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 3.1.2.
6 Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
7 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012,
8 2013 Free 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 3.1.2.
31 Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000,
32 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012,
33 2013 Free 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@inria.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@inria.fr'.
278 *Note Reporting Bugs::. Some bug fixes are available on the MPFR 3.1.2
279 web page `http://www.mpfr.org/mpfr-3.1.2/'.
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 3.1.2 web page `http://www.mpfr.org/mpfr-3.1.2/' 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 `https://sympa.inria.fr/sympa/arc/mpfr'. Otherwise, please investigate
299 and report it. We have made this library available to you, and it is
300 not to ask too much from you, to ask you to report the bugs that you
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@inria.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); to use `mpfr_set_decimal64', one should define the macro
933 `MPFR_WANT_DECIMAL_FLOATS' before including `mpfr.h'.
934 `mpfr_set_q' might fail if the numerator (or the denominator) can
935 not be represented as a `mpfr_t'.
937 Note: If you want to store a floating-point constant to a `mpfr_t',
938 you should use `mpfr_set_str' (or one of the MPFR constant
939 functions, such as `mpfr_const_pi' for Pi) instead of
940 `mpfr_set_flt', `mpfr_set_d', `mpfr_set_ld' or
941 `mpfr_set_decimal64'. Otherwise the floating-point constant will
942 be first converted into a reduced-precision (e.g., 53-bit) binary
943 (or decimal, for `mpfr_set_decimal64') number before MPFR can work
946 -- Function: int mpfr_set_ui_2exp (mpfr_t ROP, unsigned long int OP,
947 mpfr_exp_t E, mpfr_rnd_t RND)
948 -- Function: int mpfr_set_si_2exp (mpfr_t ROP, long int OP, mpfr_exp_t
950 -- Function: int mpfr_set_uj_2exp (mpfr_t ROP, uintmax_t OP, intmax_t
952 -- Function: int mpfr_set_sj_2exp (mpfr_t ROP, intmax_t OP, intmax_t
954 -- Function: int mpfr_set_z_2exp (mpfr_t ROP, mpz_t OP, mpfr_exp_t E,
956 Set the value of ROP from OP multiplied by two to the power E,
957 rounded toward the given direction RND. Note that the input 0 is
960 -- Function: int mpfr_set_str (mpfr_t ROP, const char *S, int BASE,
962 Set ROP to the value of the string S in base BASE, rounded in the
963 direction RND. See the documentation of `mpfr_strtofr' for a
964 detailed description of the valid string formats. Contrary to
965 `mpfr_strtofr', `mpfr_set_str' requires the _whole_ string to
966 represent a valid floating-point number.
968 The meaning of the return value differs from other MPFR functions:
969 it is 0 if the entire string up to the final null character is a
970 valid number in base BASE; otherwise it is -1, and ROP may have
971 changed (users interested in the *note ternary value:: should use
972 `mpfr_strtofr' instead).
974 Note: it is preferable to use `mpfr_set_str' if one wants to
975 distinguish between an infinite ROP value coming from an infinite
976 S or from an overflow.
978 -- Function: int mpfr_strtofr (mpfr_t ROP, const char *NPTR, char
979 **ENDPTR, int BASE, mpfr_rnd_t RND)
980 Read a floating-point number from a string NPTR in base BASE,
981 rounded in the direction RND; BASE must be either 0 (to detect the
982 base, as described below) or a number from 2 to 62 (otherwise the
983 behavior is undefined). If NPTR starts with valid data, the result
984 is stored in ROP and `*ENDPTR' points to the character just after
985 the valid data (if ENDPTR is not a null pointer); otherwise ROP is
986 set to zero (for consistency with `strtod') and the value of NPTR
987 is stored in the location referenced by ENDPTR (if ENDPTR is not a
988 null pointer). The usual ternary value is returned.
990 Parsing follows the standard C `strtod' function with some
991 extensions. After optional leading whitespace, one has a subject
992 sequence consisting of an optional sign (`+' or `-'), and either
993 numeric data or special data. The subject sequence is defined as
994 the longest initial subsequence of the input string, starting with
995 the first non-whitespace character, that is of the expected form.
997 The form of numeric data is a non-empty sequence of significand
998 digits with an optional decimal point, and an optional exponent
999 consisting of an exponent prefix followed by an optional sign and
1000 a non-empty sequence of decimal digits. A significand digit is
1001 either a decimal digit or a Latin letter (62 possible characters),
1002 with `A' = 10, `B' = 11, ..., `Z' = 35; case is ignored in bases
1003 less or equal to 36, in bases larger than 36, `a' = 36, `b' = 37,
1004 ..., `z' = 61. The value of a significand digit must be strictly
1005 less than the base. The decimal point can be either the one
1006 defined by the current locale or the period (the first one is
1007 accepted for consistency with the C standard and the practice, the
1008 second one is accepted to allow the programmer to provide MPFR
1009 numbers from strings in a way that does not depend on the current
1010 locale). The exponent prefix can be `e' or `E' for bases up to
1011 10, or `@' in any base; it indicates a multiplication by a power
1012 of the base. In bases 2 and 16, the exponent prefix can also be
1013 `p' or `P', in which case the exponent, called _binary exponent_,
1014 indicates a multiplication by a power of 2 instead of the base
1015 (there is a difference only for base 16); in base 16 for example
1016 `1p2' represents 4 whereas `1@2' represents 256. The value of an
1017 exponent is always written in base 10.
1019 If the argument BASE is 0, then the base is automatically detected
1020 as follows. If the significand starts with `0b' or `0B', base 2 is
1021 assumed. If the significand starts with `0x' or `0X', base 16 is
1022 assumed. Otherwise base 10 is assumed.
1024 Note: The exponent (if present) must contain at least a digit.
1025 Otherwise the possible exponent prefix and sign are not part of
1026 the number (which ends with the significand). Similarly, if `0b',
1027 `0B', `0x' or `0X' is not followed by a binary/hexadecimal digit,
1028 then the subject sequence stops at the character `0', thus 0 is
1031 Special data (for infinities and NaN) can be `@inf@' or
1032 `@nan@(n-char-sequence-opt)', and if BASE <= 16, it can also be
1033 `infinity', `inf', `nan' or `nan(n-char-sequence-opt)', all case
1034 insensitive. A `n-char-sequence-opt' is a possibly empty string
1035 containing only digits, Latin letters and the underscore (0, 1, 2,
1036 ..., 9, a, b, ..., z, A, B, ..., Z, _). Note: one has an optional
1037 sign for all data, even NaN. For example,
1038 `-@nAn@(This_Is_Not_17)' is a valid representation for NaN in base
1042 -- Function: void mpfr_set_nan (mpfr_t X)
1043 -- Function: void mpfr_set_inf (mpfr_t X, int SIGN)
1044 -- Function: void mpfr_set_zero (mpfr_t X, int SIGN)
1045 Set the variable X to NaN (Not-a-Number), infinity or zero
1046 respectively. In `mpfr_set_inf' or `mpfr_set_zero', X is set to
1047 plus infinity or plus zero iff SIGN is nonnegative; in
1048 `mpfr_set_nan', the sign bit of the result is unspecified.
1050 -- Function: void mpfr_swap (mpfr_t X, mpfr_t Y)
1051 Swap the values X and Y efficiently. Warning: the precisions are
1052 exchanged too; in case the precisions are different, `mpfr_swap'
1053 is thus not equivalent to three `mpfr_set' calls using a third
1057 File: mpfr.info, Node: Combined Initialization and Assignment Functions, Next: Conversion Functions, Prev: Assignment Functions, Up: MPFR Interface
1059 5.3 Combined Initialization and Assignment Functions
1060 ====================================================
1062 -- Macro: int mpfr_init_set (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1063 -- Macro: int mpfr_init_set_ui (mpfr_t ROP, unsigned long int OP,
1065 -- Macro: int mpfr_init_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t
1067 -- Macro: int mpfr_init_set_d (mpfr_t ROP, double OP, mpfr_rnd_t RND)
1068 -- Macro: int mpfr_init_set_ld (mpfr_t ROP, long double OP, mpfr_rnd_t
1070 -- Macro: int mpfr_init_set_z (mpfr_t ROP, mpz_t OP, mpfr_rnd_t RND)
1071 -- Macro: int mpfr_init_set_q (mpfr_t ROP, mpq_t OP, mpfr_rnd_t RND)
1072 -- Macro: int mpfr_init_set_f (mpfr_t ROP, mpf_t OP, mpfr_rnd_t RND)
1073 Initialize ROP and set its value from OP, rounded in the direction
1074 RND. The precision of ROP will be taken from the active default
1075 precision, as set by `mpfr_set_default_prec'.
1077 -- Function: int mpfr_init_set_str (mpfr_t X, const char *S, int BASE,
1079 Initialize X and set its value from the string S in base BASE,
1080 rounded in the direction RND. See `mpfr_set_str'.
1083 File: mpfr.info, Node: Conversion Functions, Next: Basic Arithmetic Functions, Prev: Combined Initialization and Assignment Functions, Up: MPFR Interface
1085 5.4 Conversion Functions
1086 ========================
1088 -- Function: float mpfr_get_flt (mpfr_t OP, mpfr_rnd_t RND)
1089 -- Function: double mpfr_get_d (mpfr_t OP, mpfr_rnd_t RND)
1090 -- Function: long double mpfr_get_ld (mpfr_t OP, mpfr_rnd_t RND)
1091 -- Function: _Decimal64 mpfr_get_decimal64 (mpfr_t OP, mpfr_rnd_t RND)
1092 Convert OP to a `float' (respectively `double', `long double' or
1093 `_Decimal64'), using the rounding mode RND. If OP is NaN, some
1094 fixed NaN (either quiet or signaling) or the result of 0.0/0.0 is
1095 returned. If OP is ±Inf, an infinity of the same sign or the
1096 result of ±1.0/0.0 is returned. If OP is zero, these functions
1097 return a zero, trying to preserve its sign, if possible. The
1098 `mpfr_get_decimal64' function is built only under some conditions:
1099 see the documentation of `mpfr_set_decimal64'.
1101 -- Function: long mpfr_get_si (mpfr_t OP, mpfr_rnd_t RND)
1102 -- Function: unsigned long mpfr_get_ui (mpfr_t OP, mpfr_rnd_t RND)
1103 -- Function: intmax_t mpfr_get_sj (mpfr_t OP, mpfr_rnd_t RND)
1104 -- Function: uintmax_t mpfr_get_uj (mpfr_t OP, mpfr_rnd_t RND)
1105 Convert OP to a `long', an `unsigned long', an `intmax_t' or an
1106 `uintmax_t' (respectively) after rounding it with respect to RND.
1107 If OP is NaN, 0 is returned and the _erange_ flag is set. If OP
1108 is too big for the return type, the function returns the maximum
1109 or the minimum of the corresponding C type, depending on the
1110 direction of the overflow; the _erange_ flag is set too. See also
1111 `mpfr_fits_slong_p', `mpfr_fits_ulong_p', `mpfr_fits_intmax_p' and
1112 `mpfr_fits_uintmax_p'.
1114 -- Function: double mpfr_get_d_2exp (long *EXP, mpfr_t OP, mpfr_rnd_t
1116 -- Function: long double mpfr_get_ld_2exp (long *EXP, mpfr_t OP,
1118 Return D and set EXP (formally, the value pointed to by EXP) such
1119 that 0.5<=abs(D)<1 and D times 2 raised to EXP equals OP rounded
1120 to double (resp. long double) precision, using the given rounding
1121 mode. If OP is zero, then a zero of the same sign (or an unsigned
1122 zero, if the implementation does not have signed zeros) is
1123 returned, and EXP is set to 0. If OP is NaN or an infinity, then
1124 the corresponding double precision (resp. long-double precision)
1125 value is returned, and EXP is undefined.
1127 -- Function: int mpfr_frexp (mpfr_exp_t *EXP, mpfr_t Y, mpfr_t X,
1129 Set EXP (formally, the value pointed to by EXP) and Y such that
1130 0.5<=abs(Y)<1 and Y times 2 raised to EXP equals X rounded to the
1131 precision of Y, using the given rounding mode. If X is zero, then
1132 Y is set to a zero of the same sign and EXP is set to 0. If X is
1133 NaN or an infinity, then Y is set to the same value and EXP is
1136 -- Function: mpfr_exp_t mpfr_get_z_2exp (mpz_t ROP, mpfr_t OP)
1137 Put the scaled significand of OP (regarded as an integer, with the
1138 precision of OP) into ROP, and return the exponent EXP (which may
1139 be outside the current exponent range) such that OP exactly equals
1140 ROP times 2 raised to the power EXP. If OP is zero, the minimal
1141 exponent `emin' is returned. If OP is NaN or an infinity, the
1142 _erange_ flag is set, ROP is set to 0, and the the minimal
1143 exponent `emin' is returned. The returned exponent may be less
1144 than the minimal exponent `emin' of MPFR numbers in the current
1145 exponent range; in case the exponent is not representable in the
1146 `mpfr_exp_t' type, the _erange_ flag is set and the minimal value
1147 of the `mpfr_exp_t' type is returned.
1149 -- Function: int mpfr_get_z (mpz_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1150 Convert OP to a `mpz_t', after rounding it with respect to RND. If
1151 OP is NaN or an infinity, the _erange_ flag is set, ROP is set to
1152 0, and 0 is returned.
1154 -- Function: int mpfr_get_f (mpf_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1155 Convert OP to a `mpf_t', after rounding it with respect to RND.
1156 The _erange_ flag is set if OP is NaN or an infinity, which do not
1157 exist in MPF. If OP is NaN, then ROP is undefined. If OP is an
1158 +Inf (resp. -Inf), then ROP is set to the maximum (resp. minimum)
1159 value in the precision of the MPF number; if a future MPF version
1160 supports infinities, this behavior will be considered incorrect
1161 and will change (portable programs should assume that ROP is set
1162 either to this finite number or to an infinite number). Note that
1163 since MPFR currently has the same exponent type as MPF (but not
1164 with the same radix), the range of values is much larger in MPF
1165 than in MPFR, so that an overflow or underflow is not possible.
1167 -- Function: char * mpfr_get_str (char *STR, mpfr_exp_t *EXPPTR, int
1168 B, size_t N, mpfr_t OP, mpfr_rnd_t RND)
1169 Convert OP to a string of digits in base B, with rounding in the
1170 direction RND, where N is either zero (see below) or the number of
1171 significant digits output in the string; in the latter case, N
1172 must be greater or equal to 2. The base may vary from 2 to 62. If
1173 the input number is an ordinary number, the exponent is written
1174 through the pointer EXPPTR (for input 0, the current minimal
1175 exponent is written).
1177 The generated string is a fraction, with an implicit radix point
1178 immediately to the left of the first digit. For example, the
1179 number -3.1416 would be returned as "-31416" in the string and 1
1180 written at EXPPTR. If RND is to nearest, and OP is exactly in the
1181 middle of two consecutive possible outputs, the one with an even
1182 significand is chosen, where both significands are considered with
1183 the exponent of OP. Note that for an odd base, this may not
1184 correspond to an even last digit: for example with 2 digits in
1185 base 7, (14) and a half is rounded to (15) which is 12 in decimal,
1186 (16) and a half is rounded to (20) which is 14 in decimal, and
1187 (26) and a half is rounded to (26) which is 20 in decimal.
1189 If N is zero, the number of digits of the significand is chosen
1190 large enough so that re-reading the printed value with the same
1191 precision, assuming both output and input use rounding to nearest,
1192 will recover the original value of OP. More precisely, in most
1193 cases, the chosen precision of STR is the minimal precision m
1194 depending only on P = PREC(OP) and B that satisfies the above
1195 property, i.e., m = 1 + ceil(P*log(2)/log(B)), with P replaced by
1196 P-1 if B is a power of 2, but in some very rare cases, it might be
1197 m+1 (the smallest case for bases up to 62 is when P equals
1198 186564318007 for bases 7 and 49).
1200 If STR is a null pointer, space for the significand is allocated
1201 using the current allocation function, and a pointer to the string
1202 is returned. To free the returned string, you must use
1205 If STR is not a null pointer, it should point to a block of storage
1206 large enough for the significand, i.e., at least `max(N + 2, 7)'.
1207 The extra two bytes are for a possible minus sign, and for the
1208 terminating null character, and the value 7 accounts for `-@Inf@'
1209 plus the terminating null character.
1211 A pointer to the string is returned, unless there is an error, in
1212 which case a null pointer is returned.
1214 -- Function: void mpfr_free_str (char *STR)
1215 Free a string allocated by `mpfr_get_str' using the current
1216 unallocation function. The block is assumed to be `strlen(STR)+1'
1217 bytes. For more information about how it is done: *note Custom
1218 Allocation: (gmp.info)Custom Allocation.
1220 -- Function: int mpfr_fits_ulong_p (mpfr_t OP, mpfr_rnd_t RND)
1221 -- Function: int mpfr_fits_slong_p (mpfr_t OP, mpfr_rnd_t RND)
1222 -- Function: int mpfr_fits_uint_p (mpfr_t OP, mpfr_rnd_t RND)
1223 -- Function: int mpfr_fits_sint_p (mpfr_t OP, mpfr_rnd_t RND)
1224 -- Function: int mpfr_fits_ushort_p (mpfr_t OP, mpfr_rnd_t RND)
1225 -- Function: int mpfr_fits_sshort_p (mpfr_t OP, mpfr_rnd_t RND)
1226 -- Function: int mpfr_fits_uintmax_p (mpfr_t OP, mpfr_rnd_t RND)
1227 -- Function: int mpfr_fits_intmax_p (mpfr_t OP, mpfr_rnd_t RND)
1228 Return non-zero if OP would fit in the respective C data type,
1229 respectively `unsigned long', `long', `unsigned int', `int',
1230 `unsigned short', `short', `uintmax_t', `intmax_t', when rounded
1231 to an integer in the direction RND.
