded in the direction RND.
1535 -- Function: int mpfr_cos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1536 -- Function: int mpfr_sin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1537 -- Function: int mpfr_tan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1538 Set ROP to the cosine of OP, sine of OP, tangent of OP, rounded in
1541 -- Function: int mpfr_sin_cos (mpfr_t SOP, mpfr_t COP, mpfr_t OP,
1543 Set simultaneously SOP to the sine of OP and COP to the cosine of
1544 OP, rounded in the direction RND with the corresponding precisions
1545 of SOP and COP, which must be different variables. Return 0 iff
1546 both results are exact, more precisely it returns s+4c where s=0
1547 if SOP is exact, s=1 if SOP is larger than the sine of OP, s=2 if
1548 SOP is smaller than the sine of OP, and similarly for c and the
1551 -- Function: int mpfr_sec (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1552 -- Function: int mpfr_csc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1553 -- Function: int mpfr_cot (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1554 Set ROP to the secant of OP, cosecant of OP, cotangent of OP,
1555 rounded in the direction RND.
1557 -- Function: int mpfr_acos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1558 -- Function: int mpfr_asin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1559 -- Function: int mpfr_atan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1560 Set ROP to the arc-cosine, arc-sine or arc-tangent of OP, rounded
1561 in the direction RND. Note that since `acos(-1)' returns the
1562 floating-point number closest to Pi according to the given
1563 rounding mode, this number might not be in the output range 0 <=
1564 ROP < \pi of the arc-cosine function; still, the result lies in
1565 the image of the output range by the rounding function. The same
1566 holds for `asin(-1)', `asin(1)', `atan(-Inf)', `atan(+Inf)' or for
1567 `atan(op)' with large OP and small precision of ROP.
1569 -- Function: int mpfr_atan2 (mpfr_t ROP, mpfr_t Y, mpfr_t X,
1571 Set ROP to the arc-tangent2 of Y and X, rounded in the direction
1572 RND: if `x > 0', `atan2(y, x) = atan (y/x)'; if `x < 0', `atan2(y,
1573 x) = sign(y)*(Pi - atan (abs(y/x)))', thus a number from -Pi to Pi.
1574 As for `atan', in case the exact mathematical result is +Pi or -Pi,
1575 its rounded result might be outside the function output range.
1577 `atan2(y, 0)' does not raise any floating-point exception.
1578 Special values are handled as described in the ISO C99 and IEEE
1579 754-2008 standards for the `atan2' function:
1580 * `atan2(+0, -0)' returns +Pi.
1582 * `atan2(-0, -0)' returns -Pi.
1584 * `atan2(+0, +0)' returns +0.
1586 * `atan2(-0, +0)' returns -0.
1588 * `atan2(+0, x)' returns +Pi for x < 0.
1590 * `atan2(-0, x)' returns -Pi for x < 0.
1592 * `atan2(+0, x)' returns +0 for x > 0.
1594 * `atan2(-0, x)' returns -0 for x > 0.
1596 * `atan2(y, 0)' returns -Pi/2 for y < 0.
1598 * `atan2(y, 0)' returns +Pi/2 for y > 0.
1600 * `atan2(+Inf, -Inf)' returns +3*Pi/4.
1602 * `atan2(-Inf, -Inf)' returns -3*Pi/4.
1604 * `atan2(+Inf, +Inf)' returns +Pi/4.
1606 * `atan2(-Inf, +Inf)' returns -Pi/4.
1608 * `atan2(+Inf, x)' returns +Pi/2 for finite x.
1610 * `atan2(-Inf, x)' returns -Pi/2 for finite x.
1612 * `atan2(y, -Inf)' returns +Pi for finite y > 0.
1614 * `atan2(y, -Inf)' returns -Pi for finite y < 0.
1616 * `atan2(y, +Inf)' returns +0 for finite y > 0.
1618 * `atan2(y, +Inf)' returns -0 for finite y < 0.
1620 -- Function: int mpfr_cosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1621 -- Function: int mpfr_sinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1622 -- Function: int mpfr_tanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1623 Set ROP to the hyperbolic cosine, sine or tangent of OP, rounded
1624 in the direction RND.
1626 -- Function: int mpfr_sinh_cosh (mpfr_t SOP, mpfr_t COP, mpfr_t OP,
1628 Set simultaneously SOP to the hyperbolic sine of OP and COP to the
1629 hyperbolic cosine of OP, rounded in the direction RND with the
1630 corresponding precision of SOP and COP, which must be different
1631 variables. Return 0 iff both results are exact (see
1632 `mpfr_sin_cos' for a more detailed description of the return
1635 -- Function: int mpfr_sech (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1636 -- Function: int mpfr_csch (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1637 -- Function: int mpfr_coth (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1638 Set ROP to the hyperbolic secant of OP, cosecant of OP, cotangent
1639 of OP, rounded in the direction RND.
1641 -- Function: int mpfr_acosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1642 -- Function: int mpfr_asinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1643 -- Function: int mpfr_atanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1644 Set ROP to the inverse hyperbolic cosine, sine or tangent of OP,
1645 rounded in the direction RND.
1647 -- Function: int mpfr_fac_ui (mpfr_t ROP, unsigned long int OP,
1649 Set ROP to the factorial of OP, rounded in the direction RND.
1651 -- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1652 Set ROP to the logarithm of one plus OP, rounded in the direction
1655 -- Function: int mpfr_expm1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1656 Set ROP to the exponential of OP followed by a subtraction by one,
1657 rounded in the direction RND.
1659 -- Function: int mpfr_eint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1660 Set ROP to the exponential integral of OP, rounded in the
1661 direction RND. For positive OP, the exponential integral is the
1662 sum of Euler's constant, of the logarithm of OP, and of the sum
1663 for k from 1 to infinity of OP to the power k, divided by k and
1664 factorial(k). For negative OP, ROP is set to NaN (this definition
1665 for negative argument follows formula 5.1.2 from the Handbook of
1666 Mathematical Functions from Abramowitz and Stegun, a future
1667 version might use another definition).
1669 -- Function: int mpfr_li2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1670 Set ROP to real part of the dilogarithm of OP, rounded in the
1671 direction RND. MPFR defines the dilogarithm function as the
1672 integral of -log(1-t)/t from 0 to OP.
1674 -- Function: int mpfr_gamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1675 Set ROP to the value of the Gamma function on OP, rounded in the
1676 direction RND. When OP is a negative integer, ROP is set to NaN.
1678 -- Function: int mpfr_lngamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1679 Set ROP to the value of the logarithm of the Gamma function on OP,
1680 rounded in the direction RND. When -2K-1 <= OP <= -2K, K being a
1681 non-negative integer, ROP is set to NaN. See also `mpfr_lgamma'.
1683 -- Function: int mpfr_lgamma (mpfr_t ROP, int *SIGNP, mpfr_t OP,
1685 Set ROP to the value of the logarithm of the absolute value of the
1686 Gamma function on OP, rounded in the direction RND. The sign (1 or
1687 -1) of Gamma(OP) is returned in the object pointed to by SIGNP.
1688 When OP is an infinity or a non-positive integer, set ROP to +Inf.
1689 When OP is NaN, -Inf or a negative integer, *SIGNP is undefined,
1690 and when OP is ±0, *SIGNP is the sign of the zero.
1692 -- Function: int mpfr_digamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1693 Set ROP to the value of the Digamma (sometimes also called Psi)
1694 function on OP, rounded in the direction RND. When OP is a
1695 negative integer, set ROP to NaN.
1697 -- Function: int mpfr_zeta (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1698 -- Function: int mpfr_zeta_ui (mpfr_t ROP, unsigned long OP,
1700 Set ROP to the value of the Riemann Zeta function on OP, rounded
1701 in the direction RND.
1703 -- Function: int mpfr_erf (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1704 -- Function: int mpfr_erfc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1705 Set ROP to the value of the error function on OP (resp. the
1706 complementary error function on OP) rounded in the direction RND.
1708 -- Function: int mpfr_j0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1709 -- Function: int mpfr_j1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1710 -- Function: int mpfr_jn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t
1712 Set ROP to the value of the first kind Bessel function of order 0,
1713 (resp. 1 and N) on OP, rounded in the direction RND. When OP is
1714 NaN, ROP is always set to NaN. When OP is plus or minus Infinity,
1715 ROP is set to +0. When OP is zero, and N is not zero, ROP is set
1716 to +0 or -0 depending on the parity and sign of N, and the sign of
1719 -- Function: int mpfr_y0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1720 -- Function: int mpfr_y1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1721 -- Function: int mpfr_yn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t
1723 Set ROP to the value of the second kind Bessel function of order 0
1724 (resp. 1 and N) on OP, rounded in the direction RND. When OP is
1725 NaN or negative, ROP is always set to NaN. When OP is +Inf, ROP is
1726 set to +0. When OP is zero, ROP is set to +Inf or -Inf depending
1727 on the parity and sign of N.
1729 -- Function: int mpfr_fma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t
1730 OP3, mpfr_rnd_t RND)
1731 -- Function: int mpfr_fms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t
1732 OP3, mpfr_rnd_t RND)
1733 Set ROP to (OP1 times OP2) + OP3 (resp. (OP1 times OP2) - OP3)
1734 rounded in the direction RND.
1736 -- Function: int mpfr_agm (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1738 Set ROP to the arithmetic-geometric mean of OP1 and OP2, rounded
1739 in the direction RND. The arithmetic-geometric mean is the common
1740 limit of the sequences U_N and V_N, where U_0=OP1, V_0=OP2,
1741 U_(N+1) is the arithmetic mean of U_N and V_N, and V_(N+1) is the
1742 geometric mean of U_N and V_N. If any operand is negative, set
1745 -- Function: int mpfr_hypot (mpfr_t ROP, mpfr_t X, mpfr_t Y,
1747 Set ROP to the Euclidean norm of X and Y, i.e., the square root of
1748 the sum of the squares of X and Y, rounded in the direction RND.
1749 Special values are handled as described in Section F.9.4.3 of the
1750 ISO C99 and IEEE 754-2008 standards: If X or Y is an infinity,
1751 then +Inf is returned in ROP, even if the other number is NaN.
1753 -- Function: int mpfr_ai (mpfr_t ROP, mpfr_t X, mpfr_rnd_t RND)
1754 Set ROP to the value of the Airy function Ai on X, rounded in the
1755 direction RND. When X is NaN, ROP is always set to NaN. When X is
1756 +Inf or -Inf, ROP is +0. The current implementation is not
1757 intended to be used with large arguments. It works with abs(X)
1758 typically smaller than 500. For larger arguments, other methods
1759 should be used and will be implemented in a future version.
1761 -- Function: int mpfr_const_log2 (mpfr_t ROP, mpfr_rnd_t RND)
1762 -- Function: int mpfr_const_pi (mpfr_t ROP, mpfr_rnd_t RND)
1763 -- Function: int mpfr_const_euler (mpfr_t ROP, mpfr_rnd_t RND)
1764 -- Function: int mpfr_const_catalan (mpfr_t ROP, mpfr_rnd_t RND)
1765 Set ROP to the logarithm of 2, the value of Pi, of Euler's
1766 constant 0.577..., of Catalan's constant 0.915..., respectively,
1767 rounded in the direction RND. These functions cache the computed
1768 values to avoid other calculations if a lower or equal precision
1769 is requested. To free these caches, use `mpfr_free_cache'.
1771 -- Function: void mpfr_free_cache (void)
1772 Free various caches used by MPFR internally, in particular the
1773 caches used by the functions computing constants
1774 (`mpfr_const_log2', `mpfr_const_pi', `mpfr_const_euler' and
1775 `mpfr_const_catalan'). You should call this function before
1776 terminating a thread, even if you did not call these functions
1777 directly (they could have been called internally).
1779 -- Function: int mpfr_sum (mpfr_t ROP, mpfr_ptr const TAB[], unsigned
1780 long int N, mpfr_rnd_t RND)
1781 Set ROP to the sum of all elements of TAB, whose size is N,
1782 rounded in the direction RND. Warning: for efficiency reasons, TAB
1783 is an array of pointers to `mpfr_t', not an array of `mpfr_t'. If
1784 the returned `int' value is zero, ROP is guaranteed to be the
1785 exact sum; otherwise ROP might be smaller than, equal to, or
1786 larger than the exact sum (in accordance to the rounding mode).
1787 However, `mpfr_sum' does guarantee the result is correctly rounded.
1790 File: mpfr.info, Node: Input and Output Functions, Next: Formatted Output Functions, Prev: Special Functions, Up: MPFR Interface
1792 5.8 Input and Output Functions
1793 ==============================
1795 This section describes functions that perform input from an input/output
1796 stream, and functions that output to an input/output stream. Passing a
1797 null pointer for a `stream' to any of these functions will make them
1798 read from `stdin' and write to `stdout', respectively.
1800 When using any of these functions, you must include the `<stdio.h>'
1801 standard header before `mpfr.h', to allow `mpfr.h' to define prototypes
1802 for these functions.
1804 -- Function: size_t mpfr_out_str (FILE *STREAM, int BASE, size_t N,
1805 mpfr_t OP, mpfr_rnd_t RND)
1806 Output OP on stream STREAM, as a string of digits in base BASE,
1807 rounded in the direction RND. The base may vary from 2 to 62.
