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660 lines
18 KiB
V
660 lines
18 KiB
V
// Provides the [error](https://en.wikipedia.org/wiki/Error_function) and related functions
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// based on https://github.com/unovor/frame/blob/master/statrs-0.10.0/src/function/erf.rs
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//
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// NOTE: This impl does not have the same precision as glibc impl of erf,erfc and others, we should fix this
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// in the future.
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module math
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// Coefficients for erf_impl polynominal
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const (
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// Polynomial coefficients for a numerator of `erf_impl`
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// in the interval [1e-10, 0.5].
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erf_impl_an = [0.00337916709551257388990745, -0.00073695653048167948530905,
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-0.374732337392919607868241, 0.0817442448733587196071743, -0.0421089319936548595203468,
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0.0070165709512095756344528, -0.00495091255982435110337458, 0.000871646599037922480317225]
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// Polynomial coefficients for a denominator of `erf_impl`
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// in the interval [1e-10, 0.5]
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erf_impl_ad = [1.0, -0.218088218087924645390535, 0.412542972725442099083918,
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-0.0841891147873106755410271, 0.0655338856400241519690695, -0.0120019604454941768171266,
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0.00408165558926174048329689, -0.000615900721557769691924509]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [0.5, 0.75].
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erf_impl_bn = [-0.0361790390718262471360258, 0.292251883444882683221149,
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0.281447041797604512774415, 0.125610208862766947294894, 0.0274135028268930549240776,
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0.00250839672168065762786937,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [0.5, 0.75].
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erf_impl_bd = [1.0, 1.8545005897903486499845, 1.43575803037831418074962,
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0.582827658753036572454135, 0.124810476932949746447682, 0.0113724176546353285778481]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [0.75, 1.25].
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erf_impl_cn = [
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-0.0397876892611136856954425,
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0.153165212467878293257683,
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0.191260295600936245503129,
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0.10276327061989304213645,
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0.029637090615738836726027,
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0.0046093486780275489468812,
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0.000307607820348680180548455,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [0.75, 1.25].
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erf_impl_cd = [
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1.0,
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1.95520072987627704987886,
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1.64762317199384860109595,
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0.768238607022126250082483,
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0.209793185936509782784315,
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0.0319569316899913392596356,
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0.00213363160895785378615014,
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]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [1.25, 2.25].
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erf_impl_dn = [
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-0.0300838560557949717328341,
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0.0538578829844454508530552,
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0.0726211541651914182692959,
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0.0367628469888049348429018,
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0.00964629015572527529605267,
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0.00133453480075291076745275,
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0.778087599782504251917881e-4,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [1.25, 2.25].
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erf_impl_dd = [
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1.0,
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1.75967098147167528287343,
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1.32883571437961120556307,
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0.552528596508757581287907,
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0.133793056941332861912279,
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0.0179509645176280768640766,
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0.00104712440019937356634038,
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-0.106640381820357337177643e-7,
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]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [2.25, 3.5].
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erf_impl_en = [
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-0.0117907570137227847827732,
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0.014262132090538809896674,
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0.0202234435902960820020765,
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0.00930668299990432009042239,
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0.00213357802422065994322516,
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0.00025022987386460102395382,
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0.120534912219588189822126e-4,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [2.25, 3.5].
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erf_impl_ed = [
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1.0,
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1.50376225203620482047419,
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0.965397786204462896346934,
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0.339265230476796681555511,
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0.0689740649541569716897427,
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0.00771060262491768307365526,
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0.000371421101531069302990367,
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]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [3.5, 5.25].
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erf_impl_fn = [
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-0.00546954795538729307482955,
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0.00404190278731707110245394,
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0.0054963369553161170521356,
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0.00212616472603945399437862,
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0.000394984014495083900689956,
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0.365565477064442377259271e-4,
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0.135485897109932323253786e-5,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [3.5, 5.25].
