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4 <title>Comparison of Nth-root Finding Algorithms</title>
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24 </div>
25 <div class="section">
26 <div class="titlepage"><div><div><h4 class="title">
27 <a name="math_toolkit.roots.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 </h4></div></div></div>
30 <p>
31 A second example compares four generalized nth-root finding algorithms
32 for 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>.
36 In each case the target accuracy was set using our "recomended"
37 accuracy limits (or at least limits that make a good starting point - which
38 is likely to give close to full accuracy without resorting to unnecessary
39 iterations).
40 </p>
41 <div class="informaltable"><table class="table">
42 <colgroup>
43 <col>
44 <col>
45 </colgroup>
46 <thead><tr>
47 <th>
48 <p>
49 Function
50 </p>
51 </th>
52 <th>
53 <p>
54 Precision Requested
55 </p>
56 </th>
57 </tr></thead>
58 <tbody>
59 <tr>
60 <td>
61 <p>
62 TOMS748
63 </p>
64 </td>
65 <td>
66 <p>
67 numeric_limits&lt;T&gt;::digits - 2
68 </p>
69 </td>
70 </tr>
71 <tr>
72 <td>
73 <p>
74 Newton
75 </p>
76 </td>
77 <td>
78 <p>
79 floor(numeric_limits&lt;T&gt;::digits * 0.6)
80 </p>
81 </td>
82 </tr>
83 <tr>
84 <td>
85 <p>
86 Halley
87 </p>
88 </td>
89 <td>
90 <p>
91 floor(numeric_limits&lt;T&gt;::digits * 0.4)
92 </p>
93 </td>
94 </tr>
95 <tr>
96 <td>
97 <p>
98 Schr&#246;der
99 </p>
100 </td>
101 <td>
102 <p>
103 floor(numeric_limits&lt;T&gt;::digits * 0.4)
104 </p>
105 </td>
106 </tr>
107 </tbody>
108 </table></div>
109 <p>
110 Tests used Microsoft Visual Studio 2013 (Update 1) and GCC 4.9.1 using
111 source code <a href="../../../../../example/root_n_finding_algorithms.cpp" target="_top">root_n_finding_algorithms.cpp</a>.
112 </p>
113 <p>
114 The timing uncertainty (especially using MSVC) is at least 5% of normalized
115 time 'Norm'.
116 </p>
117 <p>
118 To pick out the 'best' and 'worst' algorithms are highlighted in blue and
119 red. More than one result can be 'best' when normalized times are indistinguishable
120 within the uncertainty.
121 </p>
122 <h5>
123 <a name="math_toolkit.roots.root_comparison.root_n_comparison.h0"></a>
124 <span class="phrase"><a name="math_toolkit.roots.root_comparison.root_n_comparison.program_root_n_finding_algorithm"></a></span><a class="link" href="root_n_comparison.html#math_toolkit.roots.root_comparison.root_n_comparison.program_root_n_finding_algorithm">Program
125 root_n_finding_algorithms.cpp, Microsoft Visual C++ version 12.0, Dinkumware
126 standard library version 610, Win32 Compiled in optimise mode., _X86_SSE2</a>
127 </h5>
128 <p>
129 Fraction of full accuracy 1
130 </p>
131 <div class="table">
132 <a name="math_toolkit.roots.root_comparison.root_n_comparison.root_5"></a><p class="title"><b>Table&#160;12.3.&#160;5th root(28) for float, double, long double and cpp_bin_float_50
133 types, 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
135 types, 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>
322 320
323 </p>
324 </td>
325 <td>
326 <p>
327 1.53
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>
344 576
345 </p>
346 </td>
347 <td>
348 <p>
349 2.61
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>
366 557
367 </p>
368 </td>
369 <td>
370 <p>
371 2.48
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 119843
389 </p>
390 </td>
391 <td>
392 <p>
393 <span class="red">7.52</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>
417 209
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>
439 221
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>
461 225
462 </p>
463 </td>
464 <td>
465 <p>
466 <span class="blue">1.00</span>
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 15937
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>
512 214
513 </p>
514 </td>
515 <td>
516 <p>
517 <span class="blue">1.02</span>
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>
534 256
535 </p>
536 </td>
537 <td>
538 <p>
539 1.16
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>
556 243
557 </p>
558 </td>
559 <td>
560 <p>
561 1.08
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 28437
579 </p>
580 </td>
581 <td>
582 <p>
583 1.78
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>
607 218
608 </p>
609 </td>
610 <td>
611 <p>
612 <span class="blue">1.04</span>
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>
629 245
630 </p>
631 </td>
632 <td>
633 <p>
634 1.11
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>
651 245
652 </p>
653 </td>
654 <td>
655 <p>
656 1.09
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 35625
674 </p>
675 </td>
676 <td>
677 <p>
678 2.24
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.roots.root_comparison.root_n_comparison.root_7"></a><p class="title"><b>Table&#160;12.4.&#160;7th root(28) for float, double, long double and cpp_bin_float_50
694 types, 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
696 types, 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>
883 493
884 </p>
885 </td>
886 <td>
887 <p>
888 2.18
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>
905 762
906 </p>
907 </td>
908 <td>
909 <p>
910 3.05
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>
927 765
928 </p>
929 </td>
930 <td>
931 <p>
932 3.08
