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vlib/math: Add a pure V backend for vlib/math (#11267)
This commit is contained in:
Ulises Jeremias Cornejo Fandos
committed by
GitHub
GitHub
parent
dd486bb0fb
commit
1cfc4198f5
950
vlib/math/erf.v
950
vlib/math/erf.v
@@ -1,659 +1,327 @@
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// 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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/*
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* x
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* 2 |\
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* erf(x) = --------- | exp(-t*t)dt
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* sqrt(pi) \|
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* 0
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*
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* erfc(x) = 1-erf(x)
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* Note that
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* erf(-x) = -erf(x)
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* erfc(-x) = 2 - erfc(x)
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*
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* Method:
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* 1. For |x| in [0, 0.84375]
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* erf(x) = x + x*R(x**2)
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* erfc(x) = 1 - erf(x) if x in [-.84375,0.25]
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* = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375]
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* where R = P/Q where P is an odd poly of degree 8 and
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* Q is an odd poly of degree 10.
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* -57.90
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* | R - (erf(x)-x)/x | <= 2
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*
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*
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* Remark. The formula is derived by noting
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* erf(x) = (2/sqrt(pi))*(x - x**3/3 + x**5/10 - x**7/42 + ....)
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* and that
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* 2/sqrt(pi) = 1.128379167095512573896158903121545171688
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* is close to one. The interval is chosen because the fix
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* point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is
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* near 0.6174), and by some experiment, 0.84375 is chosen to
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* guarantee the error is less than one ulp for erf.
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*
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* 2. For |x| in [0.84375,1.25], let s_ = |x| - 1, and
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* c = 0.84506291151 rounded to single (24 bits)
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* erf(x) = sign(x) * (c + P1(s_)/Q1(s_))
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* erfc(x) = (1-c) - P1(s_)/Q1(s_) if x > 0
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* 1+(c+P1(s_)/Q1(s_)) if x < 0
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* |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06
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* Remark: here we use the taylor series expansion at x=1.
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* erf(1+s_) = erf(1) + s_*Poly(s_)
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* = 0.845.. + P1(s_)/Q1(s_)
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* That is, we use rational approximation to approximate
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* erf(1+s_) - (c = (single)0.84506291151)
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* Note that |P1/Q1|< 0.078 for x in [0.84375,1.25]
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* where
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* P1(s_) = degree 6 poly in s_
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* Q1(s_) = degree 6 poly in s_
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*
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* 3. For x in [1.25,1/0.35(~2.857143)],
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* erfc(x) = (1/x)*exp(-x*x-0.5625+R1/s1)
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* erf(x) = 1 - erfc(x)
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* where
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* R1(z) = degree 7 poly in z, (z=1/x**2)
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* s1(z) = degree 8 poly in z
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*
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* 4. For x in [1/0.35,28]
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* erfc(x) = (1/x)*exp(-x*x-0.5625+R2/s2) if x > 0
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* = 2.0 - (1/x)*exp(-x*x-0.5625+R2/s2) if -6<x<0
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* = 2.0 - tiny (if x <= -6)
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* erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else
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* erf(x) = sign(x)*(1.0 - tiny)
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* where
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* R2(z) = degree 6 poly in z, (z=1/x**2)
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* s2(z) = degree 7 poly in z
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*
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* Note1:
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* To compute exp(-x*x-0.5625+R/s), let s_ be a single
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* precision number and s_ := x; then
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* -x*x = -s_*s_ + (s_-x)*(s_+x)
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* exp(-x*x-0.5626+R/s) =
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* exp(-s_*s_-0.5625)*exp((s_-x)*(s_+x)+R/s);
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* Note2:
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* Here 4 and 5 make use of the asymptotic series
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* exp(-x*x)
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* erfc(x) ~ ---------- * ( 1 + Poly(1/x**2) )
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* x*sqrt(pi)
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* We use rational approximation to approximate
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* g(s_)=f(1/x**2) = log(erfc(x)*x) - x*x + 0.5625
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* Here is the error bound for R1/s1 and R2/s2
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* |R1/s1 - f(x)| < 2**(-62.57)
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* |R2/s2 - f(x)| < 2**(-61.52)
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*
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* 5. For inf > x >= 28
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* erf(x) = sign(x) *(1 - tiny) (raise inexact)
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* erfc(x) = tiny*tiny (raise underflow) if x > 0
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* = 2 - tiny if x<0
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*
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* 7. special case:
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* erf(0) = 0, erf(inf) = 1, erf(-inf) = -1,
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* erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2,
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* erfc/erf(nan) is nan
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*/
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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,
|
||||
0.0375328846356293715248719,
|
||||
0.000467919535974625308126054,
|
||||
0.193847039275845656900547e-5,
|
||||
]
|
||||
|
||||
// **********************************************************
|
||||
// ********** Coefficients for erf_inv_impl polynomial ******
|
||||
// **********************************************************
|
||||
// Polynomial coefficients for a numerator of `erf_inv_impl`
|
||||
// in the interval [0, 0.5].
