1519:
1191:
2901:
1514:{\displaystyle {\begin{aligned}{\tilde {x}}-{\tilde {y}}&=x(1+\delta _{x})-y(1+\delta _{y})=x-y+x\delta _{x}-y\delta _{y}\\&=x-y+(x-y){\frac {x\delta _{x}-y\delta _{y}}{x-y}}\\&=(x-y){\biggr (}1+{\frac {x\delta _{x}-y\delta _{y}}{x-y}}{\biggr )}.\end{aligned}}}
3979:
411:
2050:
1889:
2285:
3591:
487:
2787:
4530:
2438:
2588:
1667:
3439:
4337:
310:
4396:
4589:
2149:
2101:
3726:
3684:
3247:
1196:
1074:
988:
213:
165:
3828:
3024:
899:
842:
4650:
2961:
4052:
algorithms both rely on such cancellation after a rounding error in order to exactly compute what the error was in a floating-point addition operation as a floating-point number itself.
3593:. Although the radix conversion from decimal floating-point to binary floating-point only incurs a small relative error, catastrophic cancellation may amplify it into a much larger one:
2691:
3813:
1565:
1158:
117:
78:
4820:
4181:
4761:
3361:
758:
732:
706:
680:
3207:
4038:
654:
628:
3774:
3750:
3539:
2780:
539:
4688:
2484:
763:
Catastrophic cancellation may happen even if the difference is computed exactly, as in the example above—it is not a property of any particular kind of arithmetic like
602:
576:
3097:
513:
2929:
4123:
3117:
2743:
3489:
3149:
2376:
2189:
1813:
3181:
2336:
1976:
4434:
4219:
4091:
322:
4717:
3069:
3305:
2645:
2723:
4002:
3279:
2619:
1591:
1184:
771:
are approximations themselves. Indeed, in floating-point arithmetic, when the inputs are close enough, the floating-point difference is computed exactly, by the
3049:
1981:
4243:
4143:
1929:
1909:
1818:
1773:
1753:
1709:
1689:
1114:
1094:
544:
Catastrophic cancellation is not affected by how large the inputs are—it applies just as much to large and small inputs. It depends only on how large the
2503:
2194:
3548:
1598:
416:
38:
is the phenomenon that subtracting good approximations to two nearby numbers may yield a very bad approximation to the difference of the original numbers.
4040:
of the original values as written in decimal: catastrophic cancellation amplified a tiny error in radix conversion into a large error in the output.
4439:
2381:
3210:
2288:
4839:
Muller, Jean-Michel; Brunie, Nicolas; de
Dinechin, Florent; Jeannerod, Claude-Pierre; Joldes, Mioara; Lefèvre, Vincent; Melquiond, Guillaume;
4862:
4248:
2896:{\displaystyle \operatorname {fl} {\Bigl (}{\sqrt {\operatorname {fl} (1-\operatorname {fl} (z^{2}))}}{\Bigr )}={\sqrt {1-z^{2}}}(1+\delta )}
1719:
Subtracting nearby numbers in floating-point arithmetic does not always cause catastrophic cancellation, or even any error—by the
2968:
3366:
3822:
But even though the inputs are good approximations, and even though the subtraction is computed exactly, the difference of the
2495:
222:
4342:
4535:
2106:
2058:
119:
long, and they are measured with a ruler that is good only to the centimeter, then the approximations could come out to be
3689:
1931:
are close in magnitude, because the subtraction can expose the rounding errors in the squaring. The alternative factoring
3974:{\displaystyle {\tilde {y}}-{\tilde {x}}=(1+9\cdot 2^{-52})-(1+5\cdot 2^{-52})=4\cdot 2^{-52}\approx 8.88\times 10^{-16}}
3663:
3216:
4878:
993:
907:
170:
122:
4956:
847:
790:
4594:
2934:
2650:
764:
3779:
1526:
1119:
83:
44:
4766:
4148:
2052:, avoids catastrophic cancellation because it avoids introducing rounding error leading into the subtraction.
4722:
3310:
2440:, of which less than half the digits are correct and the other (underlined) digits reflect the missing terms
737:
711:
685:
659:
3186:
4007:
633:
607:
3759:
3735:
3524:
2748:
518:
4655:
2443:
581:
555:
3074:
492:
2908:
4898:
4096:
3102:
2728:
406:{\displaystyle {\tilde {L}}_{1}-{\tilde {L}}_{2}=254\,{\text{cm}}-252\,{\text{cm}}=2\,{\text{cm}}}
4926:
4872:
3499:
Numerical constants in software programs are often written in decimal, such as in the C fragment
3444:
3122:
2341:
2154:
1778:
31:
3154:
2294:
1934:
4401:
4186:
4058:
1723:, if the numbers are close enough the floating-point difference is exact. But cancellation may
4918:
4858:
4693:
3249:, with only five out of sixteen digits correct and the remainder (underlined) all incorrect.
3054:
4910:
4850:
4222:
3284:
2624:
219:, to the true lengths: the approximations are in error by less than 2% of the true lengths,
2696:
2045:{\displaystyle \operatorname {fl} (\operatorname {fl} (x+y)\cdot \operatorname {fl} (x-y))}
3984:
3255:
2595:
1884:{\displaystyle \operatorname {fl} (\operatorname {fl} (x^{2})-\operatorname {fl} (y^{2}))}
784:
775:—there is no rounding error introduced by the floating-point subtraction operation.
4048:
Cancellation is sometimes useful and desirable in numerical algorithms. For example, the
2338:
gives the correct result exactly (with no rounding), but evaluating the naive expression
1570:
1163:
3031:
2280:{\displaystyle 2^{-29}\cdot (1+2^{-30}+2^{-31})\approx 1.8626451518330422\times 10^{-9}}
17:
4840:
4228:
4128:
3586:{\displaystyle 1.0000000000000011102230246251565404236316680908203125=1+5\cdot 2^{-52}}
3491:, so the subtraction is effectively addition with the same sign which does not cancel.
1914:
1894:
1758:
1738:
1720:
1694:
1674:
1099:
1079:
902:
772:
216:
4950:
4930:
515:, is in error by almost 100% of the magnitude of the difference of the true values,
482:{\displaystyle L_{1}-L_{2}=253.51\,{\text{cm}}-252.49\,{\text{cm}}=1.02\,{\text{cm}}}
4849:(2nd ed.). Gewerbestrasse 11, 6330 Cham, Switzerland: Birkhäuser. p. 102.
1186:
of the true values is inversely proportional to the difference of the true values:
1727:
errors in the inputs that arose from rounding in other floating-point arithmetic.
4854:
4922:
4909:(1). New York, NY, United States: Association for Computing Machinery: 5–48.
4844:
4899:"What every computer scientist should know about floating-point arithmetic"
4525:{\displaystyle \log(\operatorname {fl} (1+x))={\hat {x}}+O({\hat {x}}^{2})}
2433:{\displaystyle 2^{-29}=1.8626451{\underline {4923095703125}}\times 10^{-9}}
4914:
4436:
evaluated directly. This works because the cancellation in the numerator
2486:, lost due to rounding when calculating the intermediate squared values.