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 157343
950 </p>
951 </td>
952 <td>
953 <p>
954 <span class="red">7.09</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>
978 226
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 250
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 248
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 22187
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 257
1074 </p>
1075 </td>
1076 <td>
1077 <p>
1078 1.14
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 293
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 293
1118 </p>
1119 </td>
1120 <td>
1121 <p>
1122 1.18
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 44062
1140 </p>
1141 </td>
1142 <td>
1143 <p>
1144 1.99
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 285
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 317
1191 </p>
1192 </td>
1193 <td>
1194 <p>
1195 1.27
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 317
1213 </p>
1214 </td>
1215 <td>
1216 <p>
1217 1.28
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 61406
1235 </p>
1236 </td>
1237 <td>
1238 <p>
1239 2.77
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.roots.root_comparison.root_n_comparison.root_11"></a><p class="title"><b>Table&#160;12.5.&#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 556
1445 </p>
1446 </td>
1447 <td>
1448 <p>
1449 2.24
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 784
1467 </p>
1468 </td>
1469 <td>
1470 <p>
1471 2.94
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 2.94
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 235781
1511 </p>
1512 </td>
1513 <td>
1514 <p>
1515 <span class="red">8.88</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 248
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 267
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 270
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 26562
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 254
1635 </p>
1636 </td>
1637 <td>
1638 <p>
1639 <span class="blue">1.02</span>
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 290
1657 </p>
1658 </td>
1659 <td>
1660 <p>
1661 1.09
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 293
1679 </p>
1680 </td>
1681 <td>
1682 <p>
1683 1.09
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 46406
1701 </p>
1702 </td>
1703 <td>
1704 <p>
1705 1.75
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 312
1730 </p>
1731 </td>
1732 <td>
1733 <p>
1734 1.26
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 351
1752 </p>
1753 </td>
1754 <td>
1755 <p>
1756 1.31
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 356
1774 </p>
1775 </td>
1776 <td>
1777 <p>
1778 1.32
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 76250
1796 </p>
1797 </td>
1798 <td>
1799 <p>
1800 2.87
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"><h5>
1815 <a name="math_toolkit.roots.root_comparison.root_n_comparison.h1"></a>
1816 <span class="phrase"><a name="math_toolkit.roots.root_comparison.root_n_comparison.program_root_n_finding_algorith0"></a></span><a class="link" href="root_n_comparison.html#math_toolkit.roots.root_comparison.root_n_comparison.program_root_n_finding_algorith0">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 </h5>
1820 <p>
1821 Fraction of full accuracy 1
1822 </p>
1823 <div class="table">
1824 <a name="math_toolkit.roots.root_comparison.root_n_comparison.root_5_0"></a><p class="title"><b>Table&#160;12.6.&#160;5th root(28) for float, double, long double and cpp_bin_float_50
1825 types, 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
1827 types, 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.roots.root_comparison.root_n_comparison.root_7_0"></a><p class="title"><b>Table&#160;12.7.&#160;7th root(28) for float, double, long double and cpp_bin_float_50
2386 types, 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
2388 types, 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.roots.root_comparison.root_n_comparison.root_11_0"></a><p class="title"><b>Table&#160;12.8.&#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"><h5>
3507 <a name="math_toolkit.roots.root_comparison.root_n_comparison.h2"></a>
3508 <span class="phrase"><a name="math_toolkit.roots.root_comparison.root_n_comparison.program_root_n_finding_algorith1"></a></span><a class="link" href="root_n_comparison.html#math_toolkit.roots.root_comparison.root_n_comparison.program_root_n_finding_algorith1">Program
3509 root_n_finding_algorithms.cpp, GNU C++ version 4.9.2, GNU libstdc++ version
3510 20141030, Win32 Compiled in optimise mode., _X64_SSE2</a>
3511 </h5>
3512 <p>
3513 Fraction of full accuracy 1
3514 </p>
3515 <div class="table">
3516 <a name="math_toolkit.roots.root_comparison.root_n_comparison.root_5_1"></a><p class="title"><b>Table&#160;12.9.&#160;5th root(28) for float, double, long double and cpp_bin_float_50
3517 types, 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
3519 types, 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 193
3707 </p>
3708 </td>
3709 <td>
3710 <p>
3711 2.14
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 432
3729 </p>
3730 </td>
3731 <td>
3732 <p>
3733 3.86
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 579
3751 </p>
3752 </td>
3753 <td>
3754 <p>
3755 3.83
