1This is mpfr.info, produced by makeinfo version 4.13 from mpfr.texi. 2 3This manual documents how to install and use the Multiple Precision 4Floating-Point Reliable Library, version 3.1.2. 5 6 Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 72001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 82013 Free Software Foundation, Inc. 9 10 Permission is granted to copy, distribute and/or modify this 11document under the terms of the GNU Free Documentation License, Version 121.2 or any later version published by the Free Software Foundation; 13with no Invariant Sections, with no Front-Cover Texts, and with no 14Back-Cover Texts. A copy of the license is included in *note GNU Free 15Documentation License::. 16 17INFO-DIR-SECTION Software libraries 18START-INFO-DIR-ENTRY 19* mpfr: (mpfr). Multiple Precision Floating-Point Reliable Library. 20END-INFO-DIR-ENTRY 21 22 23File: mpfr.info, Node: Top, Next: Copying, Prev: (dir), Up: (dir) 24 25GNU MPFR 26******** 27 28 This manual documents how to install and use the Multiple Precision 29Floating-Point Reliable Library, version 3.1.2. 30 31 Copyright 1991, 1993, 1994, 1995, 1996, 1997, 1998, 1999, 2000, 322001, 2002, 2003, 2004, 2005, 2006, 2007, 2008, 2009, 2010, 2011, 2012, 332013 Free Software Foundation, Inc. 34 35 Permission is granted to copy, distribute and/or modify this 36document under the terms of the GNU Free Documentation License, Version 371.2 or any later version published by the Free Software Foundation; 38with no Invariant Sections, with no Front-Cover Texts, and with no 39Back-Cover Texts. A copy of the license is included in *note GNU Free 40Documentation License::. 41 42 43* Menu: 44 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. 52* Contributors:: 53* References:: 54* GNU Free Documentation License:: 55* Concept Index:: 56* Function and Type Index:: 57 58 59File: mpfr.info, Node: Copying, Next: Introduction to MPFR, Prev: Top, Up: Top 60 61MPFR Copying Conditions 62*********************** 63 64The GNU MPFR library (or MPFR for short) is "free"; this means that 65everyone is free to use it and free to redistribute it on a free basis. 66The library is not in the public domain; it is copyrighted and there 67are restrictions on its distribution, but these restrictions are 68designed to permit everything that a good cooperating citizen would 69want to do. What is not allowed is to try to prevent others from 70further sharing any version of this library that they might get from 71you. 72 73 Specifically, we want to make sure that you have the right to give 74away copies of the library, that you receive source code or else can 75get it if you want it, that you can change this library or use pieces 76of it in new free programs, and that you know you can do these things. 77 78 To make sure that everyone has such rights, we have to forbid you to 79deprive anyone else of these rights. For example, if you distribute 80copies of the GNU MPFR library, you must give the recipients all the 81rights that you have. You must make sure that they, too, receive or 82can get the source code. And you must tell them their rights. 83 84 Also, for our own protection, we must make certain that everyone 85finds out that there is no warranty for the GNU MPFR library. If it is 86modified by someone else and passed on, we want their recipients to 87know that what they have is not what we distributed, so that any 88problems introduced by others will not reflect on our reputation. 89 90 The precise conditions of the license for the GNU MPFR library are 91found in the Lesser General Public License that accompanies the source 92code. See the file COPYING.LESSER. 93 94 95File: mpfr.info, Node: Introduction to MPFR, Next: Installing MPFR, Prev: Copying, Up: Top 96 971 Introduction to MPFR 98********************** 99 100MPFR is a portable library written in C for arbitrary precision 101arithmetic on floating-point numbers. It is based on the GNU MP library. 102It aims to provide a class of floating-point numbers with precise 103semantics. The main characteristics of MPFR, which make it differ from 104most arbitrary precision floating-point software tools, are: 105 106 * 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 108 current processors); 109 110 * the precision in bits can be set _exactly_ to any valid value for 111 each variable (including very small precision); 112 113 * 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. 116 117 In particular, with a precision of 53 bits, MPFR is able to exactly 118reproduce all computations with double-precision machine floating-point 119numbers (e.g., `double' type in C, with a C implementation that 120rigorously follows Annex F of the ISO C99 standard and `FP_CONTRACT' 121pragma set to `OFF') on the four arithmetic operations and the square 122root, except the default exponent range is much wider and subnormal 123numbers are not implemented (but can be emulated). 124 125 This version of MPFR is released under the GNU Lesser General Public 126License, version 3 or any later version. It is permitted to link MPFR 127to most non-free programs, as long as when distributing them the MPFR 128source code and a means to re-link with a modified MPFR library is 129provided. 130 1311.1 How to Use This Manual 132========================== 133 134Everyone should read *note MPFR Basics::. If you need to install the 135library yourself, you need to read *note Installing MPFR::, too. To 136use the library you will need to refer to *note MPFR Interface::. 137 138 The rest of the manual can be used for later reference, although it 139is probably a good idea to glance through it. 140 141 142File: mpfr.info, Node: Installing MPFR, Next: Reporting Bugs, Prev: Introduction to MPFR, Up: Top 143 1442 Installing MPFR 145***************** 146 147The MPFR library is already installed on some GNU/Linux distributions, 148but the development files necessary to the compilation such as `mpfr.h' 149are not always present. To check that MPFR is fully installed on your 150computer, 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 153instance, you can try to compile: 154 155 #include <stdio.h> 156 #include <mpfr.h> 157 int main (void) 158 { 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); 162 return 0; 163 } 164 165with 166 167 cc -o version version.c -lmpfr -lgmp 168 169and if you get errors whose first line looks like 170 171 version.c:2:19: error: mpfr.h: No such file or directory 172 173then MPFR is probably not installed. Running this program will give you 174the MPFR version. 175 176 If MPFR is not installed on your computer, or if you want to install 177a different version, please follow the steps below. 178 1792.1 How to Install 180================== 181 182Here are the steps needed to install the library on Unix systems (more 183details are provided in the `INSTALL' file): 184 185 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 189 utility commands. 190 191 Then, in the MPFR build directory, type the following commands. 192 193 2. `./configure' 194 195 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 200 get error messages. 201 202 3. `make' 203 204 This will compile MPFR, and create a library archive file 205 `libmpfr.a'. On most platforms, a dynamic library will be 206 produced too. 207 208 4. `make check' 209 210 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.) 214 215 5. `make install' 216 217 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'). 224 2252.2 Other `make' Targets 226======================== 227 228There are some other useful make targets: 229 230 * `mpfr.info' or `info' 231 232 Create or update an info version of the manual, in `mpfr.info'. 233 234 This file is already provided in the MPFR archives. 235 236 * `mpfr.pdf' or `pdf' 237 238 Create a PDF version of the manual, in `mpfr.pdf'. 239 240 * `mpfr.dvi' or `dvi' 241 242 Create a DVI version of the manual, in `mpfr.dvi'. 243 244 * `mpfr.ps' or `ps' 245 246 Create a Postscript version of the manual, in `mpfr.ps'. 247 248 * `mpfr.html' or `html' 249 250 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' 253 directory instead. 254 255 * `clean' 256 257 Delete all object files and archive files, but not the 258 configuration files. 259 260 * `distclean' 261 262 Delete all generated files not included in the distribution. 263 264 * `uninstall' 265 266 Delete all files copied by `make install'. 267 2682.3 Build Problems 269================== 270 271In case of problem, please read the `INSTALL' file carefully before 272reporting a bug, in particular section "In case of problem". Some 273problems are due to bad configuration on the user side (not specific to 274MPFR). Problems are also mentioned in the FAQ 275`http://www.mpfr.org/faq.html'. 276 277 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 279web page `http://www.mpfr.org/mpfr-3.1.2/'. 280 2812.4 Getting the Latest Version of MPFR 282====================================== 283 284The latest version of MPFR is available from 285`ftp://ftp.gnu.org/gnu/mpfr/' or `http://www.mpfr.org/'. 286 287 288File: mpfr.info, Node: Reporting Bugs, Next: MPFR Basics, Prev: Installing MPFR, Up: Top 289 2903 Reporting Bugs 291**************** 292 293If you think you have found a bug in the MPFR library, first have a look 294on the MPFR 3.1.2 web page `http://www.mpfr.org/mpfr-3.1.2/' and the 295FAQ `http://www.mpfr.org/faq.html': perhaps this bug is already known, 296in which case you may find there a workaround for it. You might also 297look in the archives of the MPFR mailing-list: 298`https://sympa.inria.fr/sympa/arc/mpfr'. Otherwise, please investigate 299and report it. We have made this library available to you, and it is 300not to ask too much from you, to ask you to report the bugs that you 301find. 302 303 There are a few things you should think about when you put your bug 304report together. 305 306 You have to send us a test case that makes it possible for us to 307reproduce the bug, i.e., a small self-content program, using no other 308library than MPFR. Include instructions on how to run the test case. 309 310 You also have to explain what is wrong; if you get a crash, or if 311the results you get are incorrect and in that case, in what way. 312 313 Please include compiler version information in your bug report. This 314can be extracted using `cc -V' on some machines, or, if you're using 315GCC, `gcc -v'. Also, include the output from `uname -a' and the MPFR 316version (the GMP version may be useful too). If you get a failure 317while running `make' or `make check', please include the `config.log' 318file in your bug report. 319 320 If your bug report is good, we will do our best to help you to get a 321corrected version of the library; if the bug report is poor, we will 322not do anything about it (aside of chiding you to send better bug 323reports). 324 325 Send your bug report to the MPFR mailing-list `mpfr@inria.fr'. 326 327 If you think something in this manual is unclear, or downright 328incorrect, or if the language needs to be improved, please send a note 329to the same address. 330 331 332File: mpfr.info, Node: MPFR Basics, Next: MPFR Interface, Prev: Reporting Bugs, Up: Top 333 3344 MPFR Basics 335************* 336 337* Menu: 338 339* Headers and Libraries:: 340* Nomenclature and Types:: 341* MPFR Variable Conventions:: 342* Rounding Modes:: 343* Floating-Point Values on Special Numbers:: 344* Exceptions:: 345* Memory Handling:: 346 347 348File: mpfr.info, Node: Headers and Libraries, Next: Nomenclature and Types, Prev: MPFR Basics, Up: MPFR Basics 349 3504.1 Headers and Libraries 351========================= 352 353All 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 355should include that file in any program using the MPFR library: 356 357 #include <mpfr.h> 358 359 Note however that prototypes for MPFR functions with `FILE *' 360parameters are provided only if `<stdio.h>' is included too (before 361`mpfr.h'): 362 363 #include <stdio.h> 364 #include <mpfr.h> 365 366 Likewise `<stdarg.h>' (or `<varargs.h>') is required for prototypes 367with `va_list' parameters, such as `mpfr_vprintf'. 368 369 And for any functions using `intmax_t', you must include 370`<stdint.h>' or `<inttypes.h>' before `mpfr.h', to allow `mpfr.h' to 371define prototypes for these functions. Moreover, users of C++ compilers 372under some platforms may need to define `MPFR_USE_INTMAX_T' (and should 373do it for portability) before `mpfr.h' has been included; of course, it 374is possible to do that on the command line, e.g., with 375`-DMPFR_USE_INTMAX_T'. 376 377 Note: If `mpfr.h' and/or `gmp.h' (used by `mpfr.h') are included 378several times (possibly from another header file), `<stdio.h>' and/or 379`<stdarg.h>' (or `<varargs.h>') should be included *before the first 380inclusion* 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 382MPFR functions with `va_list' parameters) anywhere before the last 383inclusion of `mpfr.h'. As a consequence, if your file is a public 384header that includes `mpfr.h', you need to use the latter method. 385 386 When calling a MPFR macro, it is not allowed to have previously 387defined a macro with the same name as some keywords (currently `do', 388`while' and `sizeof'). 389 390 You can avoid the use of MPFR macros encapsulating functions by 391defining the `MPFR_USE_NO_MACRO' macro before `mpfr.h' is included. In 392general this should not be necessary, but this can be useful when 393debugging user code: with some macros, the compiler may emit spurious 394warnings with some warning options, and macros can prevent some 395prototype checking. 396 397 All programs using MPFR must link against both `libmpfr' and 398`libgmp' libraries. On a typical Unix-like system this can be done 399with `-lmpfr -lgmp' (in that order), for example: 400 401 gcc myprogram.c -lmpfr -lgmp 402 403 MPFR is built using Libtool and an application can use that to link 404if desired, *note GNU Libtool: (libtool.info)Top. 405 406 If MPFR has been installed to a non-standard location, then it may be 407necessary to set up environment variables such as `C_INCLUDE_PATH' and 408`LIBRARY_PATH', or use `-I' and `-L' compiler options, in order to 409point to the right directories. For a shared library, it may also be 410necessary 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 412additional information. 