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1 | <html> |
2 | <head> | |
3 | <meta http-equiv="Content-Type" content="text/html; charset=US-ASCII"> | |
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<T>::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<T>::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<T>::digits * 0.4) | |
92 | </p> | |
93 | </td> | |
94 | </tr> | |
95 | <tr> | |
96 | <td> | |
97 | <p> | |
98 | Schröder | |
99 | </p> | |
100 | </td> | |
101 | <td> | |
102 | <p> | |
103 | floor(numeric_limits<T>::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 12.3. 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"> </td> | |
211 | <td class="auto-generated"> </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 | Schröder | |
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 12.4. 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"> </td> | |
772 | <td class="auto-generated"> </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 | Schröder | |
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 12.5. 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"> </td> | |
1333 | <td class="auto-generated"> </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 | Schröder | |
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 12.6. 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"> </td> | |
1903 | <td class="auto-generated"> </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 | Schröder | |
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 12.7. 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"> </td> | |
2464 | <td class="auto-generated"> </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 | Schröder | |
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 12.8. 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"> </td> | |
3025 | <td class="auto-generated"> </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 | Schröder | |
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 12.9. 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"> </td> | |
3595 | <td class="auto-generated"> </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 | Schröder | |
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 12.10. 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"> </td> | |
4156 | <td class="auto-generated"> </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 | Schröder | |
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 12.11. 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"> </td> | |
4717 | <td class="auto-generated"> </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 | Schröder | |
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">Schröder</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"><</span><span class="keyword">double</span><span class="special">></span></code> | |
5244 | or , if more precise <code class="computeroutput"><span class="identifier">pow</span><span class="special"><</span><span class="keyword">long</span> <span class="keyword">double</span><span class="special">></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 © 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 Råde, 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> |