|
||||
erf_inv_impl_an = [
|
||||
-0.000508781949658280665617,
|
||||
-0.00836874819741736770379,
|
||||
0.0334806625409744615033,
|
||||
-0.0126926147662974029034,
|
||||
-0.0365637971411762664006,
|
||||
0.0219878681111168899165,
|
||||
0.00822687874676915743155,
|
||||
-0.00538772965071242932965,
|
||||
]
|
||||
|
||||
// Polynomial coefficients for a denominator of `erf_inv_impl`
|
||||
// in the interval [0, 0.5].
|
||||
erf_inv_impl_ad = [
|
||||
1.0,
|
||||
-0.970005043303290640362,
|
||||
-1.56574558234175846809,
|
||||
1.56221558398423026363,
|
||||
0.662328840472002992063,
|
||||
-0.71228902341542847553,
|
||||
-0.0527396382340099713954,
|
||||
0.0795283687341571680018,
|
||||
-0.00233393759374190016776,
|
||||
0.000886216390456424707504,
|
||||
]
|
||||
|
||||
// Polynomial coefficients for a numerator of `erf_inv_impl`
|
||||
// in the interval [0.5, 0.75].
|
||||
erf_inv_impl_bn = [
|
||||
-0.202433508355938759655,
|
||||
0.105264680699391713268,
|
||||
8.37050328343119927838,
|
||||
17.6447298408374015486,
|
||||
-18.8510648058714251895,
|
||||
-44.6382324441786960818,
|
||||
17.445385985570866523,
|
||||
21.1294655448340526258,
|
||||
-3.67192254707729348546,
|
||||
]
|
||||
|
||||
// Polynomial coefficients for a denominator of `erf_inv_impl`
|
||||
// in the interval [0.5, 0.75].
|
||||
erf_inv_impl_bd = [
|
||||
1.0,
|
||||
6.24264124854247537712,
|
||||
3.9713437953343869095,
|
||||
-28.6608180499800029974,
|
||||
-20.1432634680485188801,
|
||||
48.5609213108739935468,
|
||||
10.8268667355460159008,
|
||||
-22.6436933413139721736,
|
||||
1.72114765761200282724,
|
||||
]
|
||||
|
||||
// Polynomial coefficients for a numerator of `erf_inv_impl`
|
||||
// in the interval [0.75, 1] with x less than 3.
|
||||
erf_inv_impl_cn = [
|
||||
-0.131102781679951906451,
|
||||
-0.163794047193317060787,
|
||||
0.117030156341995252019,
|
||||
0.387079738972604337464,
|
||||
0.337785538912035898924,
|
||||
0.142869534408157156766,
|
||||
0.0290157910005329060432,
|
||||
0.00214558995388805277169,
|
||||
-0.679465575181126350155e-6,
|
||||
0.285225331782217055858e-7,
|
||||
-0.681149956853776992068e-9,
|
||||
]
|
||||
|
||||
// Polynomial coefficients for a denominator of `erf_inv_impl`
|
||||
// in the interval [0.75, 1] with x less than 3.
|
||||
erf_inv_impl_cd = [
|
||||
1.0,
|
||||
3.46625407242567245975,
|
||||
5.38168345707006855425,
|
||||
4.77846592945843778382,
|
||||
2.59301921623620271374,
|
||||
0.848854343457902036425,
|
||||
0.152264338295331783612,
|
||||
0.01105924229346489121,
|
||||
]
|
||||
|
||||
// Polynomial coefficients for a numerator of `erf_inv_impl`
|
||||
// in the interval [0.75, 1] with x between 3 and 6.