413 414 415File: mpfr.info, Node: Nomenclature and Types, Next: MPFR Variable Conventions, Prev: Headers and Libraries, Up: MPFR Basics 416 4174.2 Nomenclature and Types 418========================== 419 420A "floating-point number", or "float" for short, is an arbitrary 421precision significand (also called mantissa) with a limited precision 422exponent. The C data type for such objects is `mpfr_t' (internally 423defined as a one-element array of a structure, and `mpfr_ptr' is the C 424data type representing a pointer to this structure). A floating-point 425number can have three special values: Not-a-Number (NaN) or plus or 426minus Infinity. NaN represents an uninitialized object, the result of 427an invalid operation (like 0 divided by 0), or a value that cannot be 428determined (like +Infinity minus +Infinity). Moreover, like in the IEEE 429754 standard, zero is signed, i.e., there are both +0 and -0; the 430behavior is the same as in the IEEE 754 standard and it is generalized 431to the other functions supported by MPFR. Unless documented otherwise, 432the sign bit of a NaN is unspecified. 433 434The "precision" is the number of bits used to represent the significand 435of 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. 439 440 Warning! MPFR needs to increase the precision internally, in order to 441provide accurate results (and in particular, correct rounding). Do not 442attempt to set the precision to any value near `MPFR_PREC_MAX', 443otherwise MPFR will abort due to an assertion failure. Moreover, you 444may reach some memory limit on your platform, in which case the program 445may abort, crash or have undefined behavior (depending on your C 446implementation). 447 448The "rounding mode" specifies the way to round the result of a 449floating-point operation, in case the exact result can not be 450represented exactly in the destination significand; the corresponding C 451data type is `mpfr_rnd_t'. 452 453 454File: mpfr.info, Node: MPFR Variable Conventions, Next: Rounding Modes, Prev: Nomenclature and Types, Up: MPFR Basics 455 4564.3 MPFR Variable Conventions 457============================= 458 459Before you can assign to an MPFR variable, you need to initialize it by 460calling one of the special initialization functions. When you're done 461with a variable, you need to clear it out, using one of the functions 462for that purpose. A variable should only be initialized once, or at 463least cleared out between each initialization. After a variable has 464been initialized, it may be assigned to any number of times. For 465efficiency reasons, avoid to initialize and clear out a variable in 466loops. Instead, initialize it before entering the loop, and clear it 467out after the loop has exited. You do not need to be concerned about 468allocating additional space for MPFR variables, since any variable has 469a significand of fixed size. Hence unless you change its precision, or 470clear and reinitialize it, a floating-point variable will have the same 471allocated space during all its life. 472 473 As a general rule, all MPFR functions expect output arguments before 474input arguments. This notation is based on an analogy with the 475assignment operator. MPFR allows you to use the same variable for both 476input and output in the same expression. For example, the main 477function for floating-point multiplication, `mpfr_mul', can be used 478like this: `mpfr_mul (x, x, x, rnd)'. This computes the square of X 479with rounding mode `rnd' and puts the result back in X. 480 481 482File: mpfr.info, Node: Rounding Modes, Next: Floating-Point Values on Special Numbers, Prev: MPFR Variable Conventions, Up: MPFR Basics 483 4844.4 Rounding Modes 485================== 486 487The following five rounding modes are supported: 488 489 * `MPFR_RNDN': round to nearest (roundTiesToEven in IEEE 754-2008), 490 491 * `MPFR_RNDZ': round toward zero (roundTowardZero in IEEE 754-2008), 492 493 * `MPFR_RNDU': round toward plus infinity (roundTowardPositive in 494 IEEE 754-2008), 495 496 * `MPFR_RNDD': round toward minus infinity (roundTowardNegative in 497 IEEE 754-2008), 498 499 * `MPFR_RNDA': round away from zero. 500 501 The `round to nearest' mode works as in the IEEE 754 standard: in 502case the number to be rounded lies exactly in the middle of two 503representable numbers, it is rounded to the one with the least 504significant bit set to zero. For example, the number 2.5, which is 505represented by (10.1) in binary, is rounded to (10.0)=2 with a 506precision 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 508Computer Programming (Section 4.2.2). 509 510 Most MPFR functions take as first argument the destination variable, 511as second and following arguments the input variables, as last argument 512a rounding mode, and have a return value of type `int', called the 513"ternary value". The value stored in the destination variable is 514correctly rounded, i.e., MPFR behaves as if it computed the result with 515an infinite precision, then rounded it to the precision of this 516variable. The input variables are regarded as exact (in particular, 517their precision does not affect the result). 518 519 As a consequence, in case of a non-zero real rounded result, the 520error on the result is less or equal to 1/2 ulp (unit in the last 521place) of that result in the rounding to nearest mode, and less than 1 522ulp of that result in the directed rounding modes (a ulp is the weight 523of the least significant represented bit of the result after rounding). 524 525 Unless documented otherwise, functions returning an `int' return a 526ternary value. If the ternary value is zero, it means that the value 527stored in the destination variable is the exact result of the 528corresponding mathematical function. If the ternary value is positive 529(resp. negative), it means the value stored in the destination variable 530is greater (resp. lower) than the exact result. For example with the 531`MPFR_RNDU' rounding mode, the ternary value is usually positive, 532except when the result is exact, in which case it is zero. In the case 533of an infinite result, it is considered as inexact when it was obtained 534by overflow, and exact otherwise. A NaN result (Not-a-Number) always 535corresponds to an exact return value. The opposite of a returned 536ternary value is guaranteed to be representable in an `int'. 537 538 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 541representable in the current exponent range. 542 543 544File: mpfr.info, Node: Floating-Point Values on Special Numbers, Next: Exceptions, Prev: Rounding Modes, Up: MPFR Basics 545 5464.5 Floating-Point Values on Special Numbers 547============================================ 548 549This section specifies the floating-point values (of type `mpfr_t') 550returned by MPFR functions (where by "returned" we mean here the 551modified value of the destination object, which should not be mixed 552with the ternary return value of type `int' of those functions). For 553functions returning several values (like `mpfr_sin_cos'), the rules 554apply to each result separately. 555 556 Functions can have one or several input arguments. An input point is 557a mapping from these input arguments to the set of the MPFR numbers. 558When none of its components are NaN, an input point can also be seen as 559a tuple in the extended real numbers (the set of the real numbers with 560both infinities). 561 562 When the input point is in the domain of the mathematical function, 563the result is rounded as described in Section "Rounding Modes" (but see 564below for the specification of the sign of an exact zero). Otherwise 565the general rules from this section apply unless stated otherwise in 566the description of the MPFR function (*note MPFR Interface::). 567 568 When the input point is not in the domain of the mathematical 569function but is in its closure in the extended real numbers and the 570function can be extended by continuity, the result is the obtained 571limit. Examples: `mpfr_hypot' on (+Inf,0) gives +Inf. But `mpfr_pow' 572cannot be defined on (1,+Inf) using this rule, as one can find 573sequences (X_N,Y_N) such that X_N goes to 1, Y_N goes to +Inf and X_N 574to the Y_N goes to any positive value when N goes to the infinity. 575 576 When the input point is in the closure of the domain of the 577mathematical function and an input argument is +0 (resp. -0), one 578considers the limit when the corresponding argument approaches 0 from 579above (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. 581 582 When the result is equal to 0, its sign is determined by considering 583the limit as if the input point were not in the domain: If one 584approaches 0 from above (resp. below), the result is +0 (resp. -0); for 585example, `mpfr_sin' on +0 gives +0. In the other cases, the sign is 586specified in the description of the MPFR function; for example 587`mpfr_max' on -0 and +0 gives +0. 588 589 When the input point is not in the closure of the domain of the 590function, the result is NaN. Example: `mpfr_sqrt' on -17 gives NaN. 591 592 When an input argument is NaN, the result is NaN, possibly except 593when a partial function is constant on the finite floating-point 594numbers; such a case is always explicitly specified in *note MPFR 595Interface::. Example: `mpfr_hypot' on (NaN,0) gives NaN, but 596`mpfr_hypot' on (NaN,+Inf) gives +Inf (as specified in *note Special 597Functions::), since for any finite input X, `mpfr_hypot' on (X,+Inf) 598gives +Inf. 599 600 601File: mpfr.info, Node: Exceptions, Next: Memory Handling, Prev: Floating-Point Values on Special Numbers, Up: MPFR Basics 602 6034.6 Exceptions 604============== 605 606MPFR supports 6 exception types: 607 608 * 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.) 614 615 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. 629 630 * 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. 637 638 * Divide-by-zero: An exact infinite result is obtained from finite 639 inputs. 640 641 * NaN: A NaN exception occurs when the result of a function is NaN. 642 643 * Inexact: An inexact exception occurs when the result of a function 644 cannot be represented exactly and must be rounded. 645 646 * 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 650 the target type). 651 652 653 MPFR has a global flag for each exception, which can be cleared, set 654or tested by functions described in *note Exception Related Functions::. 655 656 Differences with the ISO C99 standard: 657 658 * 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 662 signaling. 663 664 * An invalid exception in C corresponds to either a NaN exception or 665 a range error in MPFR. 666 667 668 669File: mpfr.info, Node: Memory Handling, Prev: Exceptions, Up: MPFR Basics 670 6714.7 Memory Handling 672=================== 673 674MPFR functions may create caches, e.g., when computing constants such 675as Pi, either because the user has called a function like 676`mpfr_const_pi' directly or because such a function was called 677internally by the MPFR library itself to compute some other function. 678 679 At any time, the user can free the various caches with 680`mpfr_free_cache'. It is strongly advised to do that before terminating 681a thread, or before exiting when using tools like `valgrind' (to avoid 682memory leaks being reported). 683 684 MPFR internal data such as flags, the exponent range, the default 685precision and rounding mode, and caches (i.e., data that are not 686accessed via parameters) are either global (if MPFR has not been 687compiled as thread safe) or per-thread (thread local storage, TLS). 688The initial values of TLS data after a thread is created entirely 689depend on the compiler and thread implementation (MPFR simply does a 690conventional variable initialization, the variables being declared with 691an implementation-defined TLS specifier). 692 693 694File: mpfr.info, Node: MPFR Interface, Next: API Compatibility, Prev: MPFR Basics, Up: Top 695 6965 MPFR Interface 697**************** 698 699The floating-point functions expect arguments of type `mpfr_t'. 700 701 The MPFR floating-point functions have an interface that is similar 702to the GNU MP functions. The function prefix for floating-point 703operations is `mpfr_'. 704 705 The user has to specify the precision of each variable. A 706computation that assigns a variable will take place with the precision 707of the assigned variable; the cost of that computation should not 708depend on the precision of variables used as input (on average). 709 710 The semantics of a calculation in MPFR is specified as follows: 711Compute the requested operation exactly (with "infinite accuracy"), and 712round the result to the precision of the destination variable, with the 713given rounding mode. The MPFR floating-point functions are intended to 714be a smooth extension of the IEEE 754 arithmetic. The results obtained 715on a given computer are identical to those obtained on a computer with 716a different word size, or with a different compiler or operating system. 717 718 MPFR _does not keep track_ of the accuracy of a computation. This is 719left to the user or to a higher layer (for example the MPFI library for 720interval arithmetic). As a consequence, if two variables are used to 721store only a few significant bits, and their product is stored in a 722variable with large precision, then MPFR will still compute the result 723with full precision. 724 725 The value of the standard C macro `errno' may be set to non-zero by 726any MPFR function or macro, whether or not there is an error. 727 728* Menu: 729 730* 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:: 744* Custom Interface:: 745* Internals:: 746 747 748File: mpfr.info, Node: Initialization Functions, Next: Assignment Functions, Prev: MPFR Interface, Up: MPFR Interface 749 7505.1 Initialization Functions 751============================ 752 753An `mpfr_t' object must be initialized before storing the first value in 754it. The functions `mpfr_init' and `mpfr_init2' are used for that 755purpose. 756 757 -- 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 760 to zero instead.) 761 762 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). 768 769 -- 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'). 