1<html> 2<head> 3<meta http-equiv="Content-Type" content="text/html; charset=UTF-8"> 4<title>Conceptual Requirements for Real Number Types</title> 5<link rel="stylesheet" href="../math.css" type="text/css"> 6<meta name="generator" content="DocBook XSL Stylesheets V1.79.1"> 7<link rel="home" href="../index.html" title="Math Toolkit 2.12.0"> 8<link rel="up" href="../using_udt.html" title="Chapter 20. Use with User-Defined Floating-Point Types - Boost.Multiprecision and others"> 9<link rel="prev" href="high_precision/using_test.html" title="Using without expression templates for Boost.Test and others"> 10<link rel="next" href="dist_concept.html" title="Conceptual Requirements for Distribution Types"> 11</head> 12<body bgcolor="white" text="black" link="#0000FF" vlink="#840084" alink="#0000FF"> 13<table cellpadding="2" width="100%"><tr> 14<td valign="top"><img alt="Boost C++ Libraries" width="277" height="86" src="../../../../../boost.png"></td> 15<td align="center"><a href="../../../../../index.html">Home</a></td> 16<td align="center"><a href="../../../../../libs/libraries.htm">Libraries</a></td> 17<td align="center"><a href="http://www.boost.org/users/people.html">People</a></td> 18<td align="center"><a href="http://www.boost.org/users/faq.html">FAQ</a></td> 19<td align="center"><a href="../../../../../more/index.htm">More</a></td> 20</tr></table> 21<hr> 22<div class="spirit-nav"> 23<a accesskey="p" href="high_precision/using_test.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../using_udt.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="dist_concept.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 24</div> 25<div class="section"> 26<div class="titlepage"><div><div><h2 class="title" style="clear: both"> 27<a name="math_toolkit.real_concepts"></a><a class="link" href="real_concepts.html" title="Conceptual Requirements for Real Number Types">Conceptual Requirements for 28 Real Number Types</a> 29</h2></div></div></div> 30<p> 31 The functions and statistical distributions in this library can be used with 32 any type <span class="emphasis"><em>RealType</em></span> that meets the conceptual requirements 33 given below. All the built-in floating-point types like <code class="computeroutput"><span class="keyword">double</span></code> 34 will meet these requirements. (Built-in types are also called <a href="http://en.cppreference.com/w/cpp/language/types" target="_top">fundamental 35 (built-in) types</a>). 36 </p> 37<p> 38 User-defined types that meet the conceptual requirements can also be used. 39 For example, with <a class="link" href="high_precision/use_ntl.html" title="Using NTL Library">a thin 40 wrapper class</a> one of the types provided with <a href="http://shoup.net/ntl/" target="_top">NTL 41 (RR)</a> can be used. But now that <a href="../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a> 42 library is available, this has become the preferred real-number type, typically 43 <a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_dec_float.html" target="_top">cpp_dec_float</a> 44 or <a href="../../../../../libs/multiprecision/doc/html/boost_multiprecision/tut/floats/cpp_bin_float.html" target="_top">cpp_bin_float</a>. 45 </p> 46<p> 47 Submissions of binding to other extended precision types would also still be 48 welcome. 49 </p> 50<p> 51 The guiding principal behind these requirements is that a <span class="emphasis"><em>RealType</em></span> 52 behaves just like a built-in floating-point type. 53 </p> 54<h5> 55<a name="math_toolkit.real_concepts.h0"></a> 56 <span class="phrase"><a name="math_toolkit.real_concepts.basic_arithmetic_requirements"></a></span><a class="link" href="real_concepts.html#math_toolkit.real_concepts.basic_arithmetic_requirements">Basic Arithmetic 57 Requirements</a> 58 </h5> 59<p> 60 These requirements are common to all of the functions in this library. 61 </p> 62<p> 63 In the following table <span class="emphasis"><em>r</em></span> is an object of type <code class="computeroutput"><span class="identifier">RealType</span></code>, <span class="emphasis"><em>cr</em></span> and <span class="emphasis"><em>cr2</em></span> 64 are objects of type <code class="computeroutput"><span class="keyword">const</span> <span class="identifier">RealType</span></code>, 65 and <span class="emphasis"><em>ca</em></span> is an object of type <code class="computeroutput"><span class="keyword">const</span> 66 <span class="identifier">arithmetic</span><span class="special">-</span><span class="identifier">type</span></code> (arithmetic types include all the built 67 in integers and floating point types). 