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   4 <title>Comparison of Nth-root Finding Algorithms</title>
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  25 <div class="section">
  26 <div class="titlepage"><div><div><h3 class="title">
  27 <a name="math_toolkit.root_comparison.root_n_comparison"></a><a class="link" href="root_n_comparison.html" title="Comparison of Nth-root Finding Algorithms">Comparison
  28       of Nth-root Finding Algorithms</a>
  29 </h3></div></div></div>
  30 <p>
  31         A second example compares four generalized nth-root finding algorithms for
  32         various n-th roots (5, 7 and 13) of a single value 28.0, for four floating-point
  33         types, <code class="computeroutput"><span class="keyword">float</span></code>, <code class="computeroutput"><span class="keyword">double</span></code>, <code class="computeroutput"><span class="keyword">long</span>
  34         <span class="keyword">double</span></code> and a <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
  35         type <code class="computeroutput"><span class="identifier">cpp_bin_float_50</span></code>. In
  36         each case the target accuracy was set using our "recomended" accuracy
  37         limits (or at least limits that make a good starting point - which is likely
  38         to give close to full accuracy without resorting to unnecessary iterations).
  39       </p>
  40 <div class="informaltable"><table class="table">
  41 <colgroup>
  42 <col>
  43 <col>
  44 </colgroup>
  45 <thead><tr>
  46 <th>
  47                 <p>
  48                   Function
  49                 </p>
  50               </th>
  51 <th>
  52                 <p>
  53                   Precision Requested
  54                 </p>
  55               </th>
  56 </tr></thead>
  57 <tbody>
  58 <tr>
  59 <td>
  60                 <p>
  61                   TOMS748
  62                 </p>
  63               </td>
  64 <td>
  65                 <p>
  66                   numeric_limits&lt;T&gt;::digits - 2
  67                 </p>
  68               </td>
  69 </tr>
  70 <tr>
  71 <td>
  72                 <p>
  73                   Newton
  74                 </p>
  75               </td>
  76 <td>
  77                 <p>
  78                   floor(numeric_limits&lt;T&gt;::digits * 0.6)
  79                 </p>
  80               </td>
  81 </tr>
  82 <tr>
  83 <td>
  84                 <p>
  85                   Halley
  86                 </p>
  87               </td>
  88 <td>
  89                 <p>
  90                   floor(numeric_limits&lt;T&gt;::digits * 0.4)
  91                 </p>
  92               </td>
  93 </tr>
  94 <tr>
  95 <td>
  96                 <p>
  97                   Schr&#246;der
  98                 </p>
  99               </td>
 100 <td>
 101                 <p>
 102                   floor(numeric_limits&lt;T&gt;::digits * 0.4)
 103                 </p>
 104               </td>
 105 </tr>
 106 </tbody>
 107 </table></div>
 108 <p>
 109         Tests used Microsoft Visual Studio 2013 (Update 1) and GCC 4.9.1 using source
 110         code <a href="../../../../example/root_n_finding_algorithms.cpp" target="_top">root_n_finding_algorithms.cpp</a>.
 111       </p>
 112 <p>
 113         The timing uncertainty (especially using MSVC) is at least 5% of normalized
 114         time 'Norm'.
 115       </p>
 116 <p>
 117         To pick out the 'best' and 'worst' algorithms are highlighted in blue and
 118         red. More than one result can be 'best' when normalized times are indistinguishable
 119         within the uncertainty.
 120       </p>
 121 <h4>
 122 <a name="math_toolkit.root_comparison.root_n_comparison.h0"></a>
 123         <span class="phrase"><a name="math_toolkit.root_comparison.root_n_comparison.program_example_root_n_finding_a"></a></span><a class="link" href="root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.program_example_root_n_finding_a">Program
 124         ..\example\root_n_finding_algorithms.cpp, Microsoft Visual C++ version 14.1,
 125         Dinkumware standard library version 650, Win32 Compiled in optimise mode.,
 126         _X86_SSE2</a>
 127       </h4>
 128 <p>
 129         Fraction of full accuracy 1
 130       </p>
 131 <div class="table">
 132 <a name="math_toolkit.root_comparison.root_n_comparison.root_5"></a><p class="title"><b>Table&#160;10.3.&#160;5th root(28) for float, double, long double and cpp_bin_float_50 types,
 133         using _X86_SSE2</b></p>
 134 <div class="table-contents"><table class="table" summary="5th root(28) for float, double, long double and cpp_bin_float_50 types,
 135         using _X86_SSE2">
 136 <colgroup>
 137 <col>
 138 <col>
 139 <col>
 140 <col>
 141 <col>
 142 <col>
 143 <col>
 144 <col>
 145 <col>
 146 <col>
 147 <col>
 148 <col>
 149 <col>
 150 <col>
 151 <col>
 152 <col>
 153 <col>
 154 <col>
 155 <col>
 156 <col>
 157 <col>
 158 </colgroup>
 159 <thead><tr>
 160 <th>
 161               </th>
 162 <th>
 163                 <p>
 164                   float
 165                 </p>
 166               </th>
 167 <th>
 168               </th>
 169 <th>
 170               </th>
 171 <th>
 172               </th>
 173 <th>
 174               </th>
 175 <th>
 176                 <p>
 177                   double
 178                 </p>
 179               </th>
 180 <th>
 181               </th>
 182 <th>
 183               </th>
 184 <th>
 185               </th>
 186 <th>
 187               </th>
 188 <th>
 189                 <p>
 190                   long d
 191                 </p>
 192               </th>
 193 <th>
 194               </th>
 195 <th>
 196               </th>
 197 <th>
 198               </th>
 199 <th>
 200               </th>
 201 <th>
 202                 <p>
 203                   cpp50
 204                 </p>
 205               </th>
 206 <th>
 207               </th>
 208 <th>
 209               </th>
 210 <td class="auto-generated">&#160;</td>
 211 <td class="auto-generated">&#160;</td>
 212 </tr></thead>
 213 <tbody>
 214 <tr>
 215 <td>
 216                 <p>
 217                   Algo
 218                 </p>
 219               </td>
 220 <td>
 221                 <p>
 222                   Its
 223                 </p>
 224               </td>
 225 <td>
 226                 <p>
 227                   Times
 228                 </p>
 229               </td>
 230 <td>
 231                 <p>
 232                   Norm
 233                 </p>
 234               </td>
 235 <td>
 236                 <p>
 237                   Dis
 238                 </p>
 239               </td>
 240 <td>
 241               </td>
 242 <td>
 243                 <p>
 244                   Its
 245                 </p>
 246               </td>
 247 <td>
 248                 <p>
 249                   Times
 250                 </p>
 251               </td>
 252 <td>
 253                 <p>
 254                   Norm
 255                 </p>
 256               </td>
 257 <td>
 258                 <p>
 259                   Dis
 260                 </p>
 261               </td>
 262 <td>
 263               </td>
 264 <td>
 265                 <p>
 266                   Its
 267                 </p>
 268               </td>
 269 <td>
 270                 <p>
 271                   Times
 272                 </p>
 273               </td>
 274 <td>
 275                 <p>
 276                   Norm
 277                 </p>
 278               </td>
 279 <td>
 280                 <p>
 281                   Dis
 282                 </p>
 283               </td>
 284 <td>
 285               </td>
 286 <td>
 287                 <p>
 288                   Its
 289                 </p>
 290               </td>
 291 <td>
 292                 <p>
 293                   Times
 294                 </p>
 295               </td>
 296 <td>
 297                 <p>
 298                   Norm
 299                 </p>
 300               </td>
 301 <td>
 302                 <p>
 303                   Dis
 304                 </p>
 305               </td>
 306 <td>
 307               </td>
 308 </tr>
 309 <tr>
 310 <td>
 311                 <p>
 312                   TOMS748
 313                 </p>
 314               </td>
 315 <td>
 316                 <p>
 317                   7
 318                 </p>
 319               </td>
 320 <td>
 321                 <p>
 322                   457
 323                 </p>
 324               </td>
 325 <td>
 326                 <p>
 327                   2.00
 328                 </p>
 329               </td>
 330 <td>
 331                 <p>
 332                   0
 333                 </p>
 334               </td>
 335 <td>
 336               </td>
 337 <td>
 338                 <p>
 339                   11
 340                 </p>
 341               </td>
 342 <td>
 343                 <p>
 344                   860
 345                 </p>
 346               </td>
 347 <td>
 348                 <p>
 349                   3.54
 350                 </p>
 351               </td>
 352 <td>
 353                 <p>
 354                   1
 355                 </p>
 356               </td>
 357 <td>
 358               </td>
 359 <td>
 360                 <p>
 361                   11
 362                 </p>
 363               </td>
 364 <td>
 365                 <p>
 366                   806
 367                 </p>
 368               </td>
 369 <td>
 370                 <p>
 371                   3.02
 372                 </p>
 373               </td>
 374 <td>
 375                 <p>
 376                   1
 377                 </p>
 378               </td>
 379 <td>
 380               </td>
 381 <td>
 382                 <p>
 383                   12
 384                 </p>
 385               </td>
 386 <td>
 387                 <p>
 388                   226875
 389                 </p>
 390               </td>
 391 <td>
 392                 <p>
 393                   <span class="red">8.11</span>
 394                 </p>
 395               </td>
 396 <td>
 397                 <p>
 398                   0
 399                 </p>
 400               </td>
 401 <td>
 402               </td>
 403 </tr>
 404 <tr>
 405 <td>
 406                 <p>
 407                   Newton
 408                 </p>
 409               </td>
 410 <td>
 411                 <p>
 412                   3
 413                 </p>
 414               </td>
 415 <td>
 416                 <p>
 417                   228
 418                 </p>
 419               </td>
 420 <td>
 421                 <p>
 422                   <span class="blue">1.00</span>
 423                 </p>
 424               </td>
 425 <td>
 426                 <p>
 427                   0
 428                 </p>
 429               </td>
 430 <td>
 431               </td>
 432 <td>
 433                 <p>
 434                   4
 435                 </p>
 436               </td>
 437 <td>
 438                 <p>
 439                   243
 440                 </p>
 441               </td>
 442 <td>
 443                 <p>
 444                   <span class="blue">1.00</span>
 445                 </p>
 446               </td>
 447 <td>
 448                 <p>
 449                   -1
 450                 </p>
 451               </td>
 452 <td>
 453               </td>
 454 <td>
 455                 <p>
 456                   4
 457                 </p>
 458               </td>
 459 <td>
 460                 <p>
 461                   298
 462                 </p>
 463               </td>
 464 <td>
 465                 <p>
 466                   1.12
 467                 </p>
 468               </td>
 469 <td>
 470                 <p>
 471                   -1
 472                 </p>
 473               </td>
 474 <td>
 475               </td>
 476 <td>
 477                 <p>
 478                   6
 479                 </p>
 480               </td>
 481 <td>
 482                 <p>
 483                   27968
 484                 </p>
 485               </td>
 486 <td>
 487                 <p>
 488                   <span class="blue">1.00</span>
 489                 </p>
 490               </td>
 491 <td>
 492                 <p>
 493                   0
 494                 </p>
 495               </td>
 496 <td>
 497               </td>
 498 </tr>
 499 <tr>
 500 <td>
 501                 <p>
 502                   Halley
 503                 </p>
 504               </td>
 505 <td>
 506                 <p>
 507                   2
 508                 </p>
 509               </td>
 510 <td>
 511                 <p>
 512                   250
 513                 </p>
 514               </td>
 515 <td>
 516                 <p>
 517                   1.10
 518                 </p>
 519               </td>
 520 <td>
 521                 <p>
 522                   0
 523                 </p>
 524               </td>
 525 <td>
 526               </td>
 527 <td>
 528                 <p>
 529                   3
 530                 </p>
 531               </td>
 532 <td>
 533                 <p>
 534                   268
 535                 </p>
 536               </td>
 537 <td>
 538                 <p>
 539                   1.10
 540                 </p>
 541               </td>
 542 <td>
 543                 <p>
 544                   0
 545                 </p>
 546               </td>
 547 <td>
 548               </td>
 549 <td>
 550                 <p>
 551                   3
 552                 </p>
 553               </td>
 554 <td>
 555                 <p>
 556                   267
 557                 </p>
 558               </td>
 559 <td>
 560                 <p>
 561                   <span class="blue">1.00</span>
 562                 </p>
 563               </td>
 564 <td>
 565                 <p>
 566                   0
 567                 </p>
 568               </td>
 569 <td>
 570               </td>
 571 <td>
 572                 <p>
 573                   4
 574                 </p>
 575               </td>
 576 <td>
 577                 <p>
 578                   52812
 579                 </p>
 580               </td>
 581 <td>
 582                 <p>
 583                   1.89
 584                 </p>
 585               </td>
 586 <td>
 587                 <p>
 588                   0
 589                 </p>
 590               </td>
 591 <td>
 592               </td>
 593 </tr>
 594 <tr>
 595 <td>
 596                 <p>
 597                   Schr&#246;der
 598                 </p>
 599               </td>
 600 <td>
 601                 <p>
 602                   2
 603                 </p>
 604               </td>
 605 <td>
 606                 <p>
 607                   256
 608                 </p>
 609               </td>
 610 <td>
 611                 <p>
 612                   1.12
 613                 </p>
 614               </td>
 615 <td>
 616                 <p>
 617                   0
 618                 </p>
 619               </td>
 620 <td>
 621               </td>
 622 <td>
 623                 <p>
 624                   3
 625                 </p>
 626               </td>
 627 <td>
 628                 <p>
 629                   271
 630                 </p>
 631               </td>
 632 <td>
 633                 <p>
 634                   1.12
