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README.md

Accuracy of provided algorithms

This file is generated using estimate_accuracy.py. Do not edit!

The reference values are obtained by evaluating MPMath functions using multi-precision arithmetic.

The following table shows the counts of samples that produce function values being

  • different from expected values by the given ULP difference (dULP):

    ulp_diff(func(sample), reference(sample)) == dULP

  • out of the reference values ulp-range:

    not (lower <= func(sample) <= upper)
lower = minimal(reference(s) for s in surrounding(sample) if diff_ulp(s, sample) <= ulp_width)
upper = maximal(reference(s) for s in surrounding(sample) if diff_ulp(s, sample) <= ulp_width)

When a counts value is attributed with a superscript, this indicates the number of samples that lead to out-of-ulp-range results. When dULP <= 3, out-of-ulp-range counts are acceptable as it typically indicates that reference function is not sensitive to input perturbations, that is, lower == reference(sample) == upper holds. On the other hand, when the out-of-ulp-range counts is zero, dULP > 3 counts are acceptable as it indicates that function's variability is very high with respect to minimal variations in its input.

When ulp_width is specified, its value is indicated as a superscript in function name. Notice the specified ulp_width is not the upper limit in general: there may exist function-function dependent regions in complex plane where ulp_width needs to be larger to pass the "out-of-ulp-range counts is zero" test.

Finally,

  • "using native " means "using the corresponding numpy ",
  • "using upcast " means that the function arguments are upcasted to a dtype with bits doubled, and the function results are downcasted to a dtype with bits split half.
Function dtype dULP=0 (exact) dULP=1 dULP=2 dULP=3 dULP>3 Notes
absolute complex64 967745 33704 552 - - -
absolute2 complex64 962177 39264 552 - 8 using native absolute
absolute complex128 991861 9996 144 - - -
acos float32 961608 38291 99 3 - -
acos2 float32 961878 38119 4 - - using native acos
acos float64 992599 7396 6 - - -
acos complex64 810406 190999 588 8 - -
acos2 complex64 678274 323467 260 - - using native acos
acos complex128 750619 251104 278 - - -
acosh float32 988269 11704 28 - - -
acosh2 float32 999248 753 - - - using native acosh
acosh float64 946280 53715 6 - - -
acosh complex64 810406 190999 588 8 - -
acosh2 complex64 678274 323467 260 - - using native acosh
acosh complex128 750619 251104 278 - - -
asin float32 974679 24368 942 12 - -
asin2 float32 996779 3010 212 - - using native asin
asin float64 995203 4772 26 - - -
asin complex64 808231 192742 996 32 - -
asin2 complex64 806919 194826 252 4 - using native asin
asin complex128 750491 250830 680 - - -
asinh float32 916129 83790 82 - - -
asinh2 float32 999111 890 - - - using native asinh
asinh float64 825119 174798 84 - - -
asinh complex64 808231 192742 996 32 - -
asinh2 complex64 806919 194826 252 4 - using native asinh
asinh complex128 750491 250830 680 - - -
atan complex64 851639 145498 4780 84 - -
atan2 complex64 902789 97576 1636 - - using native atan
atan complex128 936441 65256 304 - - -
atanh complex64 851639 145498 4780 84 - -
atanh2 complex64 902789 97576 1636 - - using native atanh
atanh complex128 936441 65256 304 - - -
square float32 997347 2654 - - - -
square2 float32 997347 2654 - - - using native square
square float64 999553 448 - - - -
square complex64 976865 25136 - - - -
square2 complex64 940181 28836 16 - 32968 using native square
square complex128 995705 6296 - - - -
sqrt float32 1000001 - - - - using native sqrt
sqrt complex64 639573 362328 100 - - using native sqrt
angle complex64 940289 61332 376 4 - -
angle complex128 989787 12214 - - - -
log1p1 complex64 902529 9759248500 15861172 10428 19022!! -
log1p2 complex64 902529 9759242692 158678 1046 1902!! -
log1p3 complex64 902529 9759241452 158644 104 190 -
log1p1 complex64 698489 6170650263 24371108 15191505 237850237612!! using native log1p
log1p2[using={'native log1p'}] complex64 698489 6170649384 243748 1519984 237850237376!! using native log1p
log1p3[using={'native log1p'}] complex64 698489 6170649206 24378 151936 237850235730!! using native log1p
log1p1 complex128 850341 151568139004 868 4 2 -
tan float32 866723 132062 1168 48 - using native tan
tan2 float32 1000001 - - - - using upcast tan, native tan
real_naive_tan float32 819251 179168 1580 2 - -
real_naive_tan2 float32 825895 173622 484 - - using upcast sin, upcast cos
real_naive_tan3 float32 1000001 - - - - using upcast sin, upcast cos, upcast divide
tan complex64 783819 197524 19806 792 60 using native tan
tan2 complex64 1001409 592 - - - using upcast tan, native tan
tanh float32 985109 14892 - - - using native tanh
tanh2 float32 1000001 - - - - using native tanh, upcast tanh
tanh complex64 783819 197524 19806 792 60 using native tanh
tanh2 complex64 1001409 592 - - - using native tanh, upcast tanh