68 </p> 69<div class="informaltable"><table class="table"> 70<colgroup> 71<col> 72<col> 73<col> 74</colgroup> 75<thead><tr> 76<th> 77 <p> 78 Expression 79 </p> 80 </th> 81<th> 82 <p> 83 Result Type 84 </p> 85 </th> 86<th> 87 <p> 88 Notes 89 </p> 90 </th> 91</tr></thead> 92<tbody> 93<tr> 94<td> 95 <p> 96 <code class="computeroutput"><span class="identifier">RealType</span><span class="special">(</span><span class="identifier">cr</span><span class="special">)</span></code> 97 </p> 98 </td> 99<td> 100 <p> 101 RealType 102 </p> 103 </td> 104<td> 105 <p> 106 RealType is copy constructible. 107 </p> 108 </td> 109</tr> 110<tr> 111<td> 112 <p> 113 <code class="computeroutput"><span class="identifier">RealType</span><span class="special">(</span><span class="identifier">ca</span><span class="special">)</span></code> 114 </p> 115 </td> 116<td> 117 <p> 118 RealType 119 </p> 120 </td> 121<td> 122 <p> 123 RealType is copy constructible from the arithmetic types. 124 </p> 125 </td> 126</tr> 127<tr> 128<td> 129 <p> 130 <code class="computeroutput"><span class="identifier">r</span> <span class="special">=</span> 131 <span class="identifier">cr</span></code> 132 </p> 133 </td> 134<td> 135 <p> 136 RealType& 137 </p> 138 </td> 139<td> 140 <p> 141 Assignment operator. 142 </p> 143 </td> 144</tr> 145<tr> 146<td> 147 <p> 148 <code class="computeroutput"><span class="identifier">r</span> <span class="special">=</span> 149 <span class="identifier">ca</span></code> 150 </p> 151 </td> 152<td> 153 <p> 154 RealType& 155 </p> 156 </td> 157<td> 158 <p> 159 Assignment operator from the arithmetic types. 160 </p> 161 </td> 162</tr> 163<tr> 164<td> 165 <p> 166 <code class="computeroutput"><span class="identifier">r</span> <span class="special">+=</span> 167 <span class="identifier">cr</span></code> 168 </p> 169 </td> 170<td> 171 <p> 172 RealType& 173 </p> 174 </td> 175<td> 176 <p> 177 Adds cr to r. 178 </p> 179 </td> 180</tr> 181<tr> 182<td> 183 <p> 184 <code class="computeroutput"><span class="identifier">r</span> <span class="special">+=</span> 185 <span class="identifier">ca</span></code> 186 </p> 187 </td> 188<td> 189 <p> 190 RealType& 191 </p> 192 </td> 193<td> 194 <p> 195 Adds ar to r. 196 </p> 197 </td> 198</tr> 199<tr> 200<td> 201 <p> 202 <code class="computeroutput"><span class="identifier">r</span> <span class="special">-=</span> 203 <span class="identifier">cr</span></code> 204 </p> 205 </td> 206<td> 207 <p> 208 RealType& 209 </p> 210 </td> 211<td> 212 <p> 213 Subtracts cr from r. 214 </p> 215 </td> 216</tr> 217<tr> 218<td> 219 <p> 220 <code class="computeroutput"><span class="identifier">r</span> <span class="special">-=</span> 221 <span class="identifier">ca</span></code> 222 </p> 223 </td> 224<td> 225 <p> 226 RealType& 227 </p> 228 </td> 229<td> 230 <p> 231 Subtracts ca from r. 232 </p> 233 </td> 234</tr> 235<tr> 236<td> 237 <p> 238 <code class="computeroutput"><span class="identifier">r</span> <span class="special">*=</span> 239 <span class="identifier">cr</span></code> 240 </p> 241 </td> 242<td> 243 <p> 244 RealType& 245 </p> 246 </td> 247<td> 248 <p> 249 Multiplies r by cr. 250 </p> 251 </td> 252</tr> 253<tr> 254<td> 255 <p> 256 <code class="computeroutput"><span class="identifier">r</span> <span class="special">*=</span> 257 <span class="identifier">ca</span></code> 258 </p> 259 </td> 260<td> 261 <p> 262 RealType& 263 </p> 264 </td> 265<td> 266 <p> 267 Multiplies r by ca. 268 </p> 269 </td> 270</tr> 271<tr> 272<td> 273 <p> 274 <code class="computeroutput"><span class="identifier">r</span> <span class="special">/=</span> 275 <span class="identifier">cr</span></code> 276 </p> 277 </td> 278<td> 279 <p> 280 RealType& 281 </p> 282 </td> 283<td> 284 <p> 285 Divides r by cr. 286 </p> 287 </td> 288</tr> 289<tr> 290<td> 291 <p> 292 <code class="computeroutput"><span class="identifier">r</span> <span class="special">/=</span> 293 <span class="identifier">ca</span></code> 294 </p> 295 </td> 296<td> 297 <p> 298 RealType& 299 </p> 300 </td> 301<td> 302 <p> 303 Divides r by ca. 304 </p> 305 </td> 306</tr> 307<tr> 308<td> 309 <p> 310 <code class="computeroutput"><span class="special">-</span><span class="identifier">r</span></code> 311 </p> 312 </td> 313<td> 314 <p> 315 RealType 316 </p> 317 </td> 318<td> 319 <p> 320 Unary Negation. 