1808 Print N significant digits exactly, or if N is 0, enough digits so
1809 that OP can be read back exactly (see `mpfr_get_str').
1811 In addition to the significant digits, a decimal point (defined by
1812 the current locale) at the right of the first digit and a trailing
1813 exponent in base 10, in the form `eNNN', are printed. If BASE is
1814 greater than 10, `@' will be used instead of `e' as exponent
1817 Return the number of characters written, or if an error occurred,
1820 -- Function: size_t mpfr_inp_str (mpfr_t ROP, FILE *STREAM, int BASE,
1822 Input a string in base BASE from stream STREAM, rounded in the
1823 direction RND, and put the read float in ROP.
1825 This function reads a word (defined as a sequence of characters
1826 between whitespace) and parses it using `mpfr_set_str'. See the
1827 documentation of `mpfr_strtofr' for a detailed description of the
1828 valid string formats.
1830 Return the number of bytes read, or if an error occurred, return 0.
1833 File: mpfr.info, Node: Formatted Output Functions, Next: Integer Related Functions, Prev: Input and Output Functions, Up: MPFR Interface
1835 5.9 Formatted Output Functions
1836 ==============================
1841 The class of `mpfr_printf' functions provides formatted output in a
1842 similar manner as the standard C `printf'. These functions are defined
1843 only if your system supports ISO C variadic functions and the
1844 corresponding argument access macros.
1846 When using any of these functions, you must include the `<stdio.h>'
1847 standard header before `mpfr.h', to allow `mpfr.h' to define prototypes
1848 for these functions.
1853 The format specification accepted by `mpfr_printf' is an extension of
1854 the `printf' one. The conversion specification is of the form:
1855 % [flags] [width] [.[precision]] [type] [rounding] conv
1856 `flags', `width', and `precision' have the same meaning as for the
1857 standard `printf' (in particular, notice that the `precision' is
1858 related to the number of digits displayed in the base chosen by `conv'
1859 and not related to the internal precision of the `mpfr_t' variable).
1860 `mpfr_printf' accepts the same `type' specifiers as GMP (except the
1861 non-standard and deprecated `q', use `ll' instead), namely the length
1862 modifiers defined in the C standard:
1866 `j' `intmax_t' or `uintmax_t'
1867 `l' `long' or `wchar_t'
1873 and the `type' specifiers defined in GMP plus `R' and `P' specific
1874 to MPFR (the second column in the table below shows the type of the
1875 argument read in the argument list and the kind of `conv' specifier to
1876 use after the `type' specifier):
1878 `F' `mpf_t', float conversions
1879 `Q' `mpq_t', integer conversions
1880 `M' `mp_limb_t', integer conversions
1881 `N' `mp_limb_t' array, integer conversions
1882 `Z' `mpz_t', integer conversions
1883 `P' `mpfr_prec_t', integer conversions
1884 `R' `mpfr_t', float conversions
1886 The `type' specifiers have the same restrictions as those mentioned
1887 in the GMP documentation: *note Formatted Output Strings:
1888 (gmp.info)Formatted Output Strings. In particular, the `type'
1889 specifiers (except `R' and `P') are supported only if they are
1890 supported by `gmp_printf' in your GMP build; this implies that the
1891 standard specifiers, such as `t', must _also_ be supported by your C
1892 library if you want to use them.
1894 The `rounding' field is specific to `mpfr_t' arguments and should
1895 not be used with other types.
1897 With conversion specification not involving `P' and `R' types,
1898 `mpfr_printf' behaves exactly as `gmp_printf'.
1900 The `P' type specifies that a following `o', `u', `x', or `X'
1901 conversion specifier applies to a `mpfr_prec_t' argument. It is needed
1902 because the `mpfr_prec_t' type does not necessarily correspond to an
1903 `unsigned int' or any fixed standard type. The `precision' field
1904 specifies the minimum number of digits to appear. The default
1905 `precision' is 1. For example:
1910 p = mpfr_get_prec (x);
1911 mpfr_printf ("variable x with %Pu bits", p);
1913 The `R' type specifies that a following `a', `A', `b', `e', `E',
1914 `f', `F', `g', `G', or `n' conversion specifier applies to a `mpfr_t'
1915 argument. The `R' type can be followed by a `rounding' specifier
1916 denoted by one of the following characters:
1918 `U' round toward plus infinity
1919 `D' round toward minus infinity
1920 `Y' round away from zero
1921 `Z' round toward zero
1922 `N' round to nearest (with ties to even)
1923 `*' rounding mode indicated by the
1924 `mpfr_rnd_t' argument just before the
1925 corresponding `mpfr_t' variable.
1927 The default rounding mode is rounding to nearest. The following
1928 three examples are equivalent:
1932 mpfr_printf ("%.128Rf", x);
1933 mpfr_printf ("%.128RNf", x);
1934 mpfr_printf ("%.128R*f", MPFR_RNDN, x);
1936 Note that the rounding away from zero mode is specified with `Y'
1937 because ISO C reserves the `A' specifier for hexadecimal output (see
1940 The output `conv' specifiers allowed with `mpfr_t' parameter are:
1942 `a' `A' hex float, C99 style
1944 `e' `E' scientific format float
1945 `f' `F' fixed point float
1946 `g' `G' fixed or scientific float
1948 The conversion specifier `b' which displays the argument in binary is
1949 specific to `mpfr_t' arguments and should not be used with other types.
1950 Other conversion specifiers have the same meaning as for a `double'
1953 In case of non-decimal output, only the significand is written in the
1954 specified base, the exponent is always displayed in decimal. Special
1955 values are always displayed as `nan', `-inf', and `inf' for `a', `b',
1956 `e', `f', and `g' specifiers and `NAN', `-INF', and `INF' for `A', `E',
1957 `F', and `G' specifiers.
1959 If the `precision' field is not empty, the `mpfr_t' number is
1960 rounded to the given precision in the direction specified by the
1961 rounding mode. If the precision is zero with rounding to nearest mode
1962 and one of the following `conv' specifiers: `a', `A', `b', `e', `E',
1963 tie case is rounded to even when it lies between two consecutive values
1964 at the wanted precision which have the same exponent, otherwise, it is
1965 rounded away from zero. For instance, 85 is displayed as "8e+1" and 95
1966 is displayed as "1e+2" with the format specification `"%.0RNe"'. This
1967 also applies when the `g' (resp. `G') conversion specifier uses the `e'
1968 (resp. `E') style. If the precision is set to a value greater than the
1969 maximum value for an `int', it will be silently reduced down to
1972 If the `precision' field is empty (as in `%Re' or `%.RE') with
1973 `conv' specifier `e' and `E', the number is displayed with enough
1974 digits so that it can be read back exactly, assuming that the input and
1975 output variables have the same precision and that the input and output
1976 rounding modes are both rounding to nearest (as for `mpfr_get_str').
1977 The default precision for an empty `precision' field with `conv'
1978 specifiers `f', `F', `g', and `G' is 6.
1983 For all the following functions, if the number of characters which
1984 ought to be written appears to exceed the maximum limit for an `int',
1985 nothing is written in the stream (resp. to `stdout', to BUF, to STR),
1986 the function returns -1, sets the _erange_ flag, and (in POSIX system
1987 only) `errno' is set to `EOVERFLOW'.
1989 -- Function: int mpfr_fprintf (FILE *STREAM, const char *TEMPLATE, ...)
1990 -- Function: int mpfr_vfprintf (FILE *STREAM, const char *TEMPLATE,
1992 Print to the stream STREAM the optional arguments under the
1993 control of the template string TEMPLATE. Return the number of
1994 characters written or a negative value if an error occurred.
1996 -- Function: int mpfr_printf (const char *TEMPLATE, ...)
1997 -- Function: int mpfr_vprintf (const char *TEMPLATE, va_list AP)
1998 Print to `stdout' the optional arguments under the control of the
1999 template string TEMPLATE. Return the number of characters written
2000 or a negative value if an error occurred.
2002 -- Function: int mpfr_sprintf (char *BUF, const char *TEMPLATE, ...)
2003 -- Function: int mpfr_vsprintf (char *BUF, const char *TEMPLATE,
2005 Form a null-terminated string corresponding to the optional
2006 arguments under the control of the template string TEMPLATE, and
2007 print it in BUF. No overlap is permitted between BUF and the other
2008 arguments. Return the number of characters written in the array
2009 BUF _not counting_ the terminating null character or a negative
2010 value if an error occurred.
2012 -- Function: int mpfr_snprintf (char *BUF, size_t N, const char
2014 -- Function: int mpfr_vsnprintf (char *BUF, size_t N, const char
2015 *TEMPLATE, va_list AP)
2016 Form a null-terminated string corresponding to the optional
2017 arguments under the control of the template string TEMPLATE, and
2018 print it in BUF. If N is zero, nothing is written and BUF may be a
2019 null pointer, otherwise, the N-1 first characters are written in
2020 BUF and the N-th is a null character. Return the number of
2021 characters that would have been written had N be sufficiently
2022 large, _not counting_ the terminating null character, or a
2023 negative value if an error occurred.
2025 -- Function: int mpfr_asprintf (char **STR, const char *TEMPLATE, ...)
2026 -- Function: int mpfr_vasprintf (char **STR, const char *TEMPLATE,
2028 Write their output as a null terminated string in a block of
2029 memory allocated using the current allocation function. A pointer
2030 to the block is stored in STR. The block of memory must be freed
2031 using `mpfr_free_str'. The return value is the number of
2032 characters written in the string, excluding the null-terminator,
2033 or a negative value if an error occurred.
2036 File: mpfr.info, Node: Integer Related Functions, Next: Rounding Related Functions, Prev: Formatted Output Functions, Up: MPFR Interface
2038 5.10 Integer and Remainder Related Functions
2039 ============================================
2041 -- Function: int mpfr_rint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
2042 -- Function: int mpfr_ceil (mpfr_t ROP, mpfr_t OP)
2043 -- Function: int mpfr_floor (mpfr_t ROP, mpfr_t OP)
2044 -- Function: int mpfr_round (mpfr_t ROP, mpfr_t OP)
2045 -- Function: int mpfr_trunc (mpfr_t ROP, mpfr_t OP)
2046 Set ROP to OP rounded to an integer. `mpfr_rint' rounds to the
2047 nearest representable integer in the given direction RND,
2048 `mpfr_ceil' rounds to the next higher or equal representable
2049 integer, `mpfr_floor' to the next lower or equal representable
2050 integer, `mpfr_round' to the nearest representable integer,
2051 rounding halfway cases away from zero (as in the roundTiesToAway
2052 mode of IEEE 754-2008), and `mpfr_trunc' to the next representable
2053 integer toward zero.
2055 The returned value is zero when the result is exact, positive when
2056 it is greater than the original value of OP, and negative when it
2057 is smaller. More precisely, the returned value is 0 when OP is an
2058 integer representable in ROP, 1 or -1 when OP is an integer that
2059 is not representable in ROP, 2 or -2 when OP is not an integer.
2061 Note that `mpfr_round' is different from `mpfr_rint' called with
2062 the rounding to nearest mode (where halfway cases are rounded to
2063 an even integer or significand). Note also that no double rounding
2064 is performed; for instance, 10.5 (1010.1 in binary) is rounded by
2065 `mpfr_rint' with rounding to nearest to 12 (1100 in binary) in
2066 2-bit precision, because the two enclosing numbers representable
2067 on two bits are 8 and 12, and the closest is 12. (If one first
2068 rounded to an integer, one would round 10.5 to 10 with even
2069 rounding, and then 10 would be rounded to 8 again with even
2072 -- Function: int mpfr_rint_ceil (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
2073 -- Function: int mpfr_rint_floor (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t
2075 -- Function: int mpfr_rint_round (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t
2077 -- Function: int mpfr_rint_trunc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t
2079 Set ROP to OP rounded to an integer. `mpfr_rint_ceil' rounds to
2080 the next higher or equal integer, `mpfr_rint_floor' to the next
2081 lower or equal integer, `mpfr_rint_round' to the nearest integer,
2082 rounding halfway cases away from zero, and `mpfr_rint_trunc' to
2083 the next integer toward zero. If the result is not representable,
2084 it is rounded in the direction RND. The returned value is the
2085 ternary value associated with the considered round-to-integer
2086 function (regarded in the same way as any other mathematical
2087 function). Contrary to `mpfr_rint', those functions do perform a
2088 double rounding: first OP is rounded to the nearest integer in the
2089 direction given by the function name, then this nearest integer
2090 (if not representable) is rounded in the given direction RND. For
2091 example, `mpfr_rint_round' with rounding to nearest and a precision
2092 of two bits rounds 6.5 to 7 (halfway cases away from zero), then 7
2093 is rounded to 8 by the round-even rule, despite the fact that 6 is
2094 also representable on two bits, and is closer to 6.5 than 8.
2096 -- Function: int mpfr_frac (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
2097 Set ROP to the fractional part of OP, having the same sign as OP,
2098 rounded in the direction RND (unlike in `mpfr_rint', RND affects
2099 only how the exact fractional part is rounded, not how the
2100 fractional part is generated).