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 59062
3773 </p>
3774 </td>
3775 <td>
3776 <p>
3777 <span class="red">7.56</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 90
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 112
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 151
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 7812
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 98
3897 </p>
3898 </td>
3899 <td>
3900 <p>
3901 1.09
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 135
3919 </p>
3920 </td>
3921 <td>
3922 <p>
3923 1.21
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 201
3941 </p>
3942 </td>
3943 <td>
3944 <p>
3945 1.33
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 13750
3963 </p>
3964 </td>
3965 <td>
3966 <p>
3967 1.76
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 112
3992 </p>
3993 </td>
3994 <td>
3995 <p>
3996 1.24
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 142
4014 </p>
4015 </td>
4016 <td>
4017 <p>
4018 1.27
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 206
4036 </p>
4037 </td>
4038 <td>
4039 <p>
4040 1.36
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 17031
4058 </p>
4059 </td>
4060 <td>
4061 <p>
4062 2.18
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.roots.root_comparison.root_n_comparison.root_7_1"></a><p class="title"><b>Table&#160;12.10.&#160;7th root(28) for float, double, long double and cpp_bin_float_50
4078 types, 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
4080 types, 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 351
4268 </p>
4269 </td>
4270 <td>
4271 <p>
4272 1.97
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 621
4290 </p>
4291 </td>
4292 <td>
4293 <p>
4294 3.18
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 906
4312 </p>
4313 </td>
4314 <td>
4315 <p>
4316 3.61
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 75468
4334 </p>
4335 </td>
4336 <td>
4337 <p>
4338 <span class="red">7.10</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 178
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 195
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 251
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 10625
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 196
4458 </p>
4459 </td>
4460 <td>
4461 <p>
4462 1.10
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 242
4480 </p>
4481 </td>
4482 <td>
4483 <p>
4484 1.24
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 345
4502 </p>
4503 </td>
4504 <td>
4505 <p>
4506 1.37
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 21093
4524 </p>
4525 </td>
4526 <td>
4527 <p>
4528 1.99
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 225
4553 </p>
4554 </td>
4555 <td>
4556 <p>
4557 1.26
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 384
4597 </p>
4598 </td>
4599 <td>
4600 <p>
4601 1.53
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 29062
4619 </p>
4620 </td>
4621 <td>
4622 <p>
4623 2.74
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.roots.root_comparison.root_n_comparison.root_11_1"></a><p class="title"><b>Table&#160;12.11.&#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 429
4829 </p>
4830 </td>
4831 <td>
4832 <p>
4833 2.22
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 679
4851 </p>
4852 </td>
4853 <td>
4854 <p>
4855 3.02
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 1098
4873 </p>
4874 </td>
4875 <td>
4876 <p>
4877 3.94
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 114531
4895 </p>
4896 </td>
4897 <td>
4898 <p>
4899 <span class="red">8.83</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 193
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 225
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 279
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 12968
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 196
5019 </p>
5020 </td>
5021 <td>
5022 <p>
5023 <span class="blue">1.02</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 248
5041 </p>
5042 </td>
5043 <td>
5044 <p>
5045 1.10
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 348
5063 </p>
5064 </td>
5065 <td>
5066 <p>
5067 1.25
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 21718
5085 </p>
5086 </td>
5087 <td>
5088 <p>
5089 1.67
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 323
5136 </p>
5137 </td>
5138 <td>
5139 <p>
5140 1.44
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 453
5158 </p>
5159 </td>
5160 <td>
5161 <p>
5162 1.62
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 35625
5180 </p>
5181 </td>
5182 <td>
5183 <p>
5184 2.75
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 types</a> floating-point types. Using the first derivatives (<a class="link" href="../roots_deriv.html#math_toolkit.roots.roots_deriv.newton">Newton-Raphson iteration</a>)
5206 is usually the best, but while the improvement over the no-derivative
5207 <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS Algorithm
5208 748: enclosing zeros of continuous functions</a> is considerable
5209 in number of iterations, but little in execution time. This reflects
5210 the fact that the function we are finding the root for is trivial to
5211 evaluate, so runtimetimes are dominated by the time taken by the boilerplate
5212 code in each method.