776 777 -- 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 780 with them. 781 782 -- 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'). 788 789 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). 792 793 { 794 mpfr_t x, y, z, t; 795 mpfr_inits2 (256, x, y, z, t, (mpfr_ptr) 0); 796 ... 797 mpfr_clears (x, y, z, t, (mpfr_ptr) 0); 798 } 799 800 -- 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'. 804 805 Warning! In a given program, some other libraries might change the 806 default precision and not restore it. Thus it is safer to use 807 `mpfr_init2'. 808 809 -- 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'). 816 817 Warning! In a given program, some other libraries might change the 818 default precision and not restore it. Thus it is safer to use 819 `mpfr_inits2'. 820 821 -- 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: 827 828 * 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). 831 832 * You *cannot* change their precision. 833 834 * You *should not* create variables with huge precision with 835 this macro. 836 837 * 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. 846 847 -- 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 854 initially. 855 856 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. 859 860 -- 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'. 863 864 Here is an example on how to initialize floating-point variables: 865 866 { 867 mpfr_t x, y; 868 mpfr_init (x); /* use default precision */ 869 mpfr_init2 (y, 256); /* precision _exactly_ 256 bits */ 870 ... 871 /* When the program is about to exit, do ... */ 872 mpfr_clear (x); 873 mpfr_clear (y); 874 mpfr_free_cache (); /* free the cache for constants like pi */ 875 } 876 877 The following functions are useful for changing the precision during 878a calculation. A typical use would be for adjusting the precision 879gradually in iterative algorithms like Newton-Raphson, making the 880computation precision closely match the actual accurate part of the 881numbers. 882 883 -- 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. 892 893 -- 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 895 its significand. 896 897 898File: mpfr.info, Node: Assignment Functions, Next: Combined Initialization and Assignment Functions, Prev: Initialization Functions, Up: MPFR Interface 899 9005.2 Assignment Functions 901======================== 902 903These functions assign new values to already initialized floats (*note 904Initialization Functions::). 905 906 -- 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, 908 mpfr_rnd_t RND) 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 915 RND) 916 -- Function: int mpfr_set_decimal64 (mpfr_t ROP, _Decimal64 OP, 917 mpfr_rnd_t RND) 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'. 936 937 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 944 with it. 945 946 -- 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 949 E, mpfr_rnd_t RND) 950 -- Function: int mpfr_set_uj_2exp (mpfr_t ROP, uintmax_t OP, intmax_t 951 E, mpfr_rnd_t RND) 952 -- Function: int mpfr_set_sj_2exp (mpfr_t ROP, intmax_t OP, intmax_t 953 E, mpfr_rnd_t RND) 954 -- Function: int mpfr_set_z_2exp (mpfr_t ROP, mpz_t OP, mpfr_exp_t E, 955 mpfr_rnd_t RND) 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 958 converted to +0. 959 960 -- Function: int mpfr_set_str (mpfr_t ROP, const char *S, int BASE, 961 mpfr_rnd_t RND) 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. 967 968 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). 973 974 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. 977 978 -- 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. 989 990 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. 996 997 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. 1018 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. 1023 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 1029 read. 1030 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 1039 17. 1040 1041 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. 1049 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 1054 auxiliary variable. 1055 1056 1057File: mpfr.info, Node: Combined Initialization and Assignment Functions, Next: Conversion Functions, Prev: Assignment Functions, Up: MPFR Interface 1058 10595.3 Combined Initialization and Assignment Functions 1060==================================================== 1061 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, 1064 mpfr_rnd_t RND) 1065 -- Macro: int mpfr_init_set_si (mpfr_t ROP, long int OP, mpfr_rnd_t 1066 RND) 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 1069 RND) 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'. 1076 1077 -- Function: int mpfr_init_set_str (mpfr_t X, const char *S, int BASE, 1078 mpfr_rnd_t RND) 1079 Initialize X and set its value from the string S in base BASE, 1080 rounded in the direction RND. See `mpfr_set_str'. 1081 1082 1083File: mpfr.info, Node: Conversion Functions, Next: Basic Arithmetic Functions, Prev: Combined Initialization and Assignment Functions, Up: MPFR Interface 1084 10855.4 Conversion Functions 1086======================== 1087 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'. 1100 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'. 1113 1114 -- Function: double mpfr_get_d_2exp (long *EXP, mpfr_t OP, mpfr_rnd_t 1115 RND) 1116 -- Function: long double mpfr_get_ld_2exp (long *EXP, mpfr_t OP, 1117 mpfr_rnd_t RND) 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. 1126 1127 -- Function: int mpfr_frexp (mpfr_exp_t *EXP, mpfr_t Y, mpfr_t X, 1128 mpfr_rnd_t RND) 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 1134 undefined. 1135 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. 1148 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. 1153 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. 1166 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). 1176 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. 1188 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). 1199 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 1203 `mpfr_free_str'. 1204 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. 1210 1211 A pointer to the string is returned, unless there is an error, in 1212 which case a null pointer is returned. 1213 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. 1219 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. 1232 1233 1234File: mpfr.info, Node: Basic Arithmetic Functions, Next: Comparison Functions, Prev: Conversion Functions, Up: MPFR Interface 1235 12365.5 Basic Arithmetic Functions 1237============================== 1238 1239 -- Function: int mpfr_add (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, 1240 mpfr_rnd_t RND) 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, 1244 mpfr_rnd_t RND) 1245 -- Function: int mpfr_add_d (mpfr_t ROP, mpfr_t OP1, double OP2, 1246 mpfr_rnd_t RND) 1247 -- Function: int mpfr_add_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, 1248 mpfr_rnd_t RND) 1249 -- Function: int mpfr_add_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, 1250 mpfr_rnd_t RND) 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). 1257 1258 -- Function: int mpfr_sub (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, 1259 mpfr_rnd_t RND) 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, 1265 mpfr_rnd_t RND) 1266 -- Function: int mpfr_sub_si (mpfr_t ROP, mpfr_t OP1, long int OP2, 1267 mpfr_rnd_t RND) 1268 -- Function: int mpfr_d_sub (mpfr_t ROP, double OP1, mpfr_t OP2, 1269 mpfr_rnd_t RND) 1270 -- Function: int mpfr_sub_d (mpfr_t ROP, mpfr_t OP1, double OP2, 1271 mpfr_rnd_t RND) 1272 -- Function: int mpfr_z_sub (mpfr_t ROP, mpz_t OP1, mpfr_t OP2, 1273 mpfr_rnd_t RND) 1274 -- Function: int mpfr_sub_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, 1275 mpfr_rnd_t RND) 1276 -- Function: int mpfr_sub_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, 1277 mpfr_rnd_t RND) 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 1282 `mpfr_sub_d'. 1283 1284 -- Function: int mpfr_mul (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, 1285 mpfr_rnd_t RND) 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, 1289 mpfr_rnd_t RND) 1290 -- Function: int mpfr_mul_d (mpfr_t ROP, mpfr_t OP1, double OP2, 1291 mpfr_rnd_t RND) 1292 -- Function: int mpfr_mul_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, 1293 mpfr_rnd_t RND) 1294 -- Function: int mpfr_mul_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, 1295 mpfr_rnd_t RND) 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 1300 `mpfr_mul_d'. 1301 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. 1304 1305 -- Function: int mpfr_div (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, 1306 mpfr_rnd_t RND) 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, 1312 mpfr_rnd_t RND) 1313 -- Function: int mpfr_div_si (mpfr_t ROP, mpfr_t OP1, long int OP2, 1314 mpfr_rnd_t RND) 1315 -- Function: int mpfr_d_div (mpfr_t ROP, double OP1, mpfr_t OP2, 1316 mpfr_rnd_t RND) 1317 -- Function: int mpfr_div_d (mpfr_t ROP, mpfr_t OP1, double OP2, 1318 mpfr_rnd_t RND) 1319 -- Function: int mpfr_div_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, 1320 mpfr_rnd_t RND) 1321 -- Function: int mpfr_div_q (mpfr_t ROP, mpfr_t OP1, mpq_t OP2, 1322 mpfr_rnd_t RND) 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 1327 `mpfr_div_d'. 1328 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, 1331 mpfr_rnd_t RND) 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. 1335 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. 1340 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 1343 K, mpfr_rnd_t RND) 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. 1348 1349 -- Function: int mpfr_pow (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, 1350 mpfr_rnd_t RND) 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, 1354 mpfr_rnd_t RND) 1355 -- Function: int mpfr_pow_z (mpfr_t ROP, mpfr_t OP1, mpz_t OP2, 1356 mpfr_rnd_t RND) 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 1365 odd integer. 1366 1367 * `pow(±0, Y)' returns plus infinity for Y negative and not an 1368 odd integer. 1369 1370 * `pow(±0, Y)' returns plus or minus zero for Y a positive odd 1371 integer. 1372 1373 * `pow(±0, Y)' returns plus zero for Y positive and not an odd 1374 integer. 1375 1376 * `pow(-1, ±Inf)' returns 1. 1377 1378 * `pow(+1, Y)' returns 1 for any Y, even a NaN. 1379 1380 * `pow(X, ±0)' returns 1 for any X, even a NaN. 1381 1382 * `pow(X, Y)' returns NaN for finite negative X and finite 1383 non-integer Y. 1384 1385 * `pow(X, -Inf)' returns plus infinity for 0 < abs(x) < 1, and 1386 plus zero for abs(x) > 1. 1387 1388 * `pow(X, +Inf)' returns plus zero for 0 < abs(x) < 1, and plus 1389 infinity for abs(x) > 1. 1390 1391 * `pow(-Inf, Y)' returns minus zero for Y a negative odd 1392 integer. 1393 1394 * `pow(-Inf, Y)' returns plus zero for Y negative and not an 1395 odd integer. 1396 1397 * `pow(-Inf, Y)' returns minus infinity for Y a positive odd 1398 integer. 1399 1400 * `pow(-Inf, Y)' returns plus infinity for Y positive and not 1401 an odd integer. 1402 1403 * `pow(+Inf, Y)' returns plus zero for Y negative, and plus 1404 infinity for Y positive. 1405 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. 1412 1413 -- Function: int mpfr_dim (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, 1414 mpfr_rnd_t RND) 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. 1418 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, 1422 mpfr_rnd_t RND) 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. 1425 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, 1429 mpfr_rnd_t RND) 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 1432 identical. 1433 1434 1435File: mpfr.info, Node: Comparison Functions, Next: Special Functions, Prev: Basic Arithmetic Functions, Up: MPFR Interface 1436 14375.6 Comparison Functions 1438======================== 1439 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. 1452 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 1459 conversion first). 1460 1461 -- Function: int mpfr_cmp_ui_2exp (mpfr_t OP1, unsigned long int OP2, 1462 mpfr_exp_t E) 1463 -- Function: int mpfr_cmp_si_2exp (mpfr_t OP1, long int OP2, 1464 mpfr_exp_t E) 1465 Compare OP1 and OP2 multiplied by two to the power E. Similar as 1466 above. 1467 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 1472 return zero. 1473 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 1482 otherwise. 1483 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 1488 more efficient. 1489 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. 1498 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). 1503 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. 1507 1508 1509File: mpfr.info, Node: Special Functions, Next: Input and Output Functions, Prev: Comparison Functions, Up: MPFR Interface 1510 15115.7 Special Functions 1512===================== 1513 1514All those functions, except explicitly stated (for example 1515`mpfr_sin_cos'), return a *note ternary value::, i.e., zero for an 1516exact return value, a positive value for a return value larger than the 1517exact result, and a negative value otherwise. 1518 1519 Important note: in some domains, computing special functions (either 1520with correct or incorrect rounding) is expensive, even for small 1521precision, for example the trigonometric and Bessel functions for large 1522argument. 1523 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). 1530 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. 1536 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 1541 the direction RND. 1542 1543 -- Function: int mpfr_sin_cos (mpfr_t SOP, mpfr_t COP, mpfr_t OP, 1544 mpfr_rnd_t RND) 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 1551 cosine of OP. 1552 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. 1558 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. 1570 1571 -- Function: int mpfr_atan2 (mpfr_t ROP, mpfr_t Y, mpfr_t X, 1572 mpfr_rnd_t RND) 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. 1578 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. 1583 1584 * `atan2(-0, -0)' returns -Pi. 1585 1586 * `atan2(+0, +0)' returns +0. 1587 1588 * `atan2(-0, +0)' returns -0. 1589 1590 * `atan2(+0, x)' returns +Pi for x < 0. 1591 1592 * `atan2(-0, x)' returns -Pi for x < 0. 