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erf_impl_fd = [
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1.0,
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1.21019697773630784832251,
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0.620914668221143886601045,
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0.173038430661142762569515,
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0.0276550813773432047594539,
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0.00240625974424309709745382,
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0.891811817251336577241006e-4,
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-0.465528836283382684461025e-11,
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]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [5.25, 8].
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erf_impl_gn = [
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-0.00270722535905778347999196,
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0.0013187563425029400461378,
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0.00119925933261002333923989,
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0.00027849619811344664248235,
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0.267822988218331849989363e-4,
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0.923043672315028197865066e-6,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [5.25, 8].
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erf_impl_gd = [
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1.0,
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0.814632808543141591118279,
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0.268901665856299542168425,
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0.0449877216103041118694989,
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0.00381759663320248459168994,
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0.000131571897888596914350697,
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0.404815359675764138445257e-11,
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]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [8, 11.5].
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erf_impl_hn = [
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-0.00109946720691742196814323,
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0.000406425442750422675169153,
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0.000274499489416900707787024,
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0.465293770646659383436343e-4,
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0.320955425395767463401993e-5,
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0.778286018145020892261936e-7,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [8, 11.5].
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erf_impl_hd = [
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1.0,
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0.588173710611846046373373,
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0.139363331289409746077541,
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0.0166329340417083678763028,
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0.00100023921310234908642639,
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0.24254837521587225125068e-4,
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]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [11.5, 17].
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erf_impl_in = [
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-0.00056907993601094962855594,
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0.000169498540373762264416984,
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0.518472354581100890120501e-4,
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0.382819312231928859704678e-5,
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0.824989931281894431781794e-7,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [11.5, 17].
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erf_impl_id = [
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1.0,
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0.339637250051139347430323,
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0.043472647870310663055044,
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0.00248549335224637114641629,
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0.535633305337152900549536e-4,
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-0.117490944405459578783846e-12,
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]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [17, 24].
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erf_impl_jn = [
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-0.000241313599483991337479091,
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0.574224975202501512365975e-4,
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0.115998962927383778460557e-4,
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0.581762134402593739370875e-6,
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0.853971555085673614607418e-8,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [17, 24].
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erf_impl_jd = [
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1.0,
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0.233044138299687841018015,
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0.0204186940546440312625597,
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0.000797185647564398289151125,
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0.117019281670172327758019e-4,
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]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [24, 38].
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erf_impl_kn = [
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-0.000146674699277760365803642,
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0.162666552112280519955647e-4,
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0.269116248509165239294897e-5,
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0.979584479468091935086972e-7,
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0.101994647625723465722285e-8,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [24, 38].
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erf_impl_kd = [
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1.0,
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0.165907812944847226546036,
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0.0103361716191505884359634,
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0.000286593026373868366935721,
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0.298401570840900340874568e-5,
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]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [38, 60].
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erf_impl_ln = [
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-0.583905797629771786720406e-4,
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0.412510325105496173512992e-5,
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0.431790922420250949096906e-6,
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0.993365155590013193345569e-8,
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0.653480510020104699270084e-10,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [38, 60].
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erf_impl_ld = [
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1.0,
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0.105077086072039915406159,
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0.00414278428675475620830226,
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0.726338754644523769144108e-4,
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0.477818471047398785369849e-6,
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]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [60, 85].
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erf_impl_mn = [
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-0.196457797609229579459841e-4,
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0.157243887666800692441195e-5,
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0.543902511192700878690335e-7,
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0.317472492369117710852685e-9,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [60, 85].
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erf_impl_md = [
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1.0,
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0.052803989240957632204885,
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0.000926876069151753290378112,
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0.541011723226630257077328e-5,
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0.535093845803642394908747e-15,
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]
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// Polynomial coefficients for a numerator in `erf_impl`
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// in the interval [85, 110].