1234 File: mpfr.info, Node: Basic Arithmetic Functions, Next: Comparison Functions, Prev: Conversion Functions, Up: MPFR Interface
1236 5.5 Basic Arithmetic Functions
1237 ==============================
1239 -- Function: int mpfr_add (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1241 -- Function: int mpfr_add_ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1242 int OP2, mpfr_rnd_t RND)
1243 -- Function: int mpfr_add_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1245 -- Function: int mpfr_add_d (mpfr_t ROP, mpfr_t OP1, double OP2,
1247 -- Function: int mpfr_add_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
1249 -- Function: int mpfr_add_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2,
1251 Set ROP to OP1 + OP2 rounded in the direction RND. For types
1252 having no signed zero, it is considered unsigned (i.e., (+0) + 0 =
1253 (+0) and (-0) + 0 = (-0)). The `mpfr_add_d' function assumes that
1254 the radix of the `double' type is a power of 2, with a precision
1255 at most that declared by the C implementation (macro
1256 `IEEE_DBL_MANT_DIG', and if not defined 53 bits).
1258 -- Function: int mpfr_sub (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1260 -- Function: int mpfr_ui_sub (mpfr_t ROP, unsigned long int OP1,
1261 mpfr_t OP2, mpfr_rnd_t RND)
1262 -- Function: int mpfr_sub_ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1263 int OP2, mpfr_rnd_t RND)
1264 -- Function: int mpfr_si_sub (mpfr_t ROP, long int OP1, mpfr_t OP2,
1266 -- Function: int mpfr_sub_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1268 -- Function: int mpfr_d_sub (mpfr_t ROP, double OP1, mpfr_t OP2,
1270 -- Function: int mpfr_sub_d (mpfr_t ROP, mpfr_t OP1, double OP2,
1272 -- Function: int mpfr_z_sub (mpfr_t ROP, mpz_t OP1, mpfr_t OP2,
1274 -- Function: int mpfr_sub_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
1276 -- Function: int mpfr_sub_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2,
1278 Set ROP to OP1 - OP2 rounded in the direction RND. For types
1279 having no signed zero, it is considered unsigned (i.e., (+0) - 0 =
1280 (+0), (-0) - 0 = (-0), 0 - (+0) = (-0) and 0 - (-0) = (+0)). The
1281 same restrictions than for `mpfr_add_d' apply to `mpfr_d_sub' and
1284 -- Function: int mpfr_mul (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1286 -- Function: int mpfr_mul_ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1287 int OP2, mpfr_rnd_t RND)
1288 -- Function: int mpfr_mul_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1290 -- Function: int mpfr_mul_d (mpfr_t ROP, mpfr_t OP1, double OP2,
1292 -- Function: int mpfr_mul_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
1294 -- Function: int mpfr_mul_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2,
1296 Set ROP to OP1 times OP2 rounded in the direction RND. When a
1297 result is zero, its sign is the product of the signs of the
1298 operands (for types having no signed zero, it is considered
1299 positive). The same restrictions than for `mpfr_add_d' apply to
1302 -- Function: int mpfr_sqr (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1303 Set ROP to the square of OP rounded in the direction RND.
1305 -- Function: int mpfr_div (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1307 -- Function: int mpfr_ui_div (mpfr_t ROP, unsigned long int OP1,
1308 mpfr_t OP2, mpfr_rnd_t RND)
1309 -- Function: int mpfr_div_ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1310 int OP2, mpfr_rnd_t RND)
1311 -- Function: int mpfr_si_div (mpfr_t ROP, long int OP1, mpfr_t OP2,
1313 -- Function: int mpfr_div_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1315 -- Function: int mpfr_d_div (mpfr_t ROP, double OP1, mpfr_t OP2,
1317 -- Function: int mpfr_div_d (mpfr_t ROP, mpfr_t OP1, double OP2,
1319 -- Function: int mpfr_div_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
1321 -- Function: int mpfr_div_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2,
1323 Set ROP to OP1/OP2 rounded in the direction RND. When a result is
1324 zero, its sign is the product of the signs of the operands (for
1325 types having no signed zero, it is considered positive). The same
1326 restrictions than for `mpfr_add_d' apply to `mpfr_d_div' and
1329 -- Function: int mpfr_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1330 -- Function: int mpfr_sqrt_ui (mpfr_t ROP, unsigned long int OP,
1332 Set ROP to the square root of OP rounded in the direction RND (set
1333 ROP to -0 if OP is -0, to be consistent with the IEEE 754
1334 standard). Set ROP to NaN if OP is negative.
1336 -- Function: int mpfr_rec_sqrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1337 Set ROP to the reciprocal square root of OP rounded in the
1338 direction RND. Set ROP to +Inf if OP is ±0, +0 if OP is +Inf, and
1339 NaN if OP is negative.
1341 -- Function: int mpfr_cbrt (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1342 -- Function: int mpfr_root (mpfr_t ROP, mpfr_t OP, unsigned long int
1344 Set ROP to the cubic root (resp. the Kth root) of OP rounded in
1345 the direction RND. For K odd (resp. even) and OP negative
1346 (including -Inf), set ROP to a negative number (resp. NaN). The
1347 Kth root of -0 is defined to be -0, whatever the parity of K.
1349 -- Function: int mpfr_pow (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1351 -- Function: int mpfr_pow_ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1352 int OP2, mpfr_rnd_t RND)
1353 -- Function: int mpfr_pow_si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1355 -- Function: int mpfr_pow_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2,
1357 -- Function: int mpfr_ui_pow_ui (mpfr_t ROP, unsigned long int OP1,
1358 unsigned long int OP2, mpfr_rnd_t RND)
1359 -- Function: int mpfr_ui_pow (mpfr_t ROP, unsigned long int OP1,
1360 mpfr_t OP2, mpfr_rnd_t RND)
1361 Set ROP to OP1 raised to OP2, rounded in the direction RND.
1362 Special values are handled as described in the ISO C99 and IEEE
1363 754-2008 standards for the `pow' function:
1364 * `pow(±0, Y)' returns plus or minus infinity for Y a negative
1367 * `pow(±0, Y)' returns plus infinity for Y negative and not an
1370 * `pow(±0, Y)' returns plus or minus zero for Y a positive odd
1373 * `pow(±0, Y)' returns plus zero for Y positive and not an odd
1376 * `pow(-1, ±Inf)' returns 1.
1378 * `pow(+1, Y)' returns 1 for any Y, even a NaN.
1380 * `pow(X, ±0)' returns 1 for any X, even a NaN.
1382 * `pow(X, Y)' returns NaN for finite negative X and finite
1385 * `pow(X, -Inf)' returns plus infinity for 0 < abs(x) < 1, and
1386 plus zero for abs(x) > 1.
1388 * `pow(X, +Inf)' returns plus zero for 0 < abs(x) < 1, and plus
1389 infinity for abs(x) > 1.
1391 * `pow(-Inf, Y)' returns minus zero for Y a negative odd
1394 * `pow(-Inf, Y)' returns plus zero for Y negative and not an
1397 * `pow(-Inf, Y)' returns minus infinity for Y a positive odd
1400 * `pow(-Inf, Y)' returns plus infinity for Y positive and not
1403 * `pow(+Inf, Y)' returns plus zero for Y negative, and plus
1404 infinity for Y positive.
1406 -- Function: int mpfr_neg (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1407 -- Function: int mpfr_abs (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1408 Set ROP to -OP and the absolute value of OP respectively, rounded
1409 in the direction RND. Just changes or adjusts the sign if ROP and
1410 OP are the same variable, otherwise a rounding might occur if the
1411 precision of ROP is less than that of OP.
1413 -- Function: int mpfr_dim (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1415 Set ROP to the positive difference of OP1 and OP2, i.e., OP1 - OP2
1416 rounded in the direction RND if OP1 > OP2, +0 if OP1 <= OP2, and
1417 NaN if OP1 or OP2 is NaN.
1419 -- Function: int mpfr_mul_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1420 int OP2, mpfr_rnd_t RND)
1421 -- Function: int mpfr_mul_2si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1423 Set ROP to OP1 times 2 raised to OP2 rounded in the direction RND.
1424 Just increases the exponent by OP2 when ROP and OP1 are identical.
1426 -- Function: int mpfr_div_2ui (mpfr_t ROP, mpfr_t OP1, unsigned long
1427 int OP2, mpfr_rnd_t RND)
1428 -- Function: int mpfr_div_2si (mpfr_t ROP, mpfr_t OP1, long int OP2,
1430 Set ROP to OP1 divided by 2 raised to OP2 rounded in the direction
1431 RND. Just decreases the exponent by OP2 when ROP and OP1 are
1435 File: mpfr.info, Node: Comparison Functions, Next: Special Functions, Prev: Basic Arithmetic Functions, Up: MPFR Interface
1437 5.6 Comparison Functions
1438 ========================
1440 -- Function: int mpfr_cmp (mpfr_t OP1, mpfr_t OP2)
1441 -- Function: int mpfr_cmp_ui (mpfr_t OP1, unsigned long int OP2)
1442 -- Function: int mpfr_cmp_si (mpfr_t OP1, long int OP2)
1443 -- Function: int mpfr_cmp_d (mpfr_t OP1, double OP2)
1444 -- Function: int mpfr_cmp_ld (mpfr_t OP1, long double OP2)
1445 -- Function: int mpfr_cmp_z (mpfr_t OP1, mpz_t OP2)
1446 -- Function: int mpfr_cmp_q (mpfr_t OP1, mpq_t OP2)
1447 -- Function: int mpfr_cmp_f (mpfr_t OP1, mpf_t OP2)
1448 Compare OP1 and OP2. Return a positive value if OP1 > OP2, zero
1449 if OP1 = OP2, and a negative value if OP1 < OP2. Both OP1 and OP2
1450 are considered to their full own precision, which may differ. If
1451 one of the operands is NaN, set the _erange_ flag and return zero.
1453 Note: These functions may be useful to distinguish the three
1454 possible cases. If you need to distinguish two cases only, it is
1455 recommended to use the predicate functions (e.g., `mpfr_equal_p'
1456 for the equality) described below; they behave like the IEEE 754
1457 comparisons, in particular when one or both arguments are NaN. But
1458 only floating-point numbers can be compared (you may need to do a
1461 -- Function: int mpfr_cmp_ui_2exp (mpfr_t OP1, unsigned long int OP2,
1463 -- Function: int mpfr_cmp_si_2exp (mpfr_t OP1, long int OP2,
1465 Compare OP1 and OP2 multiplied by two to the power E. Similar as
1468 -- Function: int mpfr_cmpabs (mpfr_t OP1, mpfr_t OP2)
1469 Compare |OP1| and |OP2|. Return a positive value if |OP1| >
1470 |OP2|, zero if |OP1| = |OP2|, and a negative value if |OP1| <
1471 |OP2|. If one of the operands is NaN, set the _erange_ flag and
1474 -- Function: int mpfr_nan_p (mpfr_t OP)
1475 -- Function: int mpfr_inf_p (mpfr_t OP)
1476 -- Function: int mpfr_number_p (mpfr_t OP)
1477 -- Function: int mpfr_zero_p (mpfr_t OP)
1478 -- Function: int mpfr_regular_p (mpfr_t OP)
1479 Return non-zero if OP is respectively NaN, an infinity, an ordinary
1480 number (i.e., neither NaN nor an infinity), zero, or a regular
1481 number (i.e., neither NaN, nor an infinity nor zero). Return zero
1484 -- Macro: int mpfr_sgn (mpfr_t OP)
1485 Return a positive value if OP > 0, zero if OP = 0, and a negative
1486 value if OP < 0. If the operand is NaN, set the _erange_ flag and
1487 return zero. This is equivalent to `mpfr_cmp_ui (op, 0)', but
1490 -- Function: int mpfr_greater_p (mpfr_t OP1, mpfr_t OP2)
1491 -- Function: int mpfr_greaterequal_p (mpfr_t OP1, mpfr_t OP2)
1492 -- Function: int mpfr_less_p (mpfr_t OP1, mpfr_t OP2)
1493 -- Function: int mpfr_lessequal_p (mpfr_t OP1, mpfr_t OP2)
1494 -- Function: int mpfr_equal_p (mpfr_t OP1, mpfr_t OP2)
1495 Return non-zero if OP1 > OP2, OP1 >= OP2, OP1 < OP2, OP1 <= OP2,
1496 OP1 = OP2 respectively, and zero otherwise. Those functions
1497 return zero whenever OP1 and/or OP2 is NaN.
1499 -- Function: int mpfr_lessgreater_p (mpfr_t OP1, mpfr_t OP2)
1500 Return non-zero if OP1 < OP2 or OP1 > OP2 (i.e., neither OP1, nor
1501 OP2 is NaN, and OP1 <> OP2), zero otherwise (i.e., OP1 and/or OP2
1502 is NaN, or OP1 = OP2).
1504 -- Function: int mpfr_unordered_p (mpfr_t OP1, mpfr_t OP2)
1505 Return non-zero if OP1 or OP2 is a NaN (i.e., they cannot be
1506 compared), zero otherwise.
1509 File: mpfr.info, Node: Special Functions, Next: Input and Output Functions, Prev: Comparison Functions, Up: MPFR Interface
1511 5.7 Special Functions
1512 =====================
1514 All those functions, except explicitly stated (for example
1515 `mpfr_sin_cos'), return a *note ternary value::, i.e., zero for an
1516 exact return value, a positive value for a return value larger than the
1517 exact result, and a negative value otherwise.
1519 Important note: in some domains, computing special functions (either
1520 with correct or incorrect rounding) is expensive, even for small
1521 precision, for example the trigonometric and Bessel functions for large
1524 -- Function: int mpfr_log (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1525 -- Function: int mpfr_log2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1526 -- Function: int mpfr_log10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1527 Set ROP to the natural logarithm of OP, log2(OP) or log10(OP),
1528 respectively, rounded in the direction RND. Set ROP to -Inf if OP
1529 is -0 (i.e., the sign of the zero has no influence on the result).
1531 -- Function: int mpfr_exp (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1532 -- Function: int mpfr_exp2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1533 -- Function: int mpfr_exp10 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1534 Set ROP to the exponential of OP, to 2 power of OP or to 10 power
1535 of OP, respectively, rounded in the direction RND.
1537 -- Function: int mpfr_cos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1538 -- Function: int mpfr_sin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1539 -- Function: int mpfr_tan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1540 Set ROP to the cosine of OP, sine of OP, tangent of OP, rounded in
1543 -- Function: int mpfr_sin_cos (mpfr_t SOP, mpfr_t COP, mpfr_t OP,
1545 Set simultaneously SOP to the sine of OP and COP to the cosine of
1546 OP, rounded in the direction RND with the corresponding precisions
1547 of SOP and COP, which must be different variables. Return 0 iff
1548 both results are exact, more precisely it returns s+4c where s=0
1549 if SOP is exact, s=1 if SOP is larger than the sine of OP, s=2 if
1550 SOP is smaller than the sine of OP, and similarly for c and the
1553 -- Function: int mpfr_sec (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1554 -- Function: int mpfr_csc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1555 -- Function: int mpfr_cot (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1556 Set ROP to the secant of OP, cosecant of OP, cotangent of OP,
1557 rounded in the direction RND.
1559 -- Function: int mpfr_acos (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1560 -- Function: int mpfr_asin (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1561 -- Function: int mpfr_atan (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1562 Set ROP to the arc-cosine, arc-sine or arc-tangent of OP, rounded
1563 in the direction RND. Note that since `acos(-1)' returns the
1564 floating-point number closest to Pi according to the given
1565 rounding mode, this number might not be in the output range 0 <=
1566 ROP < \pi of the arc-cosine function; still, the result lies in
1567 the image of the output range by the rounding function. The same
1568 holds for `asin(-1)', `asin(1)', `atan(-Inf)', `atan(+Inf)' or for
1569 `atan(op)' with large OP and small precision of ROP.
1571 -- Function: int mpfr_atan2 (mpfr_t ROP, mpfr_t Y, mpfr_t X,
1573 Set ROP to the arc-tangent2 of Y and X, rounded in the direction
1574 RND: if `x > 0', `atan2(y, x) = atan (y/x)'; if `x < 0', `atan2(y,
1575 x) = sign(y)*(Pi - atan (abs(y/x)))', thus a number from -Pi to Pi.
1576 As for `atan', in case the exact mathematical result is +Pi or -Pi,
1577 its rounded result might be outside the function output range.
1579 `atan2(y, 0)' does not raise any floating-point exception.
1580 Special values are handled as described in the ISO C99 and IEEE
1581 754-2008 standards for the `atan2' function:
1582 * `atan2(+0, -0)' returns +Pi.
1584 * `atan2(-0, -0)' returns -Pi.
1586 * `atan2(+0, +0)' returns +0.
1588 * `atan2(-0, +0)' returns -0.
1590 * `atan2(+0, x)' returns +Pi for x < 0.
1592 * `atan2(-0, x)' returns -Pi for x < 0.
1594 * `atan2(+0, x)' returns +0 for x > 0.
1596 * `atan2(-0, x)' returns -0 for x > 0.
1598 * `atan2(y, 0)' returns -Pi/2 for y < 0.