1593 1594 * `atan2(+0, x)' returns +0 for x > 0. 1595 1596 * `atan2(-0, x)' returns -0 for x > 0. 1597 1598 * `atan2(y, 0)' returns -Pi/2 for y < 0. 1599 1600 * `atan2(y, 0)' returns +Pi/2 for y > 0. 1601 1602 * `atan2(+Inf, -Inf)' returns +3*Pi/4. 1603 1604 * `atan2(-Inf, -Inf)' returns -3*Pi/4. 1605 1606 * `atan2(+Inf, +Inf)' returns +Pi/4. 1607 1608 * `atan2(-Inf, +Inf)' returns -Pi/4. 1609 1610 * `atan2(+Inf, x)' returns +Pi/2 for finite x. 1611 1612 * `atan2(-Inf, x)' returns -Pi/2 for finite x. 1613 1614 * `atan2(y, -Inf)' returns +Pi for finite y > 0. 1615 1616 * `atan2(y, -Inf)' returns -Pi for finite y < 0. 1617 1618 * `atan2(y, +Inf)' returns +0 for finite y > 0. 1619 1620 * `atan2(y, +Inf)' returns -0 for finite y < 0. 1621 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. 1627 1628 -- Function: int mpfr_sinh_cosh (mpfr_t SOP, mpfr_t COP, mpfr_t OP, 1629 mpfr_rnd_t RND) 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 1635 value). 1636 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. 1642 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. 1648 1649 -- Function: int mpfr_fac_ui (mpfr_t ROP, unsigned long int OP, 1650 mpfr_rnd_t RND) 1651 Set ROP to the factorial of OP, rounded in the direction RND. 1652 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 1655 RND. 1656 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. 1660 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). 1670 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. 1675 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. 1679 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'. 1684 1685 -- Function: int mpfr_lgamma (mpfr_t ROP, int *SIGNP, mpfr_t OP, 1686 mpfr_rnd_t RND) 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. 1693 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. 1698 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, 1701 mpfr_rnd_t RND) 1702 Set ROP to the value of the Riemann Zeta function on OP, rounded 1703 in the direction RND. 1704 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. 1709 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 1713 RND) 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 1719 OP. 1720 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 1724 RND) 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. 1730 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. 1737 1738 -- Function: int mpfr_agm (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, 1739 mpfr_rnd_t RND) 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 1745 ROP to NaN. 1746 1747 -- Function: int mpfr_hypot (mpfr_t ROP, mpfr_t X, mpfr_t Y, 1748 mpfr_rnd_t RND) 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. 1754 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. 1762 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'. 1772 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). 1780 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. 1790 1791 1792File: mpfr.info, Node: Input and Output Functions, Next: Formatted Output Functions, Prev: Special Functions, Up: MPFR Interface 1793 17945.8 Input and Output Functions 1795============================== 1796 1797This section describes functions that perform input from an input/output 1798stream, and functions that output to an input/output stream. Passing a 1799null pointer for a `stream' to any of these functions will make them 1800read from `stdin' and write to `stdout', respectively. 1801 1802 When using any of these functions, you must include the `<stdio.h>' 1803standard header before `mpfr.h', to allow `mpfr.h' to define prototypes 1804for these functions. 1805 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'). 1812 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 1817 delimiter. 1818 1819 Return the number of characters written, or if an error occurred, 1820 return 0. 1821 1822 -- Function: size_t mpfr_inp_str (mpfr_t ROP, FILE *STREAM, int BASE, 1823 mpfr_rnd_t RND) 1824 Input a string in base BASE from stream STREAM, rounded in the 1825 direction RND, and put the read float in ROP. 1826 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. 1831 1832 Return the number of bytes read, or if an error occurred, return 0. 1833 1834 1835File: mpfr.info, Node: Formatted Output Functions, Next: Integer Related Functions, Prev: Input and Output Functions, Up: MPFR Interface 1836 18375.9 Formatted Output Functions 1838============================== 1839 18405.9.1 Requirements 1841------------------ 1842 1843The class of `mpfr_printf' functions provides formatted output in a 1844similar manner as the standard C `printf'. These functions are defined 1845only if your system supports ISO C variadic functions and the 1846corresponding argument access macros. 1847 1848 When using any of these functions, you must include the `<stdio.h>' 1849standard header before `mpfr.h', to allow `mpfr.h' to define prototypes 1850for these functions. 1851 18525.9.2 Format String 1853------------------- 1854 1855The format specification accepted by `mpfr_printf' is an extension of 1856the `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 1859standard `printf' (in particular, notice that the `precision' is 1860related to the number of digits displayed in the base chosen by `conv' 1861and not related to the internal precision of the `mpfr_t' variable). 1862`mpfr_printf' accepts the same `type' specifiers as GMP (except the 1863non-standard and deprecated `q', use `ll' instead), namely the length 1864modifiers defined in the C standard: 1865 1866 `h' `short' 1867 `hh' `char' 1868 `j' `intmax_t' or `uintmax_t' 1869 `l' `long' or `wchar_t' 1870 `ll' `long long' 1871 `L' `long double' 1872 `t' `ptrdiff_t' 1873 `z' `size_t' 1874 1875 and the `type' specifiers defined in GMP plus `R' and `P' specific 1876to MPFR (the second column in the table below shows the type of the 1877argument read in the argument list and the kind of `conv' specifier to 1878use after the `type' specifier): 1879 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 1887 1888 The `type' specifiers have the same restrictions as those mentioned 1889in the GMP documentation: *note Formatted Output Strings: 1890(gmp.info)Formatted Output Strings. In particular, the `type' 1891specifiers (except `R' and `P') are supported only if they are 1892supported by `gmp_printf' in your GMP build; this implies that the 1893standard specifiers, such as `t', must _also_ be supported by your C 1894library if you want to use them. 1895 1896 The `rounding' field is specific to `mpfr_t' arguments and should 1897not be used with other types. 1898 1899 With conversion specification not involving `P' and `R' types, 1900`mpfr_printf' behaves exactly as `gmp_printf'. 1901 1902 The `P' type specifies that a following `o', `u', `x', or `X' 1903conversion specifier applies to a `mpfr_prec_t' argument. It is needed 1904because the `mpfr_prec_t' type does not necessarily correspond to an 1905`unsigned int' or any fixed standard type. The `precision' field 1906specifies the minimum number of digits to appear. The default 1907`precision' is 1. For example: 1908 mpfr_t x; 1909 mpfr_prec_t p; 1910 mpfr_init (x); 1911 ... 1912 p = mpfr_get_prec (x); 1913 mpfr_printf ("variable x with %Pu bits", p); 1914 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' 1917argument. The `R' type can be followed by a `rounding' specifier 1918denoted by one of the following characters: 1919 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. 1928 1929 The default rounding mode is rounding to nearest. The following 1930three examples are equivalent: 1931 mpfr_t x; 1932 mpfr_init (x); 1933 ... 1934 mpfr_printf ("%.128Rf", x); 1935 mpfr_printf ("%.128RNf", x); 1936 mpfr_printf ("%.128R*f", MPFR_RNDN, x); 1937 1938 Note that the rounding away from zero mode is specified with `Y' 1939because ISO C reserves the `A' specifier for hexadecimal output (see 1940below). 1941 1942 The output `conv' specifiers allowed with `mpfr_t' parameter are: 1943 1944 `a' `A' hex float, C99 style 1945 `b' binary output 1946 `e' `E' scientific format float 1947 `f' `F' fixed point float 1948 `g' `G' fixed or scientific float 1949 1950 The conversion specifier `b' which displays the argument in binary is 1951specific to `mpfr_t' arguments and should not be used with other types. 1952Other conversion specifiers have the same meaning as for a `double' 1953argument. 1954 1955 In case of non-decimal output, only the significand is written in the 1956specified base, the exponent is always displayed in decimal. Special 1957values 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. 1960 1961 If the `precision' field is not empty, the `mpfr_t' number is 1962rounded to the given precision in the direction specified by the 1963rounding mode. If the precision is zero with rounding to nearest mode 1964and one of the following `conv' specifiers: `a', `A', `b', `e', `E', 1965tie case is rounded to even when it lies between two consecutive values 1966at the wanted precision which have the same exponent, otherwise, it is 1967rounded away from zero. For instance, 85 is displayed as "8e+1" and 95 1968is displayed as "1e+2" with the format specification `"%.0RNe"'. This 1969also 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 1971maximum value for an `int', it will be silently reduced down to 1972`INT_MAX'. 1973 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 1976digits so that it can be read back exactly, assuming that the input and 1977output variables have the same precision and that the input and output 1978rounding modes are both rounding to nearest (as for `mpfr_get_str'). 1979The default precision for an empty `precision' field with `conv' 1980specifiers `f', `F', `g', and `G' is 6. 1981 19825.9.3 Functions 1983--------------- 1984 1985For all the following functions, if the number of characters which 1986ought to be written appears to exceed the maximum limit for an `int', 1987nothing is written in the stream (resp. to `stdout', to BUF, to STR), 1988the function returns -1, sets the _erange_ flag, and (in POSIX system 1989only) `errno' is set to `EOVERFLOW'. 1990 1991 -- Function: int mpfr_fprintf (FILE *STREAM, const char *TEMPLATE, ...) 1992 -- Function: int mpfr_vfprintf (FILE *STREAM, const char *TEMPLATE, 1993 va_list AP) 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. 1997 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. 2003 2004 -- Function: int mpfr_sprintf (char *BUF, const char *TEMPLATE, ...) 2005 -- Function: int mpfr_vsprintf (char *BUF, const char *TEMPLATE, 2006 va_list AP) 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. 2013 2014 -- Function: int mpfr_snprintf (char *BUF, size_t N, const char 2015 *TEMPLATE, ...) 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. 2026 2027 -- Function: int mpfr_asprintf (char **STR, const char *TEMPLATE, ...) 2028 -- Function: int mpfr_vasprintf (char **STR, const char *TEMPLATE, 2029 va_list AP) 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. 2036 2037 2038File: mpfr.info, Node: Integer Related Functions, Next: Rounding Related Functions, Prev: Formatted Output Functions, Up: MPFR Interface 2039 20405.10 Integer and Remainder Related Functions 2041============================================ 2042 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. 2056 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. 2062 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 2072 rounding.) 2073 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 2076 RND) 2077 -- Function: int mpfr_rint_round (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t 2078 RND) 2079 -- Function: int mpfr_rint_trunc (mpfr_t ROP, mpfr_t OP, mpfr_rnd_t 2080 RND) 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. 2097 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). 2103 2104 -- Function: int mpfr_modf (mpfr_t IOP, mpfr_t FOP, mpfr_t OP, 2105 mpfr_rnd_t RND) 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 2112 the return value). 2113 2114 -- Function: int mpfr_fmod (mpfr_t R, mpfr_t X, mpfr_t Y, mpfr_rnd_t 2115 RND) 2116 -- Function: int mpfr_remainder (mpfr_t R, mpfr_t X, mpfr_t Y, 2117 mpfr_rnd_t RND) 2118 -- Function: int mpfr_remquo (mpfr_t R, long* Q, mpfr_t X, mpfr_t Y, 2119 mpfr_rnd_t RND) 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 2124 `mpfr_remquo'. 2125 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. 2131 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. 2140 2141 -- Function: int mpfr_integer_p (mpfr_t OP) 2142 Return non-zero iff OP is an integer. 2143 2144 2145File: mpfr.info, Node: Rounding Related Functions, Next: Miscellaneous Functions, Prev: Integer Related Functions, Up: MPFR Interface 2146 21475.11 Rounding Related Functions 2148=============================== 2149 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. 2153 2154 -- Function: mpfr_rnd_t mpfr_get_default_rounding_mode (void) 2155 Get the default rounding mode. 2156 2157 -- Function: int mpfr_prec_round (mpfr_t X, mpfr_prec_t PREC, 2158 mpfr_rnd_t RND) 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. 2166 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 */ 2178 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. 2187 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. 2191 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))) 2195 ... 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. 2201 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'.) 2206 2207 The function name is subject to change. 2208 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. 2213 2214 2215File: mpfr.info, Node: Miscellaneous Functions, Next: Exception Related Functions, Prev: Rounding Related Functions, Up: MPFR Interface 2216 22175.12 Miscellaneous Functions 2218============================ 2219 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. 2228 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. 2232 minus infinity). 2233 2234 -- Function: int mpfr_min (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, 2235 mpfr_rnd_t RND) 2236 -- Function: int mpfr_max (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, 2237 mpfr_rnd_t RND) 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). 2242 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 2249 0). 2250 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). 