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erf_impl_nn = [
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-0.789224703978722689089794e-5,
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0.622088451660986955124162e-6,
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0.145728445676882396797184e-7,
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0.603715505542715364529243e-10,
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]
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// Polynomial coefficients for a denominator in `erf_impl`
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// in the interval [85, 110].
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erf_impl_nd = [
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1.0,
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0.0375328846356293715248719,
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0.000467919535974625308126054,
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0.193847039275845656900547e-5,
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]
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// **********************************************************
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// ********** Coefficients for erf_inv_impl polynomial ******
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// **********************************************************
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// Polynomial coefficients for a numerator of `erf_inv_impl`
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// in the interval [0, 0.5].
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erf_inv_impl_an = [
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-0.000508781949658280665617,
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-0.00836874819741736770379,
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0.0334806625409744615033,
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-0.0126926147662974029034,
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-0.0365637971411762664006,
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0.0219878681111168899165,
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0.00822687874676915743155,
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-0.00538772965071242932965,
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]
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// Polynomial coefficients for a denominator of `erf_inv_impl`
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// in the interval [0, 0.5].
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erf_inv_impl_ad = [
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1.0,
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-0.970005043303290640362,
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-1.56574558234175846809,
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1.56221558398423026363,
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0.662328840472002992063,
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-0.71228902341542847553,
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-0.0527396382340099713954,
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0.0795283687341571680018,
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-0.00233393759374190016776,
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0.000886216390456424707504,
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]
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// Polynomial coefficients for a numerator of `erf_inv_impl`
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// in the interval [0.5, 0.75].
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erf_inv_impl_bn = [
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-0.202433508355938759655,
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0.105264680699391713268,
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8.37050328343119927838,
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17.6447298408374015486,
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-18.8510648058714251895,
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-44.6382324441786960818,
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17.445385985570866523,
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21.1294655448340526258,
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-3.67192254707729348546,
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]
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// Polynomial coefficients for a denominator of `erf_inv_impl`
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// in the interval [0.5, 0.75].
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erf_inv_impl_bd = [
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1.0,
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6.24264124854247537712,
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3.9713437953343869095,
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-28.6608180499800029974,
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-20.1432634680485188801,
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48.5609213108739935468,
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10.8268667355460159008,
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-22.6436933413139721736,
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1.72114765761200282724,
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]
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// Polynomial coefficients for a numerator of `erf_inv_impl`
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// in the interval [0.75, 1] with x less than 3.
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erf_inv_impl_cn = [
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-0.131102781679951906451,
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-0.163794047193317060787,
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0.117030156341995252019,
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0.387079738972604337464,
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0.337785538912035898924,
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0.142869534408157156766,
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0.0290157910005329060432,
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0.00214558995388805277169,
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-0.679465575181126350155e-6,
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0.285225331782217055858e-7,
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-0.681149956853776992068e-9,
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]
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// Polynomial coefficients for a denominator of `erf_inv_impl`
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// in the interval [0.75, 1] with x less than 3.
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erf_inv_impl_cd = [
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1.0,
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3.46625407242567245975,
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5.38168345707006855425,
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4.77846592945843778382,
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2.59301921623620271374,
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0.848854343457902036425,
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0.152264338295331783612,
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0.01105924229346489121,
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]
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// Polynomial coefficients for a numerator of `erf_inv_impl`
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// in the interval [0.75, 1] with x between 3 and 6.
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erf_inv_impl_dn = [
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-0.0350353787183177984712,
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-0.00222426529213447927281,
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0.0185573306514231072324,
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0.00950804701325919603619,
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0.00187123492819559223345,
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0.000157544617424960554631,
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0.460469890584317994083e-5,
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-0.230404776911882601748e-9,
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0.266339227425782031962e-11,
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]
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// Polynomial coefficients for a denominator of `erf_inv_impl`
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// in the interval [0.75, 1] with x between 3 and 6.