1600 * `atan2(y, 0)' returns +Pi/2 for y > 0.
1602 * `atan2(+Inf, -Inf)' returns +3*Pi/4.
1604 * `atan2(-Inf, -Inf)' returns -3*Pi/4.
1606 * `atan2(+Inf, +Inf)' returns +Pi/4.
1608 * `atan2(-Inf, +Inf)' returns -Pi/4.
1610 * `atan2(+Inf, x)' returns +Pi/2 for finite x.
1612 * `atan2(-Inf, x)' returns -Pi/2 for finite x.
1614 * `atan2(y, -Inf)' returns +Pi for finite y > 0.
1616 * `atan2(y, -Inf)' returns -Pi for finite y < 0.
1618 * `atan2(y, +Inf)' returns +0 for finite y > 0.
1620 * `atan2(y, +Inf)' returns -0 for finite y < 0.
1622 -- Function: int mpfr_cosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1623 -- Function: int mpfr_sinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1624 -- Function: int mpfr_tanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1625 Set ROP to the hyperbolic cosine, sine or tangent of OP, rounded
1626 in the direction RND.
1628 -- Function: int mpfr_sinh_cosh (mpfr_t SOP, mpfr_t COP, mpfr_t OP,
1630 Set simultaneously SOP to the hyperbolic sine of OP and COP to the
1631 hyperbolic cosine of OP, rounded in the direction RND with the
1632 corresponding precision of SOP and COP, which must be different
1633 variables. Return 0 iff both results are exact (see
1634 `mpfr_sin_cos' for a more detailed description of the return
1637 -- Function: int mpfr_sech (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1638 -- Function: int mpfr_csch (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1639 -- Function: int mpfr_coth (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1640 Set ROP to the hyperbolic secant of OP, cosecant of OP, cotangent
1641 of OP, rounded in the direction RND.
1643 -- Function: int mpfr_acosh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1644 -- Function: int mpfr_asinh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1645 -- Function: int mpfr_atanh (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1646 Set ROP to the inverse hyperbolic cosine, sine or tangent of OP,
1647 rounded in the direction RND.
1649 -- Function: int mpfr_fac_ui (mpfr_t ROP, unsigned long int OP,
1651 Set ROP to the factorial of OP, rounded in the direction RND.
1653 -- Function: int mpfr_log1p (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1654 Set ROP to the logarithm of one plus OP, rounded in the direction
1657 -- Function: int mpfr_expm1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1658 Set ROP to the exponential of OP followed by a subtraction by one,
1659 rounded in the direction RND.
1661 -- Function: int mpfr_eint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1662 Set ROP to the exponential integral of OP, rounded in the
1663 direction RND. For positive OP, the exponential integral is the
1664 sum of Euler's constant, of the logarithm of OP, and of the sum
1665 for k from 1 to infinity of OP to the power k, divided by k and
1666 factorial(k). For negative OP, ROP is set to NaN (this definition
1667 for negative argument follows formula 5.1.2 from the Handbook of
1668 Mathematical Functions from Abramowitz and Stegun, a future
1669 version might use another definition).
1671 -- Function: int mpfr_li2 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1672 Set ROP to real part of the dilogarithm of OP, rounded in the
1673 direction RND. MPFR defines the dilogarithm function as the
1674 integral of -log(1-t)/t from 0 to OP.
1676 -- Function: int mpfr_gamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1677 Set ROP to the value of the Gamma function on OP, rounded in the
1678 direction RND. When OP is a negative integer, ROP is set to NaN.
1680 -- Function: int mpfr_lngamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1681 Set ROP to the value of the logarithm of the Gamma function on OP,
1682 rounded in the direction RND. When -2K-1 <= OP <= -2K, K being a
1683 non-negative integer, ROP is set to NaN. See also `mpfr_lgamma'.
1685 -- Function: int mpfr_lgamma (mpfr_t ROP, int *SIGNP, mpfr_t OP,
1687 Set ROP to the value of the logarithm of the absolute value of the
1688 Gamma function on OP, rounded in the direction RND. The sign (1 or
1689 -1) of Gamma(OP) is returned in the object pointed to by SIGNP.
1690 When OP is an infinity or a non-positive integer, set ROP to +Inf.
1691 When OP is NaN, -Inf or a negative integer, *SIGNP is undefined,
1692 and when OP is ±0, *SIGNP is the sign of the zero.
1694 -- Function: int mpfr_digamma (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1695 Set ROP to the value of the Digamma (sometimes also called Psi)
1696 function on OP, rounded in the direction RND. When OP is a
1697 negative integer, set ROP to NaN.
1699 -- Function: int mpfr_zeta (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1700 -- Function: int mpfr_zeta_ui (mpfr_t ROP, unsigned long OP,
1702 Set ROP to the value of the Riemann Zeta function on OP, rounded
1703 in the direction RND.
1705 -- Function: int mpfr_erf (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1706 -- Function: int mpfr_erfc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1707 Set ROP to the value of the error function on OP (resp. the
1708 complementary error function on OP) rounded in the direction RND.
1710 -- Function: int mpfr_j0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1711 -- Function: int mpfr_j1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1712 -- Function: int mpfr_jn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t
1714 Set ROP to the value of the first kind Bessel function of order 0,
1715 (resp. 1 and N) on OP, rounded in the direction RND. When OP is
1716 NaN, ROP is always set to NaN. When OP is plus or minus Infinity,
1717 ROP is set to +0. When OP is zero, and N is not zero, ROP is set
1718 to +0 or -0 depending on the parity and sign of N, and the sign of
1721 -- Function: int mpfr_y0 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1722 -- Function: int mpfr_y1 (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
1723 -- Function: int mpfr_yn (mpfr_t ROP, long N, mpfr_t OP, mpfr_rnd_t
1725 Set ROP to the value of the second kind Bessel function of order 0
1726 (resp. 1 and N) on OP, rounded in the direction RND. When OP is
1727 NaN or negative, ROP is always set to NaN. When OP is +Inf, ROP is
1728 set to +0. When OP is zero, ROP is set to +Inf or -Inf depending
1729 on the parity and sign of N.
1731 -- Function: int mpfr_fma (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t
1732 OP3, mpfr_rnd_t RND)
1733 -- Function: int mpfr_fms (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, mpfr_t
1734 OP3, mpfr_rnd_t RND)
1735 Set ROP to (OP1 times OP2) + OP3 (resp. (OP1 times OP2) - OP3)
1736 rounded in the direction RND.
1738 -- Function: int mpfr_agm (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
1740 Set ROP to the arithmetic-geometric mean of OP1 and OP2, rounded
1741 in the direction RND. The arithmetic-geometric mean is the common
1742 limit of the sequences U_N and V_N, where U_0=OP1, V_0=OP2,
1743 U_(N+1) is the arithmetic mean of U_N and V_N, and V_(N+1) is the
1744 geometric mean of U_N and V_N. If any operand is negative, set
1747 -- Function: int mpfr_hypot (mpfr_t ROP, mpfr_t X, mpfr_t Y,
1749 Set ROP to the Euclidean norm of X and Y, i.e., the square root of
1750 the sum of the squares of X and Y, rounded in the direction RND.
1751 Special values are handled as described in Section F.9.4.3 of the
1752 ISO C99 and IEEE 754-2008 standards: If X or Y is an infinity,
1753 then +Inf is returned in ROP, even if the other number is NaN.
1755 -- Function: int mpfr_ai (mpfr_t ROP, mpfr_t X, mpfr_rnd_t RND)
1756 Set ROP to the value of the Airy function Ai on X, rounded in the
1757 direction RND. When X is NaN, ROP is always set to NaN. When X is
1758 +Inf or -Inf, ROP is +0. The current implementation is not
1759 intended to be used with large arguments. It works with abs(X)
1760 typically smaller than 500. For larger arguments, other methods
1761 should be used and will be implemented in a future version.
1763 -- Function: int mpfr_const_log2 (mpfr_t ROP, mpfr_rnd_t RND)
1764 -- Function: int mpfr_const_pi (mpfr_t ROP, mpfr_rnd_t RND)
1765 -- Function: int mpfr_const_euler (mpfr_t ROP, mpfr_rnd_t RND)
1766 -- Function: int mpfr_const_catalan (mpfr_t ROP, mpfr_rnd_t RND)
1767 Set ROP to the logarithm of 2, the value of Pi, of Euler's
1768 constant 0.577..., of Catalan's constant 0.915..., respectively,
1769 rounded in the direction RND. These functions cache the computed
1770 values to avoid other calculations if a lower or equal precision
1771 is requested. To free these caches, use `mpfr_free_cache'.
1773 -- Function: void mpfr_free_cache (void)
1774 Free various caches used by MPFR internally, in particular the
1775 caches used by the functions computing constants
1776 (`mpfr_const_log2', `mpfr_const_pi', `mpfr_const_euler' and
1777 `mpfr_const_catalan'). You should call this function before
1778 terminating a thread, even if you did not call these functions
1779 directly (they could have been called internally).
1781 -- Function: int mpfr_sum (mpfr_t ROP, mpfr_ptr const TAB[], unsigned
1782 long int N, mpfr_rnd_t RND)
1783 Set ROP to the sum of all elements of TAB, whose size is N,
1784 rounded in the direction RND. Warning: for efficiency reasons, TAB
1785 is an array of pointers to `mpfr_t', not an array of `mpfr_t'. If
1786 the returned `int' value is zero, ROP is guaranteed to be the
1787 exact sum; otherwise ROP might be smaller than, equal to, or
1788 larger than the exact sum (in accordance to the rounding mode).
1789 However, `mpfr_sum' does guarantee the result is correctly rounded.
1792 File: mpfr.info, Node: Input and Output Functions, Next: Formatted Output Functions, Prev: Special Functions, Up: MPFR Interface
1794 5.8 Input and Output Functions
1795 ==============================
1797 This section describes functions that perform input from an input/output
1798 stream, and functions that output to an input/output stream. Passing a
1799 null pointer for a `stream' to any of these functions will make them
1800 read from `stdin' and write to `stdout', respectively.
1802 When using any of these functions, you must include the `<stdio.h>'
1803 standard header before `mpfr.h', to allow `mpfr.h' to define prototypes
1804 for these functions.
1806 -- Function: size_t mpfr_out_str (FILE *STREAM, int BASE, size_t N,
1807 mpfr_t OP, mpfr_rnd_t RND)
1808 Output OP on stream STREAM, as a string of digits in base BASE,
1809 rounded in the direction RND. The base may vary from 2 to 62.
1810 Print N significant digits exactly, or if N is 0, enough digits so
1811 that OP can be read back exactly (see `mpfr_get_str').
1813 In addition to the significant digits, a decimal point (defined by
1814 the current locale) at the right of the first digit and a trailing
1815 exponent in base 10, in the form `eNNN', are printed. If BASE is
1816 greater than 10, `@' will be used instead of `e' as exponent
1819 Return the number of characters written, or if an error occurred,
1822 -- Function: size_t mpfr_inp_str (mpfr_t ROP, FILE *STREAM, int BASE,
1824 Input a string in base BASE from stream STREAM, rounded in the
1825 direction RND, and put the read float in ROP.
1827 This function reads a word (defined as a sequence of characters
1828 between whitespace) and parses it using `mpfr_set_str'. See the
1829 documentation of `mpfr_strtofr' for a detailed description of the
1830 valid string formats.
1832 Return the number of bytes read, or if an error occurred, return 0.
1835 File: mpfr.info, Node: Formatted Output Functions, Next: Integer Related Functions, Prev: Input and Output Functions, Up: MPFR Interface
1837 5.9 Formatted Output Functions
1838 ==============================
1843 The class of `mpfr_printf' functions provides formatted output in a
1844 similar manner as the standard C `printf'. These functions are defined
1845 only if your system supports ISO C variadic functions and the
1846 corresponding argument access macros.
1848 When using any of these functions, you must include the `<stdio.h>'
1849 standard header before `mpfr.h', to allow `mpfr.h' to define prototypes
1850 for these functions.
1855 The format specification accepted by `mpfr_printf' is an extension of
1856 the `printf' one. The conversion specification is of the form:
1857 % [flags] [width] [.[precision]] [type] [rounding] conv
1858 `flags', `width', and `precision' have the same meaning as for the
1859 standard `printf' (in particular, notice that the `precision' is
1860 related to the number of digits displayed in the base chosen by `conv'
1861 and not related to the internal precision of the `mpfr_t' variable).
1862 `mpfr_printf' accepts the same `type' specifiers as GMP (except the
1863 non-standard and deprecated `q', use `ll' instead), namely the length
1864 modifiers defined in the C standard:
1868 `j' `intmax_t' or `uintmax_t'
1869 `l' `long' or `wchar_t'
1875 and the `type' specifiers defined in GMP plus `R' and `P' specific
1876 to MPFR (the second column in the table below shows the type of the
1877 argument read in the argument list and the kind of `conv' specifier to
1878 use after the `type' specifier):
1880 `F' `mpf_t', float conversions
1881 `Q' `mpq_t', integer conversions
1882 `M' `mp_limb_t', integer conversions
1883 `N' `mp_limb_t' array, integer conversions
1884 `Z' `mpz_t', integer conversions
1885 `P' `mpfr_prec_t', integer conversions
1886 `R' `mpfr_t', float conversions
1888 The `type' specifiers have the same restrictions as those mentioned
1889 in the GMP documentation: *note Formatted Output Strings:
1890 (gmp.info)Formatted Output Strings. In particular, the `type'
1891 specifiers (except `R' and `P') are supported only if they are
1892 supported by `gmp_printf' in your GMP build; this implies that the
1893 standard specifiers, such as `t', must _also_ be supported by your C
1894 library if you want to use them.
1896 The `rounding' field is specific to `mpfr_t' arguments and should
1897 not be used with other types.
1899 With conversion specification not involving `P' and `R' types,
1900 `mpfr_printf' behaves exactly as `gmp_printf'.
1902 The `P' type specifies that a following `o', `u', `x', or `X'
1903 conversion specifier applies to a `mpfr_prec_t' argument. It is needed
1904 because the `mpfr_prec_t' type does not necessarily correspond to an
1905 `unsigned int' or any fixed standard type. The `precision' field
1906 specifies the minimum number of digits to appear. The default
1907 `precision' is 1. For example:
1912 p = mpfr_get_prec (x);
1913 mpfr_printf ("variable x with %Pu bits", p);
1915 The `R' type specifies that a following `a', `A', `b', `e', `E',
1916 `f', `F', `g', `G', or `n' conversion specifier applies to a `mpfr_t'
1917 argument. The `R' type can be followed by a `rounding' specifier
1918 denoted by one of the following characters:
1920 `U' round toward plus infinity
1921 `D' round toward minus infinity
1922 `Y' round away from zero
1923 `Z' round toward zero
1924 `N' round to nearest (with ties to even)
1925 `*' rounding mode indicated by the
1926 `mpfr_rnd_t' argument just before the
1927 corresponding `mpfr_t' variable.
1929 The default rounding mode is rounding to nearest. The following
1930 three examples are equivalent:
1934 mpfr_printf ("%.128Rf", x);
1935 mpfr_printf ("%.128RNf", x);
1936 mpfr_printf ("%.128R*f", MPFR_RNDN, x);
1938 Note that the rounding away from zero mode is specified with `Y'
1939 because ISO C reserves the `A' specifier for hexadecimal output (see
1942 The output `conv' specifiers allowed with `mpfr_t' parameter are:
1944 `a' `A' hex float, C99 style
1946 `e' `E' scientific format float
1947 `f' `F' fixed point float
1948 `g' `G' fixed or scientific float
1950 The conversion specifier `b' which displays the argument in binary is
1951 specific to `mpfr_t' arguments and should not be used with other types.
1952 Other conversion specifiers have the same meaning as for a `double'
1955 In case of non-decimal output, only the significand is written in the
1956 specified base, the exponent is always displayed in decimal. Special
1957 values are always displayed as `nan', `-inf', and `inf' for `a', `b',
1958 `e', `f', and `g' specifiers and `NAN', `-INF', and `INF' for `A', `E',
1959 `F', and `G' specifiers.
1961 If the `precision' field is not empty, the `mpfr_t' number is
1962 rounded to the given precision in the direction specified by the
1963 rounding mode. If the precision is zero with rounding to nearest mode
1964 and one of the following `conv' specifiers: `a', `A', `b', `e', `E',
1965 tie case is rounded to even when it lies between two consecutive values
1966 at the wanted precision which have the same exponent, otherwise, it is
1967 rounded away from zero. For instance, 85 is displayed as "8e+1" and 95
1968 is displayed as "1e+2" with the format specification `"%.0RNe"'. This
1969 also applies when the `g' (resp. `G') conversion specifier uses the `e'
1970 (resp. `E') style. If the precision is set to a value greater than the
1971 maximum value for an `int', it will be silently reduced down to
1974 If the `precision' field is empty (as in `%Re' or `%.RE') with
1975 `conv' specifier `e' and `E', the number is displayed with enough
1976 digits so that it can be read back exactly, assuming that the input and
1977 output variables have the same precision and that the input and output
1978 rounding modes are both rounding to nearest (as for `mpfr_get_str').
1979 The default precision for an empty `precision' field with `conv'
1980 specifiers `f', `F', `g', and `G' is 6.
1985 For all the following functions, if the number of characters which
1986 ought to be written appears to exceed the maximum limit for an `int',
1987 nothing is written in the stream (resp. to `stdout', to BUF, to STR),
1988 the function returns -1, sets the _erange_ flag, and (in POSIX system
1989 only) `errno' is set to `EOVERFLOW'.