2257 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. 2261 2262 -- Function: int mpfr_urandom (mpfr_t ROP, gmp_randstate_t STATE, 2263 mpfr_rnd_t RND) 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. 2268 2269 The second argument is a `gmp_randstate_t' structure which should 2270 be created using the GMP `gmp_randinit' function (see the GMP 2271 manual). 2272 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. 2276 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. 2282 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. 2286 2287 The third argument is a `gmp_randstate_t' structure, which should 2288 be created using the GMP `gmp_randinit' function (see the GMP 2289 manual). 2290 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. 2297 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. 2301 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. 2306 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 2311 [1/2,1). 2312 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). 2316 2317 -- Function: int mpfr_setsign (mpfr_t ROP, mpfr_t OP, int S, 2318 mpfr_rnd_t RND) 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. 2322 2323 -- Function: int mpfr_copysign (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, 2324 mpfr_rnd_t RND) 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)'. 2329 2330 -- Function: const char * mpfr_get_version (void) 2331 Return the MPFR version, as a null-terminated string. 2332 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). 2352 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." 2359 #endif 2360 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). 2369 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. 2375 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. 2380 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. 2385 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 2389 type. 2390 2391 2392File: mpfr.info, Node: Exception Related Functions, Next: Compatibility with MPF, Prev: Miscellaneous Functions, Up: MPFR Interface 2393 23945.13 Exception Related Functions 2395================================ 2396 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. 2405 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. 2417 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. 2427 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. 2444 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. 2450 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. 2462 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. 2471 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. 2475 2476 This is an example of how to emulate binary double IEEE 754 2477arithmetic (binary64 in IEEE 754-2008) using MPFR: 2478 2479 { 2480 mpfr_t xa, xb; int i; volatile double a, b; 2481 2482 mpfr_set_default_prec (53); 2483 mpfr_set_emin (-1073); mpfr_set_emax (1024); 2484 2485 mpfr_init (xa); mpfr_init (xb); 2486 2487 b = 34.3; mpfr_set_d (xb, b, MPFR_RNDN); 2488 a = 0x1.1235P-1021; mpfr_set_d (xa, a, MPFR_RNDN); 2489 2490 a /= b; 2491 i = mpfr_div (xa, xa, xb, MPFR_RNDN); 2492 i = mpfr_subnormalize (xa, i, MPFR_RNDN); /* new ternary value */ 2493 2494 mpfr_clear (xa); mpfr_clear (xb); 2495 } 2496 2497 Warning: this emulates a double IEEE 754 arithmetic with correct 2498rounding in the subnormal range, which may not be the case for your 2499hardware. 2500 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 2508 and _erange_ flags. 2509 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 2517 _erange_ flags. 2518 2519 -- Function: void mpfr_clear_flags (void) 2520 Clear all global flags (underflow, overflow, divide-by-zero, 2521 invalid, inexact, _erange_). 2522 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 2531 is set. 2532 2533 2534File: mpfr.info, Node: Compatibility with MPF, Next: Custom Interface, Prev: Exception Related Functions, Up: MPFR Interface 2535 25365.14 Compatibility With MPF 2537=========================== 2538 2539A header file `mpf2mpfr.h' is included in the distribution of MPFR for 2540compatibility with the GNU MP class MPF. By inserting the following 2541two lines after the `#include <gmp.h>' line, 2542#include <mpfr.h> 2543#include <mpf2mpfr.h> 2544 any program written for MPF can be compiled directly with MPFR without 2545any changes (except the `gmp_printf' functions will not work for 2546arguments of type `mpfr_t'). All operations are then performed with 2547the default MPFR rounding mode, which can be reset with 2548`mpfr_set_default_rounding_mode'. 2549 2550 Warning: the `mpf_init' and `mpf_init2' functions initialize to 2551zero, whereas the corresponding MPFR functions initialize to NaN: this 2552is useful to detect uninitialized values, but is slightly incompatible 2553with MPF. 2554 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. 2560 2561 -- Function: int mpfr_eq (mpfr_t OP1, mpfr_t OP2, unsigned long int 2562 OP3) 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 2570 OP3 larger than 1. 2571 2572 -- Function: void mpfr_reldiff (mpfr_t ROP, mpfr_t OP1, mpfr_t OP2, 2573 mpfr_rnd_t RND) 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. 2579 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' 2587 otherwise. 2588 2589 2590File: mpfr.info, Node: Custom Interface, Next: Internals, Prev: Compatibility with MPF, Up: MPFR Interface 2591 25925.15 Custom Interface 2593===================== 2594 2595Some applications use a stack to handle the memory and their objects. 2596However, the MPFR memory design is not well suited for such a thing. So 2597that such applications are able to use MPFR, an auxiliary memory 2598interface has been created: the Custom Interface. 2599 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 2602 stack. 2603 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 2607garbaging the used memory: all the memory management (allocating, 2608destroying, garbaging) is left to the application. 2609 2610 Each function in this interface is also implemented as a macro for 2611efficiency reasons: for example `mpfr_custom_init (s, p)' uses the 2612macro, while `(mpfr_custom_init) (s, p)' uses the function. 2613 2614 Note 1: MPFR functions may still initialize temporary floating-point 2615numbers using `mpfr_init' and similar functions. See Custom Allocation 2616(GNU MP). 2617 2618 Note 2: MPFR functions may use the cached functions (`mpfr_const_pi' 2619for 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', 2621through GMP Custom Allocation, allocates its memory on the application 2622stack. 2623 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. 2627 2628 -- Function: void mpfr_custom_init (void *SIGNIFICAND, mpfr_prec_t 2629 PREC) 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 2633 Internals::). 2634 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; 2639 2640 * if `ABS(kind) == MPFR_INF_KIND', X is set to the infinity of 2641 sign `sign(kind)'; 2642 2643 * if `ABS(kind) == MPFR_ZERO_KIND', X is set to the zero of 2644 sign `sign(kind)'; 2645 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. 2654 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. 2659 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 2664 undefined. 2665 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 2671 undefined. 2672 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. 2679 2680 2681File: mpfr.info, Node: Internals, Prev: Custom Interface, Up: MPFR Interface 2682 26835.16 Internals 2684============== 2685 2686A "limb" means the part of a multi-precision number that fits in a 2687single word. Usually a limb contains 32 or 64 bits. The C data type 2688for a limb is `mp_limb_t'. 2689 2690 The `mpfr_t' type is internally defined as a one-element array of a 2691structure, and `mpfr_ptr' is the C data type representing a pointer to 2692this structure. The `mpfr_t' type consists of four fields: 2693 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'. 2696 2697 * The `_mpfr_sign' field is used to store the sign of the variable. 2698 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 2703 field. 2704 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. 2713 2714 2715 2716File: mpfr.info, Node: API Compatibility, Next: Contributors, Prev: MPFR Interface, Up: Top 2717 27186 API Compatibility 2719******************* 2720 2721The goal of this section is to describe some API changes that occurred 2722from one version of MPFR to another, and how to write code that can be 2723compiled and run with older MPFR versions. The minimum MPFR version 2724that is considered here is 2.2.0 (released on 20 September 2005). 2725 2726 API changes can only occur between major or minor versions. Thus the 2727patchlevel (the third number in the MPFR version) will be ignored in 2728the following. If a program does not use MPFR internals, changes in 2729the behavior between two versions differing only by the patchlevel 2730should only result from what was regarded as a bug or unspecified 2731behavior. 2732 2733 As a general rule, a program written for some MPFR version should 2734work with later versions, possibly except at a new major version, where 2735some features (described as obsolete for some time) can be removed. In 2736such a case, a failure should occur during compilation or linking. If 2737a result becomes incorrect because of such a change, please look at the 2738various changes below (they are minimal, and most software should be 2739unaffected), at the FAQ and at the MPFR web page for your version (a 2740bug could have been introduced and be already fixed); and if the 2741problem is not mentioned, please send us a bug report (*note Reporting 2742Bugs::). 2743 2744 However, a program written for the current MPFR version (as 2745documented by this manual) may not necessarily work with previous 2746versions of MPFR. This section should help developers to write 2747portable code. 2748 2749 Note: Information given here may be incomplete. API changes are 2750also described in the NEWS file (for each version, instead of being 2751classified like here), together with other changes. 2752 2753* Menu: 2754 2755* Type and Macro Changes:: 2756* Added Functions:: 2757* Changed Functions:: 2758* Removed Functions:: 2759* Other Changes:: 2760 2761 2762File: mpfr.info, Node: Type and Macro Changes, Next: Added Functions, Prev: API Compatibility, Up: API Compatibility 2763 27646.1 Type and Macro Changes 2765========================== 2766 2767The 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 2769it comes from GMP (with a different meaning). These types are 2770currently 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 2772change in the future. Alternatively, using the following code after 2773including `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; 2777 #endif 2778 2779 The official types for precision values and for rounding modes 2780respectively changed from `mp_prec_t' and `mp_rnd_t' to `mpfr_prec_t' 2781and `mpfr_rnd_t' in MPFR 3.0. This change was actually done a long 2782time ago in MPFR, at least since MPFR 2.2.0, with the following code in 2783`mpfr.h': 2784 #ifndef mp_rnd_t 2785 # define mp_rnd_t mpfr_rnd_t 2786 #endif 2787 #ifndef mp_prec_t 2788 # define mp_prec_t mpfr_prec_t 2789 #endif 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' 2792and `mp_rnd_t' (defined in MPFR only) may be removed in the future, as 2793the prefix `mp_' is reserved by GMP. 2794 2795 The precision type `mpfr_prec_t' (`mp_prec_t') was unsigned before 2796MPFR 3.0; it is now signed. `MPFR_PREC_MAX' has not changed, though. 2797Indeed the MPFR code requires that `MPFR_PREC_MAX' be representable in 2798the exponent type, which may have the same size as `mpfr_prec_t' but 2799has always been signed. The consequence is that valid code that does 2800not assume anything about the signedness of `mpfr_prec_t' should work 2801with past and new MPFR versions. This change was useful as the use of 2802unsigned types tends to convert signed values to unsigned ones in 2803expressions due to the usual arithmetic conversions, which can yield 2804incorrect results if a negative value is converted in such a way. 2805Warning! A program assuming (intentionally or not) that `mpfr_prec_t' 2806is signed may be affected by this problem when it is built and run 2807against MPFR 2.x. 2808 2809 The rounding modes `GMP_RNDx' were renamed to `MPFR_RNDx' in MPFR 28103.0. 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 2817MPFR 3.0 (however no rounding mode `GMP_RNDA' exists). 2818 2819 2820File: mpfr.info, Node: Added Functions, Next: Changed Functions, Prev: Type and Macro Changes, Up: API Compatibility 2821 28226.2 Added Functions 2823=================== 2824 2825We give here in alphabetical order the functions that were added after 2826MPFR 2.2, and in which MPFR version. 2827 2828 * `mpfr_add_d' in MPFR 2.4. 2829 2830 * `mpfr_ai' in MPFR 3.0 (incomplete, experimental). 2831 2832 * `mpfr_asprintf' in MPFR 2.4. 2833 2834 * `mpfr_buildopt_decimal_p' and `mpfr_buildopt_tls_p' in MPFR 3.0. 2835 2836 * `mpfr_buildopt_gmpinternals_p' and `mpfr_buildopt_tune_case' in 2837 MPFR 3.1. 2838 2839 * `mpfr_clear_divby0' in MPFR 3.1 (new divide-by-zero exception). 2840 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 2844 NaN). 2845 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 2849 `mpfr.h': 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'. 2853 2854 * `mpfr_d_div' and `mpfr_d_sub' in MPFR 2.4. 2855 2856 * `mpfr_digamma' in MPFR 3.0. 2857 2858 * `mpfr_divby0_p' in MPFR 3.1 (new divide-by-zero exception). 2859 2860 * `mpfr_div_d' in MPFR 2.4. 2861 2862 * `mpfr_fmod' in MPFR 2.4. 2863 2864 * `mpfr_fms' in MPFR 2.3. 2865 2866 * `mpfr_fprintf' in MPFR 2.4. 2867 2868 * `mpfr_frexp' in MPFR 3.1. 2869 2870 * `mpfr_get_flt' in MPFR 3.0. 2871 2872 * `mpfr_get_patches' in MPFR 2.3. 2873 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'. 2880 2881 * `mpfr_grandom' in MPFR 3.1. 2882 2883 * `mpfr_j0', `mpfr_j1' and `mpfr_jn' in MPFR 2.3. 2884 2885 * `mpfr_lgamma' in MPFR 2.3. 2886 2887 * `mpfr_li2' in MPFR 2.4. 2888 2889 * `mpfr_min_prec' in MPFR 3.0. 2890 2891 * `mpfr_modf' in MPFR 2.4. 2892 2893 * `mpfr_mul_d' in MPFR 2.4. 2894 2895 * `mpfr_printf' in MPFR 2.4. 2896 2897 * `mpfr_rec_sqrt' in MPFR 2.4. 2898 2899 * `mpfr_regular_p' in MPFR 3.0. 2900 2901 * `mpfr_remainder' and `mpfr_remquo' in MPFR 2.3. 