321 </p> 322 </td> 323</tr> 324<tr> 325<td> 326 <p> 327 <code class="computeroutput"><span class="special">+</span><span class="identifier">r</span></code> 328 </p> 329 </td> 330<td> 331 <p> 332 RealType& 333 </p> 334 </td> 335<td> 336 <p> 337 Identity Operation. 338 </p> 339 </td> 340</tr> 341<tr> 342<td> 343 <p> 344 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">+</span> 345 <span class="identifier">cr2</span></code> 346 </p> 347 </td> 348<td> 349 <p> 350 RealType 351 </p> 352 </td> 353<td> 354 <p> 355 Binary Addition 356 </p> 357 </td> 358</tr> 359<tr> 360<td> 361 <p> 362 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">+</span> 363 <span class="identifier">ca</span></code> 364 </p> 365 </td> 366<td> 367 <p> 368 RealType 369 </p> 370 </td> 371<td> 372 <p> 373 Binary Addition 374 </p> 375 </td> 376</tr> 377<tr> 378<td> 379 <p> 380 <code class="computeroutput"><span class="identifier">ca</span> <span class="special">+</span> 381 <span class="identifier">cr</span></code> 382 </p> 383 </td> 384<td> 385 <p> 386 RealType 387 </p> 388 </td> 389<td> 390 <p> 391 Binary Addition 392 </p> 393 </td> 394</tr> 395<tr> 396<td> 397 <p> 398 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">-</span> 399 <span class="identifier">cr2</span></code> 400 </p> 401 </td> 402<td> 403 <p> 404 RealType 405 </p> 406 </td> 407<td> 408 <p> 409 Binary Subtraction 410 </p> 411 </td> 412</tr> 413<tr> 414<td> 415 <p> 416 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">-</span> 417 <span class="identifier">ca</span></code> 418 </p> 419 </td> 420<td> 421 <p> 422 RealType 423 </p> 424 </td> 425<td> 426 <p> 427 Binary Subtraction 428 </p> 429 </td> 430</tr> 431<tr> 432<td> 433 <p> 434 <code class="computeroutput"><span class="identifier">ca</span> <span class="special">-</span> 435 <span class="identifier">cr</span></code> 436 </p> 437 </td> 438<td> 439 <p> 440 RealType 441 </p> 442 </td> 443<td> 444 <p> 445 Binary Subtraction 446 </p> 447 </td> 448</tr> 449<tr> 450<td> 451 <p> 452 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">*</span> 453 <span class="identifier">cr2</span></code> 454 </p> 455 </td> 456<td> 457 <p> 458 RealType 459 </p> 460 </td> 461<td> 462 <p> 463 Binary Multiplication 464 </p> 465 </td> 466</tr> 467<tr> 468<td> 469 <p> 470 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">*</span> 471 <span class="identifier">ca</span></code> 472 </p> 473 </td> 474<td> 475 <p> 476 RealType 477 </p> 478 </td> 479<td> 480 <p> 481 Binary Multiplication 482 </p> 483 </td> 484</tr> 485<tr> 486<td> 487 <p> 488 <code class="computeroutput"><span class="identifier">ca</span> <span class="special">*</span> 489 <span class="identifier">cr</span></code> 490 </p> 491 </td> 492<td> 493 <p> 494 RealType 495 </p> 496 </td> 497<td> 498 <p> 499 Binary Multiplication 500 </p> 501 </td> 502</tr> 503<tr> 504<td> 505 <p> 506 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">/</span> 507 <span class="identifier">cr2</span></code> 508 </p> 509 </td> 510<td> 511 <p> 512 RealType 513 </p> 514 </td> 515<td> 516 <p> 517 Binary Subtraction 518 </p> 519 </td> 520</tr> 521<tr> 522<td> 523 <p> 524 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">/</span> 525 <span class="identifier">ca</span></code> 526 </p> 527 </td> 528<td> 529 <p> 530 RealType 531 </p> 532 </td> 533<td> 534 <p> 535 Binary Subtraction 536 </p> 537 </td> 538</tr> 539<tr> 540<td> 541 <p> 542 <code class="computeroutput"><span class="identifier">ca</span> <span class="special">/</span> 543 <span class="identifier">cr</span></code> 544 </p> 545 </td> 546<td> 547 <p> 548 RealType 549 </p> 550 </td> 551<td> 552 <p> 553 Binary Subtraction 554 </p> 555 </td> 556</tr> 557<tr> 558<td> 559 <p> 560 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">==</span> 561 <span class="identifier">cr2</span></code> 562 </p> 563 </td> 564<td> 565 <p> 566 bool 567 </p> 568 </td> 569<td> 570 <p> 571 Equality Comparison 572 </p> 573 </td> 574</tr> 575<tr> 576<td> 577 <p> 578 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">==</span> 579 <span class="identifier">ca</span></code> 580 </p> 581 </td> 582<td> 583 <p> 584 bool 585 </p> 586 </td> 587<td> 588 <p> 589 Equality Comparison 590 </p> 591 </td> 592</tr> 593<tr> 594<td> 595 <p> 596 <code class="computeroutput"><span class="identifier">ca</span> <span class="special">==</span> 597 <span class="identifier">cr</span></code> 598 </p> 599 </td> 600<td> 601 <p> 602 bool 603 </p> 604 </td> 605<td> 606 <p> 607 Equality Comparison 608 </p> 609 </td> 610</tr> 611<tr> 612<td> 613 <p> 614 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">!