1991 -- Function: int mpfr_fprintf (FILE *STREAM, const char *TEMPLATE, ...)
1992 -- Function: int mpfr_vfprintf (FILE *STREAM, const char *TEMPLATE,
1994 Print to the stream STREAM the optional arguments under the
1995 control of the template string TEMPLATE. Return the number of
1996 characters written or a negative value if an error occurred.
1998 -- Function: int mpfr_printf (const char *TEMPLATE, ...)
1999 -- Function: int mpfr_vprintf (const char *TEMPLATE, va_list AP)
2000 Print to `stdout' the optional arguments under the control of the
2001 template string TEMPLATE. Return the number of characters written
2002 or a negative value if an error occurred.
2004 -- Function: int mpfr_sprintf (char *BUF, const char *TEMPLATE, ...)
2005 -- Function: int mpfr_vsprintf (char *BUF, const char *TEMPLATE,
2007 Form a null-terminated string corresponding to the optional
2008 arguments under the control of the template string TEMPLATE, and
2009 print it in BUF. No overlap is permitted between BUF and the other
2010 arguments. Return the number of characters written in the array
2011 BUF _not counting_ the terminating null character or a negative
2012 value if an error occurred.
2014 -- Function: int mpfr_snprintf (char *BUF, size_t N, const char
2016 -- Function: int mpfr_vsnprintf (char *BUF, size_t N, const char
2017 *TEMPLATE, va_list AP)
2018 Form a null-terminated string corresponding to the optional
2019 arguments under the control of the template string TEMPLATE, and
2020 print it in BUF. If N is zero, nothing is written and BUF may be a
2021 null pointer, otherwise, the N-1 first characters are written in
2022 BUF and the N-th is a null character. Return the number of
2023 characters that would have been written had N be sufficiently
2024 large, _not counting_ the terminating null character, or a
2025 negative value if an error occurred.
2027 -- Function: int mpfr_asprintf (char **STR, const char *TEMPLATE, ...)
2028 -- Function: int mpfr_vasprintf (char **STR, const char *TEMPLATE,
2030 Write their output as a null terminated string in a block of
2031 memory allocated using the current allocation function. A pointer
2032 to the block is stored in STR. The block of memory must be freed
2033 using `mpfr_free_str'. The return value is the number of
2034 characters written in the string, excluding the null-terminator,
2035 or a negative value if an error occurred.
2038 File: mpfr.info, Node: Integer Related Functions, Next: Rounding Related Functions, Prev: Formatted Output Functions, Up: MPFR Interface
2040 5.10 Integer and Remainder Related Functions
2041 ============================================
2043 -- Function: int mpfr_rint (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
2044 -- Function: int mpfr_ceil (mpfr_t ROP, mpfr_t OP)
2045 -- Function: int mpfr_floor (mpfr_t ROP, mpfr_t OP)
2046 -- Function: int mpfr_round (mpfr_t ROP, mpfr_t OP)
2047 -- Function: int mpfr_trunc (mpfr_t ROP, mpfr_t OP)
2048 Set ROP to OP rounded to an integer. `mpfr_rint' rounds to the
2049 nearest representable integer in the given direction RND,
2050 `mpfr_ceil' rounds to the next higher or equal representable
2051 integer, `mpfr_floor' to the next lower or equal representable
2052 integer, `mpfr_round' to the nearest representable integer,
2053 rounding halfway cases away from zero (as in the roundTiesToAway
2054 mode of IEEE 754-2008), and `mpfr_trunc' to the next representable
2055 integer toward zero.
2057 The returned value is zero when the result is exact, positive when
2058 it is greater than the original value of OP, and negative when it
2059 is smaller. More precisely, the returned value is 0 when OP is an
2060 integer representable in ROP, 1 or -1 when OP is an integer that
2061 is not representable in ROP, 2 or -2 when OP is not an integer.
2063 Note that `mpfr_round' is different from `mpfr_rint' called with
2064 the rounding to nearest mode (where halfway cases are rounded to
2065 an even integer or significand). Note also that no double rounding
2066 is performed; for instance, 10.5 (1010.1 in binary) is rounded by
2067 `mpfr_rint' with rounding to nearest to 12 (1100 in binary) in
2068 2-bit precision, because the two enclosing numbers representable
2069 on two bits are 8 and 12, and the closest is 12. (If one first
2070 rounded to an integer, one would round 10.5 to 10 with even
2071 rounding, and then 10 would be rounded to 8 again with even
2074 -- Function: int mpfr_rint_ceil (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
2075 -- Function: int mpfr_rint_floor (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t
2077 -- Function: int mpfr_rint_round (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t
2079 -- Function: int mpfr_rint_trunc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t
2081 Set ROP to OP rounded to an integer. `mpfr_rint_ceil' rounds to
2082 the next higher or equal integer, `mpfr_rint_floor' to the next
2083 lower or equal integer, `mpfr_rint_round' to the nearest integer,
2084 rounding halfway cases away from zero, and `mpfr_rint_trunc' to
2085 the next integer toward zero. If the result is not representable,
2086 it is rounded in the direction RND. The returned value is the
2087 ternary value associated with the considered round-to-integer
2088 function (regarded in the same way as any other mathematical
2089 function). Contrary to `mpfr_rint', those functions do perform a
2090 double rounding: first OP is rounded to the nearest integer in the
2091 direction given by the function name, then this nearest integer
2092 (if not representable) is rounded in the given direction RND. For
2093 example, `mpfr_rint_round' with rounding to nearest and a precision
2094 of two bits rounds 6.5 to 7 (halfway cases away from zero), then 7
2095 is rounded to 8 by the round-even rule, despite the fact that 6 is
2096 also representable on two bits, and is closer to 6.5 than 8.
2098 -- Function: int mpfr_frac (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t RND)
2099 Set ROP to the fractional part of OP, having the same sign as OP,
2100 rounded in the direction RND (unlike in `mpfr_rint', RND affects
2101 only how the exact fractional part is rounded, not how the
2102 fractional part is generated).
2104 -- Function: int mpfr_modf (mpfr_t IOP, mpfr_t FOP, mpfr_t OP,
2106 Set simultaneously IOP to the integral part of OP and FOP to the
2107 fractional part of OP, rounded in the direction RND with the
2108 corresponding precision of IOP and FOP (equivalent to
2109 `mpfr_trunc(IOP, OP, RND)' and `mpfr_frac(FOP, OP, RND)'). The
2110 variables IOP and FOP must be different. Return 0 iff both results
2111 are exact (see `mpfr_sin_cos' for a more detailed description of
2114 -- Function: int mpfr_fmod (mpfr_t R, mpfr_t X, mpfr_t Y, mpfr_rnd_t
2116 -- Function: int mpfr_remainder (mpfr_t R, mpfr_t X, mpfr_t Y,
2118 -- Function: int mpfr_remquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y,
2120 Set R to the value of X - NY, rounded according to the direction
2121 RND, where N is the integer quotient of X divided by Y, defined as
2122 follows: N is rounded toward zero for `mpfr_fmod', and to the
2123 nearest integer (ties rounded to even) for `mpfr_remainder' and
2126 Special values are handled as described in Section F.9.7.1 of the
2127 ISO C99 standard: If X is infinite or Y is zero, R is NaN. If Y
2128 is infinite and X is finite, R is X rounded to the precision of R.
2129 If R is zero, it has the sign of X. The return value is the
2130 ternary value corresponding to R.
2132 Additionally, `mpfr_remquo' stores the low significant bits from
2133 the quotient N in *Q (more precisely the number of bits in a
2134 `long' minus one), with the sign of X divided by Y (except if
2135 those low bits are all zero, in which case zero is returned).
2136 Note that X may be so large in magnitude relative to Y that an
2137 exact representation of the quotient is not practical. The
2138 `mpfr_remainder' and `mpfr_remquo' functions are useful for
2139 additive argument reduction.
2141 -- Function: int mpfr_integer_p (mpfr_t OP)
2142 Return non-zero iff OP is an integer.
2145 File: mpfr.info, Node: Rounding Related Functions, Next: Miscellaneous Functions, Prev: Integer Related Functions, Up: MPFR Interface
2147 5.11 Rounding Related Functions
2148 ===============================
2150 -- Function: void mpfr_set_default_rounding_mode (mpfr_rnd_t RND)
2151 Set the default rounding mode to RND. The default rounding mode
2152 is to nearest initially.
2154 -- Function: mpfr_rnd_t mpfr_get_default_rounding_mode (void)
2155 Get the default rounding mode.
2157 -- Function: int mpfr_prec_round (mpfr_t X, mpfr_prec_t PREC,
2159 Round X according to RND with precision PREC, which must be an
2160 integer between `MPFR_PREC_MIN' and `MPFR_PREC_MAX' (otherwise the
2161 behavior is undefined). If PREC is greater or equal to the
2162 precision of X, then new space is allocated for the significand,
2163 and it is filled with zeros. Otherwise, the significand is
2164 rounded to precision PREC with the given direction. In both cases,
2165 the precision of X is changed to PREC.
2167 Here is an example of how to use `mpfr_prec_round' to implement
2168 Newton's algorithm to compute the inverse of A, assuming X is
2169 already an approximation to N bits:
2170 mpfr_set_prec (t, 2 * n);
2171 mpfr_set (t, a, MPFR_RNDN); /* round a to 2n bits */
2172 mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to 2n bits */
2173 mpfr_ui_sub (t, 1, t, MPFR_RNDN); /* high n bits cancel with 1 */
2174 mpfr_prec_round (t, n, MPFR_RNDN); /* t is correct to n bits */
2175 mpfr_mul (t, t, x, MPFR_RNDN); /* t is correct to n bits */
2176 mpfr_prec_round (x, 2 * n, MPFR_RNDN); /* exact */
2177 mpfr_add (x, x, t, MPFR_RNDN); /* x is correct to 2n bits */
2179 -- Function: int mpfr_can_round (mpfr_t B, mpfr_exp_t ERR, mpfr_rnd_t
2180 RND1, mpfr_rnd_t RND2, mpfr_prec_t PREC)
2181 Assuming B is an approximation of an unknown number X in the
2182 direction RND1 with error at most two to the power E(b)-ERR where
2183 E(b) is the exponent of B, return a non-zero value if one is able
2184 to round correctly X to precision PREC with the direction RND2,
2185 and 0 otherwise (including for NaN and Inf). This function *does
2186 not modify* its arguments.
2188 If RND1 is `MPFR_RNDN', then the sign of the error is unknown, but
2189 its absolute value is the same, so that the possible range is
2190 twice as large as with a directed rounding for RND1.
2192 Note: if one wants to also determine the correct *note ternary
2193 value:: when rounding B to precision PREC with rounding mode RND,
2194 a useful trick is the following: if (mpfr_can_round (b, err, MPFR_RNDN, MPFR_RNDZ, prec + (rnd == MPFR_RNDN)))
2196 Indeed, if RND is `MPFR_RNDN', this will check if one can round
2197 to PREC+1 bits with a directed rounding: if so, one can surely
2198 round to nearest to PREC bits, and in addition one can determine
2199 the correct ternary value, which would not be the case when B is
2200 near from a value exactly representable on PREC bits.
2202 -- Function: mpfr_prec_t mpfr_min_prec (mpfr_t X)
2203 Return the minimal number of bits required to store the
2204 significand of X, and 0 for special values, including 0. (Warning:
2205 the returned value can be less than `MPFR_PREC_MIN'.)
2207 The function name is subject to change.
2209 -- Function: const char * mpfr_print_rnd_mode (mpfr_rnd_t RND)
2210 Return a string ("MPFR_RNDD", "MPFR_RNDU", "MPFR_RNDN",
2211 "MPFR_RNDZ", "MPFR_RNDA") corresponding to the rounding mode RND,
2212 or a null pointer if RND is an invalid rounding mode.
2215 File: mpfr.info, Node: Miscellaneous Functions, Next: Exception Related Functions, Prev: Rounding Related Functions, Up: MPFR Interface
2217 5.12 Miscellaneous Functions
2218 ============================
2220 -- Function: void mpfr_nexttoward (mpfr_t X, mpfr_t Y)
2221 If X or Y is NaN, set X to NaN. If X and Y are equal, X is
2222 unchanged. Otherwise, if X is different from Y, replace X by the
2223 next floating-point number (with the precision of X and the
2224 current exponent range) in the direction of Y (the infinite values
2225 are seen as the smallest and largest floating-point numbers). If
2226 the result is zero, it keeps the same sign. No underflow or
2227 overflow is generated.
2229 -- Function: void mpfr_nextabove (mpfr_t X)
2230 -- Function: void mpfr_nextbelow (mpfr_t X)
2231 Equivalent to `mpfr_nexttoward' where Y is plus infinity (resp.
2234 -- Function: int mpfr_min (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
2236 -- Function: int mpfr_max (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
2238 Set ROP to the minimum (resp. maximum) of OP1 and OP2. If OP1 and
2239 OP2 are both NaN, then ROP is set to NaN. If OP1 or OP2 is NaN,
2240 then ROP is set to the numeric value. If OP1 and OP2 are zeros of
2241 different signs, then ROP is set to -0 (resp. +0).
2243 -- Function: int mpfr_urandomb (mpfr_t ROP, gmp_randstate_t STATE)
2244 Generate a uniformly distributed random float in the interval 0 <=
2245 ROP < 1. More precisely, the number can be seen as a float with a
2246 random non-normalized significand and exponent 0, which is then
2247 normalized (thus if E denotes the exponent after normalization,
2248 then the least -E significant bits of the significand are always
2251 Return 0, unless the exponent is not in the current exponent
2252 range, in which case ROP is set to NaN and a non-zero value is
2253 returned (this should never happen in practice, except in very
2254 specific cases). The second argument is a `gmp_randstate_t'
2255 structure which should be created using the GMP `gmp_randinit'
2256 function (see the GMP manual).
2258 Note: for a given version of MPFR, the returned value of ROP and
2259 the new value of STATE (which controls further random values) do
2260 not depend on the machine word size.
2262 -- Function: int mpfr_urandom (mpfr_t ROP, gmp_randstate_t STATE,
2264 Generate a uniformly distributed random float. The floating-point
2265 number ROP can be seen as if a random real number is generated
2266 according to the continuous uniform distribution on the interval
2267 [0, 1] and then rounded in the direction RND.
2269 The second argument is a `gmp_randstate_t' structure which should
2270 be created using the GMP `gmp_randinit' function (see the GMP
2273 Note: the note for `mpfr_urandomb' holds too. In addition, the
2274 exponent range and the rounding mode might have a side effect on
2275 the next random state.
2277 -- Function: int mpfr_grandom (mpfr_t ROP1, mpfr_t ROP2,
2278 gmp_randstate_t STATE, mpfr_rnd_t RND)
2279 Generate two random floats according to a standard normal gaussian
2280 distribution. If ROP2 is a null pointer, then only one value is
2281 generated and stored in ROP1.
2283 The floating-point number ROP1 (and ROP2) can be seen as if a
2284 random real number were generated according to the standard normal
2285 gaussian distribution and then rounded in the direction RND.
2287 The third argument is a `gmp_randstate_t' structure, which should
2288 be created using the GMP `gmp_randinit' function (see the GMP
2291 The combination of the ternary values is returned like with
2292 `mpfr_sin_cos'. If ROP2 is a null pointer, the second ternary
2293 value is assumed to be 0 (note that the encoding of the only
2294 ternary value is not the same as the usual encoding for functions
2295 that return only one result). Otherwise the ternary value of a
2296 random number is always non-zero.
2298 Note: the note for `mpfr_urandomb' holds too. In addition, the
2299 exponent range and the rounding mode might have a side effect on
2300 the next random state.
2302 -- Function: mpfr_exp_t mpfr_get_exp (mpfr_t X)
2303 Return the exponent of X, assuming that X is a non-zero ordinary
2304 number and the significand is considered in [1/2,1). The behavior
2305 for NaN, infinity or zero is undefined.
2307 -- Function: int mpfr_set_exp (mpfr_t X, mpfr_exp_t E)
2308 Set the exponent of X if E is in the current exponent range, and
2309 return 0 (even if X is not a non-zero ordinary number); otherwise,
2310 return a non-zero value. The significand is assumed to be in
2313 -- Function: int mpfr_signbit (mpfr_t OP)
2314 Return a non-zero value iff OP has its sign bit set (i.e., if it is
2315 negative, -0, or a NaN whose representation has its sign bit set).
2317 -- Function: int mpfr_setsign (mpfr_t ROP, mpfr_t OP, int S,
2319 Set the value of ROP from OP, rounded toward the given direction
2320 RND, then set (resp. clear) its sign bit if S is non-zero (resp.
2321 zero), even when OP is a NaN.
2323 -- Function: int mpfr_copysign (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
2325 Set the value of ROP from OP1, rounded toward the given direction
2326 RND, then set its sign bit to that of OP2 (even when OP1 or OP2 is
2327 a NaN). This function is equivalent to `mpfr_setsign (ROP, OP1,
2328 mpfr_signbit (OP2), RND)'.
2330 -- Function: const char * mpfr_get_version (void)
2331 Return the MPFR version, as a null-terminated string.
2333 -- Macro: MPFR_VERSION
2334 -- Macro: MPFR_VERSION_MAJOR
2335 -- Macro: MPFR_VERSION_MINOR
2336 -- Macro: MPFR_VERSION_PATCHLEVEL
2337 -- Macro: MPFR_VERSION_STRING
2338 `MPFR_VERSION' is the version of MPFR as a preprocessing constant.