2902 2903 * `mpfr_set_divby0' in MPFR 3.1 (new divide-by-zero exception). 2904 2905 * `mpfr_set_flt' in MPFR 3.0. 2906 2907 * `mpfr_set_z_2exp' in MPFR 3.0. 2908 2909 * `mpfr_set_zero' in MPFR 3.0. 2910 2911 * `mpfr_setsign' in MPFR 2.3. 2912 2913 * `mpfr_signbit' in MPFR 2.3. 2914 2915 * `mpfr_sinh_cosh' in MPFR 2.4. 2916 2917 * `mpfr_snprintf' and `mpfr_sprintf' in MPFR 2.4. 2918 2919 * `mpfr_sub_d' in MPFR 2.4. 2920 2921 * `mpfr_urandom' in MPFR 3.0. 2922 2923 * `mpfr_vasprintf', `mpfr_vfprintf', `mpfr_vprintf', 2924 `mpfr_vsprintf' and `mpfr_vsnprintf' in MPFR 2.4. 2925 2926 * `mpfr_y0', `mpfr_y1' and `mpfr_yn' in MPFR 2.3. 2927 2928 * `mpfr_z_sub' in MPFR 3.1. 2929 2930 2931 2932File: mpfr.info, Node: Changed Functions, Next: Removed Functions, Prev: Added Functions, Up: API Compatibility 2933 29346.3 Changed Functions 2935===================== 2936 2937The following functions have changed after MPFR 2.2. Changes can affect 2938the behavior of code written for some MPFR version when built and run 2939against another MPFR version (older or newer), as described below. 2940 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. 2947 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. 2952 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. 2956 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 2966 with MPFR 3.0: 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. 2973 2974 Moreover the cases where the _erange_ flag is set were unspecified 2975 in MPFR 2.x. 2976 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. 2981 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). 2989 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). 2996 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. 3003 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. 3012 3013 3014 3015File: mpfr.info, Node: Removed Functions, Next: Other Changes, Prev: Changed Functions, Up: API Compatibility 3016 30176.4 Removed Functions 3018===================== 3019 3020Functions `mpfr_random' and `mpfr_random2' have been removed in MPFR 30213.0 (this only affects old code built against MPFR 3.0 or later). (The 3022function `mpfr_random' had been deprecated since at least MPFR 2.2.0, 3023and `mpfr_random2' since MPFR 2.4.0.) 3024 3025 3026File: mpfr.info, Node: Other Changes, Prev: Removed Functions, Up: API Compatibility 3027 30286.5 Other Changes 3029================= 3030 3031For users of a C++ compiler, the way how the availability of `intmax_t' 3032is 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. 3036However this was not always the case (more precisely, `intmax_t' can be 3037defined only in the namespace `std', as with Boost), so that 3038compilations could fail. Thus the check for `INTMAX_C' or `UINTMAX_C' 3039is now disabled for C++ compilers, with the following consequences: 3040 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. 3044 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. 3049 3050 3051 The divide-by-zero exception is new in MPFR 3.1. However it should 3052not introduce incompatible changes for programs that strictly follow 3053the MPFR API since the exception can only be seen via new functions. 3054 3055 As of MPFR 3.1, the `mpfr.h' header can be included several times, 3056while still supporting optional functions (*note Headers and 3057Libraries::). 3058 3059 3060File: mpfr.info, Node: Contributors, Next: References, Prev: API Compatibility, Up: Top 3061 3062Contributors 3063************ 3064 3065The main developers of MPFR are Guillaume Hanrot, Vincent Lefèvre, 3066Patrick Pélissier, Philippe Théveny and Paul Zimmermann. 3067 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 3071hyperbolic functions, the base-2 exponential, and the factorial 3072function. 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. 3076Emmanuel Jeandel, from ENS-Lyon too, contributed the generic 3077hypergeometric code, as well as the internal function `mpfr_exp3', a 3078first implementation of the sine and cosine, and improved versions of 3079`mpfr_const_log2' and `mpfr_const_pi'. Ludovic Meunier helped in the 3080design 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. 3083Damien Stehlé contributed the `mpfr_get_ld_2exp' function. 3084 3085 We would like to thank Jean-Michel Muller and Joris van der Hoeven 3086for very fruitful discussions at the beginning of that project, 3087Torbjörn Granlund and Kevin Ryde for their help about design issues, 3088and Nathalie Revol for her careful reading of a previous version of 3089this documentation. In particular Kevin Ryde did a tremendous job for 3090the portability of MPFR in 2002-2004. 3091 3092 The development of the MPFR library would not have been possible 3093without the continuous support of INRIA, and of the LORIA (Nancy, 3094France) and LIP (Lyon, France) laboratories. In particular the main 3095authors were or are members of the PolKA, Spaces, Cacao and Caramel 3096project-teams at LORIA and of the Arénaire and AriC project-teams at 3097LIP. This project was started during the Fiable (reliable in French) 3098action supported by INRIA, and continued during the AOC action. The 3099development of MPFR was also supported by a grant (202F0659 00 MPN 121) 3100from the Conseil Régional de Lorraine in 2002, from INRIA by an 3101"associate engineer" grant (2003-2005), an "opération de développement 3102logiciel" grant (2007-2009), and the post-doctoral grant of Sylvain 3103Chevillard in 2009-2010. The MPFR-MPC workshop in June 2012 was partly 3104supported by the ERC grant ANTICS of Andreas Enge. 3105 3106 3107File: mpfr.info, Node: References, Next: GNU Free Documentation License, Prev: Contributors, Up: Top 3108 3109References 3110********** 3111 3112 * Richard Brent and Paul Zimmermann, "Modern Computer Arithmetic", 3113 Cambridge University Press (to appear), also available from the 3114 authors' web pages. 3115 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'. 3121 3122 * Torbjörn Granlund, "GNU MP: The GNU Multiple Precision Arithmetic 3123 Library", version 5.0.1, 2010, `http://gmplib.org'. 3124 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. 3129 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. 3133 3134 * Donald E. Knuth, "The Art of Computer Programming", vol 2, 3135 "Seminumerical Algorithms", 2nd edition, Addison-Wesley, 1981. 3136 3137 * Jean-Michel Muller, "Elementary Functions, Algorithms and 3138 Implementation", Birkhäuser, Boston, 2nd edition, 2006. 3139 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. 3144 3145 3146 3147File: mpfr.info, Node: GNU Free Documentation License, Next: Concept Index, Prev: References, Up: Top 3148 3149Appendix A GNU Free Documentation License 3150***************************************** 3151 3152 Version 1.2, November 2002 3153 3154 Copyright (C) 2000,2001,2002 Free Software Foundation, Inc. 3155 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA 3156 3157 Everyone is permitted to copy and distribute verbatim copies 3158 of this license document, but changing it is not allowed. 3159 3160 0. PREAMBLE 3161 3162 The purpose of this License is to make a manual, textbook, or other 3163 functional and useful document "free" in the sense of freedom: to 3164 assure everyone the effective freedom to copy and redistribute it, 3165 with or without modifying it, either commercially or 3166 noncommercially. Secondarily, this License preserves for the 3167 author and publisher a way to get credit for their work, while not 3168 being considered responsible for modifications made by others. 3169 3170 This License is a kind of "copyleft", which means that derivative 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 3173 license designed for free software. 3174 3175 We have designed this License in order to use it for manuals for 3176 free software, because free software needs free documentation: a 3177 free program should come with manuals providing the same freedoms 3178 that the software does. But this License is not limited to 3179 software manuals; it can be used for any textual work, regardless 3180 of subject matter or whether it is published as a printed book. 3181 We recommend this License principally for works whose purpose is 3182 instruction or reference. 3183 3184 1. APPLICABILITY AND DEFINITIONS 3185 3186 This License applies to any manual or other work, in any medium, 3187 that contains a notice placed by the copyright holder saying it 3188 can be distributed under the terms of this License. Such a notice 3189 grants a world-wide, royalty-free license, unlimited in duration, 3190 to use that work under the conditions stated herein. The 3191 "Document", below, refers to any such manual or work. Any member 3192 of the public is a licensee, and is addressed as "you". You 3193 accept the license if you copy, modify or distribute the work in a 3194 way requiring permission under copyright law. 3195 3196 A "Modified Version" of the Document means any work containing the 3197 Document or a portion of it, either copied verbatim, or with 3198 modifications and/or translated into another language. 3199 3200 A "Secondary Section" is a named appendix or a front-matter section 3201 of the Document that deals exclusively with the relationship of the 3202 publishers or authors of the Document to the Document's overall 3203 subject (or to related matters) and contains nothing that could 3204 fall directly within that overall subject. (Thus, if the Document 3205 is in part a textbook of mathematics, a Secondary Section may not 3206 explain any mathematics.) The relationship could be a matter of 3207 historical connection with the subject or with related matters, or 3208 of legal, commercial, philosophical, ethical or political position 3209 regarding them. 3210 3211 The "Invariant Sections" are certain Secondary Sections whose 3212 titles are designated, as being those of Invariant Sections, in 3213 the notice that says that the Document is released under this 3214 License. If a section does not fit the above definition of 3215 Secondary then it is not allowed to be designated as Invariant. 3216 The Document may contain zero Invariant Sections. If the Document 3217 does not identify any Invariant Sections then there are none. 3218 3219 The "Cover Texts" are certain short passages of text that are 3220 listed, as Front-Cover Texts or Back-Cover Texts, in the notice 3221 that says that the Document is released under this License. A 3222 Front-Cover Text may be at most 5 words, and a Back-Cover Text may 3223 be at most 25 words. 3224 3225 A "Transparent" copy of the Document means a machine-readable copy, 3226 represented in a format whose specification is available to the 3227 general public, that is suitable for revising the document 3228 straightforwardly with generic text editors or (for images 3229 composed of pixels) generic paint programs or (for drawings) some 3230 widely available drawing editor, and that is suitable for input to 3231 text formatters or for automatic translation to a variety of 3232 formats suitable for input to text formatters. A copy made in an 3233 otherwise Transparent file format whose markup, or absence of 3234 markup, has been arranged to thwart or discourage subsequent 3235 modification by readers is not Transparent. An image format is 3236 not Transparent if used for any substantial amount of text. A 3237 copy that is not "Transparent" is called "Opaque". 3238 3239 Examples of suitable formats for Transparent copies include plain 3240 ASCII without markup, Texinfo input format, LaTeX input format, 3241 SGML or XML using a publicly available DTD, and 3242 standard-conforming simple HTML, PostScript or PDF designed for 3243 human modification. Examples of transparent image formats include 3244 PNG, XCF and JPG. Opaque formats include proprietary formats that 3245 can be read and edited only by proprietary word processors, SGML or 3246 XML for which the DTD and/or processing tools are not generally 3247 available, and the machine-generated HTML, PostScript or PDF 3248 produced by some word processors for output purposes only. 3249 3250 The "Title Page" means, for a printed book, the title page itself, 3251 plus such following pages as are needed to hold, legibly, the 3252 material this License requires to appear in the title page. For 3253 works in formats which do not have any title page as such, "Title 3254 Page" means the text near the most prominent appearance of the 3255 work's title, preceding the beginning of the body of the text. 3256 3257 A section "Entitled XYZ" means a named subunit of the Document 3258 whose title either is precisely XYZ or contains XYZ in parentheses 3259 following text that translates XYZ in another language. (Here XYZ 3260 stands for a specific section name mentioned below, such as 3261 "Acknowledgements", "Dedications", "Endorsements", or "History".) 3262 To "Preserve the Title" of such a section when you modify the 3263 Document means that it remains a section "Entitled XYZ" according 3264 to this definition. 3265 3266 The Document may include Warranty Disclaimers next to the notice 3267 which states that this License applies to the Document. These 3268 Warranty Disclaimers are considered to be included by reference in 3269 this License, but only as regards disclaiming warranties: any other 3270 implication that these Warranty Disclaimers may have is void and 3271 has no effect on the meaning of this License. 3272 3273 2. VERBATIM COPYING 3274 3275 You may copy and distribute the Document in any medium, either 3276 commercially or noncommercially, provided that this License, the 3277 copyright notices, and the license notice saying this License 3278 applies to the Document are reproduced in all copies, and that you 3279 add no other conditions whatsoever to those of this License. You 3280 may not use technical measures to obstruct or control the reading 3281 or further copying of the copies you make or distribute. However, 3282 you may accept compensation in exchange for copies. If you 3283 distribute a large enough number of copies you must also follow 3284 the conditions in section 3. 3285 3286 You may also lend copies, under the same conditions stated above, 3287 and you may publicly display copies. 