 635                 </p>
 636               </td>
 637 <td>
 638                 <p>
 639                   -1
 640                 </p>
 641               </td>
 642 <td>
 643               </td>
 644 <td>
 645                 <p>
 646                   3
 647                 </p>
 648               </td>
 649 <td>
 650                 <p>
 651                   270
 652                 </p>
 653               </td>
 654 <td>
 655                 <p>
 656                   <span class="blue">1.01</span>
 657                 </p>
 658               </td>
 659 <td>
 660                 <p>
 661                   -1
 662                 </p>
 663               </td>
 664 <td>
 665               </td>
 666 <td>
 667                 <p>
 668                   4
 669                 </p>
 670               </td>
 671 <td>
 672                 <p>
 673                   61406
 674                 </p>
 675               </td>
 676 <td>
 677                 <p>
 678                   2.20
 679                 </p>
 680               </td>
 681 <td>
 682                 <p>
 683                   0
 684                 </p>
 685               </td>
 686 <td>
 687               </td>
 688 </tr>
 689 </tbody>
 690 </table></div>
 691 </div>
 692 <br class="table-break"><div class="table">
 693 <a name="math_toolkit.root_comparison.root_n_comparison.root_7"></a><p class="title"><b>Table&#160;10.4.&#160;7th root(28) for float, double, long double and cpp_bin_float_50 types,
 694         using _X86_SSE2</b></p>
 695 <div class="table-contents"><table class="table" summary="7th root(28) for float, double, long double and cpp_bin_float_50 types,
 696         using _X86_SSE2">
 697 <colgroup>
 698 <col>
 699 <col>
 700 <col>
 701 <col>
 702 <col>
 703 <col>
 704 <col>
 705 <col>
 706 <col>
 707 <col>
 708 <col>
 709 <col>
 710 <col>
 711 <col>
 712 <col>
 713 <col>
 714 <col>
 715 <col>
 716 <col>
 717 <col>
 718 <col>
 719 </colgroup>
 720 <thead><tr>
 721 <th>
 722               </th>
 723 <th>
 724                 <p>
 725                   float
 726                 </p>
 727               </th>
 728 <th>
 729               </th>
 730 <th>
 731               </th>
 732 <th>
 733               </th>
 734 <th>
 735               </th>
 736 <th>
 737                 <p>
 738                   double
 739                 </p>
 740               </th>
 741 <th>
 742               </th>
 743 <th>
 744               </th>
 745 <th>
 746               </th>
 747 <th>
 748               </th>
 749 <th>
 750                 <p>
 751                   long d
 752                 </p>
 753               </th>
 754 <th>
 755               </th>
 756 <th>
 757               </th>
 758 <th>
 759               </th>
 760 <th>
 761               </th>
 762 <th>
 763                 <p>
 764                   cpp50
 765                 </p>
 766               </th>
 767 <th>
 768               </th>
 769 <th>
 770               </th>
 771 <td class="auto-generated">&#160;</td>
 772 <td class="auto-generated">&#160;</td>
 773 </tr></thead>
 774 <tbody>
 775 <tr>
 776 <td>
 777                 <p>
 778                   Algo
 779                 </p>
 780               </td>
 781 <td>
 782                 <p>
 783                   Its
 784                 </p>
 785               </td>
 786 <td>
 787                 <p>
 788                   Times
 789                 </p>
 790               </td>
 791 <td>
 792                 <p>
 793                   Norm
 794                 </p>
 795               </td>
 796 <td>
 797                 <p>
 798                   Dis
 799                 </p>
 800               </td>
 801 <td>
 802               </td>
 803 <td>
 804                 <p>
 805                   Its
 806                 </p>
 807               </td>
 808 <td>
 809                 <p>
 810                   Times
 811                 </p>
 812               </td>
 813 <td>
 814                 <p>
 815                   Norm
 816                 </p>
 817               </td>
 818 <td>
 819                 <p>
 820                   Dis
 821                 </p>
 822               </td>
 823 <td>
 824               </td>
 825 <td>
 826                 <p>
 827                   Its
 828                 </p>
 829               </td>
 830 <td>
 831                 <p>
 832                   Times
 833                 </p>
 834               </td>
 835 <td>
 836                 <p>
 837                   Norm
 838                 </p>
 839               </td>
 840 <td>
 841                 <p>
 842                   Dis
 843                 </p>
 844               </td>
 845 <td>
 846               </td>
 847 <td>
 848                 <p>
 849                   Its
 850                 </p>
 851               </td>
 852 <td>
 853                 <p>
 854                   Times
 855                 </p>
 856               </td>
 857 <td>
 858                 <p>
 859                   Norm
 860                 </p>
 861               </td>
 862 <td>
 863                 <p>
 864                   Dis
 865                 </p>
 866               </td>
 867 <td>
 868               </td>
 869 </tr>
 870 <tr>
 871 <td>
 872                 <p>
 873                   TOMS748
 874                 </p>
 875               </td>
 876 <td>
 877                 <p>
 878                   12
 879                 </p>
 880               </td>
 881 <td>
 882                 <p>
 883                   825
 884                 </p>
 885               </td>
 886 <td>
 887                 <p>
 888                   3.06
 889                 </p>
 890               </td>
 891 <td>
 892                 <p>
 893                   1
 894                 </p>
 895               </td>
 896 <td>
 897               </td>
 898 <td>
 899                 <p>
 900                   15
 901                 </p>
 902               </td>
 903 <td>
 904                 <p>
 905                   1145
 906                 </p>
 907               </td>
 908 <td>
 909                 <p>
 910                   <span class="red">4.06</span>
 911                 </p>
 912               </td>
 913 <td>
 914                 <p>
 915                   2
 916                 </p>
 917               </td>
 918 <td>
 919               </td>
 920 <td>
 921                 <p>
 922                   15
 923                 </p>
 924               </td>
 925 <td>
 926                 <p>
 927                   1159
 928                 </p>
 929               </td>
 930 <td>
 931                 <p>
 932                   <span class="red">4.17</span>
 933                 </p>
 934               </td>
 935 <td>
 936                 <p>
 937                   2
 938                 </p>
 939               </td>
 940 <td>
 941               </td>
 942 <td>
 943                 <p>
 944                   14
 945                 </p>
 946               </td>
 947 <td>
 948                 <p>
 949                   295781
 950                 </p>
 951               </td>
 952 <td>
 953                 <p>
 954                   <span class="red">8.12</span>
 955                 </p>
 956               </td>
 957 <td>
 958                 <p>
 959                   0
 960                 </p>
 961               </td>
 962 <td>
 963               </td>
 964 </tr>
 965 <tr>
 966 <td>
 967                 <p>
 968                   Newton
 969                 </p>
 970               </td>
 971 <td>
 972                 <p>
 973                   5
 974                 </p>
 975               </td>
 976 <td>
 977                 <p>
 978                   270
 979                 </p>
 980               </td>
 981 <td>
 982                 <p>
 983                   <span class="blue">1.00</span>
 984                 </p>
 985               </td>
 986 <td>
 987                 <p>
 988                   0
 989                 </p>
 990               </td>
 991 <td>
 992               </td>
 993 <td>
 994                 <p>
 995                   6
 996                 </p>
 997               </td>
 998 <td>
 999                 <p>
1000                   282
1001                 </p>
1002               </td>
1003 <td>
1004                 <p>
1005                   <span class="blue">1.00</span>
1006                 </p>
1007               </td>
1008 <td>
1009                 <p>
1010                   0
1011                 </p>
1012               </td>
1013 <td>
1014               </td>
1015 <td>
1016                 <p>
1017                   6
1018                 </p>
1019               </td>
1020 <td>
1021                 <p>
1022                   278
1023                 </p>
1024               </td>
1025 <td>
1026                 <p>
1027                   <span class="blue">1.00</span>
1028                 </p>
1029               </td>
1030 <td>
1031                 <p>
1032                   0
1033                 </p>
1034               </td>
1035 <td>
1036               </td>
1037 <td>
1038                 <p>
1039                   8
1040                 </p>
1041               </td>
1042 <td>
1043                 <p>
1044                   36406
1045                 </p>
1046               </td>
1047 <td>
1048                 <p>
1049                   <span class="blue">1.00</span>
1050                 </p>
1051               </td>
1052 <td>
1053                 <p>
1054                   0
1055                 </p>
1056               </td>
1057 <td>
1058               </td>
1059 </tr>
1060 <tr>
1061 <td>
1062                 <p>
1063                   Halley
1064                 </p>
1065               </td>
1066 <td>
1067                 <p>
1068                   4
1069                 </p>
1070               </td>
1071 <td>
1072                 <p>
1073                   303
1074                 </p>
1075               </td>
1076 <td>
1077                 <p>
1078                   1.12
1079                 </p>
1080               </td>
1081 <td>
1082                 <p>
1083                   0
1084                 </p>
1085               </td>
1086 <td>
1087               </td>
1088 <td>
1089                 <p>
1090                   5
1091                 </p>
1092               </td>
1093 <td>
1094                 <p>
1095                   329
1096                 </p>
1097               </td>
1098 <td>
1099                 <p>
1100                   1.17
1101                 </p>
1102               </td>
1103 <td>
1104                 <p>
1105                   0
1106                 </p>
1107               </td>
1108 <td>
1109               </td>
1110 <td>
1111                 <p>
1112                   5
1113                 </p>
1114               </td>
1115 <td>
1116                 <p>
1117                   335
1118                 </p>
1119               </td>
1120 <td>
1121                 <p>
1122                   1.21
1123                 </p>
1124               </td>
1125 <td>
1126                 <p>
1127                   0
1128                 </p>
1129               </td>
1130 <td>
1131               </td>
1132 <td>
1133                 <p>
1134                   6
1135                 </p>
1136               </td>
1137 <td>
1138                 <p>
1139                   78281
1140                 </p>
1141               </td>
1142 <td>
1143                 <p>
1144                   2.15
1145                 </p>
1146               </td>
1147 <td>
1148                 <p>
1149                   0
1150                 </p>
1151               </td>
1152 <td>
1153               </td>
1154 </tr>
1155 <tr>
1156 <td>
1157                 <p>
1158                   Schr&#246;der
1159                 </p>
1160               </td>
1161 <td>
1162                 <p>
1163                   5
1164                 </p>
1165               </td>
1166 <td>
1167                 <p>
1168                   340
1169                 </p>
1170               </td>
1171 <td>
1172                 <p>
1173                   1.26
1174                 </p>
1175               </td>
1176 <td>
1177                 <p>
1178                   0
1179                 </p>
1180               </td>
1181 <td>
1182               </td>
1183 <td>
1184                 <p>
1185                   6
1186                 </p>
1187               </td>
1188 <td>
1189                 <p>
1190                   432
1191                 </p>
1192               </td>
1193 <td>
1194                 <p>
1195                   1.53
1196                 </p>
1197               </td>
1198 <td>
1199                 <p>
1200                   0
1201                 </p>
1202               </td>
1203 <td>
1204               </td>
1205 <td>
1206                 <p>
1207                   6
1208                 </p>
1209               </td>
1210 <td>
1211                 <p>
1212                   367
1213                 </p>
1214               </td>
1215 <td>
1216                 <p>
1217                   1.32
1218                 </p>
1219               </td>
1220 <td>
1221                 <p>
1222                   0
1223                 </p>
1224               </td>
1225 <td>
1226               </td>
1227 <td>
1228                 <p>
1229                   7
1230                 </p>
1231               </td>
1232 <td>
1233                 <p>
1234                   85156
1235                 </p>
1236               </td>
1237 <td>
1238                 <p>
1239                   2.34
1240                 </p>
1241               </td>
1242 <td>
1243                 <p>
1244                   0
1245                 </p>
1246               </td>
1247 <td>
1248               </td>
1249 </tr>
1250 </tbody>
1251 </table></div>
1252 </div>
1253 <br class="table-break"><div class="table">
1254 <a name="math_toolkit.root_comparison.root_n_comparison.root_11"></a><p class="title"><b>Table&#160;10.5.&#160;11th root(28) for float, double, long double and cpp_bin_float_50
1255         types, using _X86_SSE2</b></p>
1256 <div class="table-contents"><table class="table" summary="11th root(28) for float, double, long double and cpp_bin_float_50
1257         types, using _X86_SSE2">
1258 <colgroup>
1259 <col>
1260 <col>
1261 <col>
1262 <col>
1263 <col>
1264 <col>
1265 <col>
1266 <col>
1267 <col>
1268 <col>
1269 <col>
1270 <col>
1271 <col>
1272 <col>
1273 <col>
1274 <col>
1275 <col>
1276 <col>
1277 <col>
1278 <col>
1279 <col>
1280 </colgroup>
1281 <thead><tr>
1282 <th>
1283               </th>
1284 <th>
1285                 <p>
1286                   float
1287                 </p>
1288               </th>
1289 <th>
1290               </th>
1291 <th>
1292               </th>
1293 <th>
1294               </th>
1295 <th>
1296               </th>
1297 <th>
1298                 <p>
1299                   double
1300                 </p>
1301               </th>
1302 <th>
1303               </th>
1304 <th>
1305               </th>
1306 <th>
1307               </th>
1308 <th>
1309               </th>
1310 <th>
1311                 <p>
1312                   long d
1313                 </p>
1314               </th>
1315 <th>
1316               </th>
1317 <th>
1318               </th>
1319 <th>
1320               </th>
1321 <th>
1322               </th>
1323 <th>
1324                 <p>
1325                   cpp50
1326                 </p>
1327               </th>
1328 <th>
1329               </th>
1330 <th>
1331               </th>