|
||||
erf_inv_impl_dn = [
|
||||
-0.0350353787183177984712,
|
||||
-0.00222426529213447927281,
|
||||
0.0185573306514231072324,
|
||||
0.00950804701325919603619,
|
||||
0.00187123492819559223345,
|
||||
0.000157544617424960554631,
|
||||
0.460469890584317994083e-5,
|
||||
-0.230404776911882601748e-9,
|
||||
0.266339227425782031962e-11,
|
||||
]
|
||||
|
||||
// Polynomial coefficients for a denominator of `erf_inv_impl`
|
||||
// in the interval [0.75, 1] with x between 3 and 6.
|
||||
erf_inv_impl_dd = [
|
||||
1.0,
|
||||
1.3653349817554063097,
|
||||
0.762059164553623404043,
|
||||
0.220091105764131249824,
|
||||
0.0341589143670947727934,
|
||||
0.00263861676657015992959,
|
||||
0.764675292302794483503e-4,
|
||||
]
|
||||
|
||||
// Polynomial coefficients for a numerator of `erf_inv_impl`
|
||||
// in the interval [0.75, 1] with x between 6 and 18.
|
||||
erf_inv_impl_en = [
|
||||
-0.0167431005076633737133,
|
||||
-0.00112951438745580278863,
|
||||
0.00105628862152492910091,
|
||||
0.000209386317487588078668,
|
||||
0.149624783758342370182e-4,
|
||||
0.449696789927706453732e-6,
|
||||
0.462596163522878599135e-8,
|
||||
-0.281128735628831791805e-13,
|
||||
0.99055709973310326855e-16,
|
||||
]
|
||||
|
||||
// Polynomial coefficients for a denominator of `erf_inv_impl`
|
||||
// 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,
|
||||
]
|
||||
erx = 8.45062911510467529297e-01 // 0x3FEB0AC160000000
|
||||
// Coefficients for approximation to erf in [0, 0.84375]
|
||||
efx = 1.28379167095512586316e-01 // 0x3FC06EBA8214DB69
|
||||
efx8 = 1.02703333676410069053e+00 // 0x3FF06EBA8214DB69
|
||||
pp0 = 1.28379167095512558561e-01 // 0x3FC06EBA8214DB68
|
||||
pp1 = -3.25042107247001499370e-01 // 0xBFD4CD7D691CB913
|
||||
pp2 = -2.84817495755985104766e-02 // 0xBF9D2A51DBD7194F
|
||||
pp3 = -5.77027029648944159157e-03 // 0xBF77A291236668E4
|
||||
pp4 = -2.37630166566501626084e-05 // 0xBEF8EAD6120016AC
|
||||
qq1 = 3.97917223959155352819e-01 // 0x3FD97779CDDADC09
|
||||
qq2 = 6.50222499887672944485e-02 // 0x3FB0A54C5536CEBA
|
||||
qq3 = 5.08130628187576562776e-03 // 0x3F74D022C4D36B0F
|
||||
qq4 = 1.32494738004321644526e-04 // 0x3F215DC9221C1A10
|
||||
qq5 = -3.96022827877536812320e-06 // 0xBED09C4342A26120
|
||||
// Coefficients for approximation to erf in [0.84375, 1.25]
|
||||
pa0 = -2.36211856075265944077e-03 // 0xBF6359B8BEF77538
|
||||
pa1 = 4.14856118683748331666e-01 // 0x3FDA8D00AD92B34D
|
||||
pa2 = -3.72207876035701323847e-01 // 0xBFD7D240FBB8C3F1
|
||||
pa3 = 3.18346619901161753674e-01 // 0x3FD45FCA805120E4
|
||||
pa4 = -1.10894694282396677476e-01 // 0xBFBC63983D3E28EC
|