2102 -- Function: int mpfr_modf (mpfr_t IOP, mpfr_t FOP, mpfr_t OP,
2104 Set simultaneously IOP to the integral part of OP and FOP to the
2105 fractional part of OP, rounded in the direction RND with the
2106 corresponding precision of IOP and FOP (equivalent to
2107 `mpfr_trunc(IOP, OP, RND)' and `mpfr_frac(FOP, OP, RND)'). The
2108 variables IOP and FOP must be different. Return 0 iff both results
2109 are exact (see `mpfr_sin_cos' for a more detailed description of
2112 -- Function: int mpfr_fmod (mpfr_t R, mpfr_t X, mpfr_t Y, mpfr_rnd_t
2114 -- Function: int mpfr_remainder (mpfr_t R, mpfr_t X, mpfr_t Y,
2116 -- Function: int mpfr_remquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y,
2118 Set R to the value of X - NY, rounded according to the direction
2119 RND, where N is the integer quotient of X divided by Y, defined as
2120 follows: N is rounded toward zero for `mpfr_fmod', and to the
2121 nearest integer (ties rounded to even) for `mpfr_remainder' and
2124 Special values are handled as described in Section F.9.7.1 of the
2125 ISO C99 standard: If X is infinite or Y is zero, R is NaN. If Y
2126 is infinite and X is finite, R is X rounded to the precision of R.
2127 If R is zero, it has the sign of X. The return value is the
2128 ternary value corresponding to R.
2130 Additionally, `mpfr_remquo' stores the low significant bits from
2131 the quotient N in *Q (more precisely the number of bits in a
2132 `long' minus one), with the sign of X divided by Y (except if
2133 those low bits are all zero, in which case zero is returned).
2134 Note that X may be so large in magnitude relative to Y that an
2135 exact representation of the quotient is not practical. The
2136 `mpfr_remainder' and `mpfr_remquo' functions are useful for
2137 additive argument reduction.
2139 -- Function: int mpfr_integer_p (mpfr_t OP)
2140 Return non-zero iff OP is an integer.
2143 File: mpfr.info, Node: Rounding Related Functions, Next: Miscellaneous Functions, Prev: Integer Related Functions, Up: MPFR Interface
2145 5.11 Rounding Related Functions
2146 ===============================
2148 -- Function: void mpfr_set_default_rounding_mode (mpfr_rnd_t RND)
2149 Set the default rounding mode to RND. The default rounding mode
2150 is to nearest initially.
2152 -- Function: mpfr_rnd_t mpfr_get_default_rounding_mode (void)
2153 Get the default rounding mode.
2155 -- Function: int mpfr_prec_round (mpfr_t X, mpfr_prec_t PREC,
2157 Round X according to RND with precision PREC, which must be an
2158 integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX' (otherwise the
2159 behavior is undefined). If PREC is greater or equal to the
2160 precision of X, then new space is allocated for the significand,
2161 and it is filled with zeros. Otherwise, the significand is
2162 rounded to precision PREC with the given direction. In both cases,
2163 the precision of X is changed to PREC.
2165 Here is an example of how to use `mpfr_prec_round' to implement
2166 Newton's algorithm to compute the inverse of A, assuming X is
2167 already an approximation to N bits:
2168 mpfr_set_prec (t, 2 * n);
2169 mpfr_set (t, a, MPFR_RNDN); /* round a to 2n bits */
2170 mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to 2n bits */
2171 mpfr_ui_sub (t, 1, t, MPFR_RNDN); /* high n bits cancel with 1 */
2172 mpfr_prec_round (t, n, MPFR_RNDN); /* t is correct to n bits */
2173 mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to n bits */
2174 mpfr_prec_round (x, 2 * n, MPFR_RNDN); /* exact */
2175 mpfr_add (x, x, t, MPFR_RNDN); /* x is correct to 2n bits */
2177 -- Function: int mpfr_can_round (mpfr_t B, mpfr_exp_t ERR, mpfr_rnd_t
2178 RND1, mpfr_rnd_t RND2, mpfr_prec_t PREC)
2179 Assuming B is an approximation of an unknown number X in the
2180 direction RND1 with error at most two to the power E(b)-ERR where
2181 E(b) is the exponent of B, return a non-zero value if one is able
2182 to round correctly X to precision PREC with the direction RND2,
2183 and 0 otherwise (including for NaN and Inf). This function *does
2184 not modify* its arguments.
2186 If RND1 is `MPFR_RNDN', then the sign of the error is unknown, but
2187 its absolute value is the same, so that the possible range is
2188 twice as large as with a directed rounding for RND1.
2190 Note: if one wants to also determine the correct *note ternary
2191 value:: when rounding B to precision PREC with rounding mode RND,
2192 a useful trick is the following: if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ, prec + (rnd == MPFR_RNDN)))
2194 Indeed, if RND is `MPFR_RNDN', this will check if one can round
2195 to PREC+1 bits with a directed rounding: if so, one can surely
2196 round to nearest to PREC bits, and in addition one can determine
2197 the correct ternary value, which would not be the case when B is
2198 near from a value exactly representable on PREC bits.
2200 -- Function: mpfr_prec_t mpfr_min_prec (mpfr_t X)
2201 Return the minimal number of bits required to store the
2202 significand of X, and 0 for special values, including 0. (Warning:
2203 the returned value can be less than `MPFR_PREC_MIN'.)
2205 The function name is subject to change.
2207 -- Function: const char * mpfr_print_rnd_mode (mpfr_rnd_t RND)
2208 Return a string ("MPFR_RNDD", "MPFR_RNDU", "MPFR_RNDN",
2209 "MPFR_RNDZ", "MPFR_RNDA") corresponding to the rounding mode RND,
2210 or a null pointer if RND is an invalid rounding mode.
2213 File: mpfr.info, Node: Miscellaneous Functions, Next: Exception Related Functions, Prev: Rounding Related Functions, Up: MPFR Interface
2215 5.12 Miscellaneous Functions
2216 ============================
2218 -- Function: void mpfr_nexttoward (mpfr_t X, mpfr_t Y)
2219 If X or Y is NaN, set X to NaN. If X and Y are equal, X is
2220 unchanged. Otherwise, if X is different from Y, replace X by the
2221 next floating-point number (with the precision of X and the
2222 current exponent range) in the direction of Y (the infinite values
2223 are seen as the smallest and largest floating-point numbers). If
2224 the result is zero, it keeps the same sign. No underflow or
2225 overflow is generated.
2227 -- Function: void mpfr_nextabove (mpfr_t X)
2228 -- Function: void mpfr_nextbelow (mpfr_t X)
2229 Equivalent to `mpfr_nexttoward' where Y is plus infinity (resp.
2232 -- Function: int mpfr_min (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
2234 -- Function: int mpfr_max (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
2236 Set ROP to the minimum (resp. maximum) of OP1 and OP2. If OP1 and
2237 OP2 are both NaN, then ROP is set to NaN. If OP1 or OP2 is NaN,
2238 then ROP is set to the numeric value. If OP1 and OP2 are zeros of
2239 different signs, then ROP is set to -0 (resp. +0).
2241 -- Function: int mpfr_urandomb (mpfr_t ROP, gmp_randstate_t STATE)
2242 Generate a uniformly distributed random float in the interval 0 <=
2243 ROP < 1. More precisely, the number can be seen as a float with a
2244 random non-normalized significand and exponent 0, which is then
2245 normalized (thus if E denotes the exponent after normalization,
2246 then the least -E significant bits of the significand are always
2249 Return 0, unless the exponent is not in the current exponent
2250 range, in which case ROP is set to NaN and a non-zero value is
2251 returned (this should never happen in practice, except in very
2252 specific cases). The second argument is a `gmp_randstate_t'
2253 structure which should be created using the GMP `gmp_randinit'
2254 function (see the GMP manual).
2256 Note: for a given version of MPFR, the returned value of ROP and
2257 the new value of STATE (which controls further random values) do
2258 not depend on the machine word size.
2260 -- Function: int mpfr_urandom (mpfr_t ROP, gmp_randstate_t STATE,
2262 Generate a uniformly distributed random float. The floating-point
2263 number ROP can be seen as if a random real number is generated
2264 according to the continuous uniform distribution on the interval
2265 [0, 1] and then rounded in the direction RND.
2267 The second argument is a `gmp_randstate_t' structure which should
2268 be created using the GMP `gmp_randinit' function (see the GMP
2271 Note: the note for `mpfr_urandomb' holds too. In addition, the
2272 exponent range and the rounding mode might have a side effect on
2273 the next random state.
2275 -- Function: int mpfr_grandom (mpfr_t ROP1, mpfr_t ROP2,
2276 gmp_randstate_t STATE, mpfr_rnd_t RND)
2277 Generate two random floats according to a standard normal gaussian
2278 distribution. If ROP2 is a null pointer, then only one value is
2279 generated and stored in ROP1.
2281 The floating-point number ROP1 (and ROP2) can be seen as if a
2282 random real number were generated according to the standard normal
2283 gaussian distribution and then rounded in the direction RND.
2285 The third argument is a `gmp_randstate_t' structure, which should
2286 be created using the GMP `gmp_randinit' function (see the GMP
2289 Note: the note for `mpfr_urandomb' holds too. In addition, the
2290 exponent range and the rounding mode might have a side effect on
2291 the next random state.
2293 -- Function: mpfr_exp_t mpfr_get_exp (mpfr_t X)
2294 Return the exponent of X, assuming that X is a non-zero ordinary
2295 number and the significand is considered in [1/2,1). The behavior
2296 for NaN, infinity or zero is undefined.
2298 -- Function: int mpfr_set_exp (mpfr_t X, mpfr_exp_t E)
2299 Set the exponent of X if E is in the current exponent range, and
2300 return 0 (even if X is not a non-zero ordinary number); otherwise,
2301 return a non-zero value. The significand is assumed to be in
2304 -- Function: int mpfr_signbit (mpfr_t OP)
2305 Return a non-zero value iff OP has its sign bit set (i.e., if it is
2306 negative, -0, or a NaN whose representation has its sign bit set).
2308 -- Function: int mpfr_setsign (mpfr_t ROP, mpfr_t OP, int S,
2310 Set the value of ROP from OP, rounded toward the given direction
2311 RND, then set (resp. clear) its sign bit if S is non-zero (resp.
2312 zero), even when OP is a NaN.
2314 -- Function: int mpfr_copysign (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
2316 Set the value of ROP from OP1, rounded toward the given direction
2317 RND, then set its sign bit to that of OP2 (even when OP1 or OP2 is
2318 a NaN). This function is equivalent to `mpfr_setsign (ROP, OP1,
2319 mpfr_signbit (OP2), RND)'.
2321 -- Function: const char * mpfr_get_version (void)
2322 Return the MPFR version, as a null-terminated string.
2324 -- Macro: MPFR_VERSION
2325 -- Macro: MPFR_VERSION_MAJOR
2326 -- Macro: MPFR_VERSION_MINOR
2327 -- Macro: MPFR_VERSION_PATCHLEVEL
2328 -- Macro: MPFR_VERSION_STRING
2329 `MPFR_VERSION' is the version of MPFR as a preprocessing constant.
2330 `MPFR_VERSION_MAJOR', `MPFR_VERSION_MINOR' and
2331 `MPFR_VERSION_PATCHLEVEL' are respectively the major, minor and
2332 patch level of MPFR version, as preprocessing constants.
2333 `MPFR_VERSION_STRING' is the version (with an optional suffix, used
2334 in development and pre-release versions) as a string constant,
2335 which can be compared to the result of `mpfr_get_version' to check
2336 at run time the header file and library used match:
2337 if (strcmp (mpfr_get_version (), MPFR_VERSION_STRING))
2338 fprintf (stderr, "Warning: header and library do not match\n");
2339 Note: Obtaining different strings is not necessarily an error, as
2340 in general, a program compiled with some old MPFR version can be
2341 dynamically linked with a newer MPFR library version (if allowed
2342 by the library versioning system).
2344 -- Macro: long MPFR_VERSION_NUM (MAJOR, MINOR, PATCHLEVEL)
2345 Create an integer in the same format as used by `MPFR_VERSION'
2346 from the given MAJOR, MINOR and PATCHLEVEL. Here is an example of
2347 how to check the MPFR version at compile time:
2348 #if (!defined(MPFR_VERSION) || (MPFR_VERSION<MPFR_VERSION_NUM(3,0,0)))
2349 # error "Wrong MPFR version."
2352 -- Function: const char * mpfr_get_patches (void)
2353 Return a null-terminated string containing the ids of the patches
2354 applied to the MPFR library (contents of the `PATCHES' file),
2355 separated by spaces. Note: If the program has been compiled with
2356 an older MPFR version and is dynamically linked with a new MPFR
2357 library version, the identifiers of the patches applied to the old
2358 (compile-time) MPFR version are not available (however this
2359 information should not have much interest in general).
2361 -- Function: int mpfr_buildopt_tls_p (void)
2362 Return a non-zero value if MPFR was compiled as thread safe using
2363 compiler-level Thread Local Storage (that is, MPFR was built with
2364 the `--enable-thread-safe' configure option, see `INSTALL' file),
2365 return zero otherwise.