2339 `MPFR_VERSION_MAJOR', `MPFR_VERSION_MINOR' and
2340 `MPFR_VERSION_PATCHLEVEL' are respectively the major, minor and
2341 patch level of MPFR version, as preprocessing constants.
2342 `MPFR_VERSION_STRING' is the version (with an optional suffix, used
2343 in development and pre-release versions) as a string constant,
2344 which can be compared to the result of `mpfr_get_version' to check
2345 at run time the header file and library used match:
2346 if (strcmp (mpfr_get_version (), MPFR_VERSION_STRING))
2347 fprintf (stderr, "Warning: header and library do not match\n");
2348 Note: Obtaining different strings is not necessarily an error, as
2349 in general, a program compiled with some old MPFR version can be
2350 dynamically linked with a newer MPFR library version (if allowed
2351 by the library versioning system).
2353 -- Macro: long MPFR_VERSION_NUM (MAJOR, MINOR, PATCHLEVEL)
2354 Create an integer in the same format as used by `MPFR_VERSION'
2355 from the given MAJOR, MINOR and PATCHLEVEL. Here is an example of
2356 how to check the MPFR version at compile time:
2357 #if (!defined(MPFR_VERSION) || (MPFR_VERSION<MPFR_VERSION_NUM(3,0,0)))
2358 # error "Wrong MPFR version."
2361 -- Function: const char * mpfr_get_patches (void)
2362 Return a null-terminated string containing the ids of the patches
2363 applied to the MPFR library (contents of the `PATCHES' file),
2364 separated by spaces. Note: If the program has been compiled with
2365 an older MPFR version and is dynamically linked with a new MPFR
2366 library version, the identifiers of the patches applied to the old
2367 (compile-time) MPFR version are not available (however this
2368 information should not have much interest in general).
2370 -- Function: int mpfr_buildopt_tls_p (void)
2371 Return a non-zero value if MPFR was compiled as thread safe using
2372 compiler-level Thread Local Storage (that is, MPFR was built with
2373 the `--enable-thread-safe' configure option, see `INSTALL' file),
2374 return zero otherwise.
2376 -- Function: int mpfr_buildopt_decimal_p (void)
2377 Return a non-zero value if MPFR was compiled with decimal float
2378 support (that is, MPFR was built with the `--enable-decimal-float'
2379 configure option), return zero otherwise.
2381 -- Function: int mpfr_buildopt_gmpinternals_p (void)
2382 Return a non-zero value if MPFR was compiled with GMP internals
2383 (that is, MPFR was built with either `--with-gmp-build' or
2384 `--enable-gmp-internals' configure option), return zero otherwise.
2386 -- Function: const char * mpfr_buildopt_tune_case (void)
2387 Return a string saying which thresholds file has been used at
2388 compile time. This file is normally selected from the processor
2392 File: mpfr.info, Node: Exception Related Functions, Next: Compatibility with MPF, Prev: Miscellaneous Functions, Up: MPFR Interface
2394 5.13 Exception Related Functions
2395 ================================
2397 -- Function: mpfr_exp_t mpfr_get_emin (void)
2398 -- Function: mpfr_exp_t mpfr_get_emax (void)
2399 Return the (current) smallest and largest exponents allowed for a
2400 floating-point variable. The smallest positive value of a
2401 floating-point variable is one half times 2 raised to the smallest
2402 exponent and the largest value has the form (1 - epsilon) times 2
2403 raised to the largest exponent, where epsilon depends on the
2404 precision of the considered variable.
2406 -- Function: int mpfr_set_emin (mpfr_exp_t EXP)
2407 -- Function: int mpfr_set_emax (mpfr_exp_t EXP)
2408 Set the smallest and largest exponents allowed for a
2409 floating-point variable. Return a non-zero value when EXP is not
2410 in the range accepted by the implementation (in that case the
2411 smallest or largest exponent is not changed), and zero otherwise.
2412 If the user changes the exponent range, it is her/his
2413 responsibility to check that all current floating-point variables
2414 are in the new allowed range (for example using
2415 `mpfr_check_range'), otherwise the subsequent behavior will be
2416 undefined, in the sense of the ISO C standard.
2418 -- Function: mpfr_exp_t mpfr_get_emin_min (void)
2419 -- Function: mpfr_exp_t mpfr_get_emin_max (void)
2420 -- Function: mpfr_exp_t mpfr_get_emax_min (void)
2421 -- Function: mpfr_exp_t mpfr_get_emax_max (void)
2422 Return the minimum and maximum of the exponents allowed for
2423 `mpfr_set_emin' and `mpfr_set_emax' respectively. These values
2424 are implementation dependent, thus a program using
2425 `mpfr_set_emax(mpfr_get_emax_max())' or
2426 `mpfr_set_emin(mpfr_get_emin_min())' may not be portable.
2428 -- Function: int mpfr_check_range (mpfr_t X, int T, mpfr_rnd_t RND)
2429 This function assumes that X is the correctly-rounded value of some
2430 real value Y in the direction RND and some extended exponent
2431 range, and that T is the corresponding *note ternary value::. For
2432 example, one performed `t = mpfr_log (x, u, rnd)', and Y is the
2433 exact logarithm of U. Thus T is negative if X is smaller than Y,
2434 positive if X is larger than Y, and zero if X equals Y. This
2435 function modifies X if needed to be in the current range of
2436 acceptable values: It generates an underflow or an overflow if the
2437 exponent of X is outside the current allowed range; the value of T
2438 may be used to avoid a double rounding. This function returns zero
2439 if the new value of X equals the exact one Y, a positive value if
2440 that new value is larger than Y, and a negative value if it is
2441 smaller than Y. Note that unlike most functions, the new result X
2442 is compared to the (unknown) exact one Y, not the input value X,
2443 i.e., the ternary value is propagated.
2445 Note: If X is an infinity and T is different from zero (i.e., if
2446 the rounded result is an inexact infinity), then the overflow flag
2447 is set. This is useful because `mpfr_check_range' is typically
2448 called (at least in MPFR functions) after restoring the flags that
2449 could have been set due to internal computations.
2451 -- Function: int mpfr_subnormalize (mpfr_t X, int T, mpfr_rnd_t RND)
2452 This function rounds X emulating subnormal number arithmetic: if X
2453 is outside the subnormal exponent range, it just propagates the
2454 *note ternary value:: T; otherwise, it rounds X to precision
2455 `EXP(x)-emin+1' according to rounding mode RND and previous
2456 ternary value T, avoiding double rounding problems. More
2457 precisely in the subnormal domain, denoting by E the value of
2458 `emin', X is rounded in fixed-point arithmetic to an integer
2459 multiple of two to the power E-1; as a consequence, 1.5 multiplied
2460 by two to the power E-1 when T is zero is rounded to two to the
2461 power E with rounding to nearest.
2463 `PREC(x)' is not modified by this function. RND and T must be the
2464 rounding mode and the returned ternary value used when computing X
2465 (as in `mpfr_check_range'). The subnormal exponent range is from
2466 `emin' to `emin+PREC(x)-1'. If the result cannot be represented
2467 in the current exponent range (due to a too small `emax'), the
2468 behavior is undefined. Note that unlike most functions, the
2469 result is compared to the exact one, not the input value X, i.e.,
2470 the ternary value is propagated.
2472 As usual, if the returned ternary value is non zero, the inexact
2473 flag is set. Moreover, if a second rounding occurred (because the
2474 input X was in the subnormal range), the underflow flag is set.
2476 This is an example of how to emulate binary double IEEE 754
2477 arithmetic (binary64 in IEEE 754-2008) using MPFR:
2480 mpfr_t xa, xb; int i; volatile double a, b;
2482 mpfr_set_default_prec (53);
2483 mpfr_set_emin (-1073); mpfr_set_emax (1024);
2485 mpfr_init (xa); mpfr_init (xb);
2487 b = 34.3; mpfr_set_d (xb, b, MPFR_RNDN);
2488 a = 0x1.1235P-1021; mpfr_set_d (xa, a, MPFR_RNDN);
2491 i = mpfr_div (xa, xa, xb, MPFR_RNDN);
2492 i = mpfr_subnormalize (xa, i, MPFR_RNDN); /* new ternary value */
2494 mpfr_clear (xa); mpfr_clear (xb);
2497 Warning: this emulates a double IEEE 754 arithmetic with correct
2498 rounding in the subnormal range, which may not be the case for your
2501 -- Function: void mpfr_clear_underflow (void)
2502 -- Function: void mpfr_clear_overflow (void)
2503 -- Function: void mpfr_clear_divby0 (void)
2504 -- Function: void mpfr_clear_nanflag (void)
2505 -- Function: void mpfr_clear_inexflag (void)
2506 -- Function: void mpfr_clear_erangeflag (void)
2507 Clear the underflow, overflow, divide-by-zero, invalid, inexact
2510 -- Function: void mpfr_set_underflow (void)
2511 -- Function: void mpfr_set_overflow (void)
2512 -- Function: void mpfr_set_divby0 (void)
2513 -- Function: void mpfr_set_nanflag (void)
2514 -- Function: void mpfr_set_inexflag (void)
2515 -- Function: void mpfr_set_erangeflag (void)
2516 Set the underflow, overflow, divide-by-zero, invalid, inexact and
2519 -- Function: void mpfr_clear_flags (void)
2520 Clear all global flags (underflow, overflow, divide-by-zero,
2521 invalid, inexact, _erange_).
2523 -- Function: int mpfr_underflow_p (void)
2524 -- Function: int mpfr_overflow_p (void)
2525 -- Function: int mpfr_divby0_p (void)
2526 -- Function: int mpfr_nanflag_p (void)
2527 -- Function: int mpfr_inexflag_p (void)
2528 -- Function: int mpfr_erangeflag_p (void)
2529 Return the corresponding (underflow, overflow, divide-by-zero,
2530 invalid, inexact, _erange_) flag, which is non-zero iff the flag
2534 File: mpfr.info, Node: Compatibility with MPF, Next: Custom Interface, Prev: Exception Related Functions, Up: MPFR Interface
2536 5.14 Compatibility With MPF
2537 ===========================
2539 A header file `mpf2mpfr.h' is included in the distribution of MPFR for
2540 compatibility with the GNU MP class MPF. By inserting the following
2541 two lines after the `#include <gmp.h>' line,
2543 #include <mpf2mpfr.h>
2544 any program written for MPF can be compiled directly with MPFR without
2545 any changes (except the `gmp_printf' functions will not work for
2546 arguments of type `mpfr_t'). All operations are then performed with
2547 the default MPFR rounding mode, which can be reset with
2548 `mpfr_set_default_rounding_mode'.
2550 Warning: the `mpf_init' and `mpf_init2' functions initialize to
2551 zero, whereas the corresponding MPFR functions initialize to NaN: this
2552 is useful to detect uninitialized values, but is slightly incompatible
2555 -- Function: void mpfr_set_prec_raw (mpfr_t X, mpfr_prec_t PREC)
2556 Reset the precision of X to be *exactly* PREC bits. The only
2557 difference with `mpfr_set_prec' is that PREC is assumed to be
2558 small enough so that the significand fits into the current
2559 allocated memory space for X. Otherwise the behavior is undefined.
2561 -- Function: int mpfr_eq (mpfr_t OP1, mpfr_t OP2, unsigned long int
2563 Return non-zero if OP1 and OP2 are both non-zero ordinary numbers
2564 with the same exponent and the same first OP3 bits, both zero, or
2565 both infinities of the same sign. Return zero otherwise. This
2566 function is defined for compatibility with MPF, we do not recommend
2567 to use it otherwise. Do not use it either if you want to know
2568 whether two numbers are close to each other; for instance,
2569 1.011111 and 1.100000 are regarded as different for any value of
2572 -- Function: void mpfr_reldiff (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2,
2574 Compute the relative difference between OP1 and OP2 and store the
2575 result in ROP. This function does not guarantee the correct
2576 rounding on the relative difference; it just computes
2577 |OP1-OP2|/OP1, using the precision of ROP and the rounding mode
2578 RND for all operations.
2580 -- Function: int mpfr_mul_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long
2581 int OP2, mpfr_rnd_t RND)
2582 -- Function: int mpfr_div_2exp (mpfr_t ROP, mpfr_t OP1, unsigned long
2583 int OP2, mpfr_rnd_t RND)
2584 These functions are identical to `mpfr_mul_2ui' and `mpfr_div_2ui'
2585 respectively. These functions are only kept for compatibility
2586 with MPF, one should prefer `mpfr_mul_2ui' and `mpfr_div_2ui'
2590 File: mpfr.info, Node: Custom Interface, Next: Internals, Prev: Compatibility with MPF, Up: MPFR Interface
2592 5.15 Custom Interface
2593 =====================
2595 Some applications use a stack to handle the memory and their objects.
2596 However, the MPFR memory design is not well suited for such a thing. So
2597 that such applications are able to use MPFR, an auxiliary memory
2598 interface has been created: the Custom Interface.
2600 The following interface allows one to use MPFR in two ways:
2601 * Either directly store a floating-point number as a `mpfr_t' on the
2604 * Either store its own representation on the stack and construct a
2605 new temporary `mpfr_t' each time it is needed.
2606 Nothing has to be done to destroy the floating-point numbers except
2607 garbaging the used memory: all the memory management (allocating,
2608 destroying, garbaging) is left to the application.
2610 Each function in this interface is also implemented as a macro for
2611 efficiency reasons: for example `mpfr_custom_init (s, p)' uses the
2612 macro, while `(mpfr_custom_init) (s, p)' uses the function.
2614 Note 1: MPFR functions may still initialize temporary floating-point
2615 numbers using `mpfr_init' and similar functions. See Custom Allocation
2618 Note 2: MPFR functions may use the cached functions (`mpfr_const_pi'
2619 for example), even if they are not explicitly called. You have to call
2620 `mpfr_free_cache' each time you garbage the memory iff `mpfr_init',
2621 through GMP Custom Allocation, allocates its memory on the application
2624 -- Function: size_t mpfr_custom_get_size (mpfr_prec_t PREC)
2625 Return the needed size in bytes to store the significand of a
2626 floating-point number of precision PREC.
2628 -- Function: void mpfr_custom_init (void *SIGNIFICAND, mpfr_prec_t
2630 Initialize a significand of precision PREC, where SIGNIFICAND must
2631 be an area of `mpfr_custom_get_size (prec)' bytes at least and be
2632 suitably aligned for an array of `mp_limb_t' (GMP type, *note
2635 -- Function: void mpfr_custom_init_set (mpfr_t X, int KIND, mpfr_exp_t
2636 EXP, mpfr_prec_t PREC, void *SIGNIFICAND)
2637 Perform a dummy initialization of a `mpfr_t' and set it to:
2638 * if `ABS(kind) == MPFR_NAN_KIND', X is set to NaN;
2640 * if `ABS(kind) == MPFR_INF_KIND', X is set to the infinity of
2643 * if `ABS(kind) == MPFR_ZERO_KIND', X is set to the zero of
2646 * if `ABS(kind) == MPFR_REGULAR_KIND', X is set to a regular
2647 number: `x = sign(kind)*significand*2^exp'.
2648 In all cases, it uses SIGNIFICAND directly for further computing
2649 involving X. It will not allocate anything. A floating-point
2650 number initialized with this function cannot be resized using
2651 `mpfr_set_prec' or `mpfr_prec_round', or cleared using
2652 `mpfr_clear'! The SIGNIFICAND must have been initialized with
2653 `mpfr_custom_init' using the same precision PREC.
2655 -- Function: int mpfr_custom_get_kind (mpfr_t X)
2656 Return the current kind of a `mpfr_t' as created by
2657 `mpfr_custom_init_set'. The behavior of this function for any
2658 `mpfr_t' not initialized with `mpfr_custom_init_set' is undefined.
2660 -- Function: void * mpfr_custom_get_significand (mpfr_t X)
2661 Return a pointer to the significand used by a `mpfr_t' initialized
2662 with `mpfr_custom_init_set'. The behavior of this function for
2663 any `mpfr_t' not initialized with `mpfr_custom_init_set' is
2666 -- Function: mpfr_exp_t mpfr_custom_get_exp (mpfr_t X)
2667 Return the exponent of X, assuming that X is a non-zero ordinary
2668 number. The return value for NaN, Infinity or zero is unspecified
2669 but does not produce any trap. The behavior of this function for
2670 any `mpfr_t' not initialized with `mpfr_custom_init_set' is
2673 -- Function: void mpfr_custom_move (mpfr_t X, void *NEW_POSITION)
2674 Inform MPFR that the significand of X has moved due to a garbage
2675 collect and update its new position to `new_position'. However
2676 the application has to move the significand and the `mpfr_t'
2677 itself. The behavior of this function for any `mpfr_t' not
2678 initialized with `mpfr_custom_init_set' is undefined.
2681 File: mpfr.info, Node: Internals, Prev: Custom Interface, Up: MPFR Interface
2686 A "limb" means the part of a multi-precision number that fits in a
2687 single word. Usually a limb contains 32 or 64 bits. The C data type
2688 for a limb is `mp_limb_t'.
2690 The `mpfr_t' type is internally defined as a one-element array of a
2691 structure, and `mpfr_ptr' is the C data type representing a pointer to
2692 this structure. The `mpfr_t' type consists of four fields:
2694 * The `_mpfr_prec' field is used to store the precision of the
2695 variable (in bits); this is not less than `MPFR_PREC_MIN'.
2697 * The `_mpfr_sign' field is used to store the sign of the variable.