1332 <td class="auto-generated">&#160;</td>
1333 <td class="auto-generated">&#160;</td>
1334 </tr></thead>
1335 <tbody>
1336 <tr>
1337 <td>
1338                 <p>
1339                   Algo
1340                 </p>
1341               </td>
1342 <td>
1343                 <p>
1344                   Its
1345                 </p>
1346               </td>
1347 <td>
1348                 <p>
1349                   Times
1350                 </p>
1351               </td>
1352 <td>
1353                 <p>
1354                   Norm
1355                 </p>
1356               </td>
1357 <td>
1358                 <p>
1359                   Dis
1360                 </p>
1361               </td>
1362 <td>
1363               </td>
1364 <td>
1365                 <p>
1366                   Its
1367                 </p>
1368               </td>
1369 <td>
1370                 <p>
1371                   Times
1372                 </p>
1373               </td>
1374 <td>
1375                 <p>
1376                   Norm
1377                 </p>
1378               </td>
1379 <td>
1380                 <p>
1381                   Dis
1382                 </p>
1383               </td>
1384 <td>
1385               </td>
1386 <td>
1387                 <p>
1388                   Its
1389                 </p>
1390               </td>
1391 <td>
1392                 <p>
1393                   Times
1394                 </p>
1395               </td>
1396 <td>
1397                 <p>
1398                   Norm
1399                 </p>
1400               </td>
1401 <td>
1402                 <p>
1403                   Dis
1404                 </p>
1405               </td>
1406 <td>
1407               </td>
1408 <td>
1409                 <p>
1410                   Its
1411                 </p>
1412               </td>
1413 <td>
1414                 <p>
1415                   Times
1416                 </p>
1417               </td>
1418 <td>
1419                 <p>
1420                   Norm
1421                 </p>
1422               </td>
1423 <td>
1424                 <p>
1425                   Dis
1426                 </p>
1427               </td>
1428 <td>
1429               </td>
1430 </tr>
1431 <tr>
1432 <td>
1433                 <p>
1434                   TOMS748
1435                 </p>
1436               </td>
1437 <td>
1438                 <p>
1439                   12
1440                 </p>
1441               </td>
1442 <td>
1443                 <p>
1444                   714
1445                 </p>
1446               </td>
1447 <td>
1448                 <p>
1449                   3.16
1450                 </p>
1451               </td>
1452 <td>
1453                 <p>
1454                   -2
1455                 </p>
1456               </td>
1457 <td>
1458               </td>
1459 <td>
1460                 <p>
1461                   14
1462                 </p>
1463               </td>
1464 <td>
1465                 <p>
1466                   909
1467                 </p>
1468               </td>
1469 <td>
1470                 <p>
1471                   <span class="red">4.19</span>
1472                 </p>
1473               </td>
1474 <td>
1475                 <p>
1476                   2
1477                 </p>
1478               </td>
1479 <td>
1480               </td>
1481 <td>
1482                 <p>
1483                   14
1484                 </p>
1485               </td>
1486 <td>
1487                 <p>
1488                   793
1489                 </p>
1490               </td>
1491 <td>
1492                 <p>
1493                   3.69
1494                 </p>
1495               </td>
1496 <td>
1497                 <p>
1498                   2
1499                 </p>
1500               </td>
1501 <td>
1502               </td>
1503 <td>
1504                 <p>
1505                   17
1506                 </p>
1507               </td>
1508 <td>
1509                 <p>
1510                   211718
1511                 </p>
1512               </td>
1513 <td>
1514                 <p>
1515                   <span class="red">9.28</span>
1516                 </p>
1517               </td>
1518 <td>
1519                 <p>
1520                   2
1521                 </p>
1522               </td>
1523 <td>
1524               </td>
1525 </tr>
1526 <tr>
1527 <td>
1528                 <p>
1529                   Newton
1530                 </p>
1531               </td>
1532 <td>
1533                 <p>
1534                   6
1535                 </p>
1536               </td>
1537 <td>
1538                 <p>
1539                   226
1540                 </p>
1541               </td>
1542 <td>
1543                 <p>
1544                   <span class="blue">1.00</span>
1545                 </p>
1546               </td>
1547 <td>
1548                 <p>
1549                   0
1550                 </p>
1551               </td>
1552 <td>
1553               </td>
1554 <td>
1555                 <p>
1556                   7
1557                 </p>
1558               </td>
1559 <td>
1560                 <p>
1561                   217
1562                 </p>
1563               </td>
1564 <td>
1565                 <p>
1566                   <span class="blue">1.00</span>
1567                 </p>
1568               </td>
1569 <td>
1570                 <p>
1571                   0
1572                 </p>
1573               </td>
1574 <td>
1575               </td>
1576 <td>
1577                 <p>
1578                   7
1579                 </p>
1580               </td>
1581 <td>
1582                 <p>
1583                   215
1584                 </p>
1585               </td>
1586 <td>
1587                 <p>
1588                   <span class="blue">1.00</span>
1589                 </p>
1590               </td>
1591 <td>
1592                 <p>
1593                   0
1594                 </p>
1595               </td>
1596 <td>
1597               </td>
1598 <td>
1599                 <p>
1600                   9
1601                 </p>
1602               </td>
1603 <td>
1604                 <p>
1605                   22812
1606                 </p>
1607               </td>
1608 <td>
1609                 <p>
1610                   <span class="blue">1.00</span>
1611                 </p>
1612               </td>
1613 <td>
1614                 <p>
1615                   0
1616                 </p>
1617               </td>
1618 <td>
1619               </td>
1620 </tr>
1621 <tr>
1622 <td>
1623                 <p>
1624                   Halley
1625                 </p>
1626               </td>
1627 <td>
1628                 <p>
1629                   4
1630                 </p>
1631               </td>
1632 <td>
1633                 <p>
1634                   262
1635                 </p>
1636               </td>
1637 <td>
1638                 <p>
1639                   1.16
1640                 </p>
1641               </td>
1642 <td>
1643                 <p>
1644                   -1
1645                 </p>
1646               </td>
1647 <td>
1648               </td>
1649 <td>
1650                 <p>
1651                   5
1652                 </p>
1653               </td>
1654 <td>
1655                 <p>
1656                   260
1657                 </p>
1658               </td>
1659 <td>
1660                 <p>
1661                   1.20
1662                 </p>
1663               </td>
1664 <td>
1665                 <p>
1666                   0
1667                 </p>
1668               </td>
1669 <td>
1670               </td>
1671 <td>
1672                 <p>
1673                   5
1674                 </p>
1675               </td>
1676 <td>
1677                 <p>
1678                   260
1679                 </p>
1680               </td>
1681 <td>
1682                 <p>
1683                   1.21
1684                 </p>
1685               </td>
1686 <td>
1687                 <p>
1688                   0
1689                 </p>
1690               </td>
1691 <td>
1692               </td>
1693 <td>
1694                 <p>
1695                   6
1696                 </p>
1697               </td>
1698 <td>
1699                 <p>
1700                   40781
1701                 </p>
1702               </td>
1703 <td>
1704                 <p>
1705                   1.79
1706                 </p>
1707               </td>
1708 <td>
1709                 <p>
1710                   0
1711                 </p>
1712               </td>
1713 <td>
1714               </td>
1715 </tr>
1716 <tr>
1717 <td>
1718                 <p>
1719                   Schr&#246;der
1720                 </p>
1721               </td>
1722 <td>
1723                 <p>
1724                   6
1725                 </p>
1726               </td>
1727 <td>
1728                 <p>
1729                   332
1730                 </p>
1731               </td>
1732 <td>
1733                 <p>
1734                   1.47
1735                 </p>
1736               </td>
1737 <td>
1738                 <p>
1739                   0
1740                 </p>
1741               </td>
1742 <td>
1743               </td>
1744 <td>
1745                 <p>
1746                   7
1747                 </p>
1748               </td>
1749 <td>
1750                 <p>
1751                   314
1752                 </p>
1753               </td>
1754 <td>
1755                 <p>
1756                   1.45
1757                 </p>
1758               </td>
1759 <td>
1760                 <p>
1761                   0
1762                 </p>
1763               </td>
1764 <td>
1765               </td>
1766 <td>
1767                 <p>
1768                   7
1769                 </p>
1770               </td>
1771 <td>
1772                 <p>
1773                   310
1774                 </p>
1775               </td>
1776 <td>
1777                 <p>
1778                   1.44
1779                 </p>
1780               </td>
1781 <td>
1782                 <p>
1783                   0
1784                 </p>
1785               </td>
1786 <td>
1787               </td>
1788 <td>
1789                 <p>
1790                   8
1791                 </p>
1792               </td>
1793 <td>
1794                 <p>
1795                   67187
1796                 </p>
1797               </td>
1798 <td>
1799                 <p>
1800                   2.95
1801                 </p>
1802               </td>
1803 <td>
1804                 <p>
1805                   0
1806                 </p>
1807               </td>
1808 <td>
1809               </td>
1810 </tr>
1811 </tbody>
1812 </table></div>
1813 </div>
1814 <br class="table-break"><h4>
1815 <a name="math_toolkit.root_comparison.root_n_comparison.h1"></a>
1816         <span class="phrase"><a name="math_toolkit.root_comparison.root_n_comparison.program_root_n_finding_algorithm"></a></span><a class="link" href="root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.program_root_n_finding_algorithm">Program
1817         root_n_finding_algorithms.cpp, Microsoft Visual C++ version 12.0, Dinkumware
1818         standard library version 610, Win32 Compiled in optimise mode., _X64_AVX</a>
1819       </h4>
1820 <p>
1821         Fraction of full accuracy 1
1822       </p>
1823 <div class="table">
1824 <a name="math_toolkit.root_comparison.root_n_comparison.root_5_0"></a><p class="title"><b>Table&#160;10.6.&#160;5th root(28) for float, double, long double and cpp_bin_float_50 types,
1825         using _X64_AVX</b></p>
1826 <div class="table-contents"><table class="table" summary="5th root(28) for float, double, long double and cpp_bin_float_50 types,
1827         using _X64_AVX">
1828 <colgroup>
1829 <col>
1830 <col>
1831 <col>
1832 <col>
1833 <col>
1834 <col>
1835 <col>
1836 <col>
1837 <col>
1838 <col>
1839 <col>
1840 <col>
1841 <col>
1842 <col>
1843 <col>
1844 <col>
1845 <col>
1846 <col>
1847 <col>
1848 <col>
1849 <col>
1850 </colgroup>
1851 <thead><tr>
1852 <th>
1853               </th>
1854 <th>
1855                 <p>
1856                   float
1857                 </p>
1858               </th>
1859 <th>
1860               </th>
1861 <th>
1862               </th>
1863 <th>
1864               </th>
1865 <th>
1866               </th>
1867 <th>
1868                 <p>
1869                   double
1870                 </p>
1871               </th>
1872 <th>
1873               </th>
1874 <th>
1875               </th>
1876 <th>
1877               </th>
1878 <th>
1879               </th>
1880 <th>
1881                 <p>
1882                   long d
1883                 </p>
1884               </th>
1885 <th>
1886               </th>
1887 <th>
1888               </th>
1889 <th>
1890               </th>
1891 <th>
1892               </th>
1893 <th>
1894                 <p>
1895                   cpp50
1896                 </p>
1897               </th>
1898 <th>
1899               </th>
1900 <th>
1901               </th>
1902 <td class="auto-generated">&#160;</td>
1903 <td class="auto-generated">&#160;</td>
1904 </tr></thead>
1905 <tbody>
1906 <tr>
1907 <td>
1908                 <p>
1909                   Algo
1910                 </p>
1911               </td>
1912 <td>
1913                 <p>
1914                   Its
1915                 </p>
1916               </td>
1917 <td>
1918                 <p>
1919                   Times
1920                 </p>
1921               </td>
1922 <td>
1923                 <p>
1924                   Norm
1925                 </p>
1926               </td>
1927 <td>
1928                 <p>
1929                   Dis
1930                 </p>
1931               </td>
1932 <td>
1933               </td>
1934 <td>
1935                 <p>
1936                   Its
1937                 </p>
1938               </td>
1939 <td>
1940                 <p>
1941                   Times
1942                 </p>
1943               </td>
1944 <td>
1945                 <p>
1946                   Norm
1947                 </p>
1948               </td>
1949 <td>
1950                 <p>
1951                   Dis
1952                 </p>
1953               </td>
1954 <td>
1955               </td>
1956 <td>
1957                 <p>
1958                   Its
1959                 </p>
1960               </td>
1961 <td>
1962                 <p>
1963                   Times
1964                 </p>
1965               </td>
1966 <td>
1967                 <p>
1968                   Norm
1969                 </p>
1970               </td>
1971 <td>
1972                 <p>
1973                   Dis
1974                 </p>
1975               </td>
1976 <td>
1977               </td>
1978 <td>
1979                 <p>
1980                   Its
1981                 </p>
1982               </td>
1983 <td>
1984                 <p>
1985                   Times
1986                 </p>
1987               </td>
1988 <td>
1989                 <p>
1990                   Norm
1991                 </p>
1992               </td>
1993 <td>
1994                 <p>
1995                   Dis
1996                 </p>
1997               </td>
1998 <td>
1999               </td>
2000 </tr>
2001 <tr>
2002 <td>
2003                 <p>
2004                   TOMS748
2005                 </p>
2006               </td>
2007 <td>
2008                 <p>