=</span> 615 <span class="identifier">cr2</span></code> 616 </p> 617 </td> 618<td> 619 <p> 620 bool 621 </p> 622 </td> 623<td> 624 <p> 625 Inequality Comparison 626 </p> 627 </td> 628</tr> 629<tr> 630<td> 631 <p> 632 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">!=</span> 633 <span class="identifier">ca</span></code> 634 </p> 635 </td> 636<td> 637 <p> 638 bool 639 </p> 640 </td> 641<td> 642 <p> 643 Inequality Comparison 644 </p> 645 </td> 646</tr> 647<tr> 648<td> 649 <p> 650 <code class="computeroutput"><span class="identifier">ca</span> <span class="special">!=</span> 651 <span class="identifier">cr</span></code> 652 </p> 653 </td> 654<td> 655 <p> 656 bool 657 </p> 658 </td> 659<td> 660 <p> 661 Inequality Comparison 662 </p> 663 </td> 664</tr> 665<tr> 666<td> 667 <p> 668 <code class="computeroutput"><span class="identifier">cr</span> <span class="special"><=</span> 669 <span class="identifier">cr2</span></code> 670 </p> 671 </td> 672<td> 673 <p> 674 bool 675 </p> 676 </td> 677<td> 678 <p> 679 Less than equal to. 680 </p> 681 </td> 682</tr> 683<tr> 684<td> 685 <p> 686 <code class="computeroutput"><span class="identifier">cr</span> <span class="special"><=</span> 687 <span class="identifier">ca</span></code> 688 </p> 689 </td> 690<td> 691 <p> 692 bool 693 </p> 694 </td> 695<td> 696 <p> 697 Less than equal to. 698 </p> 699 </td> 700</tr> 701<tr> 702<td> 703 <p> 704 <code class="computeroutput"><span class="identifier">ca</span> <span class="special"><=</span> 705 <span class="identifier">cr</span></code> 706 </p> 707 </td> 708<td> 709 <p> 710 bool 711 </p> 712 </td> 713<td> 714 <p> 715 Less than equal to. 716 </p> 717 </td> 718</tr> 719<tr> 720<td> 721 <p> 722 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">>=</span> 723 <span class="identifier">cr2</span></code> 724 </p> 725 </td> 726<td> 727 <p> 728 bool 729 </p> 730 </td> 731<td> 732 <p> 733 Greater than equal to. 734 </p> 735 </td> 736</tr> 737<tr> 738<td> 739 <p> 740 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">>=</span> 741 <span class="identifier">ca</span></code> 742 </p> 743 </td> 744<td> 745 <p> 746 bool 747 </p> 748 </td> 749<td> 750 <p> 751 Greater than equal to. 752 </p> 753 </td> 754</tr> 755<tr> 756<td> 757 <p> 758 <code class="computeroutput"><span class="identifier">ca</span> <span class="special">>=</span> 759 <span class="identifier">cr</span></code> 760 </p> 761 </td> 762<td> 763 <p> 764 bool 765 </p> 766 </td> 767<td> 768 <p> 769 Greater than equal to. 770 </p> 771 </td> 772</tr> 773<tr> 774<td> 775 <p> 776 <code class="computeroutput"><span class="identifier">cr</span> <span class="special"><</span> 777 <span class="identifier">cr2</span></code> 778 </p> 779 </td> 780<td> 781 <p> 782 bool 783 </p> 784 </td> 785<td> 786 <p> 787 Less than comparison. 788 </p> 789 </td> 790</tr> 791<tr> 792<td> 793 <p> 794 <code class="computeroutput"><span class="identifier">cr</span> <span class="special"><</span> 795 <span class="identifier">ca</span></code> 796 </p> 797 </td> 798<td> 799 <p> 800 bool 801 </p> 802 </td> 803<td> 804 <p> 805 Less than comparison. 806 </p> 807 </td> 808</tr> 809<tr> 810<td> 811 <p> 812 <code class="computeroutput"><span class="identifier">ca</span> <span class="special"><</span> 813 <span class="identifier">cr</span></code> 814 </p> 815 </td> 816<td> 817 <p> 818 bool 819 </p> 820 </td> 821<td> 822 <p> 823 Less than comparison. 824 </p> 825 </td> 826</tr> 827<tr> 828<td> 829 <p> 830 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">></span> 831 <span class="identifier">cr2</span></code> 832 </p> 833 </td> 834<td> 835 <p> 836 bool 837 </p> 838 </td> 839<td> 840 <p> 841 Greater than comparison. 