2699 * The `_mpfr_exp' field stores the exponent. An exponent of 0 means
2700 a radix point just above the most significant limb. Non-zero
2701 values n are a multiplier 2^n relative to that point. A NaN, an
2702 infinity and a zero are indicated by special values of the exponent
2705 * Finally, the `_mpfr_d' field is a pointer to the limbs, least
2706 significant limbs stored first. The number of limbs in use is
2707 controlled by `_mpfr_prec', namely
2708 ceil(`_mpfr_prec'/`mp_bits_per_limb'). Non-singular (i.e.,
2709 different from NaN, Infinity or zero) values always have the most
2710 significant bit of the most significant limb set to 1. When the
2711 precision does not correspond to a whole number of limbs, the
2712 excess bits at the low end of the data are zeros.
2716 File: mpfr.info, Node: API Compatibility, Next: Contributors, Prev: MPFR Interface, Up: Top
2721 The goal of this section is to describe some API changes that occurred
2722 from one version of MPFR to another, and how to write code that can be
2723 compiled and run with older MPFR versions. The minimum MPFR version
2724 that is considered here is 2.2.0 (released on 20 September 2005).
2726 API changes can only occur between major or minor versions. Thus the
2727 patchlevel (the third number in the MPFR version) will be ignored in
2728 the following. If a program does not use MPFR internals, changes in
2729 the behavior between two versions differing only by the patchlevel
2730 should only result from what was regarded as a bug or unspecified
2733 As a general rule, a program written for some MPFR version should
2734 work with later versions, possibly except at a new major version, where
2735 some features (described as obsolete for some time) can be removed. In
2736 such a case, a failure should occur during compilation or linking. If
2737 a result becomes incorrect because of such a change, please look at the
2738 various changes below (they are minimal, and most software should be
2739 unaffected), at the FAQ and at the MPFR web page for your version (a
2740 bug could have been introduced and be already fixed); and if the
2741 problem is not mentioned, please send us a bug report (*note Reporting
2744 However, a program written for the current MPFR version (as
2745 documented by this manual) may not necessarily work with previous
2746 versions of MPFR. This section should help developers to write
2749 Note: Information given here may be incomplete. API changes are
2750 also described in the NEWS file (for each version, instead of being
2751 classified like here), together with other changes.
2755 * Type and Macro Changes::
2757 * Changed Functions::
2758 * Removed Functions::
2762 File: mpfr.info, Node: Type and Macro Changes, Next: Added Functions, Prev: API Compatibility, Up: API Compatibility
2764 6.1 Type and Macro Changes
2765 ==========================
2767 The official type for exponent values changed from `mp_exp_t' to
2768 `mpfr_exp_t' in MPFR 3.0. The type `mp_exp_t' will remain available as
2769 it comes from GMP (with a different meaning). These types are
2770 currently the same (`mpfr_exp_t' is defined as `mp_exp_t' with
2771 `typedef'), so that programs can still use `mp_exp_t'; but this may
2772 change in the future. Alternatively, using the following code after
2773 including `mpfr.h' will work with official MPFR versions, as
2774 `mpfr_exp_t' was never defined in MPFR 2.x:
2775 #if MPFR_VERSION_MAJOR < 3
2776 typedef mp_exp_t mpfr_exp_t;
2779 The official types for precision values and for rounding modes
2780 respectively changed from `mp_prec_t' and `mp_rnd_t' to `mpfr_prec_t'
2781 and `mpfr_rnd_t' in MPFR 3.0. This change was actually done a long
2782 time ago in MPFR, at least since MPFR 2.2.0, with the following code in
2785 # define mp_rnd_t mpfr_rnd_t
2788 # define mp_prec_t mpfr_prec_t
2790 This means that it is safe to use the new official types
2791 `mpfr_prec_t' and `mpfr_rnd_t' in your programs. The types `mp_prec_t'
2792 and `mp_rnd_t' (defined in MPFR only) may be removed in the future, as
2793 the prefix `mp_' is reserved by GMP.
2795 The precision type `mpfr_prec_t' (`mp_prec_t') was unsigned before
2796 MPFR 3.0; it is now signed. `MPFR_PREC_MAX' has not changed, though.
2797 Indeed the MPFR code requires that `MPFR_PREC_MAX' be representable in
2798 the exponent type, which may have the same size as `mpfr_prec_t' but
2799 has always been signed. The consequence is that valid code that does
2800 not assume anything about the signedness of `mpfr_prec_t' should work
2801 with past and new MPFR versions. This change was useful as the use of
2802 unsigned types tends to convert signed values to unsigned ones in
2803 expressions due to the usual arithmetic conversions, which can yield
2804 incorrect results if a negative value is converted in such a way.
2805 Warning! A program assuming (intentionally or not) that `mpfr_prec_t'
2806 is signed may be affected by this problem when it is built and run
2809 The rounding modes `GMP_RNDx' were renamed to `MPFR_RNDx' in MPFR
2810 3.0. However the old names `GMP_RNDx' have been kept for compatibility
2811 (this might change in future versions), using:
2812 #define GMP_RNDN MPFR_RNDN
2813 #define GMP_RNDZ MPFR_RNDZ
2814 #define GMP_RNDU MPFR_RNDU
2815 #define GMP_RNDD MPFR_RNDD
2816 The rounding mode "round away from zero" (`MPFR_RNDA') was added in
2817 MPFR 3.0 (however no rounding mode `GMP_RNDA' exists).
2820 File: mpfr.info, Node: Added Functions, Next: Changed Functions, Prev: Type and Macro Changes, Up: API Compatibility
2825 We give here in alphabetical order the functions that were added after
2826 MPFR 2.2, and in which MPFR version.
2828 * `mpfr_add_d' in MPFR 2.4.
2830 * `mpfr_ai' in MPFR 3.0 (incomplete, experimental).
2832 * `mpfr_asprintf' in MPFR 2.4.
2834 * `mpfr_buildopt_decimal_p' and `mpfr_buildopt_tls_p' in MPFR 3.0.
2836 * `mpfr_buildopt_gmpinternals_p' and `mpfr_buildopt_tune_case' in
2839 * `mpfr_clear_divby0' in MPFR 3.1 (new divide-by-zero exception).
2841 * `mpfr_copysign' in MPFR 2.3. Note: MPFR 2.2 had a `mpfr_copysign'
2842 function that was available, but not documented, and with a slight
2843 difference in the semantics (when the second input operand is a
2846 * `mpfr_custom_get_significand' in MPFR 3.0. This function was
2847 named `mpfr_custom_get_mantissa' in previous versions;
2848 `mpfr_custom_get_mantissa' is still available via a macro in
2850 #define mpfr_custom_get_mantissa mpfr_custom_get_significand
2851 Thus code that needs to work with both MPFR 2.x and MPFR 3.x should
2852 use `mpfr_custom_get_mantissa'.
2854 * `mpfr_d_div' and `mpfr_d_sub' in MPFR 2.4.
2856 * `mpfr_digamma' in MPFR 3.0.
2858 * `mpfr_divby0_p' in MPFR 3.1 (new divide-by-zero exception).
2860 * `mpfr_div_d' in MPFR 2.4.
2862 * `mpfr_fmod' in MPFR 2.4.
2864 * `mpfr_fms' in MPFR 2.3.
2866 * `mpfr_fprintf' in MPFR 2.4.
2868 * `mpfr_frexp' in MPFR 3.1.
2870 * `mpfr_get_flt' in MPFR 3.0.
2872 * `mpfr_get_patches' in MPFR 2.3.
2874 * `mpfr_get_z_2exp' in MPFR 3.0. This function was named
2875 `mpfr_get_z_exp' in previous versions; `mpfr_get_z_exp' is still
2876 available via a macro in `mpfr.h':
2877 #define mpfr_get_z_exp mpfr_get_z_2exp
2878 Thus code that needs to work with both MPFR 2.x and MPFR 3.x should
2879 use `mpfr_get_z_exp'.
2881 * `mpfr_grandom' in MPFR 3.1.
2883 * `mpfr_j0', `mpfr_j1' and `mpfr_jn' in MPFR 2.3.
2885 * `mpfr_lgamma' in MPFR 2.3.
2887 * `mpfr_li2' in MPFR 2.4.
2889 * `mpfr_min_prec' in MPFR 3.0.
2891 * `mpfr_modf' in MPFR 2.4.
2893 * `mpfr_mul_d' in MPFR 2.4.
2895 * `mpfr_printf' in MPFR 2.4.
2897 * `mpfr_rec_sqrt' in MPFR 2.4.
2899 * `mpfr_regular_p' in MPFR 3.0.
2901 * `mpfr_remainder' and `mpfr_remquo' in MPFR 2.3.
2903 * `mpfr_set_divby0' in MPFR 3.1 (new divide-by-zero exception).
2905 * `mpfr_set_flt' in MPFR 3.0.
2907 * `mpfr_set_z_2exp' in MPFR 3.0.
2909 * `mpfr_set_zero' in MPFR 3.0.
2911 * `mpfr_setsign' in MPFR 2.3.
2913 * `mpfr_signbit' in MPFR 2.3.
2915 * `mpfr_sinh_cosh' in MPFR 2.4.
2917 * `mpfr_snprintf' and `mpfr_sprintf' in MPFR 2.4.
2919 * `mpfr_sub_d' in MPFR 2.4.
2921 * `mpfr_urandom' in MPFR 3.0.
2923 * `mpfr_vasprintf', `mpfr_vfprintf', `mpfr_vprintf',
2924 `mpfr_vsprintf' and `mpfr_vsnprintf' in MPFR 2.4.
2926 * `mpfr_y0', `mpfr_y1' and `mpfr_yn' in MPFR 2.3.
2928 * `mpfr_z_sub' in MPFR 3.1.
2932 File: mpfr.info, Node: Changed Functions, Next: Removed Functions, Prev: Added Functions, Up: API Compatibility
2934 6.3 Changed Functions
2935 =====================
2937 The following functions have changed after MPFR 2.2. Changes can affect
2938 the behavior of code written for some MPFR version when built and run
2939 against another MPFR version (older or newer), as described below.
2941 * `mpfr_check_range' changed in MPFR 2.3.2 and MPFR 2.4. If the
2942 value is an inexact infinity, the overflow flag is now set (in
2943 case it was lost), while it was previously left unchanged. This
2944 is really what is expected in practice (and what the MPFR code was
2945 expecting), so that the previous behavior was regarded as a bug.
2946 Hence the change in MPFR 2.3.2.
2948 * `mpfr_get_f' changed in MPFR 3.0. This function was returning
2949 zero, except for NaN and Inf, which do not exist in MPF. The
2950 _erange_ flag is now set in these cases, and `mpfr_get_f' now
2951 returns the usual ternary value.
2953 * `mpfr_get_si', `mpfr_get_sj', `mpfr_get_ui' and `mpfr_get_uj'
2954 changed in MPFR 3.0. In previous MPFR versions, the cases where
2955 the _erange_ flag is set were unspecified.
2957 * `mpfr_get_z' changed in MPFR 3.0. The return type was `void'; it
2958 is now `int', and the usual ternary value is returned. Thus
2959 programs that need to work with both MPFR 2.x and 3.x must not use
2960 the return value. Even in this case, C code using `mpfr_get_z' as
2961 the second or third term of a conditional operator may also be
2962 affected. For instance, the following is correct with MPFR 3.0,
2963 but not with MPFR 2.x:
2964 bool ? mpfr_get_z(...) : mpfr_add(...);
2965 On the other hand, the following is correct with MPFR 2.x, but not
2967 bool ? mpfr_get_z(...) : (void) mpfr_add(...);
2968 Portable code should cast `mpfr_get_z(...)' to `void' to use the
2969 type `void' for both terms of the conditional operator, as in:
2970 bool ? (void) mpfr_get_z(...) : (void) mpfr_add(...);
2971 Alternatively, `if ... else' can be used instead of the
2972 conditional operator.
2974 Moreover the cases where the _erange_ flag is set were unspecified
2977 * `mpfr_get_z_exp' changed in MPFR 3.0. In previous MPFR versions,
2978 the cases where the _erange_ flag is set were unspecified. Note:
2979 this function has been renamed to `mpfr_get_z_2exp' in MPFR 3.0,
2980 but `mpfr_get_z_exp' is still available for compatibility reasons.
2982 * `mpfr_strtofr' changed in MPFR 2.3.1 and MPFR 2.4. This was
2983 actually a bug fix since the code and the documentation did not
2984 match. But both were changed in order to have a more consistent
2985 and useful behavior. The main changes in the code are as follows.
2986 The binary exponent is now accepted even without the `0b' or `0x'
2987 prefix. Data corresponding to NaN can now have an optional sign
2988 (such data were previously invalid).
2990 * `mpfr_strtofr' changed in MPFR 3.0. This function now accepts
2991 bases from 37 to 62 (no changes for the other bases). Note: if an
2992 unsupported base is provided to this function, the behavior is
2993 undefined; more precisely, in MPFR 2.3.1 and later, providing an
2994 unsupported base yields an assertion failure (this behavior may
2995 change in the future).
2997 * `mpfr_subnormalize' changed in MPFR 3.1. This was actually
2998 regarded as a bug fix. The `mpfr_subnormalize' implementation up
2999 to MPFR 3.0.0 did not change the flags. In particular, it did not
3000 follow the generic rule concerning the inexact flag (and no
3001 special behavior was specified). The case of the underflow flag
3002 was more a lack of specification.
3004 * `mpfr_urandom' and `mpfr_urandomb' changed in MPFR 3.1. Their
3005 behavior no longer depends on the platform (assuming this is also
3006 true for GMP's random generator, which is not the case between GMP
3007 4.1 and 4.2 if `gmp_randinit_default' is used). As a consequence,
3008 the returned values can be different between MPFR 3.1 and previous
3009 MPFR versions. Note: as the reproducibility of these functions
3010 was not specified before MPFR 3.1, the MPFR 3.1 behavior is _not_
3011 regarded as backward incompatible with previous versions.
3015 File: mpfr.info, Node: Removed Functions, Next: Other Changes, Prev: Changed Functions, Up: API Compatibility
3017 6.4 Removed Functions
3018 =====================
3020 Functions `mpfr_random' and `mpfr_random2' have been removed in MPFR
3021 3.0 (this only affects old code built against MPFR 3.0 or later). (The
3022 function `mpfr_random' had been deprecated since at least MPFR 2.2.0,
3023 and `mpfr_random2' since MPFR 2.4.0.)
3026 File: mpfr.info, Node: Other Changes, Prev: Removed Functions, Up: API Compatibility
3031 For users of a C++ compiler, the way how the availability of `intmax_t'
3032 is detected has changed in MPFR 3.0. In MPFR 2.x, if a macro
3033 `INTMAX_C' or `UINTMAX_C' was defined (e.g. when the
3034 `__STDC_CONSTANT_MACROS' macro had been defined before `<stdint.h>' or
3035 `<inttypes.h>' has been included), `intmax_t' was assumed to be defined.
3036 However this was not always the case (more precisely, `intmax_t' can be
3037 defined only in the namespace `std', as with Boost), so that
3038 compilations could fail. Thus the check for `INTMAX_C' or `UINTMAX_C'
3039 is now disabled for C++ compilers, with the following consequences:
3041 * Programs written for MPFR 2.x that need `intmax_t' may no longer
3042 be compiled against MPFR 3.0: a `#define MPFR_USE_INTMAX_T' may be
3043 necessary before `mpfr.h' is included.
3045 * The compilation of programs that work with MPFR 3.0 may fail with
3046 MPFR 2.x due to the problem described above. Workarounds are
3047 possible, such as defining `intmax_t' and `uintmax_t' in the global
3048 namespace, though this is not clean.
3051 The divide-by-zero exception is new in MPFR 3.1. However it should
3052 not introduce incompatible changes for programs that strictly follow
3053 the MPFR API since the exception can only be seen via new functions.
3055 As of MPFR 3.1, the `mpfr.h' header can be included several times,
3056 while still supporting optional functions (*note Headers and
3060 File: mpfr.info, Node: Contributors, Next: References, Prev: API Compatibility, Up: Top
3065 The main developers of MPFR are Guillaume Hanrot, Vincent Lefèvre,
3066 Patrick Pélissier, Philippe Théveny and Paul Zimmermann.
3068 Sylvie Boldo from ENS-Lyon, France, contributed the functions
3069 `mpfr_agm' and `mpfr_log'. Sylvain Chevillard contributed the
3070 `mpfr_ai' function. David Daney contributed the hyperbolic and inverse
3071 hyperbolic functions, the base-2 exponential, and the factorial
3072 function. Alain Delplanque contributed the new version of the
3073 `mpfr_get_str' function. Mathieu Dutour contributed the functions
3074 `mpfr_acos', `mpfr_asin' and `mpfr_atan', and a previous version of
3075 `mpfr_gamma'. Laurent Fousse contributed the `mpfr_sum' function.
3076 Emmanuel Jeandel, from ENS-Lyon too, contributed the generic
3077 hypergeometric code, as well as the internal function `mpfr_exp3', a
3078 first implementation of the sine and cosine, and improved versions of
3079 `mpfr_const_log2' and `mpfr_const_pi'. Ludovic Meunier helped in the
3080 design of the `mpfr_erf' code. Jean-Luc Rémy contributed the
3081 `mpfr_zeta' code. Fabrice Rouillier contributed the `mpfr_xxx_z' and
3082 `mpfr_xxx_q' functions, and helped to the Microsoft Windows porting.