2009                   7
2010                 </p>
2011               </td>
2012 <td>
2013                 <p>
2014                   239
2015                 </p>
2016               </td>
2017 <td>
2018                 <p>
2019                   1.50
2020                 </p>
2021               </td>
2022 <td>
2023                 <p>
2024                   0
2025                 </p>
2026               </td>
2027 <td>
2028               </td>
2029 <td>
2030                 <p>
2031                   11
2032                 </p>
2033               </td>
2034 <td>
2035                 <p>
2036                   451
2037                 </p>
2038               </td>
2039 <td>
2040                 <p>
2041                   2.53
2042                 </p>
2043               </td>
2044 <td>
2045                 <p>
2046                   1
2047                 </p>
2048               </td>
2049 <td>
2050               </td>
2051 <td>
2052                 <p>
2053                   11
2054                 </p>
2055               </td>
2056 <td>
2057                 <p>
2058                   439
2059                 </p>
2060               </td>
2061 <td>
2062                 <p>
2063                   2.49
2064                 </p>
2065               </td>
2066 <td>
2067                 <p>
2068                   1
2069                 </p>
2070               </td>
2071 <td>
2072               </td>
2073 <td>
2074                 <p>
2075                   12
2076                 </p>
2077               </td>
2078 <td>
2079                 <p>
2080                   90312
2081                 </p>
2082               </td>
2083 <td>
2084                 <p>
2085                   <span class="red">7.51</span>
2086                 </p>
2087               </td>
2088 <td>
2089                 <p>
2090                   0
2091                 </p>
2092               </td>
2093 <td>
2094               </td>
2095 </tr>
2096 <tr>
2097 <td>
2098                 <p>
2099                   Newton
2100                 </p>
2101               </td>
2102 <td>
2103                 <p>
2104                   3
2105                 </p>
2106               </td>
2107 <td>
2108                 <p>
2109                   159
2110                 </p>
2111               </td>
2112 <td>
2113                 <p>
2114                   <span class="blue">1.00</span>
2115                 </p>
2116               </td>
2117 <td>
2118                 <p>
2119                   0
2120                 </p>
2121               </td>
2122 <td>
2123               </td>
2124 <td>
2125                 <p>
2126                   4
2127                 </p>
2128               </td>
2129 <td>
2130                 <p>
2131                   178
2132                 </p>
2133               </td>
2134 <td>
2135                 <p>
2136                   <span class="blue">1.00</span>
2137                 </p>
2138               </td>
2139 <td>
2140                 <p>
2141                   -1
2142                 </p>
2143               </td>
2144 <td>
2145               </td>
2146 <td>
2147                 <p>
2148                   4
2149                 </p>
2150               </td>
2151 <td>
2152                 <p>
2153                   176
2154                 </p>
2155               </td>
2156 <td>
2157                 <p>
2158                   <span class="blue">1.00</span>
2159                 </p>
2160               </td>
2161 <td>
2162                 <p>
2163                   -1
2164                 </p>
2165               </td>
2166 <td>
2167               </td>
2168 <td>
2169                 <p>
2170                   6
2171                 </p>
2172               </td>
2173 <td>
2174                 <p>
2175                   12031
2176                 </p>
2177               </td>
2178 <td>
2179                 <p>
2180                   <span class="blue">1.00</span>
2181                 </p>
2182               </td>
2183 <td>
2184                 <p>
2185                   0
2186                 </p>
2187               </td>
2188 <td>
2189               </td>
2190 </tr>
2191 <tr>
2192 <td>
2193                 <p>
2194                   Halley
2195                 </p>
2196               </td>
2197 <td>
2198                 <p>
2199                   2
2200                 </p>
2201               </td>
2202 <td>
2203                 <p>
2204                   168
2205                 </p>
2206               </td>
2207 <td>
2208                 <p>
2209                   1.06
2210                 </p>
2211               </td>
2212 <td>
2213                 <p>
2214                   0
2215                 </p>
2216               </td>
2217 <td>
2218               </td>
2219 <td>
2220                 <p>
2221                   3
2222                 </p>
2223               </td>
2224 <td>
2225                 <p>
2226                   203
2227                 </p>
2228               </td>
2229 <td>
2230                 <p>
2231                   1.14
2232                 </p>
2233               </td>
2234 <td>
2235                 <p>
2236                   0
2237                 </p>
2238               </td>
2239 <td>
2240               </td>
2241 <td>
2242                 <p>
2243                   3
2244                 </p>
2245               </td>
2246 <td>
2247                 <p>
2248                   198
2249                 </p>
2250               </td>
2251 <td>
2252                 <p>
2253                   1.13
2254                 </p>
2255               </td>
2256 <td>
2257                 <p>
2258                   0
2259                 </p>
2260               </td>
2261 <td>
2262               </td>
2263 <td>
2264                 <p>
2265                   4
2266                 </p>
2267               </td>
2268 <td>
2269                 <p>
2270                   20937
2271                 </p>
2272               </td>
2273 <td>
2274                 <p>
2275                   1.74
2276                 </p>
2277               </td>
2278 <td>
2279                 <p>
2280                   0
2281                 </p>
2282               </td>
2283 <td>
2284               </td>
2285 </tr>
2286 <tr>
2287 <td>
2288                 <p>
2289                   Schr&#246;der
2290                 </p>
2291               </td>
2292 <td>
2293                 <p>
2294                   2
2295                 </p>
2296               </td>
2297 <td>
2298                 <p>
2299                   173
2300                 </p>
2301               </td>
2302 <td>
2303                 <p>
2304                   1.09
2305                 </p>
2306               </td>
2307 <td>
2308                 <p>
2309                   0
2310                 </p>
2311               </td>
2312 <td>
2313               </td>
2314 <td>
2315                 <p>
2316                   3
2317                 </p>
2318               </td>
2319 <td>
2320                 <p>
2321                   206
2322                 </p>
2323               </td>
2324 <td>
2325                 <p>
2326                   1.16
2327                 </p>
2328               </td>
2329 <td>
2330                 <p>
2331                   -1
2332                 </p>
2333               </td>
2334 <td>
2335               </td>
2336 <td>
2337                 <p>
2338                   3
2339                 </p>
2340               </td>
2341 <td>
2342                 <p>
2343                   203
2344                 </p>
2345               </td>
2346 <td>
2347                 <p>
2348                   1.15
2349                 </p>
2350               </td>
2351 <td>
2352                 <p>
2353                   -1
2354                 </p>
2355               </td>
2356 <td>
2357               </td>
2358 <td>
2359                 <p>
2360                   4
2361                 </p>
2362               </td>
2363 <td>
2364                 <p>
2365                   26250
2366                 </p>
2367               </td>
2368 <td>
2369                 <p>
2370                   2.18
2371                 </p>
2372               </td>
2373 <td>
2374                 <p>
2375                   0
2376                 </p>
2377               </td>
2378 <td>
2379               </td>
2380 </tr>
2381 </tbody>
2382 </table></div>
2383 </div>
2384 <br class="table-break"><div class="table">
2385 <a name="math_toolkit.root_comparison.root_n_comparison.root_7_0"></a><p class="title"><b>Table&#160;10.7.&#160;7th root(28) for float, double, long double and cpp_bin_float_50 types,
2386         using _X64_AVX</b></p>
2387 <div class="table-contents"><table class="table" summary="7th root(28) for float, double, long double and cpp_bin_float_50 types,
2388         using _X64_AVX">
2389 <colgroup>
2390 <col>
2391 <col>
2392 <col>
2393 <col>
2394 <col>
2395 <col>
2396 <col>
2397 <col>
2398 <col>
2399 <col>
2400 <col>
2401 <col>
2402 <col>
2403 <col>
2404 <col>
2405 <col>
2406 <col>
2407 <col>
2408 <col>
2409 <col>
2410 <col>
2411 </colgroup>
2412 <thead><tr>
2413 <th>
2414               </th>
2415 <th>
2416                 <p>
2417                   float
2418                 </p>
2419               </th>
2420 <th>
2421               </th>
2422 <th>
2423               </th>
2424 <th>
2425               </th>
2426 <th>
2427               </th>
2428 <th>
2429                 <p>
2430                   double
2431                 </p>
2432               </th>
2433 <th>
2434               </th>
2435 <th>
2436               </th>
2437 <th>
2438               </th>
2439 <th>
2440               </th>
2441 <th>
2442                 <p>
2443                   long d
2444                 </p>
2445               </th>
2446 <th>
2447               </th>
2448 <th>
2449               </th>
2450 <th>
2451               </th>
2452 <th>
2453               </th>
2454 <th>
2455                 <p>
2456                   cpp50
2457                 </p>
2458               </th>
2459 <th>
2460               </th>
2461 <th>
2462               </th>
2463 <td class="auto-generated">&#160;</td>
2464 <td class="auto-generated">&#160;</td>
2465 </tr></thead>
2466 <tbody>
2467 <tr>
2468 <td>
2469                 <p>
2470                   Algo
2471                 </p>
2472               </td>
2473 <td>
2474                 <p>
2475                   Its
2476                 </p>
2477               </td>
2478 <td>
2479                 <p>
2480                   Times
2481                 </p>
2482               </td>
2483 <td>
2484                 <p>
2485                   Norm
2486                 </p>
2487               </td>
2488 <td>
2489                 <p>
2490                   Dis
2491                 </p>
2492               </td>
2493 <td>
2494               </td>
2495 <td>
2496                 <p>
2497                   Its
2498                 </p>
2499               </td>
2500 <td>
2501                 <p>
2502                   Times
2503                 </p>
2504               </td>
2505 <td>
2506                 <p>
2507                   Norm
2508                 </p>
2509               </td>
2510 <td>
2511                 <p>
2512                   Dis
2513                 </p>
2514               </td>
2515 <td>
2516               </td>
2517 <td>
2518                 <p>
2519                   Its
2520                 </p>
2521               </td>
2522 <td>
2523                 <p>
2524                   Times
2525                 </p>
2526               </td>
2527 <td>
2528                 <p>
2529                   Norm
2530                 </p>
2531               </td>
2532 <td>
2533                 <p>
2534                   Dis
2535                 </p>
2536               </td>
2537 <td>
2538               </td>
2539 <td>
2540                 <p>
2541                   Its
2542                 </p>
2543               </td>
2544 <td>
2545                 <p>
2546                   Times
2547                 </p>
2548               </td>
2549 <td>
2550                 <p>
2551                   Norm
2552                 </p>
2553               </td>
2554 <td>
2555                 <p>
2556                   Dis
2557                 </p>
2558               </td>
2559 <td>
2560               </td>
2561 </tr>
2562 <tr>
2563 <td>
2564                 <p>
2565                   TOMS748
2566                 </p>
2567               </td>
2568 <td>
2569                 <p>
2570                   12
2571                 </p>
2572               </td>
2573 <td>
2574                 <p>
2575                   385
2576                 </p>
2577               </td>
2578 <td>
2579                 <p>
2580                   2.19
2581                 </p>
2582               </td>
2583 <td>
2584                 <p>
2585                   1
2586                 </p>
2587               </td>
2588 <td>
2589               </td>
2590 <td>
2591                 <p>
2592                   15
2593                 </p>
2594               </td>
2595 <td>
2596                 <p>
2597                   635
2598                 </p>
2599               </td>
2600 <td>
2601                 <p>
2602                   3.13
2603                 </p>
2604               </td>
2605 <td>
2606                 <p>
2607                   2
2608                 </p>
2609               </td>
2610 <td>
2611               </td>
2612 <td>
2613                 <p>
2614                   15
2615                 </p>
2616               </td>
2617 <td>
2618                 <p>
2619                   621
2620                 </p>
2621               </td>
2622 <td>
2623                 <p>
2624                   3.17
2625                 </p>
2626               </td>
2627 <td>
2628                 <p>
2629                   2
2630                 </p>
2631               </td>
2632 <td>
2633               </td>
2634 <td>
2635                 <p>
2636                   14
2637                 </p>
2638               </td>
2639 <td>
2640                 <p>
2641                   114843
2642                 </p>
2643               </td>
2644 <td>
2645                 <p>
2646                   <span class="red">6.81</span>
2647                 </p>
2648               </td>
2649 <td>
2650                 <p>
2651                   0
2652                 </p>
2653               </td>
2654 <td>
2655               </td>
2656 </tr>
2657 <tr>
2658 <td>
2659                 <p>
2660                   Newton
2661                 </p>
2662               </td>
2663 <td>
2664                 <p>
2665                   5
2666                 </p>
2667               </td>
2668 <td>
2669                 <p>
2670                   176
2671                 </p>
2672               </td>
2673 <td>
2674                 <p>
2675                   <span class="blue">1.00</span>
2676                 </p>
2677               </td>
2678 <td>
2679                 <p>
2680                   0
2681                 </p>
2682               </td>
2683 <td>
2684               </td>
2685 <td>
2686                 <p>
2687                   6
2688                 </p>
2689               </td>
2690 <td>
2691                 <p>
2692                   203
2693                 </p>
2694               </td>
2695 <td>
2696                 <p>
2697                   <span class="blue">1.00</span>
2698                 </p>
2699               </td>
2700 <td>
2701                 <p>
2702                   0
2703                 </p>