3288 3289 3. COPYING IN QUANTITY 3290 3291 If you publish printed copies (or copies in media that commonly 3292 have printed covers) of the Document, numbering more than 100, and 3293 the Document's license notice requires Cover Texts, you must 3294 enclose the copies in covers that carry, clearly and legibly, all 3295 these Cover Texts: Front-Cover Texts on the front cover, and 3296 Back-Cover Texts on the back cover. Both covers must also clearly 3297 and legibly identify you as the publisher of these copies. The 3298 front cover must present the full title with all words of the 3299 title equally prominent and visible. You may add other material 3300 on the covers in addition. Copying with changes limited to the 3301 covers, as long as they preserve the title of the Document and 3302 satisfy these conditions, can be treated as verbatim copying in 3303 other respects. 3304 3305 If the required texts for either cover are too voluminous to fit 3306 legibly, you should put the first ones listed (as many as fit 3307 reasonably) on the actual cover, and continue the rest onto 3308 adjacent pages. 3309 3310 If you publish or distribute Opaque copies of the Document 3311 numbering more than 100, you must either include a 3312 machine-readable Transparent copy along with each Opaque copy, or 3313 state in or with each Opaque copy a computer-network location from 3314 which the general network-using public has access to download 3315 using public-standard network protocols a complete Transparent 3316 copy of the Document, free of added material. If you use the 3317 latter option, you must take reasonably prudent steps, when you 3318 begin distribution of Opaque copies in quantity, to ensure that 3319 this Transparent copy will remain thus accessible at the stated 3320 location until at least one year after the last time you 3321 distribute an Opaque copy (directly or through your agents or 3322 retailers) of that edition to the public. 3323 3324 It is requested, but not required, that you contact the authors of 3325 the Document well before redistributing any large number of 3326 copies, to give them a chance to provide you with an updated 3327 version of the Document. 3328 3329 4. MODIFICATIONS 3330 3331 You may copy and distribute a Modified Version of the Document 3332 under the conditions of sections 2 and 3 above, provided that you 3333 release the Modified Version under precisely this License, with 3334 the Modified Version filling the role of the Document, thus 3335 licensing distribution and modification of the Modified Version to 3336 whoever possesses a copy of it. In addition, you must do these 3337 things in the Modified Version: 3338 3339 A. Use in the Title Page (and on the covers, if any) a title 3340 distinct from that of the Document, and from those of 3341 previous versions (which should, if there were any, be listed 3342 in the History section of the Document). You may use the 3343 same title as a previous version if the original publisher of 3344 that version gives permission. 3345 3346 B. List on the Title Page, as authors, one or more persons or 3347 entities responsible for authorship of the modifications in 3348 the Modified Version, together with at least five of the 3349 principal authors of the Document (all of its principal 3350 authors, if it has fewer than five), unless they release you 3351 from this requirement. 3352 3353 C. State on the Title page the name of the publisher of the 3354 Modified Version, as the publisher. 3355 3356 D. Preserve all the copyright notices of the Document. 3357 3358 E. Add an appropriate copyright notice for your modifications 3359 adjacent to the other copyright notices. 3360 3361 F. Include, immediately after the copyright notices, a license 3362 notice giving the public permission to use the Modified 3363 Version under the terms of this License, in the form shown in 3364 the Addendum below. 3365 3366 G. Preserve in that license notice the full lists of Invariant 3367 Sections and required Cover Texts given in the Document's 3368 license notice. 3369 3370 H. Include an unaltered copy of this License. 3371 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, 3377 and publisher of the Document as given on its Title Page, 3378 then add an item describing the Modified Version as stated in 3379 the previous sentence. 3380 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. 3389 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. 3394 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 3398 titles. 3399 3400 M. Delete any section Entitled "Endorsements". Such a section 3401 may not be included in the Modified Version. 3402 3403 N. Do not retitle any existing section to be Entitled 3404 "Endorsements" or to conflict in title with any Invariant 3405 Section. 3406 3407 O. Preserve any Warranty Disclaimers. 3408 3409 If the Modified Version includes new front-matter sections or 3410 appendices that qualify as Secondary Sections and contain no 3411 material copied from the Document, you may at your option 3412 designate some or all of these sections as invariant. To do this, 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. 3416 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. 3422 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, 3429 previously added by you or by arrangement made by the same entity 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. 3433 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 3436 assert or imply endorsement of any Modified Version. 3437 3438 5. COMBINING DOCUMENTS 3439 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. 3447 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 3451 but different contents, make the title of each such section unique 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 3456 combined work. 3457 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." 3463 3464 6. COLLECTIONS OF DOCUMENTS 3465 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. 3472 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 3477 that document. 3478 3479 7. AGGREGATION WITH INDEPENDENT WORKS 3480 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 3486 works permit. When the Document is included in an aggregate, this 3487 License does not apply to the other works in the aggregate which 3488 are not themselves derivative works of the Document. 3489 3490 If the Cover Text requirement of section 3 is applicable to these 3491 copies of the Document, then if the Document is less than one half 3492 of the entire aggregate, the Document's Cover Texts may be placed 3493 on covers that bracket the Document within the aggregate, or the 3494 electronic equivalent of covers if the Document is in electronic 3495 form. Otherwise they must appear on printed covers that bracket 3496 the whole aggregate. 3497 3498 8. TRANSLATION 3499 3500 Translation is considered a kind of modification, so you may 3501 distribute translations of the Document under the terms of section 3502 4. Replacing Invariant Sections with translations requires special 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 3512 prevail. 3513 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 3517 actual title. 3518 3519 9. TERMINATION 3520 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. 3528 3529 10. FUTURE REVISIONS OF THIS LICENSE 3530 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/'. 3536 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. 3546 3547A.1 ADDENDUM: How to Use This License For Your Documents 3548======================================================== 3549 3550To use this License in a document you have written, include a copy of 3551the License in the document and put the following copyright and license 3552notices just after the title page: 3553 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''. 3561 3562 If you have Invariant Sections, Front-Cover Texts and Back-Cover 3563Texts, replace the "with...Texts." line with this: 3564 3565 with the Invariant Sections being LIST THEIR TITLES, with 3566 the Front-Cover Texts being LIST, and with the Back-Cover Texts 3567 being LIST. 3568 3569 If you have Invariant Sections without Cover Texts, or some other 3570combination of the three, merge those two alternatives to suit the 3571situation. 3572 3573 If your document contains nontrivial examples of program code, we 3574recommend releasing these examples in parallel under your choice of 3575free software license, such as the GNU General Public License, to 3576permit their use in free software. 3577 3578 3579File: mpfr.info, Node: Concept Index, Next: Function and Type Index, Prev: GNU Free Documentation License, Up: Top 3580 3581Concept Index 3582************* 3583 3584�[index�] 3585* Menu: 3586 3587* Accuracy: MPFR Interface. (line 25) 3588* Arithmetic functions: Basic Arithmetic Functions. 3589 (line 3) 3590* Assignment functions: Assignment Functions. (line 3) 3591* Basic arithmetic functions: Basic Arithmetic Functions. 3592 (line 3) 3593* Combined initialization and assignment functions: Combined Initialization and Assignment Functions. 3594 (line 3) 3595* Comparison functions: Comparison Functions. (line 3) 3596* Compatibility with MPF: Compatibility with MPF. 3597 (line 3) 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. 3603 (line 3) 3604* Float arithmetic functions: Basic Arithmetic Functions. 3605 (line 3) 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. 3609 (line 3) 3610* Float output functions: Formatted Output Functions. 3611 (line 3) 3612* Floating-point functions: MPFR Interface. (line 6) 3613* Floating-point number: Nomenclature and Types. 3614 (line 6) 3615* GNU Free Documentation License: GNU Free Documentation License. 3616 (line 6) 3617* I/O functions <1>: Formatted Output Functions. 3618 (line 3) 3619* I/O functions: Input and Output Functions. 3620 (line 3) 3621* Initialization functions: Initialization Functions. 3622 (line 3) 3623* Input functions: Input and Output Functions. 3624 (line 3) 3625* Installation: Installing MPFR. (line 6) 3626* Integer related functions: Integer Related Functions. 3627 (line 3) 3628* Internals: Internals. (line 3) 3629* intmax_t: Headers and Libraries. 3630 (line 22) 3631* inttypes.h: Headers and Libraries. 3632 (line 22) 3633* libmpfr: Headers and Libraries. 3634 (line 50) 3635* Libraries: Headers and Libraries. 3636 (line 50) 3637* Libtool: Headers and Libraries. 3638 (line 56) 3639* Limb: Internals. (line 6) 3640* Linking: Headers and Libraries. 3641 (line 50) 3642* Miscellaneous float functions: Miscellaneous Functions. 3643 (line 3) 3644* mpfr.h: Headers and Libraries. 3645 (line 6) 3646* Output functions <1>: Formatted Output Functions. 3647 (line 3) 3648* Output functions: Input and Output Functions. 3649 (line 3) 3650* Precision <1>: MPFR Interface. (line 17) 3651* Precision: Nomenclature and Types. 3652 (line 20) 3653* Reporting bugs: Reporting Bugs. (line 6) 3654* Rounding mode related functions: Rounding Related Functions. 3655 (line 3) 3656* Rounding Modes: Nomenclature and Types. 3657 (line 34) 3658* Special functions: Special Functions. (line 3) 3659* stdarg.h: Headers and Libraries. 3660 (line 19) 3661* stdint.h: Headers and Libraries. 3662 (line 22) 3663* stdio.h: Headers and Libraries. 3664 (line 12) 3665* Ternary value: Rounding Modes. (line 29) 3666* uintmax_t: Headers and Libraries. 3667 (line 22) 3668 3669 3670File: mpfr.info, Node: Function and Type Index, Prev: Concept Index, Up: Top 3671 3672Function and Type Index 3673*********************** 3674 3675�[index�] 3676* Menu: 3677 3678* mpfr_abs: Basic Arithmetic Functions. 3679 (line 175) 3680* mpfr_acos: Special Functions. (line 52) 3681* mpfr_acosh: Special Functions. (line 136) 3682* mpfr_add: Basic Arithmetic Functions. 3683 (line 8) 3684* mpfr_add_d: Basic Arithmetic Functions. 3685 (line 14) 3686* mpfr_add_q: Basic Arithmetic Functions. 3687 (line 18) 3688* mpfr_add_si: Basic Arithmetic Functions. 3689 (line 12) 3690* mpfr_add_ui: Basic Arithmetic Functions. 3691 (line 10) 3692* mpfr_add_z: Basic Arithmetic Functions. 3693 (line 16) 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. 3699 (line 194) 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. 3704 (line 163) 3705* mpfr_buildopt_gmpinternals_p: Miscellaneous Functions. 3706 (line 168) 3707* mpfr_buildopt_tls_p: Miscellaneous Functions. 3708 (line 157) 3709* mpfr_buildopt_tune_case: Miscellaneous Functions. 3710 (line 173) 3711* mpfr_can_round: Rounding Related Functions. 3712 (line 37) 3713* mpfr_cbrt: Basic Arithmetic Functions. 3714 (line 109) 3715* mpfr_ceil: Integer Related Functions. 3716 (line 8) 3717* mpfr_check_range: Exception Related Functions. 3718 (line 38) 3719* mpfr_clear: Initialization Functions. 3720 (line 31) 3721* mpfr_clear_divby0: Exception Related Functions. 3722 (line 113) 3723* mpfr_clear_erangeflag: Exception Related Functions. 3724 (line 116) 3725* mpfr_clear_flags: Exception Related Functions. 3726 (line 129) 3727* mpfr_clear_inexflag: Exception Related Functions. 3728 (line 115) 3729* mpfr_clear_nanflag: Exception Related Functions. 3730 (line 114) 3731* mpfr_clear_overflow: Exception Related Functions. 3732 (line 112) 3733* mpfr_clear_underflow: Exception Related Functions. 3734 (line 111) 3735* mpfr_clears: Initialization Functions. 3736 (line 36) 3737* mpfr_cmp: Comparison Functions. 3738 (line 7) 3739* mpfr_cmp_d: Comparison Functions. 3740 (line 10) 3741* mpfr_cmp_f: Comparison Functions. 3742 (line 14) 3743* mpfr_cmp_ld: Comparison Functions. 3744 (line 11) 3745* mpfr_cmp_q: Comparison Functions. 3746 (line 13) 3747* mpfr_cmp_si: Comparison Functions. 3748 (line 9) 3749* mpfr_cmp_si_2exp: Comparison Functions. 3750 (line 31) 3751* mpfr_cmp_ui: Comparison Functions. 3752 (line 8) 3753* mpfr_cmp_ui_2exp: Comparison Functions. 3754 (line 29) 3755* mpfr_cmp_z: Comparison Functions. 3756 (line 12) 3757* mpfr_cmpabs: Comparison Functions. 3758 (line 35) 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. 3764 (line 111) 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. 3779 (line 84) 3780* mpfr_d_sub: Basic Arithmetic Functions. 3781 (line 37) 3782* MPFR_DECL_INIT: Initialization Functions. 3783 (line 75) 3784* mpfr_digamma: Special Functions. (line 187) 3785* mpfr_dim: Basic Arithmetic Functions. 3786 (line 182) 3787* mpfr_div: Basic Arithmetic Functions. 3788 (line 74) 3789* mpfr_div_2exp: Compatibility with MPF. 3790 (line 51) 3791* mpfr_div_2si: Basic Arithmetic Functions. 