2367 -- Function: int mpfr_buildopt_decimal_p (void)
2368 Return a non-zero value if MPFR was compiled with decimal float
2369 support (that is, MPFR was built with the `--enable-decimal-float'
2370 configure option), return zero otherwise.
2372 -- Function: int mpfr_buildopt_gmpinternals_p (void)
2373 Return a non-zero value if MPFR was compiled with GMP internals
2374 (that is, MPFR was built with either `--with-gmp-build' or
2375 `--enable-gmp-internals' configure option), return zero otherwise.
2377 -- Function: const char * mpfr_buildopt_tune_case (void)
2378 Return a string saying which thresholds file has been used at
2379 compile time. This file is normally selected from the processor
2383 File: mpfr.info, Node: Exception Related Functions, Next: Compatibility with MPF, Prev: Miscellaneous Functions, Up: MPFR Interface
2385 5.13 Exception Related Functions
2386 ================================
2388 -- Function: mpfr_exp_t mpfr_get_emin (void)
2389 -- Function: mpfr_exp_t mpfr_get_emax (void)
2390 Return the (current) smallest and largest exponents allowed for a
2391 floating-point variable. The smallest positive value of a
2392 floating-point variable is one half times 2 raised to the smallest
2393 exponent and the largest value has the form (1 - epsilon) times 2
2394 raised to the largest exponent, where epsilon depends on the
2395 precision of the considered variable.
2397 -- Function: int mpfr_set_emin (mpfr_exp_t EXP)
2398 -- Function: int mpfr_set_emax (mpfr_exp_t EXP)
2399 Set the smallest and largest exponents allowed for a
2400 floating-point variable. Return a non-zero value when EXP is not
2401 in the range accepted by the implementation (in that case the
2402 smallest or largest exponent is not changed), and zero otherwise.
2403 If the user changes the exponent range, it is her/his
2404 responsibility to check that all current floating-point variables
2405 are in the new allowed range (for example using
2406 `mpfr_check_range'), otherwise the subsequent behavior will be
2407 undefined, in the sense of the ISO C standard.
2409 -- Function: mpfr_exp_t mpfr_get_emin_min (void)
2410 -- Function: mpfr_exp_t mpfr_get_emin_max (void)
2411 -- Function: mpfr_exp_t mpfr_get_emax_min (void)
2412 -- Function: mpfr_exp_t mpfr_get_emax_max (void)
2413 Return the minimum and maximum of the exponents allowed for
2414 `mpfr_set_emin' and `mpfr_set_emax' respectively. These values
2415 are implementation dependent, thus a program using
2416 `mpfr_set_emax(mpfr_get_emax_max())' or
2417 `mpfr_set_emin(mpfr_get_emin_min())' may not be portable.
2419 -- Function: int mpfr_check_range (mpfr_t X, int T, mpfr_rnd_t RND)
2420 This function assumes that X is the correctly-rounded value of some
2421 real value Y in the direction RND and some extended exponent
2422 range, and that T is the corresponding *note ternary value::. For
2423 example, one performed `t = mpfr_log (x, u, rnd)', and Y is the
2424 exact logarithm of U. Thus T is negative if X is smaller than Y,
2425 positive if X is larger than Y, and zero if X equals Y. This
2426 function modifies X if needed to be in the current range of
2427 acceptable values: It generates an underflow or an overflow if the
2428 exponent of X is outside the current allowed range; the value of T
2429 may be used to avoid a double rounding. This function returns zero
2430 if the new value of X equals the exact one Y, a positive value if
2431 that new value is larger than Y, and a negative value if it is
2432 smaller than Y. Note that unlike most functions, the new result X
2433 is compared to the (unknown) exact one Y, not the input value X,
2434 i.e., the ternary value is propagated.
2436 Note: If X is an infinity and T is different from zero (i.e., if
2437 the rounded result is an inexact infinity), then the overflow flag
2438 is set. This is useful because `mpfr_check_range' is typically
2439 called (at least in MPFR functions) after restoring the flags that
2440 could have been set due to internal computations.
2442 -- Function: int mpfr_subnormalize (mpfr_t X, int T, mpfr_rnd_t RND)
2443 This function rounds X emulating subnormal number arithmetic: if X
2444 is outside the subnormal exponent range, it just propagates the
2445 *note ternary value:: T; otherwise, it rounds X to precision
2446 `EXP(x)-emin+1' according to rounding mode RND and previous
2447 ternary value T, avoiding double rounding problems. More
2448 precisely in the subnormal domain, denoting by E the value of
2449 `emin', X is rounded in fixed-point arithmetic to an integer
2450 multiple of two to the power E-1; as a consequence, 1.5 multiplied
2451 by two to the power E-1 when T is zero is rounded to two to the
2452 power E with rounding to nearest.
2454 `PREC(x)' is not modified by this function. RND and T must be the
2455 rounding mode and the returned ternary value used when computing X
2456 (as in `mpfr_check_range'). The subnormal exponent range is from
2457 `emin' to `emin+PREC(x)-1'. If the result cannot be represented
2458 in the current exponent range (due to a too small `emax'), the
2459 behavior is undefined. Note that unlike most functions, the
2460 result is compared to the exact one, not the input value X, i.e.,
2461 the ternary value is propagated.
2463 As usual, if the returned ternary value is non zero, the inexact
2464 flag is set. Moreover, if a second rounding occurred (because the
2465 input X was in the subnormal range), the underflow flag is set.
2467 This is an example of how to emulate binary double IEEE 754
2468 arithmetic (binary64 in IEEE 754-2008) using MPFR:
2471 mpfr_t xa, xb; int i; volatile double a, b;
2473 mpfr_set_default_prec (53);
2474 mpfr_set_emin (-1073); mpfr_set_emax (1024);
2476 mpfr_init (xa); mpfr_init (xb);
2478 b = 34.3; mpfr_set_d (xb, b, MPFR_RNDN);
2479 a = 0x1.1235P-1021; mpfr_set_d (xa, a, MPFR_RNDN);
2482 i = mpfr_div (xa, xa, xb, MPFR_RNDN);
2483 i = mpfr_subnormalize (xa, i, MPFR_RNDN); /* new ternary value */
2485 mpfr_clear (xa); mpfr_clear (xb);
2488 Warning: this emulates a double IEEE 754 arithmetic with correct
2489 rounding in the subnormal range, which may not be the case for your
2492 -- Function: void mpfr_clear_underflow (void)
2493 -- Function: void mpfr_clear_overflow (void)
2494 -- Function: void mpfr_clear_divby0 (void)
2495 -- Function: void mpfr_clear_nanflag (void)
2496 -- Function: void mpfr_clear_inexflag (void)
2497 -- Function: void mpfr_clear_erangeflag (void)
2498 Clear the underflow, overflow, divide-by-zero, invalid, inexact
2501 -- Function: void mpfr_set_underflow (void)
2502 -- Function: void mpfr_set_overflow (void)
2503 -- Function: void mpfr_set_divby0 (void)
2504 -- Function: void mpfr_set_nanflag (void)
2505 -- Function: void mpfr_set_inexflag (void)
2506 -- Function: void mpfr_set_erangeflag (void)
2507 Set the underflow, overflow, divide-by-zero, invalid, inexact and
2510 -- Function: void mpfr_clear_flags (void)
2511 Clear all global flags (underflow, overflow, divide-by-zero,
2512 invalid, inexact, _erange_).
2514 -- Function: int mpfr_underflow_p (void)
2515 -- Function: int mpfr_overflow_p (void)
2516 -- Function: int mpfr_divby0_p (void)
2517 -- Function: int mpfr_nanflag_p (void)
2518 -- Function: int mpfr_inexflag_p (void)
2519 -- Function: int mpfr_erangeflag_p (void)
2520 Return the corresponding (underflow, overflow, divide-by-zero,
2521 invalid, inexact, _erange_) flag, which is non-zero iff the flag
2525 File: mpfr.info, Node: Compatibility with MPF, Next: Custom Interface, Prev: Exception Related Functions, Up: MPFR Interface
2527 5.14 Compatibility With MPF
2528 ===========================
2530 A header file `mpf2mpfr.h' is included in the distribution of MPFR for
2531 compatibility with the GNU MP class MPF. By inserting the following
2532 two lines after the `#include <gmp.h>' line,
2534 #include <mpf2mpfr.h>
2535 any program written for MPF can be compiled directly with MPFR without
2536 any changes (except the `gmp_printf' functions will not work for
2537 arguments of type `mpfr_t'). All operations are then performed with
2538 the default MPFR rounding mode, which can be reset with
2539 `mpfr_set_default_rounding_mode'.
2541 Warning: the `mpf_init' and `mpf_init2' functions initialize to
2542 zero, whereas the corresponding MPFR functions initialize to NaN: this
2543 is useful to detect uninitialized values, but is slightly incompatible
2546 -- Function: void mpfr_set_prec_raw (mpfr_t X, mpfr_prec_t PREC)
2547 Reset the precision of X to be *exactly* PREC bits. The only
2548 difference with `mpfr_set_prec' is that PREC is assumed to be
2549 small enough so that the significand fits into the current
2550 allocated memory space for X. Otherwise the behavior is undefined.
2552 -- Function: int mpfr_eq (mpfr_t OP1, mpfr_t OP2, unsigned long int
2554 Return non-zero if OP1 and OP2 are both non-zero ordinary numbers
2555 with the same exponent and the same first OP3 bits, both zero, or
2556 both infinities of the same sign. Return zero otherwise. This
2557 function is defined for compatibility with MPF, we do not recommend
2558 to use it otherwise. Do not use it either if you want to know
2559 whether two numbers are close to each other; for instance,
2560 1.011111 and 1.100000 are regarded as different for any value of
2563 -- Function: void mpfr_reldiff (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
2565 Compute the relative difference between OP1 and OP2 and store the
2566 result in ROP. This function does not guarantee the correct
2567 rounding on the relative difference; it just computes
2568 |OP1-OP2|/OP1, using the precision of ROP and the rounding mode
2569 RND for all operations.
2571 -- Function: int mpfr_mul_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long
2572 int OP2, mpfr_rnd_t RND)
2573 -- Function: int mpfr_div_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long
2574 int OP2, mpfr_rnd_t RND)
2575 These functions are identical to `mpfr_mul_2ui' and `mpfr_div_2ui'
2576 respectively. These functions are only kept for compatibility
2577 with MPF, one should prefer `mpfr_mul_2ui' and `mpfr_div_2ui'
2581 File: mpfr.info, Node: Custom Interface, Next: Internals, Prev: Compatibility with MPF, Up: MPFR Interface
2583 5.15 Custom Interface
2584 =====================
2586 Some applications use a stack to handle the memory and their objects.
2587 However, the MPFR memory design is not well suited for such a thing. So
2588 that such applications are able to use MPFR, an auxiliary memory
2589 interface has been created: the Custom Interface.
2591 The following interface allows one to use MPFR in two ways:
2592 * Either directly store a floating-point number as a `mpfr_t' on the
2595 * Either store its own representation on the stack and construct a
2596 new temporary `mpfr_t' each time it is needed.
2597 Nothing has to be done to destroy the floating-point numbers except
2598 garbaging the used memory: all the memory management (allocating,
2599 destroying, garbaging) is left to the application.
2601 Each function in this interface is also implemented as a macro for
2602 efficiency reasons: for example `mpfr_custom_init (s, p)' uses the
2603 macro, while `(mpfr_custom_init) (s, p)' uses the function.
2605 Note 1: MPFR functions may still initialize temporary floating-point
2606 numbers using `mpfr_init' and similar functions. See Custom Allocation
2609 Note 2: MPFR functions may use the cached functions (`mpfr_const_pi'
2610 for example), even if they are not explicitly called. You have to call
2611 `mpfr_free_cache' each time you garbage the memory iff `mpfr_init',
2612 through GMP Custom Allocation, allocates its memory on the application
2615 -- Function: size_t mpfr_custom_get_size (mpfr_prec_t PREC)
2616 Return the needed size in bytes to store the significand of a
2617 floating-point number of precision PREC.
2619 -- Function: void mpfr_custom_init (void *SIGNIFICAND, mpfr_prec_t
2621 Initialize a significand of precision PREC, where SIGNIFICAND must
2622 be an area of `mpfr_custom_get_size (prec)' bytes at least and be
2623 suitably aligned for an array of `mp_limb_t' (GMP type, *note
2626 -- Function: void mpfr_custom_init_set (mpfr_t X, int KIND, mpfr_exp_t
2627 EXP, mpfr_prec_t PREC, void *SIGNIFICAND)
2628 Perform a dummy initialization of a `mpfr_t' and set it to:
2629 * if `ABS(kind) == MPFR_NAN_KIND', X is set to NaN;
2631 * if `ABS(kind) == MPFR_INF_KIND', X is set to the infinity of
2634 * if `ABS(kind) == MPFR_ZERO_KIND', X is set to the zero of
2637 * if `ABS(kind) == MPFR_REGULAR_KIND', X is set to a regular
2638 number: `x = sign(kind)*significand*2^exp'.
2639 In all cases, it uses SIGNIFICAND directly for further computing
2640 involving X. It will not allocate anything. A floating-point
2641 number initialized with this function cannot be resized using
2642 `mpfr_set_prec' or `mpfr_prec_round', or cleared using
2643 `mpfr_clear'! The SIGNIFICAND must have been initialized with
2644 `mpfr_custom_init' using the same precision PREC.