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erf_inv_impl_dd = [
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1.0,
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1.3653349817554063097,
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0.762059164553623404043,
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0.220091105764131249824,
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0.0341589143670947727934,
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0.00263861676657015992959,
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0.764675292302794483503e-4,
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]
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// Polynomial coefficients for a numerator of `erf_inv_impl`
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// in the interval [0.75, 1] with x between 6 and 18.
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erf_inv_impl_en = [
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-0.0167431005076633737133,
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-0.00112951438745580278863,
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0.00105628862152492910091,
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0.000209386317487588078668,
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0.149624783758342370182e-4,
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0.449696789927706453732e-6,
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0.462596163522878599135e-8,
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-0.281128735628831791805e-13,
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0.99055709973310326855e-16,
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]
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// Polynomial coefficients for a denominator of `erf_inv_impl`
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// in the interval [0.75, 1] with x between 6 and 18.
|
|
erf_inv_impl_ed = [
|
|
1.0,
|
|
0.591429344886417493481,
|
|
0.138151865749083321638,
|
|
0.0160746087093676504695,
|
|
0.000964011807005165528527,
|
|
0.275335474764726041141e-4,
|
|
0.282243172016108031869e-6,
|
|
]
|
|
|
|
// Polynomial coefficients for a numerator of `erf_inv_impl`
|
|
// in the interval [0.75, 1] with x between 18 and 44.
|
|
erf_inv_impl_fn = [
|
|
-0.0024978212791898131227,
|
|
-0.779190719229053954292e-5,
|
|
0.254723037413027451751e-4,
|
|
0.162397777342510920873e-5,
|
|
0.396341011304801168516e-7,
|
|
0.411632831190944208473e-9,
|
|
0.145596286718675035587e-11,
|
|
-0.116765012397184275695e-17,
|
|
]
|
|
|
|
// Polynomial coefficients for a denominator of `erf_inv_impl`
|
|
// in the interval [0.75, 1] with x between 18 and 44.
|
|
erf_inv_impl_fd = [
|
|
1.0,
|
|
0.207123112214422517181,
|
|
0.0169410838120975906478,
|
|
0.000690538265622684595676,
|
|
0.145007359818232637924e-4,
|
|
0.144437756628144157666e-6,
|
|
0.509761276599778486139e-9,
|
|
]
|
|
|
|
// Polynomial coefficients for a numerator of `erf_inv_impl`
|
|
// in the interval [0.75, 1] with x greater than 44.
|
|
erf_inv_impl_gn = [
|
|
-0.000539042911019078575891,
|
|
-0.28398759004727721098e-6,
|
|
0.899465114892291446442e-6,
|
|
0.229345859265920864296e-7,
|
|
0.225561444863500149219e-9,
|
|
0.947846627503022684216e-12,
|
|
0.135880130108924861008e-14,
|
|
-0.348890393399948882918e-21,
|
|
]
|
|
|
|
// Polynomial coefficients for a denominator of `erf_inv_impl`
|
|
// in the interval [0.75, 1] with x greater than 44.