2704               </td>
2705 <td>
2706               </td>
2707 <td>
2708                 <p>
2709                   6
2710                 </p>
2711               </td>
2712 <td>
2713                 <p>
2714                   196
2715                 </p>
2716               </td>
2717 <td>
2718                 <p>
2719                   <span class="blue">1.00</span>
2720                 </p>
2721               </td>
2722 <td>
2723                 <p>
2724                   0
2725                 </p>
2726               </td>
2727 <td>
2728               </td>
2729 <td>
2730                 <p>
2731                   8
2732                 </p>
2733               </td>
2734 <td>
2735                 <p>
2736                   16875
2737                 </p>
2738               </td>
2739 <td>
2740                 <p>
2741                   <span class="blue">1.00</span>
2742                 </p>
2743               </td>
2744 <td>
2745                 <p>
2746                   0
2747                 </p>
2748               </td>
2749 <td>
2750               </td>
2751 </tr>
2752 <tr>
2753 <td>
2754                 <p>
2755                   Halley
2756                 </p>
2757               </td>
2758 <td>
2759                 <p>
2760                   4
2761                 </p>
2762               </td>
2763 <td>
2764                 <p>
2765                   209
2766                 </p>
2767               </td>
2768 <td>
2769                 <p>
2770                   1.19
2771                 </p>
2772               </td>
2773 <td>
2774                 <p>
2775                   0
2776                 </p>
2777               </td>
2778 <td>
2779               </td>
2780 <td>
2781                 <p>
2782                   5
2783                 </p>
2784               </td>
2785 <td>
2786                 <p>
2787                   254
2788                 </p>
2789               </td>
2790 <td>
2791                 <p>
2792                   1.25
2793                 </p>
2794               </td>
2795 <td>
2796                 <p>
2797                   0
2798                 </p>
2799               </td>
2800 <td>
2801               </td>
2802 <td>
2803                 <p>
2804                   5
2805                 </p>
2806               </td>
2807 <td>
2808                 <p>
2809                   246
2810                 </p>
2811               </td>
2812 <td>
2813                 <p>
2814                   1.26
2815                 </p>
2816               </td>
2817 <td>
2818                 <p>
2819                   0
2820                 </p>
2821               </td>
2822 <td>
2823               </td>
2824 <td>
2825                 <p>
2826                   6
2827                 </p>
2828               </td>
2829 <td>
2830                 <p>
2831                   32343
2832                 </p>
2833               </td>
2834 <td>
2835                 <p>
2836                   1.92
2837                 </p>
2838               </td>
2839 <td>
2840                 <p>
2841                   0
2842                 </p>
2843               </td>
2844 <td>
2845               </td>
2846 </tr>
2847 <tr>
2848 <td>
2849                 <p>
2850                   Schr&#246;der
2851                 </p>
2852               </td>
2853 <td>
2854                 <p>
2855                   5
2856                 </p>
2857               </td>
2858 <td>
2859                 <p>
2860                   223
2861                 </p>
2862               </td>
2863 <td>
2864                 <p>
2865                   1.27
2866                 </p>
2867               </td>
2868 <td>
2869                 <p>
2870                   0
2871                 </p>
2872               </td>
2873 <td>
2874               </td>
2875 <td>
2876                 <p>
2877                   6
2878                 </p>
2879               </td>
2880 <td>
2881                 <p>
2882                   273
2883                 </p>
2884               </td>
2885 <td>
2886                 <p>
2887                   1.34
2888                 </p>
2889               </td>
2890 <td>
2891                 <p>
2892                   0
2893                 </p>
2894               </td>
2895 <td>
2896               </td>
2897 <td>
2898                 <p>
2899                   6
2900                 </p>
2901               </td>
2902 <td>
2903                 <p>
2904                   275
2905                 </p>
2906               </td>
2907 <td>
2908                 <p>
2909                   1.40
2910                 </p>
2911               </td>
2912 <td>
2913                 <p>
2914                   0
2915                 </p>
2916               </td>
2917 <td>
2918               </td>
2919 <td>
2920                 <p>
2921                   7
2922                 </p>
2923               </td>
2924 <td>
2925                 <p>
2926                   45156
2927                 </p>
2928               </td>
2929 <td>
2930                 <p>
2931                   2.68
2932                 </p>
2933               </td>
2934 <td>
2935                 <p>
2936                   0
2937                 </p>
2938               </td>
2939 <td>
2940               </td>
2941 </tr>
2942 </tbody>
2943 </table></div>
2944 </div>
2945 <br class="table-break"><div class="table">
2946 <a name="math_toolkit.root_comparison.root_n_comparison.root_11_0"></a><p class="title"><b>Table&#160;10.8.&#160;11th root(28) for float, double, long double and cpp_bin_float_50
2947         types, using _X64_AVX</b></p>
2948 <div class="table-contents"><table class="table" summary="11th root(28) for float, double, long double and cpp_bin_float_50
2949         types, using _X64_AVX">
2950 <colgroup>
2951 <col>
2952 <col>
2953 <col>
2954 <col>
2955 <col>
2956 <col>
2957 <col>
2958 <col>
2959 <col>
2960 <col>
2961 <col>
2962 <col>
2963 <col>
2964 <col>
2965 <col>
2966 <col>
2967 <col>
2968 <col>
2969 <col>
2970 <col>
2971 <col>
2972 </colgroup>
2973 <thead><tr>
2974 <th>
2975               </th>
2976 <th>
2977                 <p>
2978                   float
2979                 </p>
2980               </th>
2981 <th>
2982               </th>
2983 <th>
2984               </th>
2985 <th>
2986               </th>
2987 <th>
2988               </th>
2989 <th>
2990                 <p>
2991                   double
2992                 </p>
2993               </th>
2994 <th>
2995               </th>
2996 <th>
2997               </th>
2998 <th>
2999               </th>
3000 <th>
3001               </th>
3002 <th>
3003                 <p>
3004                   long d
3005                 </p>
3006               </th>
3007 <th>
3008               </th>
3009 <th>
3010               </th>
3011 <th>
3012               </th>
3013 <th>
3014               </th>
3015 <th>
3016                 <p>
3017                   cpp50
3018                 </p>
3019               </th>
3020 <th>
3021               </th>
3022 <th>
3023               </th>
3024 <td class="auto-generated">&#160;</td>
3025 <td class="auto-generated">&#160;</td>
3026 </tr></thead>
3027 <tbody>
3028 <tr>
3029 <td>
3030                 <p>
3031                   Algo
3032                 </p>
3033               </td>
3034 <td>
3035                 <p>
3036                   Its
3037                 </p>
3038               </td>
3039 <td>
3040                 <p>
3041                   Times
3042                 </p>
3043               </td>
3044 <td>
3045                 <p>
3046                   Norm
3047                 </p>
3048               </td>
3049 <td>
3050                 <p>
3051                   Dis
3052                 </p>
3053               </td>
3054 <td>
3055               </td>
3056 <td>
3057                 <p>
3058                   Its
3059                 </p>
3060               </td>
3061 <td>
3062                 <p>
3063                   Times
3064                 </p>
3065               </td>
3066 <td>
3067                 <p>
3068                   Norm
3069                 </p>
3070               </td>
3071 <td>
3072                 <p>
3073                   Dis
3074                 </p>
3075               </td>
3076 <td>
3077               </td>
3078 <td>
3079                 <p>
3080                   Its
3081                 </p>
3082               </td>
3083 <td>
3084                 <p>
3085                   Times
3086                 </p>
3087               </td>
3088 <td>
3089                 <p>
3090                   Norm
3091                 </p>
3092               </td>
3093 <td>
3094                 <p>
3095                   Dis
3096                 </p>
3097               </td>
3098 <td>
3099               </td>
3100 <td>
3101                 <p>
3102                   Its
3103                 </p>
3104               </td>
3105 <td>
3106                 <p>
3107                   Times
3108                 </p>
3109               </td>
3110 <td>
3111                 <p>
3112                   Norm
3113                 </p>
3114               </td>
3115 <td>
3116                 <p>
3117                   Dis
3118                 </p>
3119               </td>
3120 <td>
3121               </td>
3122 </tr>
3123 <tr>
3124 <td>
3125                 <p>
3126                   TOMS748
3127                 </p>
3128               </td>
3129 <td>
3130                 <p>
3131                   12
3132                 </p>
3133               </td>
3134 <td>
3135                 <p>
3136                   467
3137                 </p>
3138               </td>
3139 <td>
3140                 <p>
3141                   2.42
3142                 </p>
3143               </td>
3144 <td>
3145                 <p>
3146                   -2
3147                 </p>
3148               </td>
3149 <td>
3150               </td>
3151 <td>
3152                 <p>
3153                   14
3154                 </p>
3155               </td>
3156 <td>
3157                 <p>
3158                   648
3159                 </p>
3160               </td>
3161 <td>
3162                 <p>
3163                   3.06
3164                 </p>
3165               </td>
3166 <td>
3167                 <p>
3168                   2
3169                 </p>
3170               </td>
3171 <td>
3172               </td>
3173 <td>
3174                 <p>
3175                   14
3176                 </p>
3177               </td>
3178 <td>
3179                 <p>
3180                   640
3181                 </p>
3182               </td>
3183 <td>
3184                 <p>
3185                   2.99
3186                 </p>
3187               </td>
3188 <td>
3189                 <p>
3190                   2
3191                 </p>
3192               </td>
3193 <td>
3194               </td>
3195 <td>
3196                 <p>
3197                   17
3198                 </p>
3199               </td>
3200 <td>
3201                 <p>
3202                   170000
3203                 </p>
3204               </td>
3205 <td>
3206                 <p>
3207                   <span class="red">8.85</span>
3208                 </p>
3209               </td>
3210 <td>
3211                 <p>
3212                   2
3213                 </p>
3214               </td>
3215 <td>
3216               </td>
3217 </tr>
3218 <tr>
3219 <td>
3220                 <p>
3221                   Newton
3222                 </p>
3223               </td>
3224 <td>
3225                 <p>
3226                   6
3227                 </p>
3228               </td>
3229 <td>
3230                 <p>
3231                   193
3232                 </p>
3233               </td>
3234 <td>
3235                 <p>
3236                   <span class="blue">1.00</span>
3237                 </p>
3238               </td>
3239 <td>
3240                 <p>
3241                   0
3242                 </p>
3243               </td>
3244 <td>
3245               </td>
3246 <td>
3247                 <p>
3248                   7
3249                 </p>
3250               </td>
3251 <td>
3252                 <p>
3253                   212
3254                 </p>
3255               </td>
3256 <td>
3257                 <p>
3258                   <span class="blue">1.00</span>
3259                 </p>
3260               </td>
3261 <td>
3262                 <p>
3263                   0
3264                 </p>
3265               </td>
3266 <td>
3267               </td>
3268 <td>
3269                 <p>
3270                   7
3271                 </p>
3272               </td>
3273 <td>
3274                 <p>
3275                   214
3276                 </p>
3277               </td>
3278 <td>
3279                 <p>
3280                   <span class="blue">1.00</span>
3281                 </p>
3282               </td>
3283 <td>
3284                 <p>
3285                   0
3286                 </p>
3287               </td>
3288 <td>
3289               </td>
3290 <td>
3291                 <p>
3292                   9
3293                 </p>
3294               </td>
3295 <td>
3296                 <p>
3297                   19218
3298                 </p>
3299               </td>
3300 <td>
3301                 <p>
3302                   <span class="blue">1.00</span>
3303                 </p>
3304               </td>
3305 <td>
3306                 <p>
3307                   0
3308                 </p>
3309               </td>
3310 <td>
3311               </td>
3312 </tr>
3313 <tr>
3314 <td>
3315                 <p>
3316                   Halley
3317                 </p>
3318               </td>
3319 <td>
3320                 <p>
3321                   4
3322                 </p>
3323               </td>
3324 <td>
3325                 <p>
3326                   209
3327                 </p>
3328               </td>
3329 <td>
3330                 <p>
3331                   1.08
3332                 </p>
3333               </td>
3334 <td>
3335                 <p>
3336                   -1
3337                 </p>
3338               </td>
3339 <td>
3340               </td>
3341 <td>
3342                 <p>
3343                   5
3344                 </p>
3345               </td>
3346 <td>
3347                 <p>
3348                   256
3349                 </p>
3350               </td>
3351 <td>
3352                 <p>
3353                   1.21
3354                 </p>
3355               </td>
3356 <td>
3357                 <p>
3358                   0
3359                 </p>
3360               </td>
3361 <td>
3362               </td>
3363 <td>
3364                 <p>
3365                   5
3366                 </p>
3367               </td>
3368 <td>
3369                 <p>
3370                   250
3371                 </p>
3372               </td>
3373 <td>
3374                 <p>
3375                   1.17
3376                 </p>
3377               </td>
3378 <td>
3379                 <p>
3380                   0
3381                 </p>
3382               </td>
3383 <td>
3384               </td>
3385 <td>
3386                 <p>
3387                   6
3388                 </p>
3389               </td>
3390 <td>
3391                 <p>
3392                   32656
3393                 </p>
3394               </td>
3395 <td>
3396                 <p>
3397                   1.70
3398                 </p>
3399               </td>