2646 -- Function: int mpfr_custom_get_kind (mpfr_t X)
2647 Return the current kind of a `mpfr_t' as created by
2648 `mpfr_custom_init_set'. The behavior of this function for any
2649 `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined.
2651 -- Function: void * mpfr_custom_get_significand (mpfr_t X)
2652 Return a pointer to the significand used by a `mpfr_t' initialized
2653 with `mpfr_custom_init_set'. The behavior of this function for
2654 any `mpfr_t' not initialized with `mpfr_custom_init_set' is
2657 -- Function: mpfr_exp_t mpfr_custom_get_exp (mpfr_t X)
2658 Return the exponent of X, assuming that X is a non-zero ordinary
2659 number. The return value for NaN, Infinity or zero is unspecified
2660 but does not produce any trap. The behavior of this function for
2661 any `mpfr_t' not initialized with `mpfr_custom_init_set' is
2664 -- Function: void mpfr_custom_move (mpfr_t X, void *NEW_POSITION)
2665 Inform MPFR that the significand of X has moved due to a garbage
2666 collect and update its new position to `new_position'. However
2667 the application has to move the significand and the `mpfr_t'
2668 itself. The behavior of this function for any `mpfr_t' not
2669 initialized with `mpfr_custom_init_set' is undefined.
2672 File: mpfr.info, Node: Internals, Prev: Custom Interface, Up: MPFR Interface
2677 A "limb" means the part of a multi-precision number that fits in a
2678 single word. Usually a limb contains 32 or 64 bits. The C data type
2679 for a limb is `mp_limb_t'.
2681 The `mpfr_t' type is internally defined as a one-element array of a
2682 structure, and `mpfr_ptr' is the C data type representing a pointer to
2683 this structure. The `mpfr_t' type consists of four fields:
2685 * The `_mpfr_prec' field is used to store the precision of the
2686 variable (in bits); this is not less than `MPFR_PREC_MIN'.
2688 * The `_mpfr_sign' field is used to store the sign of the variable.
2690 * The `_mpfr_exp' field stores the exponent. An exponent of 0 means
2691 a radix point just above the most significant limb. Non-zero
2692 values n are a multiplier 2^n relative to that point. A NaN, an
2693 infinity and a zero are indicated by special values of the exponent
2696 * Finally, the `_mpfr_d' field is a pointer to the limbs, least
2697 significant limbs stored first. The number of limbs in use is
2698 controlled by `_mpfr_prec', namely
2699 ceil(`_mpfr_prec'/`mp_bits_per_limb'). Non-singular (i.e.,
2700 different from NaN, Infinity or zero) values always have the most
2701 significant bit of the most significant limb set to 1. When the
2702 precision does not correspond to a whole number of limbs, the
2703 excess bits at the low end of the data are zeros.
2707 File: mpfr.info, Node: API Compatibility, Next: Contributors, Prev: MPFR Interface, Up: Top
2712 The goal of this section is to describe some API changes that occurred
2713 from one version of MPFR to another, and how to write code that can be
2714 compiled and run with older MPFR versions. The minimum MPFR version
2715 that is considered here is 2.2.0 (released on 20 September 2005).
2717 API changes can only occur between major or minor versions. Thus the
2718 patchlevel (the third number in the MPFR version) will be ignored in
2719 the following. If a program does not use MPFR internals, changes in
2720 the behavior between two versions differing only by the patchlevel
2721 should only result from what was regarded as a bug or unspecified
2724 As a general rule, a program written for some MPFR version should
2725 work with later versions, possibly except at a new major version, where
2726 some features (described as obsolete for some time) can be removed. In
2727 such a case, a failure should occur during compilation or linking. If
2728 a result becomes incorrect because of such a change, please look at the
2729 various changes below (they are minimal, and most software should be
2730 unaffected), at the FAQ and at the MPFR web page for your version (a
2731 bug could have been introduced and be already fixed); and if the
2732 problem is not mentioned, please send us a bug report (*note Reporting
2735 However, a program written for the current MPFR version (as
2736 documented by this manual) may not necessarily work with previous
2737 versions of MPFR. This section should help developers to write
2740 Note: Information given here may be incomplete. API changes are
2741 also described in the NEWS file (for each version, instead of being
2742 classified like here), together with other changes.
2746 * Type and Macro Changes::
2748 * Changed Functions::
2749 * Removed Functions::
2753 File: mpfr.info, Node: Type and Macro Changes, Next: Added Functions, Prev: API Compatibility, Up: API Compatibility
2755 6.1 Type and Macro Changes
2756 ==========================
2758 The official type for exponent values changed from `mp_exp_t' to
2759 `mpfr_exp_t' in MPFR 3.0. The type `mp_exp_t' will remain available as
2760 it comes from GMP (with a different meaning). These types are
2761 currently the same (`mpfr_exp_t' is defined as `mp_exp_t' with
2762 `typedef'), so that programs can still use `mp_exp_t'; but this may
2763 change in the future. Alternatively, using the following code after
2764 including `mpfr.h' will work with official MPFR versions, as
2765 `mpfr_exp_t' was never defined in MPFR 2.x:
2766 #if MPFR_VERSION_MAJOR < 3
2767 typedef mp_exp_t mpfr_exp_t;
2770 The official types for precision values and for rounding modes
2771 respectively changed from `mp_prec_t' and `mp_rnd_t' to `mpfr_prec_t'
2772 and `mpfr_rnd_t' in MPFR 3.0. This change was actually done a long
2773 time ago in MPFR, at least since MPFR 2.2.0, with the following code in
2776 # define mp_rnd_t mpfr_rnd_t
2779 # define mp_prec_t mpfr_prec_t
2781 This means that it is safe to use the new official types
2782 `mpfr_prec_t' and `mpfr_rnd_t' in your programs. The types `mp_prec_t'
2783 and `mp_rnd_t' (defined in MPFR only) may be removed in the future, as
2784 the prefix `mp_' is reserved by GMP.
2786 The precision type `mpfr_prec_t' (`mp_prec_t') was unsigned before
2787 MPFR 3.0; it is now signed. `MPFR_PREC_MAX' has not changed, though.
2788 Indeed the MPFR code requires that `MPFR_PREC_MAX' be representable in
2789 the exponent type, which may have the same size as `mpfr_prec_t' but
2790 has always been signed. The consequence is that valid code that does
2791 not assume anything about the signedness of `mpfr_prec_t' should work
2792 with past and new MPFR versions. This change was useful as the use of
2793 unsigned types tends to convert signed values to unsigned ones in
2794 expressions due to the usual arithmetic conversions, which can yield
2795 incorrect results if a negative value is converted in such a way.
2796 Warning! A program assuming (intentionally or not) that `mpfr_prec_t'
2797 is signed may be affected by this problem when it is built and run
2800 The rounding modes `GMP_RNDx' were renamed to `MPFR_RNDx' in MPFR
2801 3.0. However the old names `GMP_RNDx' have been kept for compatibility
2802 (this might change in future versions), using:
2803 #define GMP_RNDN MPFR_RNDN
2804 #define GMP_RNDZ MPFR_RNDZ
2805 #define GMP_RNDU MPFR_RNDU
2806 #define GMP_RNDD MPFR_RNDD
2807 The rounding mode "round away from zero" (`MPFR_RNDA') was added in
2808 MPFR 3.0 (however no rounding mode `GMP_RNDA' exists).
2811 File: mpfr.info, Node: Added Functions, Next: Changed Functions, Prev: Type and Macro Changes, Up: API Compatibility
2816 We give here in alphabetical order the functions that were added after
2817 MPFR 2.2, and in which MPFR version.
2819 * `mpfr_add_d' in MPFR 2.4.
2821 * `mpfr_ai' in MPFR 3.0 (incomplete, experimental).
2823 * `mpfr_asprintf' in MPFR 2.4.
2825 * `mpfr_buildopt_decimal_p' and `mpfr_buildopt_tls_p' in MPFR 3.0.
2827 * `mpfr_buildopt_gmpinternals_p' and `mpfr_buildopt_tune_case' in
2830 * `mpfr_clear_divby0' in MPFR 3.1 (new divide-by-zero exception).
2832 * `mpfr_copysign' in MPFR 2.3. Note: MPFR 2.2 had a `mpfr_copysign'
2833 function that was available, but not documented, and with a slight
2834 difference in the semantics (when the second input operand is a
2837 * `mpfr_custom_get_significand' in MPFR 3.0. This function was
2838 named `mpfr_custom_get_mantissa' in previous versions;
2839 `mpfr_custom_get_mantissa' is still available via a macro in
2841 #define mpfr_custom_get_mantissa mpfr_custom_get_significand
2842 Thus code that needs to work with both MPFR 2.x and MPFR 3.x should
2843 use `mpfr_custom_get_mantissa'.
2845 * `mpfr_d_div' and `mpfr_d_sub' in MPFR 2.4.
2847 * `mpfr_digamma' in MPFR 3.0.
2849 * `mpfr_divby0_p' in MPFR 3.1 (new divide-by-zero exception).
2851 * `mpfr_div_d' in MPFR 2.4.
2853 * `mpfr_fmod' in MPFR 2.4.
2855 * `mpfr_fms' in MPFR 2.3.
2857 * `mpfr_fprintf' in MPFR 2.4.
2859 * `mpfr_frexp' in MPFR 3.1.
2861 * `mpfr_get_flt' in MPFR 3.0.
2863 * `mpfr_get_patches' in MPFR 2.3.
2865 * `mpfr_get_z_2exp' in MPFR 3.0. This function was named
2866 `mpfr_get_z_exp' in previous versions; `mpfr_get_z_exp' is still
2867 available via a macro in `mpfr.h':
2868 #define mpfr_get_z_exp mpfr_get_z_2exp
2869 Thus code that needs to work with both MPFR 2.x and MPFR 3.x should
2870 use `mpfr_get_z_exp'.
2872 * `mpfr_grandom' in MPFR 3.1.
2874 * `mpfr_j0', `mpfr_j1' and `mpfr_jn' in MPFR 2.3.
2876 * `mpfr_lgamma' in MPFR 2.3.
2878 * `mpfr_li2' in MPFR 2.4.
2880 * `mpfr_min_prec' in MPFR 3.0.
2882 * `mpfr_modf' in MPFR 2.4.
2884 * `mpfr_mul_d' in MPFR 2.4.
2886 * `mpfr_printf' in MPFR 2.4.
2888 * `mpfr_rec_sqrt' in MPFR 2.4.
2890 * `mpfr_regular_p' in MPFR 3.0.
2892 * `mpfr_remainder' and `mpfr_remquo' in MPFR 2.3.
2894 * `mpfr_set_divby0' in MPFR 3.1 (new divide-by-zero exception).
2896 * `mpfr_set_flt' in MPFR 3.0.
2898 * `mpfr_set_z_2exp' in MPFR 3.0.
2900 * `mpfr_set_zero' in MPFR 3.0.
2902 * `mpfr_setsign' in MPFR 2.3.
2904 * `mpfr_signbit' in MPFR 2.3.
2906 * `mpfr_sinh_cosh' in MPFR 2.4.
2908 * `mpfr_snprintf' and `mpfr_sprintf' in MPFR 2.4.
2910 * `mpfr_sub_d' in MPFR 2.4.
2912 * `mpfr_urandom' in MPFR 3.0.
2914 * `mpfr_vasprintf', `mpfr_vfprintf', `mpfr_vprintf',
2915 `mpfr_vsprintf' and `mpfr_vsnprintf' in MPFR 2.4.
2917 * `mpfr_y0', `mpfr_y1' and `mpfr_yn' in MPFR 2.3.
2919 * `mpfr_z_sub' in MPFR 3.1.
2923 File: mpfr.info, Node: Changed Functions, Next: Removed Functions, Prev: Added Functions, Up: API Compatibility
2925 6.3 Changed Functions
2926 =====================
2928 The following functions have changed after MPFR 2.2. Changes can affect
2929 the behavior of code written for some MPFR version when built and run
2930 against another MPFR version (older or newer), as described below.
2932 * `mpfr_check_range' changed in MPFR 2.3.2 and MPFR 2.4. If the
2933 value is an inexact infinity, the overflow flag is now set (in
2934 case it was lost), while it was previously left unchanged. This
2935 is really what is expected in practice (and what the MPFR code was
2936 expecting), so that the previous behavior was regarded as a bug.
2937 Hence the change in MPFR 2.3.2.
2939 * `mpfr_get_f' changed in MPFR 3.0. This function was returning
2940 zero, except for NaN and Inf, which do not exist in MPF. The
2941 _erange_ flag is now set in these cases, and `mpfr_get_f' now
2942 returns the usual ternary value.
2944 * `mpfr_get_si', `mpfr_get_sj', `mpfr_get_ui' and `mpfr_get_uj'
2945 changed in MPFR 3.0. In previous MPFR versions, the cases where
2946 the _erange_ flag is set were unspecified.