3792 (line 197) 3793* mpfr_div_2ui: Basic Arithmetic Functions. 3794 (line 195) 3795* mpfr_div_d: Basic Arithmetic Functions. 3796 (line 86) 3797* mpfr_div_q: Basic Arithmetic Functions. 3798 (line 90) 3799* mpfr_div_si: Basic Arithmetic Functions. 3800 (line 82) 3801* mpfr_div_ui: Basic Arithmetic Functions. 3802 (line 78) 3803* mpfr_div_z: Basic Arithmetic Functions. 3804 (line 88) 3805* mpfr_divby0_p: Exception Related Functions. 3806 (line 135) 3807* mpfr_eint: Special Functions. (line 154) 3808* mpfr_eq: Compatibility with MPF. 3809 (line 30) 3810* mpfr_equal_p: Comparison Functions. 3811 (line 61) 3812* mpfr_erangeflag_p: Exception Related Functions. 3813 (line 138) 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. 3822 (line 146) 3823* mpfr_fits_sint_p: Conversion Functions. 3824 (line 142) 3825* mpfr_fits_slong_p: Conversion Functions. 3826 (line 140) 3827* mpfr_fits_sshort_p: Conversion Functions. 3828 (line 144) 3829* mpfr_fits_uint_p: Conversion Functions. 3830 (line 141) 3831* mpfr_fits_uintmax_p: Conversion Functions. 3832 (line 145) 3833* mpfr_fits_ulong_p: Conversion Functions. 3834 (line 139) 3835* mpfr_fits_ushort_p: Conversion Functions. 3836 (line 143) 3837* mpfr_floor: Integer Related Functions. 3838 (line 9) 3839* mpfr_fma: Special Functions. (line 225) 3840* mpfr_fmod: Integer Related Functions. 3841 (line 79) 3842* mpfr_fms: Special Functions. (line 227) 3843* mpfr_fprintf: Formatted Output Functions. 3844 (line 158) 3845* mpfr_frac: Integer Related Functions. 3846 (line 62) 3847* mpfr_free_cache: Special Functions. (line 266) 3848* mpfr_free_str: Conversion Functions. 3849 (line 133) 3850* mpfr_frexp: Conversion Functions. 3851 (line 47) 3852* mpfr_gamma: Special Functions. (line 169) 3853* mpfr_get_d: Conversion Functions. 3854 (line 8) 3855* mpfr_get_d_2exp: Conversion Functions. 3856 (line 34) 3857* mpfr_get_decimal64: Conversion Functions. 3858 (line 10) 3859* mpfr_get_default_prec: Initialization Functions. 3860 (line 114) 3861* mpfr_get_default_rounding_mode: Rounding Related Functions. 3862 (line 11) 3863* mpfr_get_emax: Exception Related Functions. 3864 (line 8) 3865* mpfr_get_emax_max: Exception Related Functions. 3866 (line 31) 3867* mpfr_get_emax_min: Exception Related Functions. 3868 (line 30) 3869* mpfr_get_emin: Exception Related Functions. 3870 (line 7) 3871* mpfr_get_emin_max: Exception Related Functions. 3872 (line 29) 3873* mpfr_get_emin_min: Exception Related Functions. 3874 (line 28) 3875* mpfr_get_exp: Miscellaneous Functions. 3876 (line 89) 3877* mpfr_get_f: Conversion Functions. 3878 (line 73) 3879* mpfr_get_flt: Conversion Functions. 3880 (line 7) 3881* mpfr_get_ld: Conversion Functions. 3882 (line 9) 3883* mpfr_get_ld_2exp: Conversion Functions. 3884 (line 36) 3885* mpfr_get_patches: Miscellaneous Functions. 3886 (line 148) 3887* mpfr_get_prec: Initialization Functions. 3888 (line 147) 3889* mpfr_get_si: Conversion Functions. 3890 (line 20) 3891* mpfr_get_sj: Conversion Functions. 3892 (line 22) 3893* mpfr_get_str: Conversion Functions. 3894 (line 87) 3895* mpfr_get_ui: Conversion Functions. 3896 (line 21) 3897* mpfr_get_uj: Conversion Functions. 3898 (line 23) 3899* mpfr_get_version: Miscellaneous Functions. 3900 (line 117) 3901* mpfr_get_z: Conversion Functions. 3902 (line 68) 3903* mpfr_get_z_2exp: Conversion Functions. 3904 (line 55) 3905* mpfr_grandom: Miscellaneous Functions. 3906 (line 65) 3907* mpfr_greater_p: Comparison Functions. 3908 (line 57) 3909* mpfr_greaterequal_p: Comparison Functions. 3910 (line 58) 3911* mpfr_hypot: Special Functions. (line 241) 3912* mpfr_inexflag_p: Exception Related Functions. 3913 (line 137) 3914* mpfr_inf_p: Comparison Functions. 3915 (line 42) 3916* mpfr_init: Initialization Functions. 3917 (line 54) 3918* mpfr_init2: Initialization Functions. 3919 (line 11) 3920* mpfr_init_set: Combined Initialization and Assignment Functions. 3921 (line 7) 3922* mpfr_init_set_d: Combined Initialization and Assignment Functions. 3923 (line 12) 3924* mpfr_init_set_f: Combined Initialization and Assignment Functions. 3925 (line 17) 3926* mpfr_init_set_ld: Combined Initialization and Assignment Functions. 3927 (line 14) 3928* mpfr_init_set_q: Combined Initialization and Assignment Functions. 3929 (line 16) 3930* mpfr_init_set_si: Combined Initialization and Assignment Functions. 3931 (line 11) 3932* mpfr_init_set_str: Combined Initialization and Assignment Functions. 3933 (line 23) 3934* mpfr_init_set_ui: Combined Initialization and Assignment Functions. 3935 (line 9) 3936* mpfr_init_set_z: Combined Initialization and Assignment Functions. 3937 (line 15) 3938* mpfr_inits: Initialization Functions. 3939 (line 63) 3940* mpfr_inits2: Initialization Functions. 3941 (line 23) 3942* mpfr_inp_str: Input and Output Functions. 3943 (line 33) 3944* mpfr_integer_p: Integer Related Functions. 3945 (line 105) 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. 3950 (line 59) 3951* mpfr_lessequal_p: Comparison Functions. 3952 (line 60) 3953* mpfr_lessgreater_p: Comparison Functions. 3954 (line 66) 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. 3963 (line 24) 3964* mpfr_min: Miscellaneous Functions. 3965 (line 22) 3966* mpfr_min_prec: Rounding Related Functions. 3967 (line 59) 3968* mpfr_modf: Integer Related Functions. 3969 (line 69) 3970* mpfr_mul: Basic Arithmetic Functions. 3971 (line 53) 3972* mpfr_mul_2exp: Compatibility with MPF. 3973 (line 49) 3974* mpfr_mul_2si: Basic Arithmetic Functions. 3975 (line 190) 3976* mpfr_mul_2ui: Basic Arithmetic Functions. 3977 (line 188) 3978* mpfr_mul_d: Basic Arithmetic Functions. 3979 (line 59) 3980* mpfr_mul_q: Basic Arithmetic Functions. 3981 (line 63) 3982* mpfr_mul_si: Basic Arithmetic Functions. 3983 (line 57) 3984* mpfr_mul_ui: Basic Arithmetic Functions. 3985 (line 55) 3986* mpfr_mul_z: Basic Arithmetic Functions. 3987 (line 61) 3988* mpfr_nan_p: Comparison Functions. 3989 (line 41) 3990* mpfr_nanflag_p: Exception Related Functions. 3991 (line 136) 3992* mpfr_neg: Basic Arithmetic Functions. 3993 (line 174) 3994* mpfr_nextabove: Miscellaneous Functions. 3995 (line 16) 3996* mpfr_nextbelow: Miscellaneous Functions. 3997 (line 17) 3998* mpfr_nexttoward: Miscellaneous Functions. 3999 (line 7) 4000* mpfr_number_p: Comparison Functions. 4001 (line 43) 4002* mpfr_out_str: Input and Output Functions. 4003 (line 17) 4004* mpfr_overflow_p: Exception Related Functions. 4005 (line 134) 4006* mpfr_pow: Basic Arithmetic Functions. 4007 (line 118) 4008* mpfr_pow_si: Basic Arithmetic Functions. 4009 (line 122) 4010* mpfr_pow_ui: Basic Arithmetic Functions. 4011 (line 120) 4012* mpfr_pow_z: Basic Arithmetic Functions. 4013 (line 124) 4014* mpfr_prec_round: Rounding Related Functions. 4015 (line 15) 4016* mpfr_prec_t: Nomenclature and Types. 4017 (line 20) 4018* mpfr_print_rnd_mode: Rounding Related Functions. 4019 (line 66) 4020* mpfr_printf: Formatted Output Functions. 4021 (line 165) 4022* mpfr_rec_sqrt: Basic Arithmetic Functions. 4023 (line 104) 4024* mpfr_regular_p: Comparison Functions. 4025 (line 45) 4026* mpfr_reldiff: Compatibility with MPF. 4027 (line 41) 4028* mpfr_remainder: Integer Related Functions. 4029 (line 81) 4030* mpfr_remquo: Integer Related Functions. 4031 (line 83) 4032* mpfr_rint: Integer Related Functions. 4033 (line 7) 4034* mpfr_rint_ceil: Integer Related Functions. 4035 (line 38) 4036* mpfr_rint_floor: Integer Related Functions. 4037 (line 40) 4038* mpfr_rint_round: Integer Related Functions. 4039 (line 42) 4040* mpfr_rint_trunc: Integer Related Functions. 4041 (line 44) 4042* mpfr_rnd_t: Nomenclature and Types. 4043 (line 34) 4044* mpfr_root: Basic Arithmetic Functions. 4045 (line 111) 4046* mpfr_round: Integer Related Functions. 4047 (line 10) 4048* mpfr_sec: Special Functions. (line 46) 4049* mpfr_sech: Special Functions. (line 130) 4050* mpfr_set: Assignment Functions. 4051 (line 10) 4052* mpfr_set_d: Assignment Functions. 4053 (line 17) 4054* mpfr_set_decimal64: Assignment Functions. 4055 (line 21) 4056* mpfr_set_default_prec: Initialization Functions. 4057 (line 101) 4058* mpfr_set_default_rounding_mode: Rounding Related Functions. 4059 (line 7) 4060* mpfr_set_divby0: Exception Related Functions. 4061 (line 122) 4062* mpfr_set_emax: Exception Related Functions. 4063 (line 17) 4064* mpfr_set_emin: Exception Related Functions. 4065 (line 16) 4066* mpfr_set_erangeflag: Exception Related Functions. 4067 (line 125) 4068* mpfr_set_exp: Miscellaneous Functions. 4069 (line 94) 4070* mpfr_set_f: Assignment Functions. 4071 (line 24) 4072* mpfr_set_flt: Assignment Functions. 4073 (line 16) 4074* mpfr_set_inexflag: Exception Related Functions. 4075 (line 124) 4076* mpfr_set_inf: Assignment Functions. 4077 (line 147) 4078* mpfr_set_ld: Assignment Functions. 4079 (line 19) 4080* mpfr_set_nan: Assignment Functions. 4081 (line 146) 4082* mpfr_set_nanflag: Exception Related Functions. 4083 (line 123) 4084* mpfr_set_overflow: Exception Related Functions. 4085 (line 121) 4086* mpfr_set_prec: Initialization Functions. 4087 (line 137) 4088* mpfr_set_prec_raw: Compatibility with MPF. 4089 (line 23) 4090* mpfr_set_q: Assignment Functions. 4091 (line 23) 4092* mpfr_set_si: Assignment Functions. 4093 (line 13) 4094* mpfr_set_si_2exp: Assignment Functions. 4095 (line 53) 4096* mpfr_set_sj: Assignment Functions. 4097 (line 15) 4098* mpfr_set_sj_2exp: Assignment Functions. 4099 (line 57) 4100* mpfr_set_str: Assignment Functions. 4101 (line 65) 4102* mpfr_set_ui: Assignment Functions. 4103 (line 12) 4104* mpfr_set_ui_2exp: Assignment Functions. 4105 (line 51) 4106* mpfr_set_uj: Assignment Functions. 4107 (line 14) 4108* mpfr_set_uj_2exp: Assignment Functions. 4109 (line 55) 4110* mpfr_set_underflow: Exception Related Functions. 4111 (line 120) 4112* mpfr_set_z: Assignment Functions. 4113 (line 22) 4114* mpfr_set_z_2exp: Assignment Functions. 4115 (line 59) 4116* mpfr_set_zero: Assignment Functions. 4117 (line 148) 4118* mpfr_setsign: Miscellaneous Functions. 4119 (line 105) 4120* mpfr_sgn: Comparison Functions. 4121 (line 51) 4122* mpfr_si_div: Basic Arithmetic Functions. 4123 (line 80) 4124* mpfr_si_sub: Basic Arithmetic Functions. 4125 (line 33) 4126* mpfr_signbit: Miscellaneous Functions. 4127 (line 100) 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. 4133 (line 182) 4134* mpfr_sprintf: Formatted Output Functions. 4135 (line 171) 4136* mpfr_sqr: Basic Arithmetic Functions. 4137 (line 70) 4138* mpfr_sqrt: Basic Arithmetic Functions. 4139 (line 97) 4140* mpfr_sqrt_ui: Basic Arithmetic Functions. 4141 (line 99) 4142* mpfr_strtofr: Assignment Functions. 4143 (line 83) 4144* mpfr_sub: Basic Arithmetic Functions. 4145 (line 27) 4146* mpfr_sub_d: Basic Arithmetic Functions. 4147 (line 39) 4148* mpfr_sub_q: Basic Arithmetic Functions. 4149 (line 45) 4150* mpfr_sub_si: Basic Arithmetic Functions. 4151 (line 35) 4152* mpfr_sub_ui: Basic Arithmetic Functions. 4153 (line 31) 4154* mpfr_sub_z: Basic Arithmetic Functions. 4155 (line 43) 4156* mpfr_subnormalize: Exception Related Functions. 4157 (line 61) 4158* mpfr_sum: Special Functions. (line 275) 4159* mpfr_swap: Assignment Functions. 4160 (line 154) 4161* mpfr_t: Nomenclature and Types. 4162 (line 6) 4163* mpfr_tan: Special Functions. (line 32) 4164* mpfr_tanh: Special Functions. (line 117) 4165* mpfr_trunc: Integer Related Functions. 4166 (line 11) 4167* mpfr_ui_div: Basic Arithmetic Functions. 4168 (line 76) 4169* mpfr_ui_pow: Basic Arithmetic Functions. 4170 (line 128) 4171* mpfr_ui_pow_ui: Basic Arithmetic Functions. 4172 (line 126) 4173* mpfr_ui_sub: Basic Arithmetic Functions. 4174 (line 29) 4175* mpfr_underflow_p: Exception Related Functions. 4176 (line 133) 4177* mpfr_unordered_p: Comparison Functions. 4178 (line 71) 4179* mpfr_urandom: Miscellaneous Functions. 4180 (line 50) 4181* mpfr_urandomb: Miscellaneous Functions. 4182 (line 30) 4183* mpfr_vasprintf: Formatted Output Functions. 4184 (line 196) 4185* MPFR_VERSION: Miscellaneous Functions. 4186 (line 120) 4187* MPFR_VERSION_MAJOR: Miscellaneous Functions. 4188 (line 121) 4189* MPFR_VERSION_MINOR: Miscellaneous Functions. 4190 (line 122) 4191* MPFR_VERSION_NUM: Miscellaneous Functions. 4192 (line 140) 4193* MPFR_VERSION_PATCHLEVEL: Miscellaneous Functions. 4194 (line 123) 4195* MPFR_VERSION_STRING: Miscellaneous Functions. 4196 (line 124) 4197* mpfr_vfprintf: Formatted Output Functions. 4198 (line 160) 4199* mpfr_vprintf: Formatted Output Functions. 4200 (line 166) 4201* mpfr_vsnprintf: Formatted Output Functions. 4202 (line 184) 4203* mpfr_vsprintf: Formatted Output Functions. 4204 (line 173) 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. 4209 (line 41) 4210* mpfr_zero_p: Comparison Functions. 4211 (line 44) 4212* mpfr_zeta: Special Functions. (line 192) 4213* mpfr_zeta_ui: Special Functions. (line 194) 4214 4215 4216 4217Tag Table: 4218Node: Top892 4219Node: Copying2243 4220Node: Introduction to MPFR4003 4221Node: Installing MPFR6092 4222Node: Reporting Bugs10914 4223Node: MPFR Basics12843 4224Node: Headers and Libraries13159 4225Node: Nomenclature and Types16143 4226Node: MPFR Variable Conventions18147 4227Node: Rounding Modes19677 4228Ref: ternary value20774 4229Node: Floating-Point Values on Special Numbers22727 4230Node: Exceptions25703 4231Node: Memory Handling28855 4232Node: MPFR Interface29987 4233Node: Initialization Functions32083 4234Node: Assignment Functions38997 4235Node: Combined Initialization and Assignment Functions47651 4236Node: Conversion Functions48944 4237Node: Basic Arithmetic Functions57496 4238Node: Comparison Functions66504 4239Node: Special Functions69986 4240Node: Input and Output Functions83739 4241Node: Formatted Output Functions85662 4242Node: Integer Related Functions94781 4243Node: Rounding Related Functions100543 4244Node: Miscellaneous Functions104157 4245Node: Exception Related Functions112724 4246Node: Compatibility with MPF119478 4247Node: Custom Interface122166 4248Node: Internals126411 4249Node: API Compatibility127895 4250Node: Type and Macro Changes129825 4251Node: Added Functions132546 4252Node: Changed Functions135489 4253Node: Removed Functions139770 4254Node: Other Changes140182 4255Node: Contributors141711 4256Node: References144285 4257Node: GNU Free Documentation License146026 4258Node: Concept Index168469 4259Node: Function and Type Index174388 4260 4261End Tag Table 4262 4263 4264Local Variables: 4265coding: utf-8 4266End: 4267