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Diffstat (limited to 'src/pkg/math/erf.go')
| -rw-r--r-- | src/pkg/math/erf.go | 335 |
1 files changed, 0 insertions, 335 deletions
diff --git a/src/pkg/math/erf.go b/src/pkg/math/erf.go deleted file mode 100644 index 4cd80f80c..000000000 --- a/src/pkg/math/erf.go +++ /dev/null @@ -1,335 +0,0 @@ -// 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 { - 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 { - 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 -} |