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Diffstat (limited to 'src/math/erf.go')
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diff --git a/src/math/erf.go b/src/math/erf.go new file mode 100644 index 0000000..4d6fe47 --- /dev/null +++ b/src/math/erf.go @@ -0,0 +1,349 @@ +// Copyright 2010 The Go Authors. All rights reserved. +// Use of this source code is governed by a BSD-style +// license that can be found in the LICENSE file. + +package math + +/* + Floating-point error function and complementary error function. +*/ + +// The original C code and the long comment below are +// from FreeBSD's /usr/src/lib/msun/src/s_erf.c and +// came with this notice. The go code is a simplified +// version of the original C. +// +// ==================================================== +// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. +// +// Developed at SunPro, a Sun Microsystems, Inc. business. +// Permission to use, copy, modify, and distribute this +// software is freely granted, provided that this notice +// is preserved. +// ==================================================== +// +// +// double erf(double x) +// double erfc(double x) +// x +// 2 |\ +// erf(x) = --------- | exp(-t*t)dt +// sqrt(pi) \| +// 0 +// +// erfc(x) = 1-erf(x) +// Note that +// erf(-x) = -erf(x) +// erfc(-x) = 2 - erfc(x) +// +// Method: +// 1. For |x| in [0, 0.84375] +// erf(x) = x + x*R(x**2) +// erfc(x) = 1 - erf(x) if x in [-.84375,0.25] +// = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] +// where R = P/Q where P is an odd poly of degree 8 and +// Q is an odd poly of degree 10. +// -57.90 +// | R - (erf(x)-x)/x | <= 2 +// +// +// Remark. The formula is derived by noting +// erf(x) = (2/sqrt(pi))*(x - x**3/3 + x**5/10 - x**7/42 + ....) +// and that +// 2/sqrt(pi) = 1.128379167095512573896158903121545171688 +// is close to one. The interval is chosen because the fix +// point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is +// near 0.6174), and by some experiment, 0.84375 is chosen to +// guarantee the error is less than one ulp for erf. +// +// 2. For |x| in [0.84375,1.25], let s = |x| - 1, and +// c = 0.84506291151 rounded to single (24 bits) +// erf(x) = sign(x) * (c + P1(s)/Q1(s)) +// erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 +// 1+(c+P1(s)/Q1(s)) if x < 0 +// |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 +// Remark: here we use the taylor series expansion at x=1. +// erf(1+s) = erf(1) + s*Poly(s) +// = 0.845.. + P1(s)/Q1(s) +// That is, we use rational approximation to approximate +// erf(1+s) - (c = (single)0.84506291151) +// Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] +// where +// P1(s) = degree 6 poly in s +// Q1(s) = degree 6 poly in s +// +// 3. For x in [1.25,1/0.35(~2.857143)], +// erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) +// erf(x) = 1 - erfc(x) +// where +// R1(z) = degree 7 poly in z, (z=1/x**2) +// S1(z) = degree 8 poly in z +// +// 4. For x in [1/0.35,28] +// erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 +// = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 +// = 2.0 - tiny (if x <= -6) +// erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else +// erf(x) = sign(x)*(1.0 - tiny) +// where +// R2(z) = degree 6 poly in z, (z=1/x**2) +// S2(z) = degree 7 poly in z +// +// Note1: +// To compute exp(-x*x-0.5625+R/S), let s be a single +// precision number and s := x; then +// -x*x = -s*s + (s-x)*(s+x) +// exp(-x*x-0.5626+R/S) = +// exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); +// Note2: +// Here 4 and 5 make use of the asymptotic series +// exp(-x*x) +// erfc(x) ~ ---------- * ( 1 + Poly(1/x**2) ) +// x*sqrt(pi) +// We use rational approximation to approximate +// g(s)=f(1/x**2) = log(erfc(x)*x) - x*x + 0.5625 +// Here is the error bound for R1/S1 and R2/S2 +// |R1/S1 - f(x)| < 2**(-62.57) +// |R2/S2 - f(x)| < 2**(-61.52) +// +// 5. For inf > x >= 28 +// erf(x) = sign(x) *(1 - tiny) (raise inexact) +// erfc(x) = tiny*tiny (raise underflow) if x > 0 +// = 2 - tiny if x<0 +// +// 7. Special case: +// erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, +// erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, +// erfc/erf(NaN) is NaN + +const ( + 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 +) + +// Erf returns the error function of x. +// +// Special cases are: +// Erf(+Inf) = 1 +// Erf(-Inf) = -1 +// Erf(NaN) = NaN +func Erf(x float64) float64 { + if haveArchErf { + return archErf(x) + } + return erf(x) +} + +func erf(x float64) float64 { + const ( + VeryTiny = 2.848094538889218e-306 // 0x0080000000000000 + Small = 1.0 / (1 << 28) // 2**-28 + ) + // special cases + switch { + case IsNaN(x): + return NaN() + case IsInf(x, 1): + return 1 + case IsInf(x, -1): + return -1 + } + sign := false + if x < 0 { + x = -x + sign = true + } + if x < 0.84375 { // |x| < 0.84375 + var temp float64 + if x < Small { // |x| < 2**-28 + if x < VeryTiny { + temp = 0.125 * (8.0*x + efx8*x) // avoid underflow + } else { + temp = x + efx*x + } + } else { + z := x * x + r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4))) + s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*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 := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))) + Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))) + if sign { + return -erx - P/Q + } + return erx + P/Q + } + if x >= 6 { // inf > |x| >= 6 + if sign { + return -1 + } + return 1 + } + s := 1 / (x * x) + var R, S float64 + if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143 + R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))) + S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8))))))) + } else { // |x| >= 1 / 0.35 ~ 2.857143 + R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))) + S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))) + } + z := Float64frombits(Float64bits(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 + } + return 1 - r/x +} + +// Erfc returns the complementary error function of x. +// +// Special cases are: +// Erfc(+Inf) = 0 +// Erfc(-Inf) = 2 +// Erfc(NaN) = NaN +func Erfc(x float64) float64 { + if haveArchErfc { + return archErfc(x) + } + return erfc(x) +} + +func erfc(x float64) float64 { + const Tiny = 1.0 / (1 << 56) // 2**-56 + // special cases + switch { + case IsNaN(x): + return NaN() + case IsInf(x, 1): + return 0 + case IsInf(x, -1): + return 2 + } + sign := false + if x < 0 { + x = -x + sign = true + } + if x < 0.84375 { // |x| < 0.84375 + var temp float64 + if x < Tiny { // |x| < 2**-56 + temp = x + } else { + z := x * x + r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4))) + s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5)))) + y := r / s + if x < 0.25 { // |x| < 1/4 + temp = x + x*y + } else { + temp = 0.5 + (x*y + (x - 0.5)) + } + } + if sign { + return 1 + temp + } + return 1 - temp + } + if x < 1.25 { // 0.84375 <= |x| < 1.25 + s := x - 1 + P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6))))) + Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6))))) + if sign { + return 1 + erx + P/Q + } + return 1 - erx - P/Q + + } + if x < 28 { // |x| < 28 + s := 1 / (x * x) + var R, S float64 + if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143 + R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7)))))) + S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8))))))) + } else { // |x| >= 1 / 0.35 ~ 2.857143 + if sign && x > 6 { + return 2 // x < -6 + } + R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6))))) + S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7)))))) + } + z := Float64frombits(Float64bits(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 - r/x + } + return r / x + } + if sign { + return 2 + } + return 0 +} |