||||
pa5 = 3.54783043256182359371e-02 // 0x3FA22A36599795EB
|
||||
pa6 = -2.16637559486879084300e-03 // 0xBF61BF380A96073F
|
||||
qa1 = 1.06420880400844228286e-01 // 0x3FBB3E6618EEE323
|
||||
qa2 = 5.40397917702171048937e-01 // 0x3FE14AF092EB6F33
|
||||
qa3 = 7.18286544141962662868e-02 // 0x3FB2635CD99FE9A7
|
||||
qa4 = 1.26171219808761642112e-01 // 0x3FC02660E763351F
|
||||
qa5 = 1.36370839120290507362e-02 // 0x3F8BEDC26B51DD1C
|
||||
qa6 = 1.19844998467991074170e-02 // 0x3F888B545735151D
|
||||
// Coefficients for approximation to erfc in [1.25, 1/0.35]
|
||||
ra0 = -9.86494403484714822705e-03 // 0xBF843412600D6435
|
||||
ra1 = -6.93858572707181764372e-01 // 0xBFE63416E4BA7360
|
||||
ra2 = -1.05586262253232909814e+01 // 0xC0251E0441B0E726
|
||||
ra3 = -6.23753324503260060396e+01 // 0xC04F300AE4CBA38D
|
||||
ra4 = -1.62396669462573470355e+02 // 0xC0644CB184282266
|
||||
ra5 = -1.84605092906711035994e+02 // 0xC067135CEBCCABB2
|
||||
ra6 = -8.12874355063065934246e+01 // 0xC054526557E4D2F2
|
||||
ra7 = -9.81432934416914548592e+00 // 0xC023A0EFC69AC25C
|
||||
sa1 = 1.96512716674392571292e+01 // 0x4033A6B9BD707687
|
||||
sa2 = 1.37657754143519042600e+02 // 0x4061350C526AE721
|
||||
sa3 = 4.34565877475229228821e+02 // 0x407B290DD58A1A71
|
||||
sa4 = 6.45387271733267880336e+02 // 0x40842B1921EC2868
|
||||
sa5 = 4.29008140027567833386e+02 // 0x407AD02157700314
|
||||
sa6 = 1.08635005541779435134e+02 // 0x405B28A3EE48AE2C
|
||||
sa7 = 6.57024977031928170135e+00 // 0x401A47EF8E484A93
|
||||
sa8 = -6.04244152148580987438e-02 // 0xBFAEEFF2EE749A62
|
||||
// Coefficients for approximation to erfc in [1/.35, 28]
|
||||
rb0 = -9.86494292470009928597e-03 // 0xBF84341239E86F4A
|
||||
rb1 = -7.99283237680523006574e-01 // 0xBFE993BA70C285DE
|
||||
rb2 = -1.77579549177547519889e+01 // 0xC031C209555F995A
|
||||
rb3 = -1.60636384855821916062e+02 // 0xC064145D43C5ED98
|
||||
rb4 = -6.37566443368389627722e+02 // 0xC083EC881375F228
|
||||
rb5 = -1.02509513161107724954e+03 // 0xC09004616A2E5992
|
||||
rb6 = -4.83519191608651397019e+02 // 0xC07E384E9BDC383F
|
||||
sb1 = 3.03380607434824582924e+01 // 0x403E568B261D5190
|
||||
sb2 = 3.25792512996573918826e+02 // 0x40745CAE221B9F0A
|
||||
sb3 = 1.53672958608443695994e+03 // 0x409802EB189D5118
|
||||
sb4 = 3.19985821950859553908e+03 // 0x40A8FFB7688C246A
|
||||
sb5 = 2.55305040643316442583e+03 // 0x40A3F219CEDF3BE6
|
||||
sb6 = 4.74528541206955367215e+02 // 0x407DA874E79FE763
|
||||
sb7 = -2.24409524465858183362e+01 // 0xC03670E242712D62
|
||||
)
|
||||
|
||||
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 {
|
||||
// erf returns the error function of x.