3083 Damien Stehlé contributed the `mpfr_get_ld_2exp' function.
3085 We would like to thank Jean-Michel Muller and Joris van der Hoeven
3086 for very fruitful discussions at the beginning of that project,
3087 Torbjörn Granlund and Kevin Ryde for their help about design issues,
3088 and Nathalie Revol for her careful reading of a previous version of
3089 this documentation. In particular Kevin Ryde did a tremendous job for
3090 the portability of MPFR in 2002-2004.
3092 The development of the MPFR library would not have been possible
3093 without the continuous support of INRIA, and of the LORIA (Nancy,
3094 France) and LIP (Lyon, France) laboratories. In particular the main
3095 authors were or are members of the PolKA, Spaces, Cacao and Caramel
3096 project-teams at LORIA and of the Arénaire and AriC project-teams at
3097 LIP. This project was started during the Fiable (reliable in French)
3098 action supported by INRIA, and continued during the AOC action. The
3099 development of MPFR was also supported by a grant (202F0659 00 MPN 121)
3100 from the Conseil Régional de Lorraine in 2002, from INRIA by an
3101 "associate engineer" grant (2003-2005), an "opération de développement
3102 logiciel" grant (2007-2009), and the post-doctoral grant of Sylvain
3103 Chevillard in 2009-2010. The MPFR-MPC workshop in June 2012 was partly
3104 supported by the ERC grant ANTICS of Andreas Enge.
3107 File: mpfr.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top
3112 * Richard Brent and Paul Zimmermann, "Modern Computer Arithmetic",
3113 Cambridge University Press (to appear), also available from the
3116 * Laurent Fousse, Guillaume Hanrot, Vincent Lefèvre, Patrick
3117 Pélissier and Paul Zimmermann, "MPFR: A Multiple-Precision Binary
3118 Floating-Point Library With Correct Rounding", ACM Transactions on
3119 Mathematical Software, volume 33, issue 2, article 13, 15 pages,
3120 2007, `http://doi.acm.org/10.1145/1236463.1236468'.
3122 * Torbjörn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic
3123 Library", version 5.0.1, 2010, `http://gmplib.org'.
3125 * IEEE standard for binary floating-point arithmetic, Technical
3126 Report ANSI-IEEE Standard 754-1985, New York, 1985. Approved
3127 March 21, 1985: IEEE Standards Board; approved July 26, 1985:
3128 American National Standards Institute, 18 pages.
3130 * IEEE Standard for Floating-Point Arithmetic, ANSI-IEEE Standard
3131 754-2008, 2008. Revision of ANSI-IEEE Standard 754-1985, approved
3132 June 12, 2008: IEEE Standards Board, 70 pages.
3134 * Donald E. Knuth, "The Art of Computer Programming", vol 2,
3135 "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981.
3137 * Jean-Michel Muller, "Elementary Functions, Algorithms and
3138 Implementation", Birkhäuser, Boston, 2nd edition, 2006.
3140 * Jean-Michel Muller, Nicolas Brisebarre, Florent de Dinechin,
3141 Claude-Pierre Jeannerod, Vincent Lefèvre, Guillaume Melquiond,
3142 Nathalie Revol, Damien Stehlé and Serge Torrès, "Handbook of
3143 Floating-Point Arithmetic", Birkhäuser, Boston, 2009.
3147 File: mpfr.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top
3149 Appendix A GNU Free Documentation License
3150 *****************************************
3152 Version 1.2, November 2002
3154 Copyright (C) 2000,2001,2002 Free Software Foundation, Inc.
3155 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
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3171 works of the document must themselves be free in the same sense.
3172 It complements the GNU General Public License, which is a copyleft
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3362 notice giving the public permission to use the Modified
3363 Version under the terms of this License, in the form shown in
3366 G. Preserve in that license notice the full lists of Invariant
3367 Sections and required Cover Texts given in the Document's
3370 H. Include an unaltered copy of this License.
3372 I. Preserve the section Entitled "History", Preserve its Title,
3373 and add to it an item stating at least the title, year, new
3374 authors, and publisher of the Modified Version as given on
3375 the Title Page. If there is no section Entitled "History" in
3376 the Document, create one stating the title, year, authors,
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3378 then add an item describing the Modified Version as stated in
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3381 J. Preserve the network location, if any, given in the Document
3382 for public access to a Transparent copy of the Document, and
3383 likewise the network locations given in the Document for
3384 previous versions it was based on. These may be placed in
3385 the "History" section. You may omit a network location for a
3386 work that was published at least four years before the
3387 Document itself, or if the original publisher of the version
3388 it refers to gives permission.
3390 K. For any section Entitled "Acknowledgements" or "Dedications",
3391 Preserve the Title of the section, and preserve in the
3392 section all the substance and tone of each of the contributor
3393 acknowledgements and/or dedications given therein.
3395 L. Preserve all the Invariant Sections of the Document,
3396 unaltered in their text and in their titles. Section numbers
3397 or the equivalent are not considered part of the section
3400 M. Delete any section Entitled "Endorsements". Such a section
3401 may not be included in the Modified Version.
3403 N. Do not retitle any existing section to be Entitled
3404 "Endorsements" or to conflict in title with any Invariant
3407 O. Preserve any Warranty Disclaimers.
3409 If the Modified Version includes new front-matter sections or
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3413 add their titles to the list of Invariant Sections in the Modified
3414 Version's license notice. These titles must be distinct from any
3415 other section titles.
3417 You may add a section Entitled "Endorsements", provided it contains
3418 nothing but endorsements of your Modified Version by various
3419 parties--for example, statements of peer review or that the text
3420 has been approved by an organization as the authoritative
3421 definition of a standard.
3423 You may add a passage of up to five words as a Front-Cover Text,
3424 and a passage of up to 25 words as a Back-Cover Text, to the end
3425 of the list of Cover Texts in the Modified Version. Only one
3426 passage of Front-Cover Text and one of Back-Cover Text may be
3427 added by (or through arrangements made by) any one entity. If the
3428 Document already includes a cover text for the same cover,
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3430 you are acting on behalf of, you may not add another; but you may
3431 replace the old one, on explicit permission from the previous
3432 publisher that added the old one.
3434 The author(s) and publisher(s) of the Document do not by this
3435 License give permission to use their names for publicity for or to
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3438 5. COMBINING DOCUMENTS
3440 You may combine the Document with other documents released under
3441 this License, under the terms defined in section 4 above for
3442 modified versions, provided that you include in the combination
3443 all of the Invariant Sections of all of the original documents,
3444 unmodified, and list them all as Invariant Sections of your
3445 combined work in its license notice, and that you preserve all
3446 their Warranty Disclaimers.
3448 The combined work need only contain one copy of this License, and
3449 multiple identical Invariant Sections may be replaced with a single
3450 copy. If there are multiple Invariant Sections with the same name
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3452 by adding at the end of it, in parentheses, the name of the
3453 original author or publisher of that section if known, or else a
3454 unique number. Make the same adjustment to the section titles in
3455 the list of Invariant Sections in the license notice of the
3458 In the combination, you must combine any sections Entitled
3459 "History" in the various original documents, forming one section
3460 Entitled "History"; likewise combine any sections Entitled
3461 "Acknowledgements", and any sections Entitled "Dedications". You
3462 must delete all sections Entitled "Endorsements."
3464 6. COLLECTIONS OF DOCUMENTS
3466 You may make a collection consisting of the Document and other
3467 documents released under this License, and replace the individual
3468 copies of this License in the various documents with a single copy
3469 that is included in the collection, provided that you follow the
3470 rules of this License for verbatim copying of each of the
3471 documents in all other respects.
3473 You may extract a single document from such a collection, and
3474 distribute it individually under this License, provided you insert
3475 a copy of this License into the extracted document, and follow
3476 this License in all other respects regarding verbatim copying of
3479 7. AGGREGATION WITH INDEPENDENT WORKS
3481 A compilation of the Document or its derivatives with other
3482 separate and independent documents or works, in or on a volume of
3483 a storage or distribution medium, is called an "aggregate" if the
3484 copyright resulting from the compilation is not used to limit the
3485 legal rights of the compilation's users beyond what the individual
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3488 are not themselves derivative works of the Document.
3490 If the Cover Text requirement of section 3 is applicable to these
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3494 electronic equivalent of covers if the Document is in electronic
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3496 the whole aggregate.
3500 Translation is considered a kind of modification, so you may
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3503 permission from their copyright holders, but you may include
3504 translations of some or all Invariant Sections in addition to the
3505 original versions of these Invariant Sections. You may include a
3506 translation of this License, and all the license notices in the
3507 Document, and any Warranty Disclaimers, provided that you also
3508 include the original English version of this License and the
3509 original versions of those notices and disclaimers. In case of a
3510 disagreement between the translation and the original version of
3511 this License or a notice or disclaimer, the original version will
3514 If a section in the Document is Entitled "Acknowledgements",
3515 "Dedications", or "History", the requirement (section 4) to
3516 Preserve its Title (section 1) will typically require changing the
3521 You may not copy, modify, sublicense, or distribute the Document
3522 except as expressly provided for under this License. Any other
3523 attempt to copy, modify, sublicense or distribute the Document is
3524 void, and will automatically terminate your rights under this
3525 License. However, parties who have received copies, or rights,
3526 from you under this License will not have their licenses
3527 terminated so long as such parties remain in full compliance.
3529 10. FUTURE REVISIONS OF THIS LICENSE
3531 The Free Software Foundation may publish new, revised versions of
3532 the GNU Free Documentation License from time to time. Such new
3533 versions will be similar in spirit to the present version, but may
3534 differ in detail to address new problems or concerns. See
3535 `http://www.gnu.org/copyleft/'.
3537 Each version of the License is given a distinguishing version
3538 number. If the Document specifies that a particular numbered
3539 version of this License "or any later version" applies to it, you
3540 have the option of following the terms and conditions either of
3541 that specified version or of any later version that has been
3542 published (not as a draft) by the Free Software Foundation. If
3543 the Document does not specify a version number of this License,
3544 you may choose any version ever published (not as a draft) by the
3545 Free Software Foundation.
3547 A.1 ADDENDUM: How to Use This License For Your Documents
3548 ========================================================
3550 To use this License in a document you have written, include a copy of
3551 the License in the document and put the following copyright and license
3552 notices just after the title page:
3554 Copyright (C) YEAR YOUR NAME.
3555 Permission is granted to copy, distribute and/or modify this document
3556 under the terms of the GNU Free Documentation License, Version 1.2
3557 or any later version published by the Free Software Foundation;
3558 with no Invariant Sections, no Front-Cover Texts, and no Back-Cover
3559 Texts. A copy of the license is included in the section entitled ``GNU
3560 Free Documentation License''.
3562 If you have Invariant Sections, Front-Cover Texts and Back-Cover
3563 Texts, replace the "with...Texts." line with this:
3565 with the Invariant Sections being LIST THEIR TITLES, with
3566 the Front-Cover Texts being LIST, and with the Back-Cover Texts
3569 If you have Invariant Sections without Cover Texts, or some other
3570 combination of the three, merge those two alternatives to suit the
3573 If your document contains nontrivial examples of program code, we
3574 recommend releasing these examples in parallel under your choice of
3575 free software license, such as the GNU General Public License, to
3576 permit their use in free software.
3579 File: mpfr.info, Node: Concept Index, Next: Function and Type Index, Prev: GNU Free Documentation License, Up: Top
3587 * Accuracy: MPFR Interface. (line 25)
3588 * Arithmetic functions: Basic Arithmetic Functions.
3590 * Assignment functions: Assignment Functions. (line 3)
3591 * Basic arithmetic functions: Basic Arithmetic Functions.
3593 * Combined initialization and assignment functions: Combined Initialization and Assignment Functions.
3595 * Comparison functions: Comparison Functions. (line 3)
3596 * Compatibility with MPF: Compatibility with MPF.
3598 * Conditions for copying MPFR: Copying. (line 6)
3599 * Conversion functions: Conversion Functions. (line 3)
3600 * Copying conditions: Copying. (line 6)
3601 * Custom interface: Custom Interface. (line 3)
3602 * Exception related functions: Exception Related Functions.
3604 * Float arithmetic functions: Basic Arithmetic Functions.
3606 * Float comparisons functions: Comparison Functions. (line 3)
3607 * Float functions: MPFR Interface. (line 6)
3608 * Float input and output functions: Input and Output Functions.
3610 * Float output functions: Formatted Output Functions.
3612 * Floating-point functions: MPFR Interface. (line 6)
3613 * Floating-point number: Nomenclature and Types.
3615 * GNU Free Documentation License: GNU Free Documentation License.
3617 * I/O functions <1>: Formatted Output Functions.
3619 * I/O functions: Input and Output Functions.
3621 * Initialization functions: Initialization Functions.
3623 * Input functions: Input and Output Functions.
3625 * Installation: Installing MPFR. (line 6)
3626 * Integer related functions: Integer Related Functions.
3628 * Internals: Internals. (line 3)
3629 * intmax_t: Headers and Libraries.
3631 * inttypes.h: Headers and Libraries.
3633 * libmpfr: Headers and Libraries.
3635 * Libraries: Headers and Libraries.
3637 * Libtool: Headers and Libraries.
3639 * Limb: Internals. (line 6)
3640 * Linking: Headers and Libraries.
3642 * Miscellaneous float functions: Miscellaneous Functions.
3644 * mpfr.h: Headers and Libraries.
3646 * Output functions <1>: Formatted Output Functions.
3648 * Output functions: Input and Output Functions.
3650 * Precision <1>: MPFR Interface. (line 17)
3651 * Precision: Nomenclature and Types.
3653 * Reporting bugs: Reporting Bugs. (line 6)
3654 * Rounding mode related functions: Rounding Related Functions.
3656 * Rounding Modes: Nomenclature and Types.
3658 * Special functions: Special Functions. (line 3)
3659 * stdarg.h: Headers and Libraries.
3661 * stdint.h: Headers and Libraries.
3663 * stdio.h: Headers and Libraries.
3665 * Ternary value: Rounding Modes. (line 29)
3666 * uintmax_t: Headers and Libraries.
3670 File: mpfr.info, Node: Function and Type Index, Prev: Concept Index, Up: Top
3672 Function and Type Index
3673 ***********************
3678 * mpfr_abs: Basic Arithmetic Functions.
3680 * mpfr_acos: Special Functions. (line 52)
3681 * mpfr_acosh: Special Functions. (line 136)
3682 * mpfr_add: Basic Arithmetic Functions.
3684 * mpfr_add_d: Basic Arithmetic Functions.
3686 * mpfr_add_q: Basic Arithmetic Functions.
3688 * mpfr_add_si: Basic Arithmetic Functions.
3690 * mpfr_add_ui: Basic Arithmetic Functions.
3692 * mpfr_add_z: Basic Arithmetic Functions.
3694 * mpfr_agm: Special Functions. (line 232)
3695 * mpfr_ai: Special Functions. (line 248)
3696 * mpfr_asin: Special Functions. (line 53)
3697 * mpfr_asinh: Special Functions. (line 137)
3698 * mpfr_asprintf: Formatted Output Functions.
3700 * mpfr_atan: Special Functions. (line 54)
3701 * mpfr_atan2: Special Functions. (line 65)
3702 * mpfr_atanh: Special Functions. (line 138)
3703 * mpfr_buildopt_decimal_p: Miscellaneous Functions.
3705 * mpfr_buildopt_gmpinternals_p: Miscellaneous Functions.
3707 * mpfr_buildopt_tls_p: Miscellaneous Functions.
3709 * mpfr_buildopt_tune_case: Miscellaneous Functions.
3711 * mpfr_can_round: Rounding Related Functions.
3713 * mpfr_cbrt: Basic Arithmetic Functions.
3715 * mpfr_ceil: Integer Related Functions.
3717 * mpfr_check_range: Exception Related Functions.
3719 * mpfr_clear: Initialization Functions.
3721 * mpfr_clear_divby0: Exception Related Functions.
3723 * mpfr_clear_erangeflag: Exception Related Functions.
3725 * mpfr_clear_flags: Exception Related Functions.
3727 * mpfr_clear_inexflag: Exception Related Functions.
3729 * mpfr_clear_nanflag: Exception Related Functions.
3731 * mpfr_clear_overflow: Exception Related Functions.
3733 * mpfr_clear_underflow: Exception Related Functions.
3735 * mpfr_clears: Initialization Functions.
3737 * mpfr_cmp: Comparison Functions.
3739 * mpfr_cmp_d: Comparison Functions.
3741 * mpfr_cmp_f: Comparison Functions.
3743 * mpfr_cmp_ld: Comparison Functions.
3745 * mpfr_cmp_q: Comparison Functions.
3747 * mpfr_cmp_si: Comparison Functions.
3749 * mpfr_cmp_si_2exp: Comparison Functions.
3751 * mpfr_cmp_ui: Comparison Functions.
3753 * mpfr_cmp_ui_2exp: Comparison Functions.
3755 * mpfr_cmp_z: Comparison Functions.
3757 * mpfr_cmpabs: Comparison Functions.
3759 * mpfr_const_catalan: Special Functions. (line 259)
3760 * mpfr_const_euler: Special Functions. (line 258)
3761 * mpfr_const_log2: Special Functions. (line 256)
3762 * mpfr_const_pi: Special Functions. (line 257)
3763 * mpfr_copysign: Miscellaneous Functions.