3400 <td>
3401                 <p>
3402                   0
3403                 </p>
3404               </td>
3405 <td>
3406               </td>
3407 </tr>
3408 <tr>
3409 <td>
3410                 <p>
3411                   Schr&#246;der
3412                 </p>
3413               </td>
3414 <td>
3415                 <p>
3416                   6
3417                 </p>
3418               </td>
3419 <td>
3420                 <p>
3421                   248
3422                 </p>
3423               </td>
3424 <td>
3425                 <p>
3426                   1.28
3427                 </p>
3428               </td>
3429 <td>
3430                 <p>
3431                   0
3432                 </p>
3433               </td>
3434 <td>
3435               </td>
3436 <td>
3437                 <p>
3438                   7
3439                 </p>
3440               </td>
3441 <td>
3442                 <p>
3443                   306
3444                 </p>
3445               </td>
3446 <td>
3447                 <p>
3448                   1.44
3449                 </p>
3450               </td>
3451 <td>
3452                 <p>
3453                   0
3454                 </p>
3455               </td>
3456 <td>
3457               </td>
3458 <td>
3459                 <p>
3460                   7
3461                 </p>
3462               </td>
3463 <td>
3464                 <p>
3465                   298
3466                 </p>
3467               </td>
3468 <td>
3469                 <p>
3470                   1.39
3471                 </p>
3472               </td>
3473 <td>
3474                 <p>
3475                   0
3476                 </p>
3477               </td>
3478 <td>
3479               </td>
3480 <td>
3481                 <p>
3482                   8
3483                 </p>
3484               </td>
3485 <td>
3486                 <p>
3487                   53437
3488                 </p>
3489               </td>
3490 <td>
3491                 <p>
3492                   2.78
3493                 </p>
3494               </td>
3495 <td>
3496                 <p>
3497                   0
3498                 </p>
3499               </td>
3500 <td>
3501               </td>
3502 </tr>
3503 </tbody>
3504 </table></div>
3505 </div>
3506 <br class="table-break"><h4>
3507 <a name="math_toolkit.root_comparison.root_n_comparison.h2"></a>
3508         <span class="phrase"><a name="math_toolkit.root_comparison.root_n_comparison.program_example_root_n_finding_0"></a></span><a class="link" href="root_n_comparison.html#math_toolkit.root_comparison.root_n_comparison.program_example_root_n_finding_0">Program
3509         ..\example\root_n_finding_algorithms.cpp, GNU C++ version 7.1.0, GNU libstdc++
3510         version 20170502, Win32 Compiled in optimise mode., _X64_SSE2</a>
3511       </h4>
3512 <p>
3513         Fraction of full accuracy 1
3514       </p>
3515 <div class="table">
3516 <a name="math_toolkit.root_comparison.root_n_comparison.root_5_1"></a><p class="title"><b>Table&#160;10.9.&#160;5th root(28) for float, double, long double and cpp_bin_float_50 types,
3517         using _X64_SSE2</b></p>
3518 <div class="table-contents"><table class="table" summary="5th root(28) for float, double, long double and cpp_bin_float_50 types,
3519         using _X64_SSE2">
3520 <colgroup>
3521 <col>
3522 <col>
3523 <col>
3524 <col>
3525 <col>
3526 <col>
3527 <col>
3528 <col>
3529 <col>
3530 <col>
3531 <col>
3532 <col>
3533 <col>
3534 <col>
3535 <col>
3536 <col>
3537 <col>
3538 <col>
3539 <col>
3540 <col>
3541 <col>
3542 </colgroup>
3543 <thead><tr>
3544 <th>
3545               </th>
3546 <th>
3547                 <p>
3548                   float
3549                 </p>
3550               </th>
3551 <th>
3552               </th>
3553 <th>
3554               </th>
3555 <th>
3556               </th>
3557 <th>
3558               </th>
3559 <th>
3560                 <p>
3561                   double
3562                 </p>
3563               </th>
3564 <th>
3565               </th>
3566 <th>
3567               </th>
3568 <th>
3569               </th>
3570 <th>
3571               </th>
3572 <th>
3573                 <p>
3574                   long d
3575                 </p>
3576               </th>
3577 <th>
3578               </th>
3579 <th>
3580               </th>
3581 <th>
3582               </th>
3583 <th>
3584               </th>
3585 <th>
3586                 <p>
3587                   cpp50
3588                 </p>
3589               </th>
3590 <th>
3591               </th>
3592 <th>
3593               </th>
3594 <td class="auto-generated">&#160;</td>
3595 <td class="auto-generated">&#160;</td>
3596 </tr></thead>
3597 <tbody>
3598 <tr>
3599 <td>
3600                 <p>
3601                   Algo
3602                 </p>
3603               </td>
3604 <td>
3605                 <p>
3606                   Its
3607                 </p>
3608               </td>
3609 <td>
3610                 <p>
3611                   Times
3612                 </p>
3613               </td>
3614 <td>
3615                 <p>
3616                   Norm
3617                 </p>
3618               </td>
3619 <td>
3620                 <p>
3621                   Dis
3622                 </p>
3623               </td>
3624 <td>
3625               </td>
3626 <td>
3627                 <p>
3628                   Its
3629                 </p>
3630               </td>
3631 <td>
3632                 <p>
3633                   Times
3634                 </p>
3635               </td>
3636 <td>
3637                 <p>
3638                   Norm
3639                 </p>
3640               </td>
3641 <td>
3642                 <p>
3643                   Dis
3644                 </p>
3645               </td>
3646 <td>
3647               </td>
3648 <td>
3649                 <p>
3650                   Its
3651                 </p>
3652               </td>
3653 <td>
3654                 <p>
3655                   Times
3656                 </p>
3657               </td>
3658 <td>
3659                 <p>
3660                   Norm
3661                 </p>
3662               </td>
3663 <td>
3664                 <p>
3665                   Dis
3666                 </p>
3667               </td>
3668 <td>
3669               </td>
3670 <td>
3671                 <p>
3672                   Its
3673                 </p>
3674               </td>
3675 <td>
3676                 <p>
3677                   Times
3678                 </p>
3679               </td>
3680 <td>
3681                 <p>
3682                   Norm
3683                 </p>
3684               </td>
3685 <td>
3686                 <p>
3687                   Dis
3688                 </p>
3689               </td>
3690 <td>
3691               </td>
3692 </tr>
3693 <tr>
3694 <td>
3695                 <p>
3696                   TOMS748
3697                 </p>
3698               </td>
3699 <td>
3700                 <p>
3701                   7
3702                 </p>
3703               </td>
3704 <td>
3705                 <p>
3706                   206
3707                 </p>
3708               </td>
3709 <td>
3710                 <p>
3711                   2.24
3712                 </p>
3713               </td>
3714 <td>
3715                 <p>
3716                   0
3717                 </p>
3718               </td>
3719 <td>
3720               </td>
3721 <td>
3722                 <p>
3723                   11
3724                 </p>
3725               </td>
3726 <td>
3727                 <p>
3728                   460
3729                 </p>
3730               </td>
3731 <td>
3732                 <p>
3733                   <span class="red">4.04</span>
3734                 </p>
3735               </td>
3736 <td>
3737                 <p>
3738                   1
3739                 </p>
3740               </td>
3741 <td>
3742               </td>
3743 <td>
3744                 <p>
3745                   9
3746                 </p>
3747               </td>
3748 <td>
3749                 <p>
3750                   554
3751                 </p>
3752               </td>
3753 <td>
3754                 <p>
3755                   <span class="red">4.40</span>
3756                 </p>
3757               </td>
3758 <td>
3759                 <p>
3760                   0
3761                 </p>
3762               </td>
3763 <td>
3764               </td>
3765 <td>
3766                 <p>
3767                   12
3768                 </p>
3769               </td>
3770 <td>
3771                 <p>
3772                   57656
3773                 </p>
3774               </td>
3775 <td>
3776                 <p>
3777                   <span class="red">8.39</span>
3778                 </p>
3779               </td>
3780 <td>
3781                 <p>
3782                   0
3783                 </p>
3784               </td>
3785 <td>
3786               </td>
3787 </tr>
3788 <tr>
3789 <td>
3790                 <p>
3791                   Newton
3792                 </p>
3793               </td>
3794 <td>
3795                 <p>
3796                   3
3797                 </p>
3798               </td>
3799 <td>
3800                 <p>
3801                   92
3802                 </p>
3803               </td>
3804 <td>
3805                 <p>
3806                   <span class="blue">1.00</span>
3807                 </p>
3808               </td>
3809 <td>
3810                 <p>
3811                   0
3812                 </p>
3813               </td>
3814 <td>
3815               </td>
3816 <td>
3817                 <p>
3818                   4
3819                 </p>
3820               </td>
3821 <td>
3822                 <p>
3823                   114
3824                 </p>
3825               </td>
3826 <td>
3827                 <p>
3828                   <span class="blue">1.00</span>
3829                 </p>
3830               </td>
3831 <td>
3832                 <p>
3833                   -1
3834                 </p>
3835               </td>
3836 <td>
3837               </td>
3838 <td>
3839                 <p>
3840                   5
3841                 </p>
3842               </td>
3843 <td>
3844                 <p>
3845                   126
3846                 </p>
3847               </td>
3848 <td>
3849                 <p>
3850                   <span class="blue">1.00</span>
3851                 </p>
3852               </td>
3853 <td>
3854                 <p>
3855                   0
3856                 </p>
3857               </td>
3858 <td>
3859               </td>
3860 <td>
3861                 <p>
3862                   6
3863                 </p>
3864               </td>
3865 <td>
3866                 <p>
3867                   6875
3868                 </p>
3869               </td>
3870 <td>
3871                 <p>
3872                   <span class="blue">1.00</span>
3873                 </p>
3874               </td>
3875 <td>
3876                 <p>
3877                   0
3878                 </p>
3879               </td>
3880 <td>
3881               </td>
3882 </tr>
3883 <tr>
3884 <td>
3885                 <p>
3886                   Halley
3887                 </p>
3888               </td>
3889 <td>
3890                 <p>
3891                   2
3892                 </p>
3893               </td>
3894 <td>
3895                 <p>
3896                   106
3897                 </p>
3898               </td>
3899 <td>
3900                 <p>
3901                   1.15
3902                 </p>
3903               </td>
3904 <td>
3905                 <p>
3906                   0
3907                 </p>
3908               </td>
3909 <td>
3910               </td>
3911 <td>
3912                 <p>
3913                   3
3914                 </p>
3915               </td>
3916 <td>
3917                 <p>
3918                   134
3919                 </p>
3920               </td>
3921 <td>
3922                 <p>
3923                   1.18
3924                 </p>
3925               </td>
3926 <td>
3927                 <p>
3928                   0
3929                 </p>
3930               </td>
3931 <td>
3932               </td>
3933 <td>
3934                 <p>
3935                   3
3936                 </p>
3937               </td>
3938 <td>
3939                 <p>
3940                   178
3941                 </p>
3942               </td>
3943 <td>
3944                 <p>
3945                   1.41
3946                 </p>
3947               </td>
3948 <td>
3949                 <p>
3950                   0
3951                 </p>
3952               </td>
3953 <td>
3954               </td>
3955 <td>
3956                 <p>
3957                   4
3958                 </p>
3959               </td>
3960 <td>
3961                 <p>
3962                   12500
3963                 </p>
3964               </td>
3965 <td>
3966                 <p>
3967                   1.82
3968                 </p>
3969               </td>
3970 <td>
3971                 <p>
3972                   0
3973                 </p>
3974               </td>
3975 <td>
3976               </td>
3977 </tr>
3978 <tr>
3979 <td>
3980                 <p>
3981                   Schr&#246;der
3982                 </p>
3983               </td>
3984 <td>
3985                 <p>
3986                   2
3987                 </p>
3988               </td>
3989 <td>
3990                 <p>
3991                   126
3992                 </p>
3993               </td>
3994 <td>
3995                 <p>
3996                   1.37
3997                 </p>
3998               </td>
3999 <td>
4000                 <p>
4001                   0
4002                 </p>
4003               </td>
4004 <td>
4005               </td>
4006 <td>
4007                 <p>
4008                   3
4009                 </p>
4010               </td>
4011 <td>
4012                 <p>
4013                   143
4014                 </p>
4015               </td>
4016 <td>
4017                 <p>
4018                   1.25
4019                 </p>
4020               </td>
4021 <td>
4022                 <p>
4023                   -1
4024                 </p>
4025               </td>
4026 <td>
4027               </td>
4028 <td>
4029                 <p>
4030                   3
4031                 </p>
4032               </td>
4033 <td>
4034                 <p>
4035                   198
4036                 </p>
4037               </td>
4038 <td>
4039                 <p>
4040                   1.57
4041                 </p>
4042               </td>
4043 <td>
4044                 <p>
4045                   0
4046                 </p>
4047               </td>
4048 <td>
4049               </td>
4050 <td>
4051                 <p>
4052                   4
4053                 </p>
4054               </td>
4055 <td>
4056                 <p>
4057                   15312
4058                 </p>
4059               </td>
4060 <td>
4061                 <p>
4062                   2.23
4063                 </p>
4064               </td>
4065 <td>
4066                 <p>
4067                   0
4068                 </p>
4069               </td>
4070 <td>
4071               </td>
4072 </tr>
4073 </tbody>
4074 </table></div>
4075 </div>
4076 <br class="table-break"><div class="table">