2583:{\displaystyle \arcsin(z)=i\log {\bigl (}{\sqrt {1-z^{2}}}-iz{\bigr )}.}
2498:
function, one may be tempted to use the logarithmic formula directly:
1662:{\displaystyle \left|{\frac {x\delta _{x}-y\delta _{y}}{x-y}}\right|.}
783:
Formally, catastrophic cancellation happens because subtraction is
552:
of the inputs. Exactly the same error would arise by subtracting
4049:
3541:
is not a binary64 floating-point number; the nearest one, which
2963:
denotes floating-point rounding, then computing the difference
3517:
to declare and initialize an IEEE 754 binary64 variable named
3434:{\textstyle {\sqrt {1-(-z)^{2}}}={\sqrt {1-z^{2}}}\approx -y}
1775:, the naive attempt to compute the mathematical function
413:, even though the true difference between the lengths is
4332:{\displaystyle \log(1+x)=x{\frac {\log(1+x)}{(1+x)-1}}}
305:{\displaystyle |L_{1}-{\tilde {L}}_{1}|/|L_{1}|<2\%}
4391:{\displaystyle {\hat {x}}:=\operatorname {fl} (1+x)-1}
3553:
1.0000000000000011102230246251565404236316680908203125
3369:
4769:
4725:
4696:
4658:
4597:
4584:{\displaystyle {\hat {x}}=\operatorname {fl} (1+x)-1}
4538:
4442:
4404:
4345:
4251:
4231:
4189:
4151:
4131:
4099:
4061:
4010:
3987:
3831:
3782:
3762:
3738:
3692:
3666:
3551:
3527:
3447:
3313:
3287:
3258:
3219:
3189:
3157:
3125:
3105:
3077:
3057:
3034:
2971:
2937:
2911:
2790:
2751:
2731:
2699:
2653:
2627:
2598:
2506:
2446:
2384:
2344:
2297:
2197:
2157:
2144:{\displaystyle y=1+2^{-30}\approx 1.0000000009313226}
2109:
2096:{\displaystyle x=1+2^{-29}\approx 1.0000000018626451}
2061:
1984:
1937:
1917:
1897:
1821:
1781:
1761:
1741:
1697:
1677:
1601:
1573:
1529:
1194:
1166:
1122:
1116:, respectively, the relative error of the difference
1102:
1082:
996:
910:
850:
793:
740:
714:
688:
662:
636:
610:
584:
558:
521:
495:
419:
325:
225:
173:
125:
86:
47:
3721:{\displaystyle 0.000000000000001=1.0\times 10^{-15}}
3679:{\displaystyle 1.000000000000002-1.000000000000001}
3500:
3242:{\displaystyle -14.719{\underline {644263563968}}i}
4814:
4755:
4711:
4682:
4644:
4583:
4524:
4428:
4390:
4331:
4237:
4213:
4175:
4137:
4117:
4085:
4032:
3996:
3973:
3807:
3768:
3744:
3720:
3678:
3585:
3533:
3483:
3433:
3355:
3299:
3273:
3241:
3201:
3175:
3143:
3111:
3091:
3063:
3043:
3018:
2955:
2923:
2895:
2774:
2737:
2717:
2685:
2639:
2613:
2582:
2478:
2432:
2370:
2330:
2279:
2183:
2143:
2095:
2044:
1970:
1923:
1903:
1883:
1807:
1767:
1747:
1703:
1683:
1671:which can be arbitrarily large if the true values
1661:
1585:
1559:
1513:
1178:
1152:
1108:
1088:
1069:{\displaystyle |\delta _{y}|=|y-{\tilde {y}}|/|y|}
1068:
983:{\displaystyle |\delta _{x}|=|x-{\tilde {x}}|/|x|}
982:
893:
836:
767:; rather, it is inherent to subtraction, when the
752:
726:
700:
674:
648:
622:
596:
570:
533:
507:
481:
405:
304:
207:
159:
111:
72:
2850:
2799:
2782:is evaluated in floating-point arithmetic giving
2291:arithmetic, evaluating the alternative factoring
1523:Thus, the relative error of the exact difference
1499:
1440:
208:{\displaystyle {\tilde {L}}_{2}=252\,{\text{cm}}}
160:{\displaystyle {\tilde {L}}_{1}=254\,{\text{cm}}}
2745:—a very small difference, nearly zero. If
3019:{\displaystyle {\sqrt {1-z^{2}}}(1+\delta )-iz}
319:lengths are subtracted, the difference will be
3209:, but using the naive logarithmic formula in
2572:
2536:
1978:, evaluated by the floating-point arithmetic
1891:is subject to catastrophic cancellation when
894:{\displaystyle {\tilde {y}}=y(1+\delta _{y})}
837:{\displaystyle {\tilde {x}}=x(1+\delta _{x})}
8:
4645:{\displaystyle \mu (\xi )=\log(1+\xi )/\xi }
3545:will be initialized to in this fragment, is
2956:{\displaystyle \operatorname {fl} (\cdots )}
3819:is computed exactly by the Sterbenz lemma.
2686:{\displaystyle {\sqrt {1-z^{2}}}\approx -y}
4892:
4890:
4888:
4652:is well-enough conditioned near zero that
3808:{\displaystyle 10^{-15}=0.0000000000001\%}
3028:of two nearby numbers, both very close to
1567:of the approximations from the difference
1160:of the approximations from the difference
4768:
4739:
4738:
4724:
4695:
4666:
4665:
4657:
4634:
4596:
4540:
4539:
4537:
4513:
4502:
4501:
4480:
4479:
4441:
4403:
4347:
4346:
4344:
4279:
4250:
4230:
4188:
4150:
4130:
4098:
4060:
4021:
4009:
3986:
3962:
3940:
3915:
3881:
3848:
3847:
3833:
3832:
3830:
3787:
3781:
3761:
3737:
3709:
3691:
3665:
3574:
3550:
3526:
3446:
3414:
3402:
3391:
3370:
3368:
3312:
3286:
3257:
3226:
3218:
3188:
3156:
3124:
3104:
3081:
3076:
3056:
3033:
2984:
2972:
2970:
2936:
2910:
2870:
2858:
2849:
2848:
2834:
2804:
2798:
2797:
2789:
2764:
2752:
2750:
2730:
2698:
2666:
2654:
2652:
2626:
2597:
2571:
2570:
2553:
2541:
2535:
2534:
2505:
2467:
2451:
2445:
2421:
2404:
2389:
2383:
2362:
2349:
2343:
2296:
2268:
2243:
2227:
2202:
2196:
2175:
2162:
2156:
2126:
2108:
2078:
2060:
1983:
1936:
1916:
1896:
1869:
1844:
1820:
1799:
1786:
1780:
1760:
1740:
1696:
1676:
1632:
1616:
1606:
1600:
1572:
1560:{\displaystyle {\tilde {x}}-{\tilde {y}}}
1546:
1545:
1531:
1530:
1528:
1498:
1497:
1477:
1461:
1451:
1439:
1438:
1393:
1377:
1367:
1324:
1308:
1277:
1249:
1215:
1214:
1200:
1199:
1195:
1193:
1165:
1153:{\displaystyle {\tilde {x}}-{\tilde {y}}}
1139:
1138:
1124:
1123:
1121:
1101:
1081:
1061:
1053:
1048:
1043:
1032:
1031:
1020:
1012:
1006:
997:
995:
975:
967:
962:
957:
946:
945:
934:
926:
920:
911:
909:
882:
852:
851:
849:
825:
795:
794:
792:
787:at nearby inputs: even if approximations
745:
744:
739:
719:
718:
713:
693:
692:
687:
667:
666:
661:
641:
640:
635:
615:
614:
609:
589:
588:
583:
563:
562:
557:
526:
525:
520:
500:
499:
494:
474:
473:
462:
461:
450:
449:
437:
424:
418:
398:
397:
386:
385:
374:
373:
361:
350:
349:
339:
328:
327:
324:
288:
282:
273:
268:
263:
257:
246:
245:
235:
226:
224:
200:
199:
187:
176:
175:
172:
152:
151:
139:
128:
127:
124:
112:{\displaystyle L_{2}=252.49\,{\text{cm}}}
104:
103:
91:
85:
73:{\displaystyle L_{1}=253.51\,{\text{cm}}}
65:
64:
52:
46:
41:For example, if there are two studs, one
4815:{\displaystyle x\cdot \mu (x)=\log(1+x)}
4532:and the cancellation in the denominator
4176:{\displaystyle \operatorname {fl} (1+x)}
2151:, then the true value of the difference
489:. The difference of the approximations,
4831:
4756:{\displaystyle x\cdot \mu ({\hat {x}})}
3356:{\displaystyle \arcsin(z)=-\arcsin(-z)}
215:. These may be good approximations, in
27:Loss of precision in numerical analysis
4870:
753:{\displaystyle 2.0005351\,{\text{km}}}
727:{\displaystyle 2.0005249\,{\text{km}}}
4846:Handbook of Floating-Point Arithmetic
3815:, and the floating-point subtraction
7:
4591:counteract each other; the function
701:{\displaystyle 2.00054\,{\text{km}}}
675:{\displaystyle 2.00052\,{\text{km}}}
3202:{\displaystyle -14.71937803983977i}
3099:—a very large factor because
2725:; call the difference between them
4125:, will lose most of the digits of
4033:{\displaystyle 1.0\times 10^{-15}}
3991:
3802:
3655:// difference is exactly 4*2^{-52}
3119:was nearly zero. For instance, if
649:{\displaystyle 53.51\,{\text{cm}}}
623:{\displaystyle 52.49\,{\text{cm}}}
299:
25:
4093:, if evaluated naively at points
3769:{\displaystyle 1.000000000000002}
3745:{\displaystyle 1.000000000000001}
3534:{\displaystyle 1.000000000000001}
2775:{\displaystyle {\sqrt {1-z^{2}}}}
1815:by the floating-point arithmetic
534:{\displaystyle 1.02\,{\text{cm}}}
4683:{\displaystyle \mu ({\hat {x}})}
2378:gives the floating-point number
2479:{\displaystyle 2^{-59}+2^{-60}}
597:{\displaystyle 54\,{\text{cm}}}
571:{\displaystyle 52\,{\text{cm}}}
4897:Goldberg, David (March 1991).