3765 * mpfr_cos: Special Functions. (line 30)
3766 * mpfr_cosh: Special Functions. (line 115)
3767 * mpfr_cot: Special Functions. (line 48)
3768 * mpfr_coth: Special Functions. (line 132)
3769 * mpfr_csc: Special Functions. (line 47)
3770 * mpfr_csch: Special Functions. (line 131)
3771 * mpfr_custom_get_exp: Custom Interface. (line 78)
3772 * mpfr_custom_get_kind: Custom Interface. (line 67)
3773 * mpfr_custom_get_significand: Custom Interface. (line 72)
3774 * mpfr_custom_get_size: Custom Interface. (line 36)
3775 * mpfr_custom_init: Custom Interface. (line 41)
3776 * mpfr_custom_init_set: Custom Interface. (line 48)
3777 * mpfr_custom_move: Custom Interface. (line 85)
3778 * mpfr_d_div: Basic Arithmetic Functions.
3780 * mpfr_d_sub: Basic Arithmetic Functions.
3782 * MPFR_DECL_INIT: Initialization Functions.
3784 * mpfr_digamma: Special Functions. (line 187)
3785 * mpfr_dim: Basic Arithmetic Functions.
3787 * mpfr_div: Basic Arithmetic Functions.
3789 * mpfr_div_2exp: Compatibility with MPF.
3791 * mpfr_div_2si: Basic Arithmetic Functions.
3793 * mpfr_div_2ui: Basic Arithmetic Functions.
3795 * mpfr_div_d: Basic Arithmetic Functions.
3797 * mpfr_div_q: Basic Arithmetic Functions.
3799 * mpfr_div_si: Basic Arithmetic Functions.
3801 * mpfr_div_ui: Basic Arithmetic Functions.
3803 * mpfr_div_z: Basic Arithmetic Functions.
3805 * mpfr_divby0_p: Exception Related Functions.
3807 * mpfr_eint: Special Functions. (line 154)
3808 * mpfr_eq: Compatibility with MPF.
3810 * mpfr_equal_p: Comparison Functions.
3812 * mpfr_erangeflag_p: Exception Related Functions.
3814 * mpfr_erf: Special Functions. (line 198)
3815 * mpfr_erfc: Special Functions. (line 199)
3816 * mpfr_exp: Special Functions. (line 24)
3817 * mpfr_exp10: Special Functions. (line 26)
3818 * mpfr_exp2: Special Functions. (line 25)
3819 * mpfr_expm1: Special Functions. (line 150)
3820 * mpfr_fac_ui: Special Functions. (line 143)
3821 * mpfr_fits_intmax_p: Conversion Functions.
3823 * mpfr_fits_sint_p: Conversion Functions.
3825 * mpfr_fits_slong_p: Conversion Functions.
3827 * mpfr_fits_sshort_p: Conversion Functions.
3829 * mpfr_fits_uint_p: Conversion Functions.
3831 * mpfr_fits_uintmax_p: Conversion Functions.
3833 * mpfr_fits_ulong_p: Conversion Functions.
3835 * mpfr_fits_ushort_p: Conversion Functions.
3837 * mpfr_floor: Integer Related Functions.
3839 * mpfr_fma: Special Functions. (line 225)
3840 * mpfr_fmod: Integer Related Functions.
3842 * mpfr_fms: Special Functions. (line 227)
3843 * mpfr_fprintf: Formatted Output Functions.
3845 * mpfr_frac: Integer Related Functions.
3847 * mpfr_free_cache: Special Functions. (line 266)
3848 * mpfr_free_str: Conversion Functions.
3850 * mpfr_frexp: Conversion Functions.
3852 * mpfr_gamma: Special Functions. (line 169)
3853 * mpfr_get_d: Conversion Functions.
3855 * mpfr_get_d_2exp: Conversion Functions.
3857 * mpfr_get_decimal64: Conversion Functions.
3859 * mpfr_get_default_prec: Initialization Functions.
3861 * mpfr_get_default_rounding_mode: Rounding Related Functions.
3863 * mpfr_get_emax: Exception Related Functions.
3865 * mpfr_get_emax_max: Exception Related Functions.
3867 * mpfr_get_emax_min: Exception Related Functions.
3869 * mpfr_get_emin: Exception Related Functions.
3871 * mpfr_get_emin_max: Exception Related Functions.
3873 * mpfr_get_emin_min: Exception Related Functions.
3875 * mpfr_get_exp: Miscellaneous Functions.
3877 * mpfr_get_f: Conversion Functions.
3879 * mpfr_get_flt: Conversion Functions.
3881 * mpfr_get_ld: Conversion Functions.
3883 * mpfr_get_ld_2exp: Conversion Functions.
3885 * mpfr_get_patches: Miscellaneous Functions.
3887 * mpfr_get_prec: Initialization Functions.
3889 * mpfr_get_si: Conversion Functions.
3891 * mpfr_get_sj: Conversion Functions.
3893 * mpfr_get_str: Conversion Functions.
3895 * mpfr_get_ui: Conversion Functions.
3897 * mpfr_get_uj: Conversion Functions.
3899 * mpfr_get_version: Miscellaneous Functions.
3901 * mpfr_get_z: Conversion Functions.
3903 * mpfr_get_z_2exp: Conversion Functions.
3905 * mpfr_grandom: Miscellaneous Functions.
3907 * mpfr_greater_p: Comparison Functions.
3909 * mpfr_greaterequal_p: Comparison Functions.
3911 * mpfr_hypot: Special Functions. (line 241)
3912 * mpfr_inexflag_p: Exception Related Functions.
3914 * mpfr_inf_p: Comparison Functions.
3916 * mpfr_init: Initialization Functions.
3918 * mpfr_init2: Initialization Functions.
3920 * mpfr_init_set: Combined Initialization and Assignment Functions.
3922 * mpfr_init_set_d: Combined Initialization and Assignment Functions.
3924 * mpfr_init_set_f: Combined Initialization and Assignment Functions.
3926 * mpfr_init_set_ld: Combined Initialization and Assignment Functions.
3928 * mpfr_init_set_q: Combined Initialization and Assignment Functions.
3930 * mpfr_init_set_si: Combined Initialization and Assignment Functions.
3932 * mpfr_init_set_str: Combined Initialization and Assignment Functions.
3934 * mpfr_init_set_ui: Combined Initialization and Assignment Functions.
3936 * mpfr_init_set_z: Combined Initialization and Assignment Functions.
3938 * mpfr_inits: Initialization Functions.
3940 * mpfr_inits2: Initialization Functions.
3942 * mpfr_inp_str: Input and Output Functions.
3944 * mpfr_integer_p: Integer Related Functions.
3946 * mpfr_j0: Special Functions. (line 203)
3947 * mpfr_j1: Special Functions. (line 204)
3948 * mpfr_jn: Special Functions. (line 206)
3949 * mpfr_less_p: Comparison Functions.
3951 * mpfr_lessequal_p: Comparison Functions.
3953 * mpfr_lessgreater_p: Comparison Functions.
3955 * mpfr_lgamma: Special Functions. (line 179)
3956 * mpfr_li2: Special Functions. (line 164)
3957 * mpfr_lngamma: Special Functions. (line 173)
3958 * mpfr_log: Special Functions. (line 17)
3959 * mpfr_log10: Special Functions. (line 19)
3960 * mpfr_log1p: Special Functions. (line 146)
3961 * mpfr_log2: Special Functions. (line 18)
3962 * mpfr_max: Miscellaneous Functions.
3964 * mpfr_min: Miscellaneous Functions.
3966 * mpfr_min_prec: Rounding Related Functions.
3968 * mpfr_modf: Integer Related Functions.
3970 * mpfr_mul: Basic Arithmetic Functions.
3972 * mpfr_mul_2exp: Compatibility with MPF.
3974 * mpfr_mul_2si: Basic Arithmetic Functions.
3976 * mpfr_mul_2ui: Basic Arithmetic Functions.
3978 * mpfr_mul_d: Basic Arithmetic Functions.
3980 * mpfr_mul_q: Basic Arithmetic Functions.
3982 * mpfr_mul_si: Basic Arithmetic Functions.
3984 * mpfr_mul_ui: Basic Arithmetic Functions.
3986 * mpfr_mul_z: Basic Arithmetic Functions.
3988 * mpfr_nan_p: Comparison Functions.
3990 * mpfr_nanflag_p: Exception Related Functions.
3992 * mpfr_neg: Basic Arithmetic Functions.
3994 * mpfr_nextabove: Miscellaneous Functions.
3996 * mpfr_nextbelow: Miscellaneous Functions.
3998 * mpfr_nexttoward: Miscellaneous Functions.
4000 * mpfr_number_p: Comparison Functions.
4002 * mpfr_out_str: Input and Output Functions.
4004 * mpfr_overflow_p: Exception Related Functions.
4006 * mpfr_pow: Basic Arithmetic Functions.
4008 * mpfr_pow_si: Basic Arithmetic Functions.
4010 * mpfr_pow_ui: Basic Arithmetic Functions.
4012 * mpfr_pow_z: Basic Arithmetic Functions.
4014 * mpfr_prec_round: Rounding Related Functions.
4016 * mpfr_prec_t: Nomenclature and Types.
4018 * mpfr_print_rnd_mode: Rounding Related Functions.
4020 * mpfr_printf: Formatted Output Functions.
4022 * mpfr_rec_sqrt: Basic Arithmetic Functions.
4024 * mpfr_regular_p: Comparison Functions.
4026 * mpfr_reldiff: Compatibility with MPF.
4028 * mpfr_remainder: Integer Related Functions.
4030 * mpfr_remquo: Integer Related Functions.
4032 * mpfr_rint: Integer Related Functions.
4034 * mpfr_rint_ceil: Integer Related Functions.
4036 * mpfr_rint_floor: Integer Related Functions.
4038 * mpfr_rint_round: Integer Related Functions.
4040 * mpfr_rint_trunc: Integer Related Functions.
4042 * mpfr_rnd_t: Nomenclature and Types.
4044 * mpfr_root: Basic Arithmetic Functions.
4046 * mpfr_round: Integer Related Functions.
4048 * mpfr_sec: Special Functions. (line 46)
4049 * mpfr_sech: Special Functions. (line 130)
4050 * mpfr_set: Assignment Functions.
4052 * mpfr_set_d: Assignment Functions.
4054 * mpfr_set_decimal64: Assignment Functions.
4056 * mpfr_set_default_prec: Initialization Functions.
4058 * mpfr_set_default_rounding_mode: Rounding Related Functions.
4060 * mpfr_set_divby0: Exception Related Functions.
4062 * mpfr_set_emax: Exception Related Functions.
4064 * mpfr_set_emin: Exception Related Functions.
4066 * mpfr_set_erangeflag: Exception Related Functions.
4068 * mpfr_set_exp: Miscellaneous Functions.
4070 * mpfr_set_f: Assignment Functions.
4072 * mpfr_set_flt: Assignment Functions.
4074 * mpfr_set_inexflag: Exception Related Functions.
4076 * mpfr_set_inf: Assignment Functions.
4078 * mpfr_set_ld: Assignment Functions.
4080 * mpfr_set_nan: Assignment Functions.
4082 * mpfr_set_nanflag: Exception Related Functions.
4084 * mpfr_set_overflow: Exception Related Functions.
4086 * mpfr_set_prec: Initialization Functions.
4088 * mpfr_set_prec_raw: Compatibility with MPF.
4090 * mpfr_set_q: Assignment Functions.
4092 * mpfr_set_si: Assignment Functions.
4094 * mpfr_set_si_2exp: Assignment Functions.
4096 * mpfr_set_sj: Assignment Functions.
4098 * mpfr_set_sj_2exp: Assignment Functions.
4100 * mpfr_set_str: Assignment Functions.
4102 * mpfr_set_ui: Assignment Functions.
4104 * mpfr_set_ui_2exp: Assignment Functions.
4106 * mpfr_set_uj: Assignment Functions.
4108 * mpfr_set_uj_2exp: Assignment Functions.
4110 * mpfr_set_underflow: Exception Related Functions.
4112 * mpfr_set_z: Assignment Functions.
4114 * mpfr_set_z_2exp: Assignment Functions.
4116 * mpfr_set_zero: Assignment Functions.
4118 * mpfr_setsign: Miscellaneous Functions.
4120 * mpfr_sgn: Comparison Functions.
4122 * mpfr_si_div: Basic Arithmetic Functions.
4124 * mpfr_si_sub: Basic Arithmetic Functions.
4126 * mpfr_signbit: Miscellaneous Functions.
4128 * mpfr_sin: Special Functions. (line 31)
4129 * mpfr_sin_cos: Special Functions. (line 37)
4130 * mpfr_sinh: Special Functions. (line 116)
4131 * mpfr_sinh_cosh: Special Functions. (line 122)
4132 * mpfr_snprintf: Formatted Output Functions.
4134 * mpfr_sprintf: Formatted Output Functions.
4136 * mpfr_sqr: Basic Arithmetic Functions.
4138 * mpfr_sqrt: Basic Arithmetic Functions.
4140 * mpfr_sqrt_ui: Basic Arithmetic Functions.
4142 * mpfr_strtofr: Assignment Functions.
4144 * mpfr_sub: Basic Arithmetic Functions.
4146 * mpfr_sub_d: Basic Arithmetic Functions.
4148 * mpfr_sub_q: Basic Arithmetic Functions.
4150 * mpfr_sub_si: Basic Arithmetic Functions.
4152 * mpfr_sub_ui: Basic Arithmetic Functions.
4154 * mpfr_sub_z: Basic Arithmetic Functions.
4156 * mpfr_subnormalize: Exception Related Functions.
4158 * mpfr_sum: Special Functions. (line 275)
4159 * mpfr_swap: Assignment Functions.
4161 * mpfr_t: Nomenclature and Types.
4163 * mpfr_tan: Special Functions. (line 32)
4164 * mpfr_tanh: Special Functions. (line 117)
4165 * mpfr_trunc: Integer Related Functions.
4167 * mpfr_ui_div: Basic Arithmetic Functions.
4169 * mpfr_ui_pow: Basic Arithmetic Functions.
4171 * mpfr_ui_pow_ui: Basic Arithmetic Functions.
4173 * mpfr_ui_sub: Basic Arithmetic Functions.
4175 * mpfr_underflow_p: Exception Related Functions.
4177 * mpfr_unordered_p: Comparison Functions.
4179 * mpfr_urandom: Miscellaneous Functions.
4181 * mpfr_urandomb: Miscellaneous Functions.
4183 * mpfr_vasprintf: Formatted Output Functions.
4185 * MPFR_VERSION: Miscellaneous Functions.
4187 * MPFR_VERSION_MAJOR: Miscellaneous Functions.
4189 * MPFR_VERSION_MINOR: Miscellaneous Functions.
4191 * MPFR_VERSION_NUM: Miscellaneous Functions.
4193 * MPFR_VERSION_PATCHLEVEL: Miscellaneous Functions.
4195 * MPFR_VERSION_STRING: Miscellaneous Functions.
4197 * mpfr_vfprintf: Formatted Output Functions.
4199 * mpfr_vprintf: Formatted Output Functions.
4201 * mpfr_vsnprintf: Formatted Output Functions.
4203 * mpfr_vsprintf: Formatted Output Functions.
4205 * mpfr_y0: Special Functions. (line 214)
4206 * mpfr_y1: Special Functions. (line 215)
4207 * mpfr_yn: Special Functions. (line 217)
4208 * mpfr_z_sub: Basic Arithmetic Functions.
4210 * mpfr_zero_p: Comparison Functions.
4212 * mpfr_zeta: Special Functions. (line 192)
4213 * mpfr_zeta_ui: Special Functions. (line 194)
4219 Node: Copying
\7f2243
4220 Node: Introduction to MPFR
\7f4003
4221 Node: Installing MPFR
\7f6092
4222 Node: Reporting Bugs
\7f10914
4223 Node: MPFR Basics
\7f12843
4224 Node: Headers and Libraries
\7f13159
4225 Node: Nomenclature and Types
\7f16143
4226 Node: MPFR Variable Conventions
\7f18147
4227 Node: Rounding Modes
\7f19677
4228 Ref: ternary value
\7f20774
4229 Node: Floating-Point Values on Special Numbers
\7f22727
4230 Node: Exceptions
\7f25703
4231 Node: Memory Handling
\7f28855
4232 Node: MPFR Interface
\7f29987
4233 Node: Initialization Functions
\7f32083
4234 Node: Assignment Functions
\7f38997
4235 Node: Combined Initialization and Assignment Functions
\7f47651
4236 Node: Conversion Functions
\7f48944
4237 Node: Basic Arithmetic Functions
\7f57496
4238 Node: Comparison Functions
\7f66504
4239 Node: Special Functions
\7f69986
4240 Node: Input and Output Functions
\7f83739
4241 Node: Formatted Output Functions
\7f85662
4242 Node: Integer Related Functions
\7f94781
4243 Node: Rounding Related Functions
\7f100543
4244 Node: Miscellaneous Functions
\7f104157
4245 Node: Exception Related Functions
\7f112724
4246 Node: Compatibility with MPF
\7f119478
4247 Node: Custom Interface
\7f122166
4248 Node: Internals
\7f126411
4249 Node: API Compatibility
\7f127895
4250 Node: Type and Macro Changes
\7f129825
4251 Node: Added Functions
\7f132546
4252 Node: Changed Functions
\7f135489
4253 Node: Removed Functions
\7f139770
4254 Node: Other Changes
\7f140182
4255 Node: Contributors
\7f141711
4256 Node: References
\7f144285
4257 Node: GNU Free Documentation License
\7f146026
4258 Node: Concept Index
\7f168469
4259 Node: Function and Type Index
\7f174388