4077 <a name="math_toolkit.root_comparison.root_n_comparison.root_7_1"></a><p class="title"><b>Table&#160;10.10.&#160;7th root(28) for float, double, long double and cpp_bin_float_50 types,
4078         using _X64_SSE2</b></p>
4079 <div class="table-contents"><table class="table" summary="7th root(28) for float, double, long double and cpp_bin_float_50 types,
4080         using _X64_SSE2">
4081 <colgroup>
4082 <col>
4083 <col>
4084 <col>
4085 <col>
4086 <col>
4087 <col>
4088 <col>
4089 <col>
4090 <col>
4091 <col>
4092 <col>
4093 <col>
4094 <col>
4095 <col>
4096 <col>
4097 <col>
4098 <col>
4099 <col>
4100 <col>
4101 <col>
4102 <col>
4103 </colgroup>
4104 <thead><tr>
4105 <th>
4106               </th>
4107 <th>
4108                 <p>
4109                   float
4110                 </p>
4111               </th>
4112 <th>
4113               </th>
4114 <th>
4115               </th>
4116 <th>
4117               </th>
4118 <th>
4119               </th>
4120 <th>
4121                 <p>
4122                   double
4123                 </p>
4124               </th>
4125 <th>
4126               </th>
4127 <th>
4128               </th>
4129 <th>
4130               </th>
4131 <th>
4132               </th>
4133 <th>
4134                 <p>
4135                   long d
4136                 </p>
4137               </th>
4138 <th>
4139               </th>
4140 <th>
4141               </th>
4142 <th>
4143               </th>
4144 <th>
4145               </th>
4146 <th>
4147                 <p>
4148                   cpp50
4149                 </p>
4150               </th>
4151 <th>
4152               </th>
4153 <th>
4154               </th>
4155 <td class="auto-generated">&#160;</td>
4156 <td class="auto-generated">&#160;</td>
4157 </tr></thead>
4158 <tbody>
4159 <tr>
4160 <td>
4161                 <p>
4162                   Algo
4163                 </p>
4164               </td>
4165 <td>
4166                 <p>
4167                   Its
4168                 </p>
4169               </td>
4170 <td>
4171                 <p>
4172                   Times
4173                 </p>
4174               </td>
4175 <td>
4176                 <p>
4177                   Norm
4178                 </p>
4179               </td>
4180 <td>
4181                 <p>
4182                   Dis
4183                 </p>
4184               </td>
4185 <td>
4186               </td>
4187 <td>
4188                 <p>
4189                   Its
4190                 </p>
4191               </td>
4192 <td>
4193                 <p>
4194                   Times
4195                 </p>
4196               </td>
4197 <td>
4198                 <p>
4199                   Norm
4200                 </p>
4201               </td>
4202 <td>
4203                 <p>
4204                   Dis
4205                 </p>
4206               </td>
4207 <td>
4208               </td>
4209 <td>
4210                 <p>
4211                   Its
4212                 </p>
4213               </td>
4214 <td>
4215                 <p>
4216                   Times
4217                 </p>
4218               </td>
4219 <td>
4220                 <p>
4221                   Norm
4222                 </p>
4223               </td>
4224 <td>
4225                 <p>
4226                   Dis
4227                 </p>
4228               </td>
4229 <td>
4230               </td>
4231 <td>
4232                 <p>
4233                   Its
4234                 </p>
4235               </td>
4236 <td>
4237                 <p>
4238                   Times
4239                 </p>
4240               </td>
4241 <td>
4242                 <p>
4243                   Norm
4244                 </p>
4245               </td>
4246 <td>
4247                 <p>
4248                   Dis
4249                 </p>
4250               </td>
4251 <td>
4252               </td>
4253 </tr>
4254 <tr>
4255 <td>
4256                 <p>
4257                   TOMS748
4258                 </p>
4259               </td>
4260 <td>
4261                 <p>
4262                   12
4263                 </p>
4264               </td>
4265 <td>
4266                 <p>
4267                   345
4268                 </p>
4269               </td>
4270 <td>
4271                 <p>
4272                   2.09
4273                 </p>
4274               </td>
4275 <td>
4276                 <p>
4277                   1
4278                 </p>
4279               </td>
4280 <td>
4281               </td>
4282 <td>
4283                 <p>
4284                   15
4285                 </p>
4286               </td>
4287 <td>
4288                 <p>
4289                   615
4290                 </p>
4291               </td>
4292 <td>
4293                 <p>
4294                   3.14
4295                 </p>
4296               </td>
4297 <td>
4298                 <p>
4299                   2
4300                 </p>
4301               </td>
4302 <td>
4303               </td>
4304 <td>
4305                 <p>
4306                   13
4307                 </p>
4308               </td>
4309 <td>
4310                 <p>
4311                   875
4312                 </p>
4313               </td>
4314 <td>
4315                 <p>
4316                   3.98
4317                 </p>
4318               </td>
4319 <td>
4320                 <p>
4321                   0
4322                 </p>
4323               </td>
4324 <td>
4325               </td>
4326 <td>
4327                 <p>
4328                   14
4329                 </p>
4330               </td>
4331 <td>
4332                 <p>
4333                   70937
4334                 </p>
4335               </td>
4336 <td>
4337                 <p>
4338                   <span class="red">7.32</span>
4339                 </p>
4340               </td>
4341 <td>
4342                 <p>
4343                   0
4344                 </p>
4345               </td>
4346 <td>
4347               </td>
4348 </tr>
4349 <tr>
4350 <td>
4351                 <p>
4352                   Newton
4353                 </p>
4354               </td>
4355 <td>
4356                 <p>
4357                   5
4358                 </p>
4359               </td>
4360 <td>
4361                 <p>
4362                   165
4363                 </p>
4364               </td>
4365 <td>
4366                 <p>
4367                   <span class="blue">1.00</span>
4368                 </p>
4369               </td>
4370 <td>
4371                 <p>
4372                   0
4373                 </p>
4374               </td>
4375 <td>
4376               </td>
4377 <td>
4378                 <p>
4379                   6
4380                 </p>
4381               </td>
4382 <td>
4383                 <p>
4384                   196
4385                 </p>
4386               </td>
4387 <td>
4388                 <p>
4389                   <span class="blue">1.00</span>
4390                 </p>
4391               </td>
4392 <td>
4393                 <p>
4394                   0
4395                 </p>
4396               </td>
4397 <td>
4398               </td>
4399 <td>
4400                 <p>
4401                   7
4402                 </p>
4403               </td>
4404 <td>
4405                 <p>
4406                   220
4407                 </p>
4408               </td>
4409 <td>
4410                 <p>
4411                   <span class="blue">1.00</span>
4412                 </p>
4413               </td>
4414 <td>
4415                 <p>
4416                   0
4417                 </p>
4418               </td>
4419 <td>
4420               </td>
4421 <td>
4422                 <p>
4423                   8
4424                 </p>
4425               </td>
4426 <td>
4427                 <p>
4428                   9687
4429                 </p>
4430               </td>
4431 <td>
4432                 <p>
4433                   <span class="blue">1.00</span>
4434                 </p>
4435               </td>
4436 <td>
4437                 <p>
4438                   0
4439                 </p>
4440               </td>
4441 <td>
4442               </td>
4443 </tr>
4444 <tr>
4445 <td>
4446                 <p>
4447                   Halley
4448                 </p>
4449               </td>
4450 <td>
4451                 <p>
4452                   4
4453                 </p>
4454               </td>
4455 <td>
4456                 <p>
4457                   193
4458                 </p>
4459               </td>
4460 <td>
4461                 <p>
4462                   1.17
4463                 </p>
4464               </td>
4465 <td>
4466                 <p>
4467                   0
4468                 </p>
4469               </td>
4470 <td>
4471               </td>
4472 <td>
4473                 <p>
4474                   5
4475                 </p>
4476               </td>
4477 <td>
4478                 <p>
4479                   239
4480                 </p>
4481               </td>
4482 <td>
4483                 <p>
4484                   1.22
4485                 </p>
4486               </td>
4487 <td>
4488                 <p>
4489                   0
4490                 </p>
4491               </td>
4492 <td>
4493               </td>
4494 <td>
4495                 <p>
4496                   5
4497                 </p>
4498               </td>
4499 <td>
4500                 <p>
4501                   298
4502                 </p>
4503               </td>
4504 <td>
4505                 <p>
4506                   1.35
4507                 </p>
4508               </td>
4509 <td>
4510                 <p>
4511                   0
4512                 </p>
4513               </td>
4514 <td>
4515               </td>
4516 <td>
4517                 <p>
4518                   6
4519                 </p>
4520               </td>
4521 <td>
4522                 <p>
4523                   19062
4524                 </p>
4525               </td>
4526 <td>
4527                 <p>
4528                   1.97
4529                 </p>
4530               </td>
4531 <td>
4532                 <p>
4533                   0
4534                 </p>
4535               </td>
4536 <td>
4537               </td>
4538 </tr>
4539 <tr>
4540 <td>
4541                 <p>
4542                   Schr&#246;der
4543                 </p>
4544               </td>
4545 <td>
4546                 <p>
4547                   5
4548                 </p>
4549               </td>
4550 <td>
4551                 <p>
4552                   217
4553                 </p>
4554               </td>
4555 <td>
4556                 <p>
4557                   1.32
4558                 </p>
4559               </td>
4560 <td>
4561                 <p>
4562                   0
4563                 </p>
4564               </td>
4565 <td>
4566               </td>
4567 <td>
4568                 <p>
4569                   6
4570                 </p>
4571               </td>
4572 <td>
4573                 <p>
4574                   270
4575                 </p>
4576               </td>
4577 <td>
4578                 <p>
4579                   1.38
4580                 </p>
4581               </td>
4582 <td>
4583                 <p>
4584                   0
4585                 </p>
4586               </td>
4587 <td>
4588               </td>
4589 <td>
4590                 <p>
4591                   6
4592                 </p>
4593               </td>
4594 <td>
4595                 <p>
4596                   367
4597                 </p>
4598               </td>
4599 <td>
4600                 <p>
4601                   1.67
4602                 </p>
4603               </td>
4604 <td>
4605                 <p>
4606                   0
4607                 </p>
4608               </td>
4609 <td>
4610               </td>
4611 <td>
4612                 <p>
4613                   7
4614                 </p>
4615               </td>
4616 <td>
4617                 <p>
4618                   27343
4619                 </p>
4620               </td>
4621 <td>
4622                 <p>
4623                   2.82
4624                 </p>
4625               </td>
4626 <td>
4627                 <p>
4628                   0
4629                 </p>
4630               </td>
4631 <td>
4632               </td>
4633 </tr>
4634 </tbody>
4635 </table></div>
4636 </div>
4637 <br class="table-break"><div class="table">
4638 <a name="math_toolkit.root_comparison.root_n_comparison.root_11_1"></a><p class="title"><b>Table&#160;10.11.&#160;11th root(28) for float, double, long double and cpp_bin_float_50
4639         types, using _X64_SSE2</b></p>
4640 <div class="table-contents"><table class="table" summary="11th root(28) for float, double, long double and cpp_bin_float_50
4641         types, using _X64_SSE2">
4642 <colgroup>
4643 <col>
4644 <col>
4645 <col>
4646 <col>
4647 <col>
4648 <col>
4649 <col>
4650 <col>
4651 <col>
4652 <col>
4653 <col>
4654 <col>
4655 <col>
4656 <col>
4657 <col>
4658 <col>
4659 <col>
4660 <col>
4661 <col>
4662 <col>
4663 <col>
4664 </colgroup>
4665 <thead><tr>
4666 <th>
4667               </th>
4668 <th>
4669                 <p>
4670                   float
4671                 </p>
4672               </th>
4673 <th>
4674               </th>
4675 <th>
4676               </th>
4677 <th>
4678               </th>
4679 <th>
4680               </th>
4681 <th>
4682                 <p>
4683                   double
4684                 </p>
4685               </th>
4686 <th>
4687               </th>
4688 <th>
4689               </th>
4690 <th>
4691               </th>
4692 <th>
4693               </th>
4694 <th>
4695                 <p>
4696                   long d
4697                 </p>
4698               </th>
4699 <th>
4700               </th>
4701 <th>
4702               </th>
4703 <th>
4704               </th>
4705 <th>
4706               </th>
4707 <th>
4708                 <p>
4709                   cpp50
4710                 </p>
4711               </th>
4712 <th>
4713               </th>
4714 <th>
4715               </th>
4716 <td class="auto-generated">&#160;</td>
4717 <td class="auto-generated">&#160;</td>
4718 </tr></thead>
4719 <tbody>
4720 <tr>
4721 <td>
4722                 <p>
4723                   Algo
4724                 </p>
4725               </td>
4726 <td>
4727                 <p>
4728                   Its
4729                 </p>
4730               </td>
4731 <td>
4732                 <p>
4733                   Times
4734                 </p>
4735               </td>
4736 <td>
4737                 <p>
4738                   Norm
4739                 </p>
4740               </td>
4741 <td>
4742                 <p>
4743                   Dis
4744                 </p>
4745               </td>
4746 <td>
4747               </td>
4748 <td>
4749                 <p>
4750                   Its
4751                 </p>
4752               </td>
4753 <td>
4754                 <p>
4755                   Times
4756                 </p>
4757               </td>
4758 <td>
4759                 <p>
4760                   Norm
4761                 </p>
4762               </td>
4763 <td>
4764                 <p>
4765                   Dis
4766                 </p>
4767               </td>
4768 <td>
4769               </td>
4770 <td>
4771                 <p>
4772                   Its
4773                 </p>
4774               </td>
4775 <td>
4776                 <p>