842 </p> 843 </td> 844</tr> 845<tr> 846<td> 847 <p> 848 <code class="computeroutput"><span class="identifier">cr</span> <span class="special">></span> 849 <span class="identifier">ca</span></code> 850 </p> 851 </td> 852<td> 853 <p> 854 bool 855 </p> 856 </td> 857<td> 858 <p> 859 Greater than comparison. 860 </p> 861 </td> 862</tr> 863<tr> 864<td> 865 <p> 866 <code class="computeroutput"><span class="identifier">ca</span> <span class="special">></span> 867 <span class="identifier">cr</span></code> 868 </p> 869 </td> 870<td> 871 <p> 872 bool 873 </p> 874 </td> 875<td> 876 <p> 877 Greater than comparison. 878 </p> 879 </td> 880</tr> 881<tr> 882<td> 883 <p> 884 <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">digits</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span></code> 885 </p> 886 </td> 887<td> 888 <p> 889 int 890 </p> 891 </td> 892<td> 893 <p> 894 The number of digits in the significand of RealType. 895 </p> 896 </td> 897</tr> 898<tr> 899<td> 900 <p> 901 <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">max_value</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span></code> 902 </p> 903 </td> 904<td> 905 <p> 906 RealType 907 </p> 908 </td> 909<td> 910 <p> 911 The largest representable number by type RealType. 912 </p> 913 </td> 914</tr> 915<tr> 916<td> 917 <p> 918 <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">min_value</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span></code> 919 </p> 920 </td> 921<td> 922 <p> 923 RealType 924 </p> 925 </td> 926<td> 927 <p> 928 The smallest representable number by type RealType. 929 </p> 930 </td> 931</tr> 932<tr> 933<td> 934 <p> 935 <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">log_max_value</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span></code> 936 </p> 937 </td> 938<td> 939 <p> 940 RealType 941 </p> 942 </td> 943<td> 944 <p> 945 The natural logarithm of the largest representable number by type 946 RealType. 947 </p> 948 </td> 949</tr> 950<tr> 951<td> 952 <p> 953 <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">log_min_value</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span></code> 954 </p> 955 </td> 956<td> 957 <p> 958 RealType 959 </p> 960 </td> 961<td> 962 <p> 963 The natural logarithm of the smallest representable number by type 964 RealType. 965 </p> 966 </td> 967</tr> 968<tr> 969<td> 970 <p> 971 <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">math</span><span class="special">::</span><span class="identifier">tools</span><span class="special">::</span><span class="identifier">epsilon</span><span class="special"><</span><span class="identifier">RealType</span><span class="special">>()</span></code> 972 </p> 973 </td> 974<td> 975 <p> 976 RealType 977 </p> 978 </td> 979<td> 980 <p> 981 The machine epsilon of RealType. 982 </p> 983 </td> 984</tr> 985</tbody> 986</table></div> 987<p> 988 Note that: 989 </p> 990<div class="orderedlist"><ol class="orderedlist" type="1"> 991<li class="listitem"> 992 The functions <code class="computeroutput"><span class="identifier">log_max_value</span></code> 993 and <code class="computeroutput"><span class="identifier">log_min_value</span></code> can be 994 synthesised from the others, and so no explicit specialisation is required. 995 </li> 996<li class="listitem"> 997 The function <code class="computeroutput"><span class="identifier">epsilon</span></code> can 998 be synthesised from the others, so no explicit specialisation is required 999 provided the precision of RealType does not vary at runtime (see the header 1000 <a href="../../../../../boost/math/bindings/rr.hpp" target="_top">boost/math/bindings/rr.hpp</a> 1001 for an example where the precision does vary at runtime). 1002 </li> 1003<li class="listitem"> 1004 The functions <code class="computeroutput"><span class="identifier">digits</span></code>, 1005 <code class="computeroutput"><span class="identifier">max_value</span></code> and <code class="computeroutput"><span class="identifier">min_value</span></code>, all get synthesised automatically 1006 from <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span></code>. However, if <code class="computeroutput"><span class="identifier">numeric_limits</span></code> is not specialised for 1007 type RealType, then you will get a compiler error when code tries to use 1008 these functions, <span class="emphasis"><em>unless</em></span> you explicitly specialise 1009 them. For example if the precision of RealType varies at runtime, then 1010 <code class="computeroutput"><span class="identifier">numeric_limits</span></code> support 1011 may not be appropriate, see <a href="../../../../../boost/math/bindings/rr.hpp" target="_top">boost/math/bindings/rr.hpp</a> 1012 for examples. 