|
||||
//
|
||||
// special cases are:
|
||||
// erf(+inf) = 1
|
||||
// erf(-inf) = -1
|
||||
// erf(nan) = nan
|
||||
pub fn erf(a f64) f64 {
|
||||
mut x := a
|
||||
very_tiny := 2.848094538889218e-306 // 0x0080000000000000
|
||||
small := 1.0 / f64(u64(1) << 28) // 2**-28
|
||||
if is_nan(x) {
|
||||
return nan()
|
||||
} else if is_inf(x, 1) {
|
||||
}
|
||||
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)
|
||||
}
|
||||
if is_inf(x, -1) {
|
||||
return f64(-1)
|
||||
}
|
||||
mut sign := false
|
||||
if x < 0 {
|
||||
x = -x
|
||||
sign = true
|
||||
}
|
||||
if x < 0.84375 { // |x| < 0.84375
|
||||
mut temp := 0.0
|
||||
if x < small { // |x| < 2**-28
|
||||
if x < very_tiny {
|
||||
temp = 0.125 * (8.0 * x + math.efx8 * x) // avoid underflow
|
||||
} else {
|
||||
temp = x + math.efx * x
|
||||
}
|
||||
} else {
|
||||
z := x * x
|
||||
r := math.pp0 + z * (math.pp1 + z * (math.pp2 + z * (math.pp3 + z * math.pp4)))
|
||||
s_ := 1.0 + z * (math.qq1 + z * (math.qq2 + z * (math.qq3 + z * (math.qq4 +
|
||||
z * math.qq5))))
|
||||
y := r / s_
|
||||
temp = x + x * y
|
||||
}
|
||||
if sign {
|
||||
return -temp
|
||||
}
|
||||
return temp
|
||||
}
|
||||
if x < 1.25 { // 0.84375 <= |x| < 1.25
|
||||
s_ := x - 1
|
||||
p := math.pa0 + s_ * (math.pa1 + s_ * (math.pa2 + s_ * (math.pa3 + s_ * (math.pa4 +
|
||||
s_ * (math.pa5 + s_ * math.pa6)))))
|
||||
q := 1.0 + s_ * (math.qa1 + s_ * (math.qa2 + s_ * (math.qa3 + s_ * (math.qa4 +
|
||||
s_ * (math.qa5 + s_ * math.qa6)))))
|
||||
if sign {
|
||||
return -math.erx - p / q
|
||||
}
|
||||
return math.erx + p / q
|
||||
}
|
||||
if x >= 6 { // inf > |x| >= 6
|
||||
if sign {
|
||||
return -1
|
||||
}
|
||||
return 1.0
|
||||
}
|
||||
s_ := 1.0 / (x * x)
|
||||
mut r := 0.0
|
||||
mut s := 0.0
|
||||
if x < 1.0 / 0.35 { // |x| < 1 / 0.35 ~ 2.857143
|
||||
r = math.ra0 + s_ * (math.ra1 + s_ * (math.ra2 + s_ * (math.ra3 + s_ * (math.ra4 +
|
||||
s_ * (math.ra5 + s_ * (math.ra6 + s_ * math.ra7))))))
|
||||
s = 1.0 + s_ * (math.sa1 + s_ * (math.sa2 + s_ * (math.sa3 + s_ * (math.sa4 +
|
||||
s_ * (math.sa5 + s_ * (math.sa6 + s_ * (math.sa7 + s_ * math.sa8)))))))
|
||||
} else { // |x| >= 1 / 0.35 ~ 2.857143
|
||||
r = math.rb0 + s_ * (math.rb1 + s_ * (math.rb2 + s_ * (math.rb3 + s_ * (math.rb4 +
|
||||
s_ * (math.rb5 + s_ * math.rb6)))))
|
||||
s = 1.0 + s_ * (math.sb1 + s_ * (math.sb2 + s_ * (math.sb3 + s_ * (math.sb4 +
|
||||
s_ * (math.sb5 + s_ * (math.sb6 + s_ * math.sb7))))))
|
||||
}
|
||||
z := f64_from_bits(f64_bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
|
||||
r_ := exp(-z * z - 0.5625) * exp((z - x) * (z + x) + r / s)
|
||||
if sign {
|
||||
return r_ / x - 1.0
|
||||
}
|
||||
return 1.0 - r_ / x
|
||||
}
|
||||
|
||||
// `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 {
|
||||
// erfc returns the complementary error function of x.