4809:
4797:
4785:
4779:
4763:gives a good approximation to
4750:
4744:
4735:
4706:
4700:
4690:gives a good approximation to
4677:
4671:
4662:
4631:
4619:
4607:
4601:
4572:
4560:
4545:
4519:
4507:
4497:
4485:
4473:
4470:
4458:
4449:
4423:
4411:
4379:
4367:
4352:
4317:
4305:
4300:
4288:
4270:
4258:
4208:
4196:
4170:
4158:
4080:
4068:
3924:
3896:
3890:
3862:
3853:
3838:
3460:
3451:
3388:
3378:
3350:
3341:
3326:
3320:
3170:
3164:
3092:{\displaystyle 1/\varepsilon }
3004:
2992:
2950:
2944:
2890:
2878:
2843:
2840:
2827:
2812:
2519:
2513:
2325:
2313:
2310:
2298:
2252:
2214:
2039:
2036:
2024:
2012:
2000:
1991:
1965:
1953:
1950:
1938:
1878:
1875:
1862:
1850:
1837:
1828:
1731:Example: Difference of squares
1551:
1536:
1435:
1423:
1364:
1352:
1283:
1264:
1255:
1236:
1220:
1205:
1144:
1129:
1062:
1054:
1044:
1037:
1021:
1013:
998:
976:
968:
958:
951:
935:
927:
912:
888:
869:
857:
831:
812:
800:
508:{\displaystyle 2\,{\text{cm}}}
355:
333:
289:
274:
264:
251:
227:
181:
133:
1:
3981:has a relative error of over
2924:{\displaystyle \delta \neq 0}
4118:{\displaystyle 0<x\lll 1}
3363:avoids cancellation because
3112:{\displaystyle \varepsilon }
3071:in one input by a factor of
2738:{\displaystyle \varepsilon }
3631:// rounded to 1 + 9*2^{-52}
3613:// rounded to 1 + 5*2^{-52}
3484:{\displaystyle i(-z)=-iz=y}
3144:{\displaystyle z=-1234567i}
2371:{\displaystyle x^{2}-y^{2}}
2184:{\displaystyle x^{2}-y^{2}}
1808:{\displaystyle x^{2}-y^{2}}
4973:
3176:{\displaystyle \arcsin(z)}
2331:{\displaystyle (x+y)(x-y)}
1971:{\displaystyle (x+y)(x-y)}
4855:10.1007/978-3-319-76526-6
4429:{\displaystyle \log(1+x)}
4339:exploits cancellation in
4214:{\displaystyle \log(1+x)}
4086:{\displaystyle \log(1+x)}
3728:. The relative errors of
3495:Example: Radix conversion
765:floating-point arithmetic
36:catastrophic cancellation
4877:: CS1 maint: location (
4843:; Torres, Serge (2018).
4398:to avoid the error from
4183:. However, the function
3595:
3051:, may amplify the error
2490:Example: Complex arcsine
18:Catastrophic cancelation
4712:{\displaystyle \mu (x)}
3064:{\displaystyle \delta }
1715:In numerical algorithms
4816:
4757:
4713:
4684:
4646:
4585:
4526:
4430:
4392:
4333:
4239:
4215:
4177:
4139:
4119:
4087:
4034:
3998:
3975:
3809:
3770:
3746:
3722:
3680:
3587:
3535:
3485:
3435:
3357:
3301:
3300:{\displaystyle y<0}
3275:
3243:
3203:
3177:
3145:
3113:
3093:
3065:
3045:
3020:
2957:
2925:
2897:
2776:
2739:
2719:
2687:
2641:
2640:{\displaystyle y\ll 0}
2615:
2584:
2480:
2434:
2372:
2332:
2281:
2185:
2145:
2097:
2046:
1972:
1925:
1905:
1885:
1809:
1769:
1749:
1705:
1685:
1663:
1593:of the true values is
1587:
1561:
1515:
1180:
1154:
1110:
1090:
1070:
984:
895:
838:
754:
728:
702:
676:
650:
624:
598:
572:
535:
509:
483:
407:
306:
209:
161:
113:
74:
4915:10.1145/103162.103163
4903:ACM Computing Surveys
4817:
4758:
4714:
4685:
4647:
4586:
4527:
4431:
4393:
4334:
4240:
4216:
4178:
4140:
4120:
4088:
4035:
3999:
3976:
3810:
3771:
3747:
3723:
3681:
3588:
3536:
3486:
3436:
3358:
3307:, using the identity
3302:
3276:
3244:
3204:
3178:
3146:
3114:
3094:
3066:
3046:
3021:
2958:
2926:
2898:
2777:
2740:
2720:
2718:{\displaystyle iz=-y}
2688:
2642:
2616:
2585:
2481:
2435:
2373:
2333:
2282:
2186:
2146:
2098:
2047:
1973:
1926:
1906:
1886:
1810:
1770:
1750:
1706:
1686:
1664:
1588:
1562:
1516:
1181:
1155:
1111:
1091:
1071:
985:
896:
839:
755:
729:
708:as approximations to
703:
677:
651:
625:
604:as approximations to
599:
573:
536:
510:
484:
408:
307:
210:
162:
114:
75:
4767:
4723:
4694:
4656:
4595:
4536:
4440:
4402:
4343:
4249:
4229:
4187:
4149:
4129:
4097:
4059:
4008:
4004:from the difference
3997:{\displaystyle 11\%}
3985:
3829:
3780:
3760:
3736:
3690:
3664:
3549:
3525:
3445:
3367:
3311:
3285:
3274:{\displaystyle z=iy}
3256:
3217:
3213:arithmetic may give
3187:
3155:
3151:, the true value of
3123:
3103:
3075:
3055:
3032:
2969:
2935:
2909:
2788:
2749:
2729:
2697:
2651:
2625:
2614:{\displaystyle z=iy}
2596:
2504:
2444:
2382:
2342:
2295:
2195:
2155:
2107:
2059:
1982:
1935:
1915:
1895:
1819:
1779:
1759:
1739:
1695:
1675:
1599:
1571:
1527:
1192:
1164:
1120:
1100:
1080:
994:
908:
848:
791:
738:
712:
686:
660:
656:, or by subtracting
634:
608:
582:
556:
519:
493:
417:
323:
223:
171:
123:
84:
45:
4044:Benign cancellation
2494:When computing the
1586:{\displaystyle x-y}
1179:{\displaystyle x-y}
80:long and the other
4957:Numerical analysis
4812:
4753:
4709:
4680:
4642:
4581:
4522:
4426:
4388:
4329:
4245:. Rewriting it as
4235:
4211:
4173:
4135:
4115:
4083:
4030:
3994:
3971:
3805:
3766:
3742:
3718:
3676:
3583:
3531:
3481:
3431:
3353:
3297:
3271:
3239:
3234:
3199:
3173:
3141:
3109:
3089:
3061:
3044:{\displaystyle -y}
3041:
3016:
2953:
2921:
2893:
2772:
2735:
2715:
2683:
2637:
2611:
2580:
2476:
2430:
2412:
2368:
2328:
2277:
2259:1.8626451518330422
2181:
2141:
2139:1.0000000009313226
2093:
2091:1.0000000018626451
2042:
1968:
1921:
1901:
1881:
1805:
1765:
1745:
1701:
1681:
1659:
1583:
1557:
1511:
1509:
1176:
1150:
1106:
1086:
1066:
980:
891:
834:
750:
724:
698:
672:
646:
620:
594:
568:
531:
505:
479:
403:
302:
205:
157:
109:
70:
32:numerical analysis
4864:978-3-319-76525-9
4747:
4674:
4548:
4510:
4488:
4355:
4327:
4238:{\displaystyle 0}
4138:{\displaystyle x}
4050:2Sum and Fast2Sum
3856:
3841:
3764:1.000000000000002
3740:1.000000000000001
3694:0.000000000000001
3674:1.000000000000001
3668:1.000000000000002
3625:1.000000000000002
3607:1.000000000000001
3529:1.000000000000001
3511:1.000000000000001
3420:
3397:
3227:
3211:IEEE 754 binary64
3194:14.71937803983977
3183:is approximately
2990:
2876:
2846:
2770:
2672:
2592:However, suppose
2559:
2405:
2289:IEEE 754 binary64
1924:{\displaystyle y}
1904:{\displaystyle x}
1768:{\displaystyle y}
1748:{\displaystyle x}
1704:{\displaystyle y}
1684:{\displaystyle x}
1650:
1554:
1539:
1495:
1411:
1223:
1208:
1147:
1132:
1109:{\displaystyle y}
1089:{\displaystyle x}
1076:from true values
1040:
954:
860:
803:
748:
722:
696:
670:
644:
618:
592:
566:
529:
503:
477:
465:
453:
401:
389:
377:
358:
336:
254:
203:
184:
155:
136:
107:
68:
16:(Redirected from
4964:
4941:
4940:
4938:
4937:
4894:
4883:
4882:
4876:
4868:
4836:
4821:
4819:
4818:
4813:
4762:
4760:
4759:
4754:
4749:
4748:
4740:
4718:
4716:
4715:
4710:
4689:
4687:
4686:
4681:
4676:
4675:
4667:
4651:
4649:
4648:
4643:
4638:
4590:
4588:
4587:
4582:
4550:
4549:
4541:
4531:
4529:
4528:
4523:
4518:
4517:
4512:
4511:
4503:
4490:
4489:
4481:
4435:
4433:
4432:
4427:
4397:
4395:
4394:
4389:
4357:
4356:
4348:
4338:
4336:
4335:
4330:
4328:
4326:
4303:
4280:
4244:
4242:
4241:
4236:
4223:well-conditioned
4220:
4218:
4217:
4212:
4182:
4180:
4179:
4174:
4144:
4142:
4141:
4136:
4124:
4122:
4121:
4116:
4092:
4090:
4089:
4084:
4039:
4037:
4036:
4031:
4029:
4028:
4003:
4001:
4000:
3995:
3980:
3978:
3977:
3972:
3970:
3969:
3948:
3947:
3923:
3922:
3889:
3888:
3858:
3857:
3849:
3843:
3842:
3834:
3818:
3814:
3812:
3811:
3806:
3795:
3794:
3775:
3773:
3772:
3767:
3755:
3751:
3749:
3748:
3743:
3731:
3727:
3725:
3724:
3719:
3717:
3716:
3685:
3683:
3682:
3677:
3656:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3632:
3629:
3626:
3623:
3620:
3617:
3614:
3611:
3608:
3605:
3602:
3599:
3592:
3590:
3589:
3584:
3582:
3581:
3544:
3540:
3538:
3537:
3532:
3520:
3516:
3515:
3512:
3509:
3506:
3503:
3490:
3488:
3487:
3482:
3440:
3438:
3437:
3432:
3421:
3419:
3418:
3403:
3398:
3396:
3395:
3371:
3362:
3360:
3359:
3354:
3306:
3304:
3303:
3298:
3280:
3278:
3277:
3272:
3248:
3246:
3245:
3240:
3235:
3208:
3206:
3205:
3200:
3182:
3180:
3179:
3174:
3150:
3148:
3147:
3142:
3118:
3116:
3115:
3110:
3098:
3096:
3095:
3090:
3085:
3070:
3068:
3067:
3062:
3050:
3048:
3047:
3042:
3025:
3023:
3022:
3017:
2991:
2989:
2988:
2973:
2962:
2960:
2959:
2954:
2930:
2928:
2927:
2922:
2902:
2900:
2899:
2894:
2877:
2875:
2874:
2859:
2854:
2853:
2847:
2839:
2838:
2805:
2803:
2802:
2781:
2779:
2778:
2773:
2771:
2769:
2768:
2753:
2744:
2742:
2741:
2736:
2724:
2722:
2721:
2716:
2692:
2690:
2689:
2684:
2673:
2671:
2670:
2655:
2646:
2644:
2643:
2638:
2620:
2618:
2617:
2612:
2589:
2587:
2586:
2581:
2576:
2575:
2560:
2558:
2557:
2542:
2540:
2539:
2485:
2483:
2482:
2477:
2475:
2474:
2459:
2458:
2439:
2437:
2436:
2431:
2429:
2428:
2413:
2397:
2396:
2377:
2375:
2374:
2369:
2367:
2366:
2354:
2353:
2337:
2335:
2334:
2329:
2286:
2284:
2283:
2278:
2276:
2275:
2251:
2250:
2235:
2234:
2210:
2209:
2190:
2188:
2187:
2182:
2180:
2179:
2167:
2166:
2150:
2148:
2147:
2142:
2134:
2133:
2102:
2100:
2099:
2094:
2086:
2085:
2055:For example, if
2051:
2049:
2048:
2043:
1977:
1975:
1974:
1969:
1930:
1928:
1927:
1922:
1910:
1908:
1907:
1902:
1890:
1888:
1887:
1882:
1874:
1873:
1849:
1848:
1814:
1812:
1811:
1806:
1804:
1803:
1791:
1790:
1774:
1772:
1771:
1766:
1754:
1752:
1751:
1746:
1710:
1708:
1707:
1702:
1690:
1688:
1687:
1682:
1668:
1666:
1665:
1660:
1655:
1651:
1649:
1638:
1637:
1636:
1621:
1620:
1607:
1592:
1590:
1589:
1584:
1566:
1564:
1563:
1558:
1556:
1555:
1547:
1541:
1540:
1532:
1520:
1518:
1517:
1512:
1510:
1503:
1502:
1496:
1494:
1483:
1482:
1481:
1466:
1465:
1452:
1444:
1443:
1416:
1412:
1410:
1399:
1398:
1397:
1382:
1381:
1368:
1333:
1329:
1328:
1313:
1312:
1282:
1281:
1254:
1253:
1225:
1224:
1216:
1210:
1209:
1201:
1185:
1183:
1182:
1177:
1159:
1157:
1156:
1151:
1149:
1148:
1140:
1134:
1133:
1125:
1115:
1113:
1112:
1107:
1095:
1093:
1092:
1087:
1075:
1073:
1072:
1067:
1065:
1057:
1052:
1047:
1042:
1041:
1033:
1024:
1016:
1011:
1010:
1001:
989:
987:
986:
981:
979:
971:
966:
961:
956:
955:
947:
938:
930:
925:
924:
915:
900:
898:
897:
892:
887:
886:
862:
861:
853:
843:
841:
840:
835:
830:
829:
805:
804:
796:
759:
757:
756:
751:
749:
746:
733:
731:
730:
725:
723:
720:
707:
705:
704:
699:
697:
694:
681:
679:
678:
673:
671:
668:
655:
653:
652:
647:
645:
642:
629:
627:
626:
621:
619:
616:
603:
601:
600:
595:
593:
590:
577:
575:
574:
569:
567:
564:
540:
538:
537:
532:
530:
527:
514:
512:
511:
506:
504:
501:
488:
486:
485:
480:
478:
475:
466:
463:
454:
451:
442:
441:
429:
428:
412:
410:
409:
404:
402:
399:
390:
387:
378:
375:
366:
365:
360:
359:
351:
344:
343:
338:
337:
329:
315:However, if the
311:
309:
308:
303:
292:
287:
286:
277:
272:
267:
262:
261:
256:
255:
247:
240:
239:
230:
214:
212:
211:
206:
204:
201:
192:
191:
186:
185:
177:
166:
164:
163:
158:
156:
153:
144:
143:
138:
137:
129:
118:
116:
115:
110:
108:
105:
96:
95:
79:
77:
76:
71:
69:
66:
57:
56:
21:
4972:
4971:
4967:
4966:
4965:
4963:
4962:
4961:
4947:
4946:
4945:
4944:
4935:
4933:
4896:
4895:
4886:
4869:
4865:
4841:Revol, Nathalie
4838:
4837:
4833:
4828:
4765:
4764:
4721:
4720:
4692:
4691:
4654:
4653:
4593:
4592:
4534:
4533:
4500:
4438:
4437:
4400:
4399:
4341:
4340:
4304:
4281:
4247:
4246:
4227:
4226:
4225:at inputs near
4185:
4184:
4147:
4146:
4127:
4126:
4095:
4094:
4057:
4056:
4046:
4017:
4006:
4005:
3983:
3982:
3958:
3936:
3911:
3877:
3827:
3826:
3816:
3800:0.0000000000001
3783:
3778:
3777:
3776:are both below
3758:
3757:
3753:
3734:
3733:
3729:
3705:
3688:
3687:
3662:
3661:
3660:The difference
3658:
3657:
3654:
3651:
3648:
3645:
3642:
3639:
3636:
3633:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3570:
3547:
3546:
3542:
3523:
3522:
3518:
3513:
3510:
3507:
3504:
3501:
3497:
3443:
3442:
3410:
3387:
3365:
3364:
3309:
3308:
3283:
3282:
3254:
3253:
3252:In the case of
3215:
3214:
3185:
3184:
3153:
3152:
3121:
3120:
3101:
3100:
3073:
3072:
3053:
3052:
3030:
3029:
2980:
2967:
2966:
2933:
2932:
2907:
2906:
2905:with any error
2866:
2830:
2786:
2785:
2760:
2747:
2746:
2727:
2726:
2695:
2694:
2662:
2649:
2648:
2623:
2622:
2594:
2593:
2549:
2502:
2501:
2496:complex arcsine
2492:
2463:
2447:
2442:
2441:
2417:
2385:
2380:
2379:
2358:
2345:
2340:
2339:
2293:
2292:
2264:
2239:
2223:
2198:
2193:
2192:
2171:
2158:
2153:
2152:
2122:
2105:
2104:
2074:
2057:
2056:
1980:
1979:
1933:
1932:
1913:
1912:
1893:
1892:
1865:
1840:
1817:
1816:
1795:
1782:
1777:
1776:
1757:
1756:
1737:
1736:
1733:
1717:
1693:
1692:
1673:
1672:
1639:
1628:
1612:
1608:
1602:
1597:
1596:
1569:
1568:
1525:
1524:
1508:
1507:
1484:
1473:
1457:
1453:
1414:
1413:
1400:
1389:
1373:
1369:
1331:
1330:
1320:
1304:
1273:
1245:
1226:
1190:
1189:
1162:
1161:
1118:
1117:
1098:
1097:
1078:
1077:
1002:
992:
991:
916:
906:
905:
903:relative errors
878:
846:
845:
821:
789:
788:
785:ill-conditioned
781:
779:Formal analysis
736:
735:
710:
709:
684:
683:
658:
657:
632:
631:
606:
605:
580:
579:
554:
553:
548:is, and on the
517:
516:
491:
490:
433:
420:
415:
414:
348:
326:
321:
320:
278:
244:
231:
221:
220:
174:
169:
168:
126:
121:
120:
87:
82:
81:
48:
43:
42:
28:
23:
22:
15:
12:
11:
5:
4970:
4968:
4960:
4959:
4949:
4948:
4943:
4942:
4884:
4863:
4830:
4829:
4827:
4824:
4811:
4808:
4805:
4802:
4799:
4796:
4793:
4790:
4787:
4784:
4781:
4778:
4775:
4772:
4752:
4746:
4743:
4737:
4734:
4731:
4728:
4708:
4705:
4702:
4699:
4679:
4673:
4670:
4664:
4661:
4641:
4637:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4609:
4606:
4603:
4600:
4580:
4577:
4574:
4571:
4568:
4565:
4562:
4559:
4556:
4553:
4547:
4544:
4521:
4516:
4509:
4506:
4499:
4496:
4493:
4487:
4484:
4478:
4475:
4472:
4469:
4466:
4463:
4460:
4457:
4454:
4451:
4448:
4445:
4425:
4422:
4419:
4416:
4413:
4410:
4407:
4387:
4384:
4381:
4378:
4375:
4372:
4369:
4366:
4363:
4360:
4354:
4351:
4325:
4322:
4319:
4316:
4313:
4310:
4307:
4302:
4299:
4296:
4293:
4290:
4287:
4284:
4278:
4275:
4272:
4269:
4266:
4263:
4260:
4257:
4254:
4234:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4172:
4169:
4166:
4163:
4160:
4157:
4154:
4134:
4114:
4111:
4108:
4105:
4102:
4082:
4079:
4076:
4073:
4070:
4067:
4064:
4045:
4042:
4027:
4024:
4020:
4016:
4013:
3993:
3990:
3968:
3965:
3961:
3957:
3954:
3951:
3946:
3943:
3939:
3935:
3932:
3929:
3926:
3921:
3918:
3914:
3910:
3907:
3904:
3901:
3898:
3895:
3892:
3887:
3884:
3880:
3876:
3873:
3870:
3867:
3864:
3861:
3855:
3852:
3846:
3840:
3837:
3824:approximations
3804:
3801:
3798:
3793:
3790:
3786:
3765:
3741:
3715:
3712:
3708:
3704:
3701:
3698:
3695:
3675:
3672:
3669:
3596:
3580:
3577:
3573:
3569:
3566:
3563:
3560:
3557:
3554:
3530:
3496:
3493:
3480:
3477:
3474:
3471:
3468:
3465:
3462:
3459:
3456:
3453:
3450:
3430:
3427:
3424:
3417:
3413:
3409:
3406:
3401:
3394:
3390:
3386:
3383:
3380:
3377:
3374:
3352:
3349:
3346:
3343:
3340:
3337:
3334:
3331:
3328:
3325:
3322:
3319:
3316:
3296:
3293:
3290:
3270:
3267:
3264:
3261:
3238:
3233:
3230:
3225:
3222:
3198:
3195:
3192:
3172:
3169:
3166:
3163:
3160:
3140:
3137:
3134:
3131:
3128:
3108:
3088:
3084:
3080:
3060:
3040:
3037:
3015:
3012:
3009:
3006:
3003:
3000:
2997:
2994:
2987:
2983:
2979:
2976:
2952:
2949:
2946:
2943:
2940:
2920:
2917:
2914:
2892:
2889:
2886:
2883:
2880:
2873:
2869:
2865:
2862:
2857:
2852:
2845:
2842:
2837:
2833:
2829:
2826:
2823:
2820:
2817:
2814:
2811:
2808:
2801:
2796:
2793:
2767:
2763:
2759:
2756:
2734:
2714:
2711:
2708:
2705:
2702:
2682:
2679:
2676:
2669:
2665:
2661:
2658:
2636:
2633:
2630:
2610:
2607:
2604:
2601:
2579:
2574:
2569:
2566:
2563:
2556:
2552:
2548:
2545:
2538:
2533:
2530:
2527:
2524:
2521:
2518:
2515:
2512:
2509:
2491:
2488:
2473:
2470:
2466:
2462:
2457:
2454:
2450:
2427:
2424:
2420:
2416:
2411:
2408:
2403:
2400:
2395:
2392:
2388:
2365:
2361:
2357:
2352:
2348:
2327:
2324:
2321:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2274:
2271:
2267:
2263:
2260:
2257:
2254:
2249:
2246:
2242:
2238:
2233:
2230:
2226:
2222:
2219:
2216:
2213:
2208:
2205:
2201:
2178:
2174:
2170:
2165:
2161:
2140:
2137:
2132:
2129:
2125:
2121:
2118:
2115:
2112:
2092:
2089:
2084:
2081:
2077:
2073:
2070:
2067:
2064:
2041:
2038:
2035:
2032:
2029:
2026:
2023:
2020:
2017:
2014:
2011:
2008:
2005:
2002:
1999:
1996:
1993:
1990:
1987:
1967:
1964:
1961:
1958:
1955:
1952:
1949:
1946:
1943:
1940:
1920:
1900:
1880:
1877:
1872:
1868:
1864:
1861:
1858:
1855:
1852:
1847:
1843:
1839:
1836:
1833:
1830:
1827:
1824:
1802:
1798:
1794:
1789:
1785:
1764:
1744:
1735:Given numbers
1732:
1729:
1721:Sterbenz lemma
1716:
1713:
1700:
1680:
1658:
1654:
1648:
1645:
1642:
1635:
1631:
1627:
1624:
1619:
1615:
1611:
1605:
1582:
1579:
1576:
1553:
1550:
1544:
1538:
1535:
1506:
1501:
1493:
1490:
1487:
1480:
1476:
1472:
1469:
1464:
1460:
1456:
1450:
1447:
1442:
1437:
1434:
1431:
1428:
1425:
1422:
1419:
1417:
1415:
1409:
1406:
1403:
1396:
1392:
1388:
1385:
1380:
1376:
1372:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1345:
1342:
1339:
1336:
1334:
1332:
1327:
1323:
1319:
1316:
1311:
1307:
1303:
1300:
1297:
1294:
1291:
1288:
1285:
1280:
1276:
1272:
1269:
1266:
1263:
1260:
1257:
1252:
1248:
1244:
1241:
1238:
1235:
1232:
1229:
1227:
1222:
1219:
1213:
1207:
1204:
1198:
1197:
1175:
1172:
1169:
1146:
1143:
1137:
1131:
1128:
1105:
1085:
1064:
1060:
1056:
1051:
1046:
1039:
1036:
1030:
1027:
1023:
1019:
1015:
1009:
1005:
1000:
978:
974:
970:
965:
960:
953:
950:
944:
941:
937:
933:
929:
923:
919:
914:
890:
885:
881:
877:
874:
871:
868:
865:
859:
856:
833:
828:
824:
820:
817:
814:
811:
808:
802:
799:
780:
777:
773:Sterbenz lemma
743:
717:
691:
665:
639:
613:
587:
561:
524:
498:
472:
469:
460:
457:
448:
445:
440:
436:
432:
427:
423:
396:
393:
384:
381:
372:
369:
364:
357:
354:
347:
342:
335:
332:
301:
298:
295:
291:
285:
281:
276:
271:
266:
260:
253:
250:
243:
238:
234:
229:
217:relative error
198:
195:
190:
183:
180:
150:
147:
142:
135:
132:
102:
99:
94:
90:
63:
60:
55:
51:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4969:
4958:
4955:
4954:
4952:
4932:
4928:
4924:
4920:
4916:
4912:
4908:
4904:
4900:
4893:
4891:
4889:
4885:
4880:
4874:
4866:
4860:
4856:
4852:
4848:
4847:
4842:
4835:
4832:
4825:
4823:
4806:
4803:
4800:
4794:
4791:
4788:
4782:
4776:
4773:
4770:
4741:
4732:
4729:
4726:
4703:
4697:
4668:
4659:
4639:
4635:
4628:
4625:
4622:
4616:
4613:
4610:
4604:
4598:
4578:
4575:
4569:
4566:
4563:
4557:
4554:
4551:
4542:
4514:
4504:
4494:
4491:
4482:
4476:
4467:
4464:
4461:
4455:
4452:
4446:
4443:
4420:
4417:
4414:
4408:
4405:
4385:
4382:
4376:
4373:
4370:
4364:
4361:
4358:
4349:
4323:
4320:
4314:
4311:
4308:
4297:
4294:
4291:
4285:
4282:
4276:
4273:
4267:
4264:
4261:
4255:
4252:
4232:
4224:
4205:
4202:
4199:
4193:
4190:
4167:
4164:
4161:
4155:
4152:
4132:
4112:
4109:
4106:
4103:
4100:
4077:
4074:
4071:
4065:
4062:
4055:The function
4053:
4051:
4043:
4041:
4025:
4022:
4018:
4014:
4011:
3988:
3966:
3963:
3959:
3955:
3952:
3949:
3944:
3941:
3937:
3933:
3930:
3927:
3919:
3916:
3912:
3908:
3905:
3902:
3899:
3893:
3885:
3882:
3878:
3874:
3871:
3868:
3865:
3859:
3850:
3844:
3835:
3825:
3820:
3799:
3796:
3791:
3788:
3784:
3763:
3739:
3713:
3710:
3706:
3702:
3699:
3696:
3693:
3673:
3670:
3667:
3594:
3578:
3575:
3571:
3567:
3564:
3561:
3558:
3555:
3552:
3528:
3494:
3492:
3478:
3475:
3472:
3469:
3466:
3463:
3457:
3454:
3448:
3428:
3425:
3422:
3415:
3411:
3407:
3404:
3399:
3392:
3384:
3381:
3375:
3372:
3347:
3344:
3338:
3335:
3332:
3329:
3323:
3317:
3314:
3294:
3291:
3288:
3268:
3265:
3262:
3259:
3250:
3236:
3231:
3228:
3223:
3220:
3212:
3196:
3193:
3190:
3167:
3161:
3158:
3138:
3135:
3132:
3129:
3126:
3106:
3086:
3082:
3078:
3058:
3038:
3035:
3026:
3013:
3010:
3007:
3001:
2998:
2995:
2985:
2981:
2977:
2974:
2964:
2947:
2941:
2938:
2918:
2915:
2912:
2903:
2887:
2884:
2881:
2871:
2867:
2863:
2860:
2855:
2835:
2831:
2824:
2821:
2818:
2815:
2809:
2806:
2794:
2791:
2783:
2765:
2761:
2757:
2754:
2732:
2712:
2709:
2706:
2703:
2700:
2680:
2677:
2674:
2667:
2663:
2659:
2656:
2634:
2631:
2628:
2608:
2605:
2602:
2599:
2590:
2577:
2567:
2564:
2561:
2554:
2550:
2546:
2543:
2531:
2528:
2525:
2522:
2516:
2510:
2507:
2499:
2497:
2489:
2487:
2471:
2468:
2464:
2460:
2455:
2452:
2448:
2425:
2422:
2418:
2414:
2409:
2407:4923095703125
2406:
2401:
2398:
2393:
2390:
2386:
2363:
2359:
2355:
2350:
2346:
2322:
2319:
2316:
2307:
2304:
2301:
2290:
2272:
2269:
2265:
2261:
2258:
2255:
2247:
2244:
2240:
2236:
2231:
2228:
2224:
2220:
2217:
2211:
2206:
2203:
2199:
2176:
2172:
2168:
2163:
2159:
2138:
2135:
2130:
2127:
2123:
2119:
2116:
2113:
2110:
2090:
2087:
2082:
2079:
2075:
2071:
2068:
2065:
2062:
2053:
2033:
2030:
2027:
2021:
2018:
2015:
2009:
2006:
2003:
1997:
1994:
1988:
1985:
1962:
1959:
1956:
1947:
1944:
1941:
1918:
1898:
1870:
1866:
1859:
1856:
1853:
1845:
1841:
1834:
1831:
1825:
1822:
1800:
1796:
1792:
1787:
1783:
1762:
1742:
1730:
1728:
1726:
1722:
1714:
1712:
1698:
1678:
1669:
1656:
1652:
1646:
1643:
1640:
1633:
1629:
1625:
1622:
1617:
1613:
1609:
1603:
1594:
1580:
1577:
1574:
1548:
1542:
1533:
1521:
1504:
1491:
1488:
1485:
1478:
1474:
1470:
1467:
1462:
1458:
1454:
1448:
1445:
1432:
1429:
1426:
1420:
1418:
1407:
1404:
1401:
1394:
1390:
1386:
1383:
1378:
1374:
1370:
1361:
1358:
1355:
1349:
1346:
1343:
1340:
1337:
1335:
1325:
1321:
1317:
1314:
1309:
1305:
1301:
1298:
1295:
1292:
1289:
1286:
1278:
1274:
1270:
1267:
1261:
1258:
1250:
1246:
1242:
1239:
1233:
1230:
1228:
1217:
1211:
1202:
1187:
1173:
1170:
1167:
1141:
1135:
1126:
1103:
1083:
1058:
1049:
1034:
1028:
1025:
1017:
1007:
1003:
972:
963:
948:
942:
939:
931:
921:
917:
904:
883:
879:
875:
872:
866:
863:
854:
826:
822:
818:
815:
809:
806:
797:
786:
778:
776:
774:
770:
766:
761:
741:
715:
689:
663:
637:
611:
585:
559:
551:
547:
542:
522:
496:
470:
467:
458:
455:
446:
443:
438:
434:
430:
425:
421:
394:
391:
382:
379:
370:
367:
362:
352:
345:
340:
330:
318:
313:
296:
293:
283:
279:
269:
258:
248:
241:
236:
232:
218:
196:
193:
188:
178:
148:
145:
140:
130:
100:
97:
92:
88:
61:
58:
53:
49:
39:
37:
33:
19:
4934:. Retrieved
4906:
4902:
4845:
4834:
4145:in rounding
4054:
4047:
3823:
3821:
3659:
3498:
3251:
3229:644263563968
3027:
2965:
2904:
2784:
2591:
2500:
2493:
2054:
1734:
1724:
1718:
1670:
1595:
1522:
1188:
782:
768:
762:
549:
545:
543:
316:
314:
40:
35:
29:
4719:, and thus
3521:. However,
1711:are close.
901:have small
317:approximate
4936:2020-09-17
4826:References
4221:itself is
546:difference
4931:222008826
4923:0360-0300
4873:cite book
4795:
4777:μ
4774:⋅
4745:^
4733:μ
4730:⋅
4698:μ
4672:^
4660:μ
4640:ξ
4629:ξ
4617:
4605:ξ
4599:μ
4576:−
4558:
4546:^
4508:^
4486:^
4456:
4447:
4409:
4383:−
4365:
4353:^
4321:−
4286:
4256:
4194:
4156:
4110:⋘
4066:
4023:−
4015:×
3992:%
3964:−
3956:×
3950:≈
3942:−
3934:⋅
3917:−
3909:⋅
3894:−
3883:−
3875:⋅
3854:~
3845:−
3839:~
3803:%
3789:−
3711:−
3703:×
3671:−
3576:−
3568:⋅
3467:−
3455:−
3426:−
3423:≈
3408:−
3382:−
3376:−
3345:−
3339:
3333:−
3318:
3232:_
3221:−
3191:−
3162:
3133:−
3107:ε
3087:ε
3059:δ
3036:−
3008:−
3002:δ
2978:−
2948:⋯
2942:
2916:≠
2913:δ
2888:δ
2864:−
2825:
2819:−
2810:
2795:
2758:−
2733:ε
2710:−
2678:−
2675:≈
2660:−
2632:≪
2562:−
2547:−
2532:
2511:
2469:−
2453:−
2423:−
2415:×
2410:_
2402:1.8626451
2391:−
2356:−
2320:−
2270:−
2262:×
2256:≈
2245:−
2229:−
2212:⋅
2204:−
2169:−
2136:≈
2128:−
2088:≈
2080:−
2031:−
2022:
2016:⋅
1998:
1989:
1960:−
1860:
1854:−
1835:
1826:
1793:−
1644:−
1630:δ
1623:−
1614:δ
1578:−
1552:~
1543:−
1537:~
1489:−
1475:δ
1468:−
1459:δ
1430:−
1405:−
1391:δ
1384:−
1375:δ
1359:−
1344:−
1322:δ
1315:−
1306:δ
1293:−
1275:δ
1259:−
1247:δ
1221:~
1212:−
1206:~
1171:−
1145:~
1136:−
1130:~
1038:~
1029:−
1004:δ
952:~
943:−
918:δ
880:δ
858:~
823:δ
801:~
742:2.0005351
716:2.0005249
456:−
431:−
380:−
356:~
346:−
334:~
300:%
252:~
242:−
182:~
134:~
4951:Category
2931:, where
3752:and of
3136:1234567
2647:. Then
1725:amplify
690:2.00054
664:2.00052
4929:
4921:
4861:
3634:double
3616:double
3598:double
3502:double
3336:arcsin
3315:arcsin
3224:14.719
3159:arcsin
2508:arcsin
769:inputs
459:252.49
447:253.51
101:252.49
62:253.51
4927:S2CID
3817:y - x
3756:from
3732:from
2287:. In
682:from
638:53.51
612:52.49
578:from
550:error
4919:ISSN
4879:link
4859:ISBN
4104:<
3953:8.88
3441:but
3292:<
3281:for
2693:and
2621:for
2103:and
1911:and
1755:and
1691:and
1096:and
990:and
844:and
734:and
630:and
523:1.02
471:1.02
294:<
167:and
4911:doi
4851:doi
4792:log
4614:log
4444:log
4406:log
4283:log
4253:log
4191:log
4063:log
4012:1.0
3700:1.0
3686:is
2529:log
2191:is
383:252
371:254
197:252
149:254
30:In
4953::
4925:.