|
|
erf_inv_impl_gd = [
|
|
1.0,
|
|
0.0845746234001899436914,
|
|
0.00282092984726264681981,
|
|
0.468292921940894236786e-4,
|
|
0.399968812193862100054e-6,
|
|
0.161809290887904476097e-8,
|
|
0.231558608310259605225e-11,
|
|
]
|
|
)
|
|
|
|
fn erf_inv_impl(p f64, q f64, s f64) f64 {
|
|
mut result := 0.0
|
|
if p <= 0.5 {
|
|
y := 0.0891314744949340820313
|
|
g := p * (p + 10.0)
|
|
r := polynomial(p, math.erf_inv_impl_an) / polynomial(p, math.erf_inv_impl_ad)
|
|
result = g * y + g * r
|
|
} else if q >= 0.25 {
|
|
y := 2.249481201171875
|
|
g := sqrt(-2.0 * log(q))
|
|
xs := q - 0.25
|
|
r := polynomial(xs, math.erf_inv_impl_bn) / polynomial(xs, math.erf_inv_impl_bd)
|
|
result = g / (y + r)
|
|
} else {
|
|
x := sqrt(-log(q))
|
|
if x < 3.0 {
|
|
y := 0.807220458984375
|
|
xs := x - 1.125
|
|
r := polynomial(xs, math.erf_inv_impl_cn) / polynomial(xs, math.erf_inv_impl_cd)
|
|
result = y * x + r * x
|
|
} else if x < 6.0 {
|
|
y := 0.93995571136474609375
|
|
xs := x - 3.0
|
|
r := polynomial(xs, math.erf_inv_impl_dn) / polynomial(xs, math.erf_inv_impl_dd)
|
|
result = y * x + r * x
|
|
} else if x < 18.0 {
|
|
y := 0.98362827301025390625
|
|
xs := x - 6.0
|
|
r := polynomial(xs, math.erf_inv_impl_en) / polynomial(xs, math.erf_inv_impl_ed)
|
|
result = y * x + r * x
|
|
} else if x < 44.0 {
|
|
y := 0.99714565277099609375
|
|
xs := x - 18.0
|
|
r := polynomial(xs, math.erf_inv_impl_fn) / polynomial(xs, math.erf_inv_impl_fd)
|
|
result = y * x + r * x
|
|
} else {
|
|
y := 0.99941349029541015625
|
|
xs := x - 44.0
|
|
r := polynomial(xs, math.erf_inv_impl_gn) / polynomial(xs, math.erf_inv_impl_gd)
|
|
result = y * x + r * x
|
|
}
|
|
}
|
|
|
|
return s * result
|
|
}
|
|
|
|
fn erf_impl(z f64, inv bool) f64 {
|
|
if z < 0.0 {
|
|
if !inv {
|
|
return -erf_impl(-z, false)
|
|
}
|
|
if z < -0.5 {
|
|
return 2.0 - erf_impl(-z, true)
|
|
}
|
|
return 1.0 + erf_impl(-z, false)
|
|
}
|
|
mut result := 0.0
|
|
if z < 0.5 {
|
|
if z < 1e-10 {
|
|
result = z * 1.125 + z * 0.003379167095512573896158903121545171688
|
|
} else {
|
|
result = z * 1.125 +
|
|
z * polynomial(z, math.erf_impl_an) / polynomial(z, math.erf_impl_ad)
|
|
}
|
|
} else if z < 110.0 {
|
|
mut r := 0.0
|
|
mut b := 0.0
|
|
if z < 0.75 {
|
|
r = polynomial(z - 0.5, math.erf_impl_bn) / polynomial(z - 0.5, math.erf_impl_bd)
|
|
b = 0.3440242112
|
|
} else if z < 1.25 {
|
|
r = polynomial(z - 0.75, math.erf_impl_cn) / polynomial(z - 0.75, math.erf_impl_cd)
|
|
b = 0.419990927
|
|
} else if z < 2.25 {
|
|
r = polynomial(z - 1.25, math.erf_impl_dn) / polynomial(z - 1.25, math.erf_impl_dd)
|
|
b = 0.4898625016
|
|
} else if z < 3.5 {
|
|
r = polynomial(z - 2.25, math.erf_impl_en) / polynomial(z - 2.25, math.erf_impl_ed)