2948 * `mpfr_get_z' changed in MPFR 3.0. The return type was `void'; it
2949 is now `int', and the usual ternary value is returned. Thus
2950 programs that need to work with both MPFR 2.x and 3.x must not use
2951 the return value. Even in this case, C code using `mpfr_get_z' as
2952 the second or third term of a conditional operator may also be
2953 affected. For instance, the following is correct with MPFR 3.0,
2954 but not with MPFR 2.x:
2955 bool ? mpfr_get_z(...) : mpfr_add(...);
2956 On the other hand, the following is correct with MPFR 2.x, but not
2958 bool ? mpfr_get_z(...) : (void) mpfr_add(...);
2959 Portable code should cast `mpfr_get_z(...)' to `void' to use the
2960 type `void' for both terms of the conditional operator, as in:
2961 bool ? (void) mpfr_get_z(...) : (void) mpfr_add(...);
2962 Alternatively, `if ... else' can be used instead of the
2963 conditional operator.
2965 Moreover the cases where the _erange_ flag is set were unspecified
2968 * `mpfr_get_z_exp' changed in MPFR 3.0. In previous MPFR versions,
2969 the cases where the _erange_ flag is set were unspecified. Note:
2970 this function has been renamed to `mpfr_get_z_2exp' in MPFR 3.0,
2971 but `mpfr_get_z_exp' is still available for compatibility reasons.
2973 * `mpfr_strtofr' changed in MPFR 2.3.1 and MPFR 2.4. This was
2974 actually a bug fix since the code and the documentation did not
2975 match. But both were changed in order to have a more consistent
2976 and useful behavior. The main changes in the code are as follows.
2977 The binary exponent is now accepted even without the `0b' or `0x'
2978 prefix. Data corresponding to NaN can now have an optional sign
2979 (such data were previously invalid).
2981 * `mpfr_strtofr' changed in MPFR 3.0. This function now accepts
2982 bases from 37 to 62 (no changes for the other bases). Note: if an
2983 unsupported base is provided to this function, the behavior is
2984 undefined; more precisely, in MPFR 2.3.1 and later, providing an
2985 unsupported base yields an assertion failure (this behavior may
2986 change in the future).
2988 * `mpfr_subnormalize' changed in MPFR 3.1. This was actually
2989 regarded as a bug fix. The `mpfr_subnormalize' implementation up
2990 to MPFR 3.0.0 did not change the flags. In particular, it did not
2991 follow the generic rule concerning the inexact flag (and no
2992 special behavior was specified). The case of the underflow flag
2993 was more a lack of specification.
2995 * `mpfr_urandom' and `mpfr_urandomb' changed in MPFR 3.1. Their
2996 behavior no longer depends on the platform (assuming this is also
2997 true for GMP's random generator). As a consequence, the returned
2998 values can be different between MPFR 3.1 and previous MPFR
2999 versions. Note: as the reproducibility of these functions was not
3000 specified before MPFR 3.1, the MPFR 3.1 behavior is _not_ regarded
3001 as backward incompatible with previous versions.
3005 File: mpfr.info, Node: Removed Functions, Next: Other Changes, Prev: Changed Functions, Up: API Compatibility
3007 6.4 Removed Functions
3008 =====================
3010 Functions `mpfr_random' and `mpfr_random2' have been removed in MPFR
3011 3.0 (this only affects old code built against MPFR 3.0 or later). (The
3012 function `mpfr_random' had been deprecated since at least MPFR 2.2.0,
3013 and `mpfr_random2' since MPFR 2.4.0.)
3016 File: mpfr.info, Node: Other Changes, Prev: Removed Functions, Up: API Compatibility
3021 For users of a C++ compiler, the way how the availability of `intmax_t'
3022 is detected has changed in MPFR 3.0. In MPFR 2.x, if a macro
3023 `INTMAX_C' or `UINTMAX_C' was defined (e.g. when the
3024 `__STDC_CONSTANT_MACROS' macro had been defined before `<stdint.h>' or
3025 `<inttypes.h>' has been included), `intmax_t' was assumed to be defined.
3026 However this was not always the case (more precisely, `intmax_t' can be
3027 defined only in the namespace `std', as with Boost), so that
3028 compilations could fail. Thus the check for `INTMAX_C' or `UINTMAX_C'
3029 is now disabled for C++ compilers, with the following consequences:
3031 * Programs written for MPFR 2.x that need `intmax_t' may no longer
3032 be compiled against MPFR 3.0: a `#define MPFR_USE_INTMAX_T' may be
3033 necessary before `mpfr.h' is included.
3035 * The compilation of programs that work with MPFR 3.0 may fail with
3036 MPFR 2.x due to the problem described above. Workarounds are
3037 possible, such as defining `intmax_t' and `uintmax_t' in the global
3038 namespace, though this is not clean.
3041 The divide-by-zero exception is new in MPFR 3.1. However it should
3042 not introduce incompatible changes for programs that strictly follow
3043 the MPFR API since the exception can only be seen via new functions.
3045 As of MPFR 3.1, the `mpfr.h' header can be included several times,
3046 while still supporting optional functions (*note Headers and
3050 File: mpfr.info, Node: Contributors, Next: References, Prev: API Compatibility, Up: Top
3055 The main developers of MPFR are Guillaume Hanrot, Vincent Lefèvre,
3056 Patrick Pélissier, Philippe Théveny and Paul Zimmermann.
3058 Sylvie Boldo from ENS-Lyon, France, contributed the functions
3059 `mpfr_agm' and `mpfr_log'. Sylvain Chevillard contributed the
3060 `mpfr_ai' function. David Daney contributed the hyperbolic and inverse
3061 hyperbolic functions, the base-2 exponential, and the factorial
3062 function. Alain Delplanque contributed the new version of the
3063 `mpfr_get_str' function. Mathieu Dutour contributed the functions
3064 `mpfr_acos', `mpfr_asin' and `mpfr_atan', and a previous version of
3065 `mpfr_gamma'. Laurent Fousse contributed the `mpfr_sum' function.
3066 Emmanuel Jeandel, from ENS-Lyon too, contributed the generic
3067 hypergeometric code, as well as the internal function `mpfr_exp3', a
3068 first implementation of the sine and cosine, and improved versions of
3069 `mpfr_const_log2' and `mpfr_const_pi'. Ludovic Meunier helped in the
3070 design of the `mpfr_erf' code. Jean-Luc Rémy contributed the
3071 `mpfr_zeta' code. Fabrice Rouillier contributed the `mpfr_xxx_z' and
3072 `mpfr_xxx_q' functions, and helped to the Microsoft Windows porting.
3073 Damien Stehlé contributed the `mpfr_get_ld_2exp' function.
3075 We would like to thank Jean-Michel Muller and Joris van der Hoeven
3076 for very fruitful discussions at the beginning of that project,
3077 Torbjörn Granlund and Kevin Ryde for their help about design issues,
3078 and Nathalie Revol for her careful reading of a previous version of
3079 this documentation. In particular Kevin Ryde did a tremendous job for
3080 the portability of MPFR in 2002-2004.
3082 The development of the MPFR library would not have been possible
3083 without the continuous support of INRIA, and of the LORIA (Nancy,
3084 France) and LIP (Lyon, France) laboratories. In particular the main
3085 authors were or are members of the PolKA, Spaces, Cacao and Caramel
3086 project-teams at LORIA and of the Arénaire project-team at LIP. This
3087 project was started during the Fiable (reliable in French) action
3088 supported by INRIA, and continued during the AOC action. The
3089 development of MPFR was also supported by a grant (202F0659 00 MPN 121)
3090 from the Conseil Régional de Lorraine in 2002, from INRIA by an
3091 "associate engineer" grant (2003-2005), an "opération de développement
3092 logiciel" grant (2007-2009), and the post-doctoral grant of Sylvain
3093 Chevillard in 2009-2010.
3096 File: mpfr.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top
3101 * Richard Brent and Paul Zimmermann, "Modern Computer Arithmetic",
3102 Cambridge University Press (to appear), also available from the
3105 * Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick
3106 Pélissier and Paul Zimmermann, "MPFR: A Multiple-Precision Binary
3107 Floating-Point Library With Correct Rounding", ACM Transactions on
3108 Mathematical Software, volume 33, issue 2, article 13, 15 pages,
3109 2007, `http://doi.acm.org/10.1145/1236463.1236468'.
3111 * Torbjörn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic
3112 Library", version 5.0.1, 2010, `http://gmplib.org'.
3114 * IEEE standard for binary floating-point arithmetic, Technical
3115 Report ANSI-IEEE Standard 754-1985, New York, 1985. Approved
3116 March 21, 1985: IEEE Standards Board; approved July 26, 1985:
3117 American National Standards Institute, 18 pages.
3119 * IEEE Standard for Floating-Point Arithmetic, ANSI-IEEE Standard
3120 754-2008, 2008. Revision of ANSI-IEEE Standard 754-1985, approved
3121 June 12, 2008: IEEE Standards Board, 70 pages.
3123 * Donald E. Knuth, "The Art of Computer Programming", vol 2,
3124 "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981.
3126 * Jean-Michel Muller, "Elementary Functions, Algorithms and
3127 Implementation", Birkhäuser, Boston, 2nd edition, 2006.
3129 * Jean-Michel Muller, Nicolas Brisebarre, Florent de Dinechin,
3130 Claude-Pierre Jeannerod, Vincent Lefèvre, Guillaume Melquiond,
3131 Nathalie Revol, Damien Stehlé and Serge Torrès, "Handbook of
3132 Floating-Point Arithmetic", Birkhäuser, Boston, 2009.
3136 File: mpfr.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top
3138 Appendix A GNU Free Documentation License
3139 *****************************************
3141 Version 1.2, November 2002
3143 Copyright (C) 2000,2001,2002 Free Software Foundation, Inc.
3144 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
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3374 the "History" section. You may omit a network location for a
3375 work that was published at least four years before the
3376 Document itself, or if the original publisher of the version
3377 it refers to gives permission.
3379 K. For any section Entitled "Acknowledgements" or "Dedications",
3380 Preserve the Title of the section, and preserve in the
3381 section all the substance and tone of each of the contributor
3382 acknowledgements and/or dedications given therein.
3384 L. Preserve all the Invariant Sections of the Document,
3385 unaltered in their text and in their titles. Section numbers
3386 or the equivalent are not considered part of the section
3389 M. Delete any section Entitled "Endorsements". Such a section
3390 may not be included in the Modified Version.
3392 N. Do not retitle any existing section to be Entitled
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3398 If the Modified Version includes new front-matter sections or
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3402 add their titles to the list of Invariant Sections in the Modified
3403 Version's license notice. These titles must be distinct from any
3404 other section titles.
3406 You may add a section Entitled "Endorsements", provided it contains
3407 nothing but endorsements of your Modified Version by various
3408 parties--for example, statements of peer review or that the text
3409 has been approved by an organization as the authoritative
3410 definition of a standard.
3412 You may add a passage of up to five words as a Front-Cover Text,
3413 and a passage of up to 25 words as a Back-Cover Text, to the end
3414 of the list of Cover Texts in the Modified Version. Only one
3415 passage of Front-Cover Text and one of Back-Cover Text may be
3416 added by (or through arrangements made by) any one entity. If the
3417 Document already includes a cover text for the same cover,
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3423 The author(s) and publisher(s) of the Document do not by this
3424 License give permission to use their names for publicity for or to
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3427 5. COMBINING DOCUMENTS
3429 You may combine the Document with other documents released under
3430 this License, under the terms defined in section 4 above for
3431 modified versions, provided that you include in the combination
3432 all of the Invariant Sections of all of the original documents,
3433 unmodified, and list them all as Invariant Sections of your
3434 combined work in its license notice, and that you preserve all
3435 their Warranty Disclaimers.
3437 The combined work need only contain one copy of this License, and
3438 multiple identical Invariant Sections may be replaced with a single
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3441 by adding at the end of it, in parentheses, the name of the
3442 original author or publisher of that section if known, or else a
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3444 the list of Invariant Sections in the license notice of the
3447 In the combination, you must combine any sections Entitled
3448 "History" in the various original documents, forming one section
3449 Entitled "History"; likewise combine any sections Entitled
3450 "Acknowledgements", and any sections Entitled "Dedications". You
3451 must delete all sections Entitled "Endorsements."
3453 6. COLLECTIONS OF DOCUMENTS
3455 You may make a collection consisting of the Document and other
3456 documents released under this License, and replace the individual
3457 copies of this License in the various documents with a single copy
3458 that is included in the collection, provided that you follow the
3459 rules of this License for verbatim copying of each of the
3460 documents in all other respects.
3462 You may extract a single document from such a collection, and
3463 distribute it individually under this License, provided you insert
3464 a copy of this License into the extracted document, and follow
3465 this License in all other respects regarding verbatim copying of
3468 7. AGGREGATION WITH INDEPENDENT WORKS
3470 A compilation of the Document or its derivatives with other
3471 separate and independent documents or works, in or on a volume of
3472 a storage or distribution medium, is called an "aggregate" if the
3473 copyright resulting from the compilation is not used to limit the
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3479 If the Cover Text requirement of section 3 is applicable to these
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3493 translations of some or all Invariant Sections in addition to the
3494 original versions of these Invariant Sections. You may include a
3495 translation of this License, and all the license notices in the
3496 Document, and any Warranty Disclaimers, provided that you also
3497 include the original English version of this License and the
3498 original versions of those notices and disclaimers. In case of a
3499 disagreement between the translation and the original version of
3500 this License or a notice or disclaimer, the original version will
3503 If a section in the Document is Entitled "Acknowledgements",
3504 "Dedications", or "History", the requirement (section 4) to
3505 Preserve its Title (section 1) will typically require changing the
3510 You may not copy, modify, sublicense, or distribute the Document
3511 except as expressly provided for under this License. Any other
3512 attempt to copy, modify, sublicense or distribute the Document is
3513 void, and will automatically terminate your rights under this
3514 License. However, parties who have received copies, or rights,
3515 from you under this License will not have their licenses
3516 terminated so long as such parties remain in full compliance.