4777                   Times
4778                 </p>
4779               </td>
4780 <td>
4781                 <p>
4782                   Norm
4783                 </p>
4784               </td>
4785 <td>
4786                 <p>
4787                   Dis
4788                 </p>
4789               </td>
4790 <td>
4791               </td>
4792 <td>
4793                 <p>
4794                   Its
4795                 </p>
4796               </td>
4797 <td>
4798                 <p>
4799                   Times
4800                 </p>
4801               </td>
4802 <td>
4803                 <p>
4804                   Norm
4805                 </p>
4806               </td>
4807 <td>
4808                 <p>
4809                   Dis
4810                 </p>
4811               </td>
4812 <td>
4813               </td>
4814 </tr>
4815 <tr>
4816 <td>
4817                 <p>
4818                   TOMS748
4819                 </p>
4820               </td>
4821 <td>
4822                 <p>
4823                   12
4824                 </p>
4825               </td>
4826 <td>
4827                 <p>
4828                   412
4829                 </p>
4830               </td>
4831 <td>
4832                 <p>
4833                   2.15
4834                 </p>
4835               </td>
4836 <td>
4837                 <p>
4838                   -2
4839                 </p>
4840               </td>
4841 <td>
4842               </td>
4843 <td>
4844                 <p>
4845                   14
4846                 </p>
4847               </td>
4848 <td>
4849                 <p>
4850                   646
4851                 </p>
4852               </td>
4853 <td>
4854                 <p>
4855                   2.96
4856                 </p>
4857               </td>
4858 <td>
4859                 <p>
4860                   2
4861                 </p>
4862               </td>
4863 <td>
4864               </td>
4865 <td>
4866                 <p>
4867                   14
4868                 </p>
4869               </td>
4870 <td>
4871                 <p>
4872                   1054
4873                 </p>
4874               </td>
4875 <td>
4876                 <p>
4877                   <span class="red">4.22</span>
4878                 </p>
4879               </td>
4880 <td>
4881                 <p>
4882                   1
4883                 </p>
4884               </td>
4885 <td>
4886               </td>
4887 <td>
4888                 <p>
4889                   17
4890                 </p>
4891               </td>
4892 <td>
4893                 <p>
4894                   107187
4895                 </p>
4896               </td>
4897 <td>
4898                 <p>
4899                   <span class="red">9.53</span>
4900                 </p>
4901               </td>
4902 <td>
4903                 <p>
4904                   2
4905                 </p>
4906               </td>
4907 <td>
4908               </td>
4909 </tr>
4910 <tr>
4911 <td>
4912                 <p>
4913                   Newton
4914                 </p>
4915               </td>
4916 <td>
4917                 <p>
4918                   6
4919                 </p>
4920               </td>
4921 <td>
4922                 <p>
4923                   192
4924                 </p>
4925               </td>
4926 <td>
4927                 <p>
4928                   <span class="blue">1.00</span>
4929                 </p>
4930               </td>
4931 <td>
4932                 <p>
4933                   0
4934                 </p>
4935               </td>
4936 <td>
4937               </td>
4938 <td>
4939                 <p>
4940                   7
4941                 </p>
4942               </td>
4943 <td>
4944                 <p>
4945                   218
4946                 </p>
4947               </td>
4948 <td>
4949                 <p>
4950                   <span class="blue">1.00</span>
4951                 </p>
4952               </td>
4953 <td>
4954                 <p>
4955                   0
4956                 </p>
4957               </td>
4958 <td>
4959               </td>
4960 <td>
4961                 <p>
4962                   7
4963                 </p>
4964               </td>
4965 <td>
4966                 <p>
4967                   250
4968                 </p>
4969               </td>
4970 <td>
4971                 <p>
4972                   <span class="blue">1.00</span>
4973                 </p>
4974               </td>
4975 <td>
4976                 <p>
4977                   0
4978                 </p>
4979               </td>
4980 <td>
4981               </td>
4982 <td>
4983                 <p>
4984                   9
4985                 </p>
4986               </td>
4987 <td>
4988                 <p>
4989                   11250
4990                 </p>
4991               </td>
4992 <td>
4993                 <p>
4994                   <span class="blue">1.00</span>
4995                 </p>
4996               </td>
4997 <td>
4998                 <p>
4999                   0
5000                 </p>
5001               </td>
5002 <td>
5003               </td>
5004 </tr>
5005 <tr>
5006 <td>
5007                 <p>
5008                   Halley
5009                 </p>
5010               </td>
5011 <td>
5012                 <p>
5013                   4
5014                 </p>
5015               </td>
5016 <td>
5017                 <p>
5018                   200
5019                 </p>
5020               </td>
5021 <td>
5022                 <p>
5023                   <span class="blue">1.04</span>
5024                 </p>
5025               </td>
5026 <td>
5027                 <p>
5028                   -1
5029                 </p>
5030               </td>
5031 <td>
5032               </td>
5033 <td>
5034                 <p>
5035                   5
5036                 </p>
5037               </td>
5038 <td>
5039                 <p>
5040                   243
5041                 </p>
5042               </td>
5043 <td>
5044                 <p>
5045                   1.11
5046                 </p>
5047               </td>
5048 <td>
5049                 <p>
5050                   0
5051                 </p>
5052               </td>
5053 <td>
5054               </td>
5055 <td>
5056                 <p>
5057                   5
5058                 </p>
5059               </td>
5060 <td>
5061                 <p>
5062                   345
5063                 </p>
5064               </td>
5065 <td>
5066                 <p>
5067                   1.38
5068                 </p>
5069               </td>
5070 <td>
5071                 <p>
5072                   0
5073                 </p>
5074               </td>
5075 <td>
5076               </td>
5077 <td>
5078                 <p>
5079                   6
5080                 </p>
5081               </td>
5082 <td>
5083                 <p>
5084                   19687
5085                 </p>
5086               </td>
5087 <td>
5088                 <p>
5089                   1.75
5090                 </p>
5091               </td>
5092 <td>
5093                 <p>
5094                   0
5095                 </p>
5096               </td>
5097 <td>
5098               </td>
5099 </tr>
5100 <tr>
5101 <td>
5102                 <p>
5103                   Schr&#246;der
5104                 </p>
5105               </td>
5106 <td>
5107                 <p>
5108                   6
5109                 </p>
5110               </td>
5111 <td>
5112                 <p>
5113                   254
5114                 </p>
5115               </td>
5116 <td>
5117                 <p>
5118                   1.32
5119                 </p>
5120               </td>
5121 <td>
5122                 <p>
5123                   0
5124                 </p>
5125               </td>
5126 <td>
5127               </td>
5128 <td>
5129                 <p>
5130                   7
5131                 </p>
5132               </td>
5133 <td>
5134                 <p>
5135                   321
5136                 </p>
5137               </td>
5138 <td>
5139                 <p>
5140                   1.47
5141                 </p>
5142               </td>
5143 <td>
5144                 <p>
5145                   0
5146                 </p>
5147               </td>
5148 <td>
5149               </td>
5150 <td>
5151                 <p>
5152                   7
5153                 </p>
5154               </td>
5155 <td>
5156                 <p>
5157                   471
5158                 </p>
5159               </td>
5160 <td>
5161                 <p>
5162                   1.88
5163                 </p>
5164               </td>
5165 <td>
5166                 <p>
5167                   0
5168                 </p>
5169               </td>
5170 <td>
5171               </td>
5172 <td>
5173                 <p>
5174                   8
5175                 </p>
5176               </td>
5177 <td>
5178                 <p>
5179                   33281
5180                 </p>
5181               </td>
5182 <td>
5183                 <p>
5184                   2.96
5185                 </p>
5186               </td>
5187 <td>
5188                 <p>
5189                   0
5190                 </p>
5191               </td>
5192 <td>
5193               </td>
5194 </tr>
5195 </tbody>
5196 </table></div>
5197 </div>
5198 <br class="table-break"><p>
5199         Some tentative conclusions can be drawn from this limited exercise.
5200       </p>
5201 <div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
5202 <li class="listitem">
5203             Perhaps surprisingly, there is little difference between the various
5204             algorithms for <a href="http://en.cppreference.com/w/cpp/language/types" target="_top">fundamental
5205             (built-in) types</a> floating-point types. Using the first derivatives
5206             (<a class="link" href="../roots_deriv.html#math_toolkit.roots_deriv.newton">Newton-Raphson iteration</a>)
5207             is usually the best, but while the improvement over the no-derivative
5208             <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS Algorithm
5209             748: enclosing zeros of continuous functions</a> is considerable
5210             in number of iterations, but little in execution time. This reflects
5211             the fact that the function we are finding the root for is trivial to
5212             evaluate, so runtimetimes are dominated by the time taken by the boilerplate
5213             code in each method.
5214           </li>
5215 <li class="listitem">
5216             The extra cost of evaluating the second derivatives (<a class="link" href="../roots_deriv.html#math_toolkit.roots_deriv.halley">Halley</a>
5217             or <a class="link" href="../roots_deriv.html#math_toolkit.roots_deriv.schroder">Schr&#246;der</a>)
5218             is usually too much for any net benefit: as with the cube root, these
5219             functors are so cheap to evaluate that the runtime is largely dominated
5220             by the complexity of the root finding method.
5221           </li>
5222 <li class="listitem">
5223             For a <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
5224             floating-point type, the <a class="link" href="../roots_deriv.html#math_toolkit.roots_deriv.newton">Newton-Raphson
5225             iteration</a> is a clear winner with a several-fold gain over <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS Algorithm 748:
5226             enclosing zeros of continuous functions</a>, and again no improvement
5227             from the second-derivative algorithms.
5228           </li>
5229 <li class="listitem">
5230             The run-time of 50 decimal-digit <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
5231             is about 30-fold greater than <code class="computeroutput"><span class="keyword">double</span></code>.
5232           </li>
5233 <li class="listitem">
5234             The column 'dis' showing the number of bits distance from the correct
5235             result. The Newton-Raphson algorithm shows a bit or two better accuracy
5236             than <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS Algorithm
5237             748: enclosing zeros of continuous functions</a>.
5238           </li>
5239 <li class="listitem">
5240             The goodness of the 'guess' is especially crucial for <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.
5241             Separate experiments show that evaluating the 'guess' using <code class="computeroutput"><span class="keyword">double</span></code> allows convergence to the final
5242             exact result in one or two iterations. So in this contrived example,
5243             crudely dividing the exponent by N for a 'guess', it would be far better
5244             to use a <code class="computeroutput"><span class="identifier">pow</span><span class="special">&lt;</span><span class="keyword">double</span><span class="special">&gt;</span></code>
5245             or , if more precise <code class="computeroutput"><span class="identifier">pow</span><span class="special">&lt;</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">&gt;</span></code>,
5246             function to estimate a 'guess'. The limitation of this tactic is that
5247             the range of possible (exponent) values may be less than the multiprecision
5248             type.
5249           </li>
5250 <li class="listitem">
5251             Using floating-point extension <a href="http://en.wikipedia.org/wiki/SSE2" target="_top">SSE2
5252             instructions</a> made a modest ten-percent speedup.
5253           </li>
5254 <li class="listitem">
5255             Using MSVC, there was some improvement using 64-bit, markedly for <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.
5256           </li>
5257 <li class="listitem">
5258             The GCC compiler 4.9.1 using 64-bit was at least five-folder faster that
5259             32-bit, apparently reflecting better optimization.
5260           </li>
5261 </ul></div>
5262 <p>
5263         Clearly, your mileage <span class="bold"><strong>will vary</strong></span>, but in
5264         summary, <a class="link" href="../roots_deriv.html#math_toolkit.roots_deriv.newton">Newton-Raphson iteration</a>
5265         seems the first choice of algorithm, and effort to find a good 'guess' the
5266         first speed-up target, especially for <a href="../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.
5267         And of course, compiler optimisation is crucial for speed.
5268       </p>
5269 </div>
5270 <table xmlns:rev="http://www.cs.rpi.edu/~gregod/boost/tools/doc/revision" width="100%"><tr>
5271 <td align="left"></td>
5272 <td align="right"><div class="copyright-footer">Copyright &#169; 2006-2019 Nikhar
5273       Agrawal, Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos,
5274       Hubert Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Matthew Pulver, Johan
5275       R&#229;de, Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg,
5276       Daryle Walker and Xiaogang Zhang<p>
5277         Distributed under the Boost Software License, Version 1.0. (See accompanying
5278         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>)
5279       </p>
5280 </div></td>
5281 </tr></table>
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