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jeremybenn |
// Copyright 2010 The Go Authors. All rights reserved.
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// Use of this source code is governed by a BSD-style
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// license that can be found in the LICENSE file.
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package math
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/*
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Floating-point error function and complementary error function.
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*/
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// The original C code and the long comment below are
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// from FreeBSD's /usr/src/lib/msun/src/s_erf.c and
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// came with this notice. The go code is a simplified
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// version of the original C.
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//
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// ====================================================
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// Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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//
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// Developed at SunPro, a Sun Microsystems, Inc. business.
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// Permission to use, copy, modify, and distribute this
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// software is freely granted, provided that this notice
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// is preserved.
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// ====================================================
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//
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//
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// double erf(double x)
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// double erfc(double x)
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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
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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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const (
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erx = 8.45062911510467529297e-01 // 0x3FEB0AC160000000
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// Coefficients for approximation to erf in [0, 0.84375]
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efx = 1.28379167095512586316e-01 // 0x3FC06EBA8214DB69
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efx8 = 1.02703333676410069053e+00 // 0x3FF06EBA8214DB69
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pp0 = 1.28379167095512558561e-01 // 0x3FC06EBA8214DB68
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pp1 = -3.25042107247001499370e-01 // 0xBFD4CD7D691CB913
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pp2 = -2.84817495755985104766e-02 // 0xBF9D2A51DBD7194F
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pp3 = -5.77027029648944159157e-03 // 0xBF77A291236668E4
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pp4 = -2.37630166566501626084e-05 // 0xBEF8EAD6120016AC
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qq1 = 3.97917223959155352819e-01 // 0x3FD97779CDDADC09
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qq2 = 6.50222499887672944485e-02 // 0x3FB0A54C5536CEBA
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qq3 = 5.08130628187576562776e-03 // 0x3F74D022C4D36B0F
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qq4 = 1.32494738004321644526e-04 // 0x3F215DC9221C1A10
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qq5 = -3.96022827877536812320e-06 // 0xBED09C4342A26120
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// Coefficients for approximation to erf in [0.84375, 1.25]
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pa0 = -2.36211856075265944077e-03 // 0xBF6359B8BEF77538
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pa1 = 4.14856118683748331666e-01 // 0x3FDA8D00AD92B34D
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pa2 = -3.72207876035701323847e-01 // 0xBFD7D240FBB8C3F1
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pa3 = 3.18346619901161753674e-01 // 0x3FD45FCA805120E4
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pa4 = -1.10894694282396677476e-01 // 0xBFBC63983D3E28EC
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pa5 = 3.54783043256182359371e-02 // 0x3FA22A36599795EB
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pa6 = -2.16637559486879084300e-03 // 0xBF61BF380A96073F
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qa1 = 1.06420880400844228286e-01 // 0x3FBB3E6618EEE323
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qa2 = 5.40397917702171048937e-01 // 0x3FE14AF092EB6F33
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qa3 = 7.18286544141962662868e-02 // 0x3FB2635CD99FE9A7
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qa4 = 1.26171219808761642112e-01 // 0x3FC02660E763351F
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qa5 = 1.36370839120290507362e-02 // 0x3F8BEDC26B51DD1C
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qa6 = 1.19844998467991074170e-02 // 0x3F888B545735151D
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// Coefficients for approximation to erfc in [1.25, 1/0.35]
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ra0 = -9.86494403484714822705e-03 // 0xBF843412600D6435
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ra1 = -6.93858572707181764372e-01 // 0xBFE63416E4BA7360
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ra2 = -1.05586262253232909814e+01 // 0xC0251E0441B0E726
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ra3 = -6.23753324503260060396e+01 // 0xC04F300AE4CBA38D
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ra4 = -1.62396669462573470355e+02 // 0xC0644CB184282266
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ra5 = -1.84605092906711035994e+02 // 0xC067135CEBCCABB2
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ra6 = -8.12874355063065934246e+01 // 0xC054526557E4D2F2
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ra7 = -9.81432934416914548592e+00 // 0xC023A0EFC69AC25C
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sa1 = 1.96512716674392571292e+01 // 0x4033A6B9BD707687
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sa2 = 1.37657754143519042600e+02 // 0x4061350C526AE721
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sa3 = 4.34565877475229228821e+02 // 0x407B290DD58A1A71
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sa4 = 6.45387271733267880336e+02 // 0x40842B1921EC2868
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sa5 = 4.29008140027567833386e+02 // 0x407AD02157700314
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sa6 = 1.08635005541779435134e+02 // 0x405B28A3EE48AE2C
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sa7 = 6.57024977031928170135e+00 // 0x401A47EF8E484A93
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sa8 = -6.04244152148580987438e-02 // 0xBFAEEFF2EE749A62
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// Coefficients for approximation to erfc in [1/.35, 28]
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rb0 = -9.86494292470009928597e-03 // 0xBF84341239E86F4A
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rb1 = -7.99283237680523006574e-01 // 0xBFE993BA70C285DE
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rb2 = -1.77579549177547519889e+01 // 0xC031C209555F995A
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rb3 = -1.60636384855821916062e+02 // 0xC064145D43C5ED98