3518 10. FUTURE REVISIONS OF THIS LICENSE
3520 The Free Software Foundation may publish new, revised versions of
3521 the GNU Free Documentation License from time to time. Such new
3522 versions will be similar in spirit to the present version, but may
3523 differ in detail to address new problems or concerns. See
3524 `http://www.gnu.org/copyleft/'.
3526 Each version of the License is given a distinguishing version
3527 number. If the Document specifies that a particular numbered
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3529 have the option of following the terms and conditions either of
3530 that specified version or of any later version that has been
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3532 the Document does not specify a version number of this License,
3533 you may choose any version ever published (not as a draft) by the
3534 Free Software Foundation.
3536 A.1 ADDENDUM: How to Use This License For Your Documents
3537 ========================================================
3539 To use this License in a document you have written, include a copy of
3540 the License in the document and put the following copyright and license
3541 notices just after the title page:
3543 Copyright (C) YEAR YOUR NAME.
3544 Permission is granted to copy, distribute and/or modify this document
3545 under the terms of the GNU Free Documentation License, Version 1.2
3546 or any later version published by the Free Software Foundation;
3547 with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
3548 Texts. A copy of the license is included in the section entitled ``GNU
3549 Free Documentation License''.
3551 If you have Invariant Sections, Front-Cover Texts and Back-Cover
3552 Texts, replace the "with...Texts." line with this:
3554 with the Invariant Sections being LIST THEIR TITLES, with
3555 the Front-Cover Texts being LIST, and with the Back-Cover Texts
3558 If you have Invariant Sections without Cover Texts, or some other
3559 combination of the three, merge those two alternatives to suit the
3562 If your document contains nontrivial examples of program code, we
3563 recommend releasing these examples in parallel under your choice of
3564 free software license, such as the GNU General Public License, to
3565 permit their use in free software.
3568 File: mpfr.info, Node: Concept Index, Next: Function and Type Index, Prev: GNU Free Documentation License, Up: Top
3576 * Accuracy: MPFR Interface. (line 25)
3577 * Arithmetic functions: Basic Arithmetic Functions.
3579 * Assignment functions: Assignment Functions. (line 3)
3580 * Basic arithmetic functions: Basic Arithmetic Functions.
3582 * Combined initialization and assignment functions: Combined Initialization and Assignment Functions.
3584 * Comparison functions: Comparison Functions. (line 3)
3585 * Compatibility with MPF: Compatibility with MPF.
3587 * Conditions for copying MPFR: Copying. (line 6)
3588 * Conversion functions: Conversion Functions. (line 3)
3589 * Copying conditions: Copying. (line 6)
3590 * Custom interface: Custom Interface. (line 3)
3591 * Exception related functions: Exception Related Functions.
3593 * Float arithmetic functions: Basic Arithmetic Functions.
3595 * Float comparisons functions: Comparison Functions. (line 3)
3596 * Float functions: MPFR Interface. (line 6)
3597 * Float input and output functions: Input and Output Functions.
3599 * Float output functions: Formatted Output Functions.
3601 * Floating-point functions: MPFR Interface. (line 6)
3602 * Floating-point number: Nomenclature and Types.
3604 * GNU Free Documentation License: GNU Free Documentation License.
3606 * I/O functions <1>: Formatted Output Functions.
3608 * I/O functions: Input and Output Functions.
3610 * Initialization functions: Initialization Functions.
3612 * Input functions: Input and Output Functions.
3614 * Installation: Installing MPFR. (line 6)
3615 * Integer related functions: Integer Related Functions.
3617 * Internals: Internals. (line 3)
3618 * intmax_t: Headers and Libraries.
3620 * inttypes.h: Headers and Libraries.
3622 * libmpfr: Headers and Libraries.
3624 * Libraries: Headers and Libraries.
3626 * Libtool: Headers and Libraries.
3628 * Limb: Internals. (line 6)
3629 * Linking: Headers and Libraries.
3631 * Miscellaneous float functions: Miscellaneous Functions.
3633 * mpfr.h: Headers and Libraries.
3635 * Output functions <1>: Formatted Output Functions.
3637 * Output functions: Input and Output Functions.
3639 * Precision <1>: MPFR Interface. (line 17)
3640 * Precision: Nomenclature and Types.
3642 * Reporting bugs: Reporting Bugs. (line 6)
3643 * Rounding mode related functions: Rounding Related Functions.
3645 * Rounding Modes: Nomenclature and Types.
3647 * Special functions: Special Functions. (line 3)
3648 * stdarg.h: Headers and Libraries.
3650 * stdint.h: Headers and Libraries.
3652 * stdio.h: Headers and Libraries.
3654 * Ternary value: Rounding Modes. (line 29)
3655 * uintmax_t: Headers and Libraries.
3659 File: mpfr.info, Node: Function and Type Index, Prev: Concept Index, Up: Top
3661 Function and Type Index
3662 ***********************
3667 * mpfr_abs: Basic Arithmetic Functions.
3669 * mpfr_acos: Special Functions. (line 52)
3670 * mpfr_acosh: Special Functions. (line 136)
3671 * mpfr_add: Basic Arithmetic Functions.
3673 * mpfr_add_d: Basic Arithmetic Functions.
3675 * mpfr_add_q: Basic Arithmetic Functions.
3677 * mpfr_add_si: Basic Arithmetic Functions.
3679 * mpfr_add_ui: Basic Arithmetic Functions.
3681 * mpfr_add_z: Basic Arithmetic Functions.
3683 * mpfr_agm: Special Functions. (line 232)
3684 * mpfr_ai: Special Functions. (line 248)
3685 * mpfr_asin: Special Functions. (line 53)
3686 * mpfr_asinh: Special Functions. (line 137)
3687 * mpfr_asprintf: Formatted Output Functions.
3689 * mpfr_atan: Special Functions. (line 54)
3690 * mpfr_atan2: Special Functions. (line 65)
3691 * mpfr_atanh: Special Functions. (line 138)
3692 * mpfr_buildopt_decimal_p: Miscellaneous Functions.
3694 * mpfr_buildopt_gmpinternals_p: Miscellaneous Functions.
3696 * mpfr_buildopt_tls_p: Miscellaneous Functions.
3698 * mpfr_buildopt_tune_case: Miscellaneous Functions.
3700 * mpfr_can_round: Rounding Related Functions.
3702 * mpfr_cbrt: Basic Arithmetic Functions.
3704 * mpfr_ceil: Integer Related Functions.
3706 * mpfr_check_range: Exception Related Functions.
3708 * mpfr_clear: Initialization Functions.
3710 * mpfr_clear_divby0: Exception Related Functions.
3712 * mpfr_clear_erangeflag: Exception Related Functions.
3714 * mpfr_clear_flags: Exception Related Functions.
3716 * mpfr_clear_inexflag: Exception Related Functions.
3718 * mpfr_clear_nanflag: Exception Related Functions.
3720 * mpfr_clear_overflow: Exception Related Functions.
3722 * mpfr_clear_underflow: Exception Related Functions.
3724 * mpfr_clears: Initialization Functions.
3726 * mpfr_cmp: Comparison Functions.
3728 * mpfr_cmp_d: Comparison Functions.
3730 * mpfr_cmp_f: Comparison Functions.
3732 * mpfr_cmp_ld: Comparison Functions.
3734 * mpfr_cmp_q: Comparison Functions.
3736 * mpfr_cmp_si: Comparison Functions.
3738 * mpfr_cmp_si_2exp: Comparison Functions.
3740 * mpfr_cmp_ui: Comparison Functions.
3742 * mpfr_cmp_ui_2exp: Comparison Functions.
3744 * mpfr_cmp_z: Comparison Functions.
3746 * mpfr_cmpabs: Comparison Functions.
3748 * mpfr_const_catalan: Special Functions. (line 259)
3749 * mpfr_const_euler: Special Functions. (line 258)
3750 * mpfr_const_log2: Special Functions. (line 256)
3751 * mpfr_const_pi: Special Functions. (line 257)
3752 * mpfr_copysign: Miscellaneous Functions.
3754 * mpfr_cos: Special Functions. (line 30)
3755 * mpfr_cosh: Special Functions. (line 115)
3756 * mpfr_cot: Special Functions. (line 48)
3757 * mpfr_coth: Special Functions. (line 132)
3758 * mpfr_csc: Special Functions. (line 47)
3759 * mpfr_csch: Special Functions. (line 131)
3760 * mpfr_custom_get_exp: Custom Interface. (line 78)
3761 * mpfr_custom_get_kind: Custom Interface. (line 67)
3762 * mpfr_custom_get_significand: Custom Interface. (line 72)
3763 * mpfr_custom_get_size: Custom Interface. (line 36)
3764 * mpfr_custom_init: Custom Interface. (line 41)
3765 * mpfr_custom_init_set: Custom Interface. (line 48)
3766 * mpfr_custom_move: Custom Interface. (line 85)
3767 * mpfr_d_div: Basic Arithmetic Functions.
3769 * mpfr_d_sub: Basic Arithmetic Functions.
3771 * MPFR_DECL_INIT: Initialization Functions.
3773 * mpfr_digamma: Special Functions. (line 187)
3774 * mpfr_dim: Basic Arithmetic Functions.
3776 * mpfr_div: Basic Arithmetic Functions.
3778 * mpfr_div_2exp: Compatibility with MPF.
3780 * mpfr_div_2si: Basic Arithmetic Functions.
3782 * mpfr_div_2ui: Basic Arithmetic Functions.
3784 * mpfr_div_d: Basic Arithmetic Functions.
3786 * mpfr_div_q: Basic Arithmetic Functions.
3788 * mpfr_div_si: Basic Arithmetic Functions.
3790 * mpfr_div_ui: Basic Arithmetic Functions.
3792 * mpfr_div_z: Basic Arithmetic Functions.
3794 * mpfr_divby0_p: Exception Related Functions.
3796 * mpfr_eint: Special Functions. (line 154)
3797 * mpfr_eq: Compatibility with MPF.
3799 * mpfr_equal_p: Comparison Functions.
3801 * mpfr_erangeflag_p: Exception Related Functions.
3803 * mpfr_erf: Special Functions. (line 198)
3804 * mpfr_erfc: Special Functions. (line 199)
3805 * mpfr_exp: Special Functions. (line 24)
3806 * mpfr_exp10: Special Functions. (line 26)
3807 * mpfr_exp2: Special Functions. (line 25)
3808 * mpfr_expm1: Special Functions. (line 150)
3809 * mpfr_fac_ui: Special Functions. (line 143)
3810 * mpfr_fits_intmax_p: Conversion Functions.
3812 * mpfr_fits_sint_p: Conversion Functions.
3814 * mpfr_fits_slong_p: Conversion Functions.
3816 * mpfr_fits_sshort_p: Conversion Functions.
3818 * mpfr_fits_uint_p: Conversion Functions.
3820 * mpfr_fits_uintmax_p: Conversion Functions.
3822 * mpfr_fits_ulong_p: Conversion Functions.
3824 * mpfr_fits_ushort_p: Conversion Functions.
3826 * mpfr_floor: Integer Related Functions.
3828 * mpfr_fma: Special Functions. (line 225)
3829 * mpfr_fmod: Integer Related Functions.
3831 * mpfr_fms: Special Functions. (line 227)
3832 * mpfr_fprintf: Formatted Output Functions.
3834 * mpfr_frac: Integer Related Functions.
3836 * mpfr_free_cache: Special Functions. (line 266)
3837 * mpfr_free_str: Conversion Functions.
3839 * mpfr_frexp: Conversion Functions.
3841 * mpfr_gamma: Special Functions. (line 169)
3842 * mpfr_get_d: Conversion Functions.
3844 * mpfr_get_d_2exp: Conversion Functions.
3846 * mpfr_get_decimal64: Conversion Functions.
3848 * mpfr_get_default_prec: Initialization Functions.
3850 * mpfr_get_default_rounding_mode: Rounding Related Functions.
3852 * mpfr_get_emax: Exception Related Functions.
3854 * mpfr_get_emax_max: Exception Related Functions.
3856 * mpfr_get_emax_min: Exception Related Functions.
3858 * mpfr_get_emin: Exception Related Functions.
3860 * mpfr_get_emin_max: Exception Related Functions.
3862 * mpfr_get_emin_min: Exception Related Functions.
3864 * mpfr_get_exp: Miscellaneous Functions.
3866 * mpfr_get_f: Conversion Functions.
3868 * mpfr_get_flt: Conversion Functions.
3870 * mpfr_get_ld: Conversion Functions.
3872 * mpfr_get_ld_2exp: Conversion Functions.