4917:.
4907:23
4905:.
4901:.
4887:^
4875:}}
4871:{{
4857:.
4822:.
4555:fl
4453:fl
4362:fl
4359::=
4153:fl
4026:15
4019:10
3989:11
3967:16
3960:10
3945:52
3920:52
3886:52
3792:15
3785:10
3714:15
3707:10
3579:52
2939:fl
2822:fl
2807:fl
2792:fl
2472:60
2456:59
2419:10
2394:29
2266:10
2248:31
2232:30
2207:29
2131:30
2083:29
2019:fl
1995:fl
1986:fl
1857:fl
1832:fl
1823:fl
760:.
747:km
721:km
695:km
669:km
643:cm
617:cm
591:cm
586:54
565:cm
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541:.
528:cm
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476:cm
464:cm
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400:cm
388:cm
376:cm
312:.
202:cm
154:cm
106:cm
67:cm
34:,
4939:.
4913::
4881:)
4867:.
4853::
4810:)
4807:x
4804:+
4801:1
4798:(
4789:=
4786:)
4783:x
4780:(
4771:x
4751:)
4742:x
4736:(
4727:x
4707:)
4704:x
4701:(
4678:)
4669:x
4663:(
4636:/
4632:)
4626:+
4623:1
4620:(
4611:=
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4602:(
4579:1
4573:)
4570:x
4567:+
4564:1
4561:(
4552:=
4543:x
4520:)
4515:2
4505:x
4498:(
4495:O
4492:+
4483:x
4477:=
4474:)
4471:)
4468:x
4465:+
4462:1
4459:(
4450:(
4424:)
4421:x
4418:+
4415:1
4412:(
4386:1
4380:)
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4371:1
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4350:x
4324:1
4318:)
4315:x
4312:+
4309:1
4306:(
4301:)
4298:x
4295:+
4292:1
4289:(
4277:x
4274:=
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4268:x
4265:+
4262:1
4259:(
4233:0
4209:)
4206:x
4203:+
4200:1
4197:(
4171:)
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4113:1
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4101:0
4081:)
4078:x
4075:+
4072:1
4069:(
3938:2
3931:4
3928:=
3925:)
3913:2
3906:5
3903:+
3900:1
3897:(
3891:)
3879:2
3872:9
3869:+
3866:1
3863:(
3860:=
3851:x
3836:y
3797:=
3754:y
3730:x
3697:=
3652:;
3649:x
3646:-
3643:y
3640:=
3637:z
3628:;
3622:=
3619:y
3610:;
3604:=
3601:x
3572:2
3565:5
3562:+
3559:1
3556:=
3543:x
3519:x
3514:;
3508:=
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3479:y
3476:=
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3470:i
3464:=
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3458:z
3452:(
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3429:y
3416:2
3412:z
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3400:=
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3389:)
3385:z
3379:(
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3351:)
3348:z
3342:(
3330:=
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3324:z
3321:(
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3289:y
3269:y
3266:i
3263:=
3260:z
3237:i
3197:i
3171:)
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3165:(
3139:i
3130:=
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3083:/
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3039:y
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3011:i
3005:)
2999:+
2996:1
2993:(
2986:2
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2975:1
2951:)
2945:(
2919:0
2891:)
2885:+
2882:1
2879:(
2872:2
2868:z
2861:1
2856:=
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2844:)
2841:)
2836:2
2832:z
2828:(
2816:1
2813:(
2800:(
2766:2
2762:z
2755:1
2713:y
2707:=
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2701:i
2681:y
2668:2
2664:z
2657:1
2635:0
2629:y
2609:y
2606:i
2603:=
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2578:.
2573:)
2568:z
2565:i
2555:2
2551:z
2544:1
2537:(
2526:i
2523:=
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2517:z
2514:(
2465:2
2461:+
2449:2
2426:9
2399:=
2387:2
2364:2
2360:y
2351:2
2347:x
2326:)
2323:y
2317:x
2314:(
2311:)
2308:y
2305:+
2302:x
2299:(
2273:9
2253:)
2241:2
2237:+
2225:2
2221:+
2218:1
2215:(
2200:2
2177:2
2173:y
2164:2
2160:x
2124:2
2120:+
2117:1
2114:=
2111:y
2076:2
2072:+
2069:1
2066:=
2063:x
2040:)
2037:)
2034:y
2028:x
2025:(
2013:)
2010:y
2007:+
2004:x
2001:(
1992:(
1966:)
1963:y
1957:x
1954:(
1951:)
1948:y
1945:+
1942:x
1939:(
1919:y
1899:x
1879:)
1876:)
1871:2
1867:y
1863:(
1851:)
1846:2
1842:x
1838:(
1829:(
1801:2
1797:y
1788:2
1784:x
1763:y
1743:x
1699:y
1679:x
1657:.
1653:|
1647:y
1641:x
1634:y
1626:y
1618:x
1610:x
1604:|
1581:y
1575:x
1549:y
1534:x
1505:.
1500:)
1492:y
1486:x
1479:y
1471:y
1463:x
1455:x
1449:+
1446:1
1441:(
1436:)
1433:y
1427:x
1424:(
1421:=
1408:y
1402:x
1395:y
1387:y
1379:x
1371:x
1365:)
1362:y
1356:x
1353:(
1350:+
1347:y
1341:x
1338:=
1326:y
1318:y
1310:x
1302:x
1299:+
1296:y
1290:x
1287:=
1284:)
1279:y
1271:+
1268:1
1265:(
1262:y
1256:)
1251:x
1243:+
1240:1
1237:(
1234:x
1231:=
1218:y
1203:x
1174:y
1168:x
1142:y
1127:x
1104:y
1084:x
1063:|
1059:y
1055:|
1050:/
1045:|
1035:y
1026:y
1022:|
1018:=
1014:|
1008:y
999:|
977:|
973:x
969:|
964:/
959:|
949:x
940:x
936:|
932:=
928:|
922:x
913:|
889:)
884:y
876:+
873:1
870:(
867:y
864:=
855:y
832:)
827:x
819:+
816:1
813:(
810:x
807:=
798:x
497:2
468:=
444:=
439:2
435:L
426:1
422:L
395:2
392:=
368:=
363:2
353:L
341:1
331:L
297:2
290:|
284:1
280:L
275:|
270:/
265:|
259:1
249:L
237:1
233:L
228:|
194:=
189:2
179:L
146:=
141:1
131:L
98:=
93:2
89:L
59:=
54:1
50:L
20:)
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