1013 </li> 1014</ol></div> 1015<div class="warning"><table border="0" summary="Warning"> 1016<tr> 1017<td rowspan="2" align="center" valign="top" width="25"><img alt="[Warning]" src="../../../../../doc/src/images/warning.png"></td> 1018<th align="left">Warning</th> 1019</tr> 1020<tr><td align="left" valign="top"> 1021<p> 1022 If <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span><span class="special"><></span></code> 1023 is <span class="bold"><strong>not specialized</strong></span> for type <span class="emphasis"><em>RealType</em></span> 1024 then the default float precision of 6 decimal digits will be used by other 1025 Boost programs including: 1026 </p> 1027<p> 1028 Boost.Test: giving misleading error messages like 1029 </p> 1030<p> 1031 <span class="emphasis"><em>"difference between {9.79796} and {9.79796} exceeds 5.42101e-19%".</em></span> 1032 </p> 1033<p> 1034 Boost.LexicalCast and Boost.Serialization when converting the number to a 1035 string, causing potentially serious loss of accuracy on output. 1036 </p> 1037<p> 1038 Although it might seem obvious that RealType should require <code class="computeroutput"><span class="identifier">std</span><span class="special">::</span><span class="identifier">numeric_limits</span></code> 1039 to be specialized, this is not sensible for <code class="computeroutput"><span class="identifier">NTL</span><span class="special">::</span><span class="identifier">RR</span></code> and 1040 similar classes where the <span class="bold"><strong>number of digits is a runtime 1041 parameter</strong></span> (whereas for <code class="computeroutput"><span class="identifier">numeric_limits</span></code> 1042 everything has to be fixed at compile time). 1043 </p> 1044</td></tr> 1045</table></div> 1046<h5> 1047<a name="math_toolkit.real_concepts.h1"></a> 1048 <span class="phrase"><a name="math_toolkit.real_concepts.standard_library_support_require"></a></span><a class="link" href="real_concepts.html#math_toolkit.real_concepts.standard_library_support_require">Standard 1049 Library Support Requirements</a> 1050 </h5> 1051<p> 1052 Many (though not all) of the functions in this library make calls to standard 1053 library functions, the following table summarises the requirements. Note that 1054 most of the functions in this library will only call a small subset of the 1055 functions listed here, so if in doubt whether a user-defined type has enough 1056 standard library support to be useable the best advise is to try it and see! 1057 </p> 1058<p> 1059 In the following table <span class="emphasis"><em>r</em></span> is an object of type <code class="computeroutput"><span class="identifier">RealType</span></code>, <span class="emphasis"><em>cr1</em></span> and <span class="emphasis"><em>cr2</em></span> 1060 are objects of type <code class="computeroutput"><span class="keyword">const</span> <span class="identifier">RealType</span></code>, 1061 and <span class="emphasis"><em>i</em></span> is an object of type <code class="computeroutput"><span class="keyword">int</span></code>. 1062 </p> 1063<div class="informaltable"><table class="table"> 1064<colgroup> 1065<col> 1066<col> 1067</colgroup> 1068<thead><tr> 1069<th> 1070 <p> 1071 Expression 1072 </p> 1073 </th> 1074<th> 1075 <p> 1076 Result Type 1077 </p> 1078 </th> 1079</tr></thead> 1080<tbody> 1081<tr> 1082<td> 1083 <p> 1084 <code class="computeroutput"><span class="identifier">fabs</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1085 </p> 1086 </td> 1087<td> 1088 <p> 1089 RealType 1090 </p> 1091 </td> 1092</tr> 1093<tr> 1094<td> 1095 <p> 1096 <code class="computeroutput"><span class="identifier">abs</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1097 </p> 1098 </td> 1099<td> 1100 <p> 1101 RealType 1102 </p> 1103 </td> 1104</tr> 1105<tr> 1106<td> 1107 <p> 1108 <code class="computeroutput"><span class="identifier">ceil</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1109 </p> 1110 </td> 1111<td> 1112 <p> 1113 RealType 1114 </p> 1115 </td> 1116</tr> 1117<tr> 1118<td> 1119 <p> 1120 <code class="computeroutput"><span class="identifier">floor</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1121 </p> 1122 </td> 1123<td> 1124 <p> 1125 RealType 1126 </p> 1127 </td> 1128</tr> 1129<tr> 1130<td> 1131 <p> 1132 <code class="computeroutput"><span class="identifier">exp</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1133 </p> 1134 </td> 1135<td> 1136 <p> 1137 RealType 1138 </p> 1139 </td> 1140</tr> 1141<tr> 1142<td> 1143 <p> 1144 <code class="computeroutput"><span class="identifier">pow</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">,</span> 1145 <span class="identifier">cr2</span><span class="special">)</span></code> 1146 </p> 1147 </td> 1148<td> 1149 <p> 1150 RealType 1151 </p> 1152 </td> 1153</tr> 1154<tr> 1155<td> 1156 <p> 1157 <code class="computeroutput"><span class="identifier">sqrt</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1158 </p> 1159 </td> 1160<td> 1161 <p> 1162 RealType 1163 </p> 1164 </td> 1165</tr> 1166<tr> 1167<td> 1168 <p> 1169 <code class="computeroutput"><span class="identifier">log</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1170 </p> 1171 </td> 1172<td> 1173 <p> 1174 RealType 1175 </p> 1176 </td> 1177</tr> 1178<tr> 1179<td> 1180 <p> 1181 <code class="computeroutput"><span class="identifier">frexp</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">,</span> 1182 <span class="special">&</span><span class="identifier">i</span><span class="special">)</span></code> 1183 </p> 1184 </td> 1185<td> 1186 <p> 1187 RealType 1188 </p> 1189 </td> 1190</tr> 1191<tr> 1192<td> 1193 <p> 1194 <code class="computeroutput"><span class="identifier">ldexp</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">,</span> 1195 <span class="identifier">i</span><span class="special">)</span></code> 1196 </p> 1197 </td> 1198<td> 1199 <p> 1200 RealType 1201 </p> 1202 </td> 1203</tr> 1204<tr> 1205<td> 1206 <p> 1207 <code class="computeroutput"><span class="identifier">cos</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1208 </p> 1209 </td> 1210<td> 1211 <p> 1212 RealType 1213 </p> 1214 </td> 1215</tr> 1216<tr> 1217<td> 1218 <p> 1219 <code class="computeroutput"><span class="identifier">sin</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1220 </p> 1221 </td> 1222<td> 1223 <p> 1224 RealType 1225 </p> 1226 </td> 1227</tr> 1228<tr> 1229<td> 1230 <p> 1231 <code class="computeroutput"><span class="identifier">asin</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1232 </p> 1233 </td> 1234<td> 1235 <p> 1236 RealType 1237 </p> 1238 </td> 1239</tr> 1240<tr> 1241<td> 1242 <p> 1243 <code class="computeroutput"><span class="identifier">tan</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1244 </p> 1245 </td> 1246<td> 1247 <p> 1248 RealType 1249 </p> 1250 </td> 1251</tr> 1252<tr> 1253<td> 1254 <p> 1255 <code class="computeroutput"><span class="identifier">atan</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1256 </p> 1257 </td> 1258<td> 1259 <p> 1260 RealType 1261 </p> 1262 </td> 1263</tr> 1264<tr> 1265<td> 1266 <p> 1267 <code class="computeroutput"><span class="identifier">fmod</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1268 </p> 1269 </td> 1270<td> 1271 <p> 1272 RealType 1273 </p> 1274 </td> 1275</tr> 1276<tr> 1277<td> 1278 <p> 1279 <code class="computeroutput"><span class="identifier">round</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1280 </p> 1281 </td> 1282<td> 1283 <p> 1284 RealType 1285 </p> 1286 </td> 1287</tr> 1288<tr> 1289<td> 1290 <p> 1291 <code class="computeroutput"><span class="identifier">iround</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1292 </p> 1293 </td> 1294<td> 1295 <p> 1296 int 1297 </p> 1298 </td> 1299</tr> 1300<tr> 1301<td> 1302 <p> 1303 <code class="computeroutput"><span class="identifier">trunc</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1304 </p> 1305 </td> 1306<td> 1307 <p> 1308 RealType 1309 </p> 1310 </td> 1311</tr> 1312<tr> 1313<td> 1314 <p> 1315 <code class="computeroutput"><span class="identifier">itrunc</span><span class="special">(</span><span class="identifier">cr1</span><span class="special">)</span></code> 1316 </p> 1317 </td> 1318<td> 1319 <p> 1320 int 1321 </p> 1322 </td> 1323</tr> 1324</tbody> 1325</table></div> 1326<p> 1327 Note that