5213 </li>
5214 <li class="listitem">
5215 The extra cost of evaluating the second derivatives (<a class="link" href="../roots_deriv.html#math_toolkit.roots.roots_deriv.halley">Halley</a>
5216 or <a class="link" href="../roots_deriv.html#math_toolkit.roots.roots_deriv.schroder">Schr&#246;der</a>)
5217 is usually too much for any net benefit: as with the cube root, these
5218 functors are so cheap to evaluate that the runtime is largely dominated
5219 by the complexity of the root finding method.
5220 </li>
5221 <li class="listitem">
5222 For a <a href="../../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
5223 floating-point type, the <a class="link" href="../roots_deriv.html#math_toolkit.roots.roots_deriv.newton">Newton-Raphson
5224 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:
5225 enclosing zeros of continuous functions</a>, and again no improvement
5226 from the second-derivative algorithms.
5227 </li>
5228 <li class="listitem">
5229 The run-time of 50 decimal-digit <a href="../../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>
5230 is about 30-fold greater than <code class="computeroutput"><span class="keyword">double</span></code>.
5231 </li>
5232 <li class="listitem">
5233 The column 'dis' showing the number of bits distance from the correct
5234 result. The Newton-Raphson algorithm shows a bit or two better accuracy
5235 than <a href="http://portal.acm.org/citation.cfm?id=210111" target="_top">TOMS
5236 Algorithm 748: enclosing zeros of continuous functions</a>.
5237 </li>
5238 <li class="listitem">
5239 The goodness of the 'guess' is especially crucial for <a href="../../../../../../../libs/multiprecision/doc/html/index.html" target="_top">Boost.Multiprecision</a>.
5240 Separate experiments show that evaluating the 'guess' using <code class="computeroutput"><span class="keyword">double</span></code> allows convergence to the final
5241 exact result in one or two iterations. So in this contrived example,
5242 crudely dividing the exponent by N for a 'guess', it would be far better
5243 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>
5244 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>,
5245 function to estimate a 'guess'. The limitation of this tactic is that
5246 the range of possible (exponent) values may be less than the multiprecision
5247 type.
5248 </li>
5249 <li class="listitem">
5250 Using floating-point extension <a href="http://en.wikipedia.org/wiki/SSE2" target="_top">SSE2
5251 instructions</a> made a modest ten-percent speedup.
5252 </li>
5253 <li class="listitem">
5254 Using MSVC, there was some improvement using 64-bit, markedly for
5255 <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
5259 that 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.roots_deriv.newton">Newton-Raphson
5265 iteration</a> seems the first choice of algorithm, and effort to find
5266 a good 'guess' the 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; 2006-2010, 2012-2014 Nikhar Agrawal,
5273 Anton Bikineev, Paul A. Bristow, Marco Guazzone, Christopher Kormanyos, Hubert
5274 Holin, Bruno Lalande, John Maddock, Jeremy Murphy, Johan R&#229;de, Gautam Sewani,
5275 Benjamin Sobotta, Thijs van den Berg, Daryle Walker and Xiaogang Zhang<p>
5276 Distributed under the Boost Software License, Version 1.0. (See accompanying
5277 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>)
5278 </p>
5279 </div></td>
5280 </tr></table>
5281 <hr>
5282 <div class="spirit-nav">
5283 <a accesskey="p" href="cbrt_comparison.html"><img src="../../../../../../../doc/src/images/prev.png" alt="Prev"></a><a accesskey="u" href="../root_comparison.html"><img src="../../../../../../../doc/src/images/up.png" alt="Up"></a><a accesskey="h" href="../../../index.html"><img src="../../../../../../../doc/src/images/home.png" alt="Home"></a><a accesskey="n" href="elliptic_comparison.html"><img src="../../../../../../../doc/src/images/next.png" alt="Next"></a>
5284 </div>
5285 </body>
5286 </html>
Ceph