|
||||
//
|
||||
// special cases are:
|
||||
// erfc(+inf) = 0
|
||||
// erfc(-inf) = 2
|
||||
// erfc(nan) = nan
|
||||
pub fn erfc(a f64) f64 {
|
||||
mut x := a
|
||||
tiny := 1.0 / f64(u64(1) << 56) // 2**-56
|
||||
// special cases
|
||||
if is_nan(x) {
|
||||
return nan()
|
||||
} else if is_inf(x, 1) {
|
||||
}
|
||||
if is_inf(x, 1) {
|
||||
return 0.0
|
||||
} else if is_inf(x, -1) {
|
||||
}
|
||||
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)
|
||||
mut sign := false
|
||||
if x < 0 {
|
||||
x = -x
|
||||
sign = true
|
||||
}
|
||||
if x < 0.84375 { // |x| < 0.84375
|
||||
mut temp := 0.0
|
||||
if x < tiny { // |x| < 2**-56
|
||||
temp = x
|
||||
} else {
|
||||
z := x * x
|
||||
r := math.pp0 + z * (math.pp1 + z * (math.pp2 + z * (math.pp3 + z * math.pp4)))
|
||||
s_ := 1.0 + z * (math.qq1 + z * (math.qq2 + z * (math.qq3 + z * (math.qq4 +
|
||||
z * math.qq5))))
|
||||
y := r / s_
|
||||
if x < 0.25 { // |x| < 1.0/4
|
||||
temp = x + x * y
|
||||
} else {
|
||||
temp = 0.5 + (x * y + (x - 0.5))
|
||||
}
|
||||
}
|
||||
if sign {
|
||||
return 1.0 + temp
|
||||
}
|
||||
return 1.0 - temp
|
||||
}
|
||||
if x < 1.25 { // 0.84375 <= |x| < 1.25
|
||||
s_ := x - 1
|
||||
p := math.pa0 + s_ * (math.pa1 + s_ * (math.pa2 + s_ * (math.pa3 + s_ * (math.pa4 +
|
||||
s_ * (math.pa5 + s_ * math.pa6)))))
|
||||
q := 1.0 + s_ * (math.qa1 + s_ * (math.qa2 + s_ * (math.qa3 + s_ * (math.qa4 +
|
||||
s_ * (math.qa5 + s_ * math.qa6)))))
|
||||
if sign {
|
||||
return 1.0 + math.erx + p / q
|
||||
}
|
||||
return 1.0 - math.erx - p / q
|
||||
}
|
||||
if x < 28 { // |x| < 28
|
||||
s_ := 1.0 / (x * x)
|
||||
mut r := 0.0
|
||||
mut s := 0.0
|
||||
if x < 1.0 / 0.35 { // |x| < 1 / 0.35 ~ 2.857143
|
||||
r = math.ra0 + s_ * (math.ra1 + s_ * (math.ra2 + s_ * (math.ra3 + s_ * (math.ra4 +
|
||||
s_ * (math.ra5 + s_ * (math.ra6 + s_ * math.ra7))))))
|
||||
s = 1.0 + s_ * (math.sa1 + s_ * (math.sa2 + s_ * (math.sa3 + s_ * (math.sa4 +
|
||||
s_ * (math.sa5 + s_ * (math.sa6 + s_ * (math.sa7 + s_ * math.sa8)))))))
|
||||
} else { // |x| >= 1 / 0.35 ~ 2.857143
|
||||
if sign && x > 6 {
|
||||
return 2.0 // x < -6
|
||||
}
|
||||
r = math.rb0 + s_ * (math.rb1 + s_ * (math.rb2 + s_ * (math.rb3 + s_ * (math.rb4 +
|
||||
s_ * (math.rb5 + s_ * math.rb6)))))
|
||||
s = 1.0 + s_ * (math.sb1 + s_ * (math.sb2 + s_ * (math.sb3 + s_ * (math.sb4 +
|
||||
s_ * (math.sb5 + s_ * (math.sb6 + s_ * math.sb7))))))
|
||||
}
|
||||
z := f64_from_bits(f64_bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precision x
|
||||
r_ := exp(-z * z - 0.5625) * exp((z - x) * (z + x) + r / s)
|
||||
if sign {
|
||||
return 2.0 - r_ / x
|
||||
}
|
||||
return r_ / x
|
||||
}
|
||||
if sign {
|
||||
return 2.0
|
||||
}
|
||||
return 0.0
|
||||
}
|
||||
|
||||
Reference in New Issue
Block a user