the table above lists only those standard library functions known 1328 to be used (or likely to be used in the near future) by this library. The following 1329 functions: <code class="computeroutput"><span class="identifier">acos</span></code>, <code class="computeroutput"><span class="identifier">atan2</span></code>, <code class="computeroutput"><span class="identifier">fmod</span></code>, 1330 <code class="computeroutput"><span class="identifier">cosh</span></code>, <code class="computeroutput"><span class="identifier">sinh</span></code>, 1331 <code class="computeroutput"><span class="identifier">tanh</span></code>, <code class="computeroutput"><span class="identifier">log10</span></code>, 1332 <code class="computeroutput"><span class="identifier">lround</span></code>, <code class="computeroutput"><span class="identifier">llround</span></code>, 1333 <code class="computeroutput"><span class="identifier">ltrunc</span></code>, <code class="computeroutput"><span class="identifier">lltrunc</span></code> 1334 and <code class="computeroutput"><span class="identifier">modf</span></code> are not currently 1335 used, but may be if further special functions are added. 1336 </p> 1337<p> 1338 Note that the <code class="computeroutput"><span class="identifier">round</span></code>, <code class="computeroutput"><span class="identifier">trunc</span></code> and <code class="computeroutput"><span class="identifier">modf</span></code> 1339 functions are not part of the current C++ standard: they are part of the additions 1340 added to C99 which will likely be in the next C++ standard. There are Boost 1341 versions of these provided as a backup, and the functions are always called 1342 unqualified so that argument-dependent-lookup can take place. 1343 </p> 1344<p> 1345 In addition, for efficient and accurate results, a <a class="link" href="lanczos.html" title="The Lanczos Approximation">Lanczos 1346 approximation</a> is highly desirable. You may be able to adapt an existing 1347 approximation from <a href="../../../../../boost/math/special_functions/lanczos.hpp" target="_top">boost/math/special_functions/lanczos.hpp</a> 1348 or <a href="../../../../../boost/math/bindings/detail/big_lanczos.hpp" target="_top">boost/math/bindings/detail/big_lanczos.hpp</a>: 1349 in the former case you will need change <code class="computeroutput"><span class="keyword">static_cast</span></code>'s 1350 to <code class="computeroutput"><span class="identifier">lexical_cast</span></code>'s, and the 1351 constants to <span class="emphasis"><em>strings</em></span> (in order to ensure the coefficients 1352 aren't truncated to <code class="computeroutput"><span class="keyword">long</span> <span class="keyword">double</span></code>) 1353 and then specialise <code class="computeroutput"><span class="identifier">lanczos_traits</span></code> 1354 for type T. Otherwise you may have to hack <a href="../../../tools/lanczos_generator.cpp" target="_top">libs/math/tools/lanczos_generator.cpp</a> 1355 to find a suitable approximation for your RealType. The code will still compile 1356 if you don't do this, but both accuracy and efficiency will be somewhat compromised 1357 in any function that makes use of the gamma/beta/erf family of functions. 1358 </p> 1359</div> 1360<table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr> 1361<td align="left"></td> 1362<td align="right"><div class="copyright-footer">Copyright © 2006-2019 Nikhar 1363 Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, 1364 Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan 1365 Råde, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, 1366 Daryle Walker and Xiaogang Zhang<p> 1367 Distributed under the Boost Software License, Version 1.0. (See accompanying 1368 file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>) 1369 </p> 1370</div></td> 1371</tr></table> 1372<hr> 1373<div class="spirit-nav"> 1374<a accesskey="p" href="high_precision/using_test.html"><img src="../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../using_udt.html"><img src="../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../index.html"><img src="../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="dist_concept.html"><img src="../../../../../doc/src/images/next.png" alt="Next"></a> 1375</div> 1376</body> 1377</html> 1378