3874 * mpfr_get_patches: Miscellaneous Functions.
3876 * mpfr_get_prec: Initialization Functions.
3878 * mpfr_get_si: Conversion Functions.
3880 * mpfr_get_sj: Conversion Functions.
3882 * mpfr_get_str: Conversion Functions.
3884 * mpfr_get_ui: Conversion Functions.
3886 * mpfr_get_uj: Conversion Functions.
3888 * mpfr_get_version: Miscellaneous Functions.
3890 * mpfr_get_z: Conversion Functions.
3892 * mpfr_get_z_2exp: Conversion Functions.
3894 * mpfr_grandom: Miscellaneous Functions.
3896 * mpfr_greater_p: Comparison Functions.
3898 * mpfr_greaterequal_p: Comparison Functions.
3900 * mpfr_hypot: Special Functions. (line 241)
3901 * mpfr_inexflag_p: Exception Related Functions.
3903 * mpfr_inf_p: Comparison Functions.
3905 * mpfr_init: Initialization Functions.
3907 * mpfr_init2: Initialization Functions.
3909 * mpfr_init_set: Combined Initialization and Assignment Functions.
3911 * mpfr_init_set_d: Combined Initialization and Assignment Functions.
3913 * mpfr_init_set_f: Combined Initialization and Assignment Functions.
3915 * mpfr_init_set_ld: Combined Initialization and Assignment Functions.
3917 * mpfr_init_set_q: Combined Initialization and Assignment Functions.
3919 * mpfr_init_set_si: Combined Initialization and Assignment Functions.
3921 * mpfr_init_set_str: Combined Initialization and Assignment Functions.
3923 * mpfr_init_set_ui: Combined Initialization and Assignment Functions.
3925 * mpfr_init_set_z: Combined Initialization and Assignment Functions.
3927 * mpfr_inits: Initialization Functions.
3929 * mpfr_inits2: Initialization Functions.
3931 * mpfr_inp_str: Input and Output Functions.
3933 * mpfr_integer_p: Integer Related Functions.
3935 * mpfr_j0: Special Functions. (line 203)
3936 * mpfr_j1: Special Functions. (line 204)
3937 * mpfr_jn: Special Functions. (line 206)
3938 * mpfr_less_p: Comparison Functions.
3940 * mpfr_lessequal_p: Comparison Functions.
3942 * mpfr_lessgreater_p: Comparison Functions.
3944 * mpfr_lgamma: Special Functions. (line 179)
3945 * mpfr_li2: Special Functions. (line 164)
3946 * mpfr_lngamma: Special Functions. (line 173)
3947 * mpfr_log: Special Functions. (line 17)
3948 * mpfr_log10: Special Functions. (line 19)
3949 * mpfr_log1p: Special Functions. (line 146)
3950 * mpfr_log2: Special Functions. (line 18)
3951 * mpfr_max: Miscellaneous Functions.
3953 * mpfr_min: Miscellaneous Functions.
3955 * mpfr_min_prec: Rounding Related Functions.
3957 * mpfr_modf: Integer Related Functions.
3959 * mpfr_mul: Basic Arithmetic Functions.
3961 * mpfr_mul_2exp: Compatibility with MPF.
3963 * mpfr_mul_2si: Basic Arithmetic Functions.
3965 * mpfr_mul_2ui: Basic Arithmetic Functions.
3967 * mpfr_mul_d: Basic Arithmetic Functions.
3969 * mpfr_mul_q: Basic Arithmetic Functions.
3971 * mpfr_mul_si: Basic Arithmetic Functions.
3973 * mpfr_mul_ui: Basic Arithmetic Functions.
3975 * mpfr_mul_z: Basic Arithmetic Functions.
3977 * mpfr_nan_p: Comparison Functions.
3979 * mpfr_nanflag_p: Exception Related Functions.
3981 * mpfr_neg: Basic Arithmetic Functions.
3983 * mpfr_nextabove: Miscellaneous Functions.
3985 * mpfr_nextbelow: Miscellaneous Functions.
3987 * mpfr_nexttoward: Miscellaneous Functions.
3989 * mpfr_number_p: Comparison Functions.
3991 * mpfr_out_str: Input and Output Functions.
3993 * mpfr_overflow_p: Exception Related Functions.
3995 * mpfr_pow: Basic Arithmetic Functions.
3997 * mpfr_pow_si: Basic Arithmetic Functions.
3999 * mpfr_pow_ui: Basic Arithmetic Functions.
4001 * mpfr_pow_z: Basic Arithmetic Functions.
4003 * mpfr_prec_round: Rounding Related Functions.
4005 * mpfr_prec_t: Nomenclature and Types.
4007 * mpfr_print_rnd_mode: Rounding Related Functions.
4009 * mpfr_printf: Formatted Output Functions.
4011 * mpfr_rec_sqrt: Basic Arithmetic Functions.
4013 * mpfr_regular_p: Comparison Functions.
4015 * mpfr_reldiff: Compatibility with MPF.
4017 * mpfr_remainder: Integer Related Functions.
4019 * mpfr_remquo: Integer Related Functions.
4021 * mpfr_rint: Integer Related Functions.
4023 * mpfr_rint_ceil: Integer Related Functions.
4025 * mpfr_rint_floor: Integer Related Functions.
4027 * mpfr_rint_round: Integer Related Functions.
4029 * mpfr_rint_trunc: Integer Related Functions.
4031 * mpfr_rnd_t: Nomenclature and Types.
4033 * mpfr_root: Basic Arithmetic Functions.
4035 * mpfr_round: Integer Related Functions.
4037 * mpfr_sec: Special Functions. (line 46)
4038 * mpfr_sech: Special Functions. (line 130)
4039 * mpfr_set: Assignment Functions.
4041 * mpfr_set_d: Assignment Functions.
4043 * mpfr_set_decimal64: Assignment Functions.
4045 * mpfr_set_default_prec: Initialization Functions.
4047 * mpfr_set_default_rounding_mode: Rounding Related Functions.
4049 * mpfr_set_divby0: Exception Related Functions.
4051 * mpfr_set_emax: Exception Related Functions.
4053 * mpfr_set_emin: Exception Related Functions.
4055 * mpfr_set_erangeflag: Exception Related Functions.
4057 * mpfr_set_exp: Miscellaneous Functions.
4059 * mpfr_set_f: Assignment Functions.
4061 * mpfr_set_flt: Assignment Functions.
4063 * mpfr_set_inexflag: Exception Related Functions.
4065 * mpfr_set_inf: Assignment Functions.
4067 * mpfr_set_ld: Assignment Functions.
4069 * mpfr_set_nan: Assignment Functions.
4071 * mpfr_set_nanflag: Exception Related Functions.
4073 * mpfr_set_overflow: Exception Related Functions.
4075 * mpfr_set_prec: Initialization Functions.
4077 * mpfr_set_prec_raw: Compatibility with MPF.
4079 * mpfr_set_q: Assignment Functions.
4081 * mpfr_set_si: Assignment Functions.
4083 * mpfr_set_si_2exp: Assignment Functions.
4085 * mpfr_set_sj: Assignment Functions.
4087 * mpfr_set_sj_2exp: Assignment Functions.
4089 * mpfr_set_str: Assignment Functions.
4091 * mpfr_set_ui: Assignment Functions.
4093 * mpfr_set_ui_2exp: Assignment Functions.
4095 * mpfr_set_uj: Assignment Functions.
4097 * mpfr_set_uj_2exp: Assignment Functions.
4099 * mpfr_set_underflow: Exception Related Functions.
4101 * mpfr_set_z: Assignment Functions.
4103 * mpfr_set_z_2exp: Assignment Functions.
4105 * mpfr_set_zero: Assignment Functions.
4107 * mpfr_setsign: Miscellaneous Functions.
4109 * mpfr_sgn: Comparison Functions.
4111 * mpfr_si_div: Basic Arithmetic Functions.
4113 * mpfr_si_sub: Basic Arithmetic Functions.
4115 * mpfr_signbit: Miscellaneous Functions.
4117 * mpfr_sin: Special Functions. (line 31)
4118 * mpfr_sin_cos: Special Functions. (line 37)
4119 * mpfr_sinh: Special Functions. (line 116)
4120 * mpfr_sinh_cosh: Special Functions. (line 122)
4121 * mpfr_snprintf: Formatted Output Functions.
4123 * mpfr_sprintf: Formatted Output Functions.
4125 * mpfr_sqr: Basic Arithmetic Functions.
4127 * mpfr_sqrt: Basic Arithmetic Functions.
4129 * mpfr_sqrt_ui: Basic Arithmetic Functions.
4131 * mpfr_strtofr: Assignment Functions.
4133 * mpfr_sub: Basic Arithmetic Functions.
4135 * mpfr_sub_d: Basic Arithmetic Functions.
4137 * mpfr_sub_q: Basic Arithmetic Functions.
4139 * mpfr_sub_si: Basic Arithmetic Functions.
4141 * mpfr_sub_ui: Basic Arithmetic Functions.
4143 * mpfr_sub_z: Basic Arithmetic Functions.
4145 * mpfr_subnormalize: Exception Related Functions.
4147 * mpfr_sum: Special Functions. (line 275)
4148 * mpfr_swap: Assignment Functions.
4150 * mpfr_t: Nomenclature and Types.
4152 * mpfr_tan: Special Functions. (line 32)
4153 * mpfr_tanh: Special Functions. (line 117)
4154 * mpfr_trunc: Integer Related Functions.
4156 * mpfr_ui_div: Basic Arithmetic Functions.
4158 * mpfr_ui_pow: Basic Arithmetic Functions.
4160 * mpfr_ui_pow_ui: Basic Arithmetic Functions.
4162 * mpfr_ui_sub: Basic Arithmetic Functions.
4164 * mpfr_underflow_p: Exception Related Functions.
4166 * mpfr_unordered_p: Comparison Functions.
4168 * mpfr_urandom: Miscellaneous Functions.
4170 * mpfr_urandomb: Miscellaneous Functions.
4172 * mpfr_vasprintf: Formatted Output Functions.
4174 * MPFR_VERSION: Miscellaneous Functions.
4176 * MPFR_VERSION_MAJOR: Miscellaneous Functions.
4178 * MPFR_VERSION_MINOR: Miscellaneous Functions.
4180 * MPFR_VERSION_NUM: Miscellaneous Functions.
4182 * MPFR_VERSION_PATCHLEVEL: Miscellaneous Functions.
4184 * MPFR_VERSION_STRING: Miscellaneous Functions.
4186 * mpfr_vfprintf: Formatted Output Functions.
4188 * mpfr_vprintf: Formatted Output Functions.
4190 * mpfr_vsnprintf: Formatted Output Functions.
4192 * mpfr_vsprintf: Formatted Output Functions.
4194 * mpfr_y0: Special Functions. (line 214)
4195 * mpfr_y1: Special Functions. (line 215)
4196 * mpfr_yn: Special Functions. (line 217)
4197 * mpfr_z_sub: Basic Arithmetic Functions.
4199 * mpfr_zero_p: Comparison Functions.
4201 * mpfr_zeta: Special Functions. (line 192)
4202 * mpfr_zeta_ui: Special Functions. (line 194)
4208 Node: Copying
\7f2219
4209 Node: Introduction to MPFR
\7f3979
4210 Node: Installing MPFR
\7f6068
4211 Node: Reporting Bugs
\7f10890
4212 Node: MPFR Basics
\7f12823
4213 Node: Headers and Libraries
\7f13139
4214 Node: Nomenclature and Types
\7f16123
4215 Node: MPFR Variable Conventions
\7f18127
4216 Node: Rounding Modes
\7f19657
4217 Ref: ternary value
\7f20754
4218 Node: Floating-Point Values on Special Numbers
\7f22707
4219 Node: Exceptions
\7f25683
4220 Node: Memory Handling
\7f28835
4221 Node: MPFR Interface
\7f29967
4222 Node: Initialization Functions
\7f32063
4223 Node: Assignment Functions
\7f38977
4224 Node: Combined Initialization and Assignment Functions
\7f47511
4225 Node: Conversion Functions
\7f48804
4226 Node: Basic Arithmetic Functions
\7f57356
4227 Node: Comparison Functions
\7f66364
4228 Node: Special Functions
\7f69846
4229 Node: Input and Output Functions
\7f83599
4230 Node: Formatted Output Functions
\7f85522
4231 Node: Integer Related Functions
\7f94641
4232 Node: Rounding Related Functions
\7f100403
4233 Node: Miscellaneous Functions
\7f104017
4234 Node: Exception Related Functions
\7f112207
4235 Node: Compatibility with MPF
\7f118961
4236 Node: Custom Interface
\7f121649
4237 Node: Internals
\7f125894
4238 Node: API Compatibility
\7f127378
4239 Node: Type and Macro Changes
\7f129308
4240 Node: Added Functions
\7f132029
4241 Node: Changed Functions
\7f134972
4242 Node: Removed Functions
\7f139167
4243 Node: Other Changes
\7f139579
4244 Node: Contributors
\7f141108
4245 Node: References
\7f143574
4246 Node: GNU Free Documentation License
\7f145315
4247 Node: Concept Index
\7f167758
4248 Node: Function and Type Index
\7f173677