|
|
b = 0.5317370892
|
|
} else if z < 5.25 {
|
|
r = polynomial(z - 3.5, math.erf_impl_fn) / polynomial(z - 3.5, math.erf_impl_fd)
|
|
b = 0.5489973426
|
|
} else if z < 8.0 {
|
|
r = polynomial(z - 5.25, math.erf_impl_gn) / polynomial(z - 5.25, math.erf_impl_gd)
|
|
b = 0.5571740866
|
|
} else if z < 11.5 {
|
|
r = polynomial(z - 8.0, math.erf_impl_hn) / polynomial(z - 8.0, math.erf_impl_hd)
|
|
b = 0.5609807968
|
|
} else if z < 17.0 {
|
|
r = polynomial(z - 11.5, math.erf_impl_in) / polynomial(z - 11.5, math.erf_impl_id)
|
|
b = 0.5626493692
|
|
} else if z < 24.0 {
|
|
r = polynomial(z - 17.0, math.erf_impl_jn) / polynomial(z - 17.0, math.erf_impl_jd)
|
|
b = 0.5634598136
|
|
} else if z < 38.0 {
|
|
r = polynomial(z - 24.0, math.erf_impl_kn) / polynomial(z - 24.0, math.erf_impl_kd)
|
|
b = 0.5638477802
|
|
} else if z < 60.0 {
|
|
r = polynomial(z - 38.0, math.erf_impl_ln) / polynomial(z - 38.0, math.erf_impl_ld)
|
|
b = 0.5640528202
|
|
} else if z < 85.0 {
|
|
r = polynomial(z - 60.0, math.erf_impl_mn) / polynomial(z - 60.0, math.erf_impl_md)
|
|
b = 0.5641309023
|
|
} else {
|
|
r = polynomial(z - 85.0, math.erf_impl_nn) / polynomial(z - 85.0, math.erf_impl_nd)
|
|
b = 0.5641584396
|
|
}
|
|
|
|
g := exp(-z * z) / z
|
|
result = g * b + g * r
|
|
} else {
|
|
result = 0.0
|
|
}
|
|
if inv && z >= 0.5 {
|
|
return result
|
|
} else if z >= 0.5 || inv {
|
|
return 1.0 - result
|
|
} else {
|
|
return result
|
|
}
|
|
}
|
|
|
|
/// 'erf' calculates the error function at `x`.
|
|
pub fn erf(x f64) f64 {
|
|
if is_nan(x) {
|
|
return nan()
|
|
} else if is_inf(x, 1) {
|
|
return 1.0
|
|
} else if is_inf(x, -1) {
|
|
return -1.0
|
|
} else if x == 0.0 {
|
|
return 0.0
|
|
} else {
|
|
return erf_impl(x, false)
|
|
}
|
|
}
|
|
|
|
// `erf_inv` calculates the inverse error function at `x`.
|
|
pub fn erf_inv(x f64) f64 {
|
|
if x == 0 {
|
|
return 0.0
|
|
} else if x >= 1.0 {
|
|
return inf(1)
|
|
} else if x <= -1.0 {
|
|
return inf(-1)
|
|
} else if x < 0.0 {
|
|
return erf_inv_impl(-x, 1.0 + x, -1.0)
|
|
} else {
|
|
return erf_inv_impl(x, 1.0 - x, 1.0)
|
|
}
|
|
}
|
|
|
|
// `erfc` calculates the complementary error function at `x`.
|
|
pub fn erfc(x f64) f64 {
|
|
if is_nan(x) {
|
|
return nan()
|
|
} else if is_inf(x, 1) {
|
|
return 0.0
|
|
} else if is_inf(x, -1) {
|
|
return 2.0
|
|
} else {
|
|
return erf_impl(x, true)
|
|
}
|
|
}
|
|
|
|
// `erfc_inv` calculates the complementary inverse error function at `x`.
|
|
pub fn erfc_inv(x f64) f64 {
|
|
if x <= 0.0 {
|
|
return inf(1)
|
|
} else if x >= 2.0 {
|
|
return inf(-1)
|
|
} else if is_inf(x, -1) {
|
|
return erf_inv_impl(-1.0 + x, 2.0 - x, -1.0)
|
|
} else {
|
|
return erf_inv_impl(1.0 - x, x, 1.0)
|
|
}
|
|
}
|