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rb4 = -6.37566443368389627722e+02 // 0xC083EC881375F228
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rb5 = -1.02509513161107724954e+03 // 0xC09004616A2E5992
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rb6 = -4.83519191608651397019e+02 // 0xC07E384E9BDC383F
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sb1 = 3.03380607434824582924e+01 // 0x403E568B261D5190
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sb2 = 3.25792512996573918826e+02 // 0x40745CAE221B9F0A
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sb3 = 1.53672958608443695994e+03 // 0x409802EB189D5118
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sb4 = 3.19985821950859553908e+03 // 0x40A8FFB7688C246A
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sb5 = 2.55305040643316442583e+03 // 0x40A3F219CEDF3BE6
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sb6 = 4.74528541206955367215e+02 // 0x407DA874E79FE763
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sb7 = -2.24409524465858183362e+01 // 0xC03670E242712D62
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)
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// Erf(x) returns the error function of x.
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//
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// Special cases are:
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// Erf(+Inf) = 1
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// Erf(-Inf) = -1
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// Erf(NaN) = NaN
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func Erf(x float64) float64 {
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const (
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VeryTiny = 2.848094538889218e-306 // 0x0080000000000000
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Small = 1.0 / (1 << 28) // 2**-28
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)
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// special cases
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switch {
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case IsNaN(x):
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return NaN()
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case IsInf(x, 1):
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return 1
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case IsInf(x, -1):
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return -1
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}
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sign := false
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if x < 0 {
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x = -x
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sign = true
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}
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if x < 0.84375 { // |x| < 0.84375
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var temp float64
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if x < Small { // |x| < 2**-28
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if x < VeryTiny {
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temp = 0.125 * (8.0*x + efx8*x) // avoid underflow
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} else {
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temp = x + efx*x
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}
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} else {
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z := x * x
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r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
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s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
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y := r / s
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temp = x + x*y
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}
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if sign {
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return -temp
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}
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return temp
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}
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if x < 1.25 { // 0.84375 <= |x| < 1.25
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s := x - 1
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P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
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Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
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if sign {
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return -erx - P/Q
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}
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return erx + P/Q
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}
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if x >= 6 { // inf > |x| >= 6
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if sign {
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return -1
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}
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return 1
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}
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s := 1 / (x * x)
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var R, S float64
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if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
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R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
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S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
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} else { // |x| >= 1 / 0.35 ~ 2.857143
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R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
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S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
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}
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z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precison x
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r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S)
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if sign {
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return r/x - 1
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}
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return 1 - r/x
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}
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// Erfc(x) returns the complementary error function of x.
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//
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// Special cases are:
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// Erfc(+Inf) = 0
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// Erfc(-Inf) = 2
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// Erfc(NaN) = NaN
|
| 265 |
|
|
func Erfc(x float64) float64 {
|
| 266 |
|
|
const Tiny = 1.0 / (1 << 56) // 2**-56
|
| 267 |
|
|
// special cases
|
| 268 |
|
|
switch {
|
| 269 |
|
|
case IsNaN(x):
|
| 270 |
|
|
return NaN()
|
| 271 |
|
|
case IsInf(x, 1):
|
| 272 |
|
|
return 0
|
| 273 |
|
|
case IsInf(x, -1):
|
| 274 |
|
|
return 2
|
| 275 |
|
|
}
|
| 276 |
|
|
sign := false
|
| 277 |
|
|
if x < 0 {
|
| 278 |
|
|
x = -x
|
| 279 |
|
|
sign = true
|
| 280 |
|
|
}
|
| 281 |
|
|
if x < 0.84375 { // |x| < 0.84375
|
| 282 |
|
|
var temp float64
|
| 283 |
|
|
if x < Tiny { // |x| < 2**-56
|
| 284 |
|
|
temp = x
|
| 285 |
|
|
} else {
|
| 286 |
|
|
z := x * x
|
| 287 |
|
|
r := pp0 + z*(pp1+z*(pp2+z*(pp3+z*pp4)))
|
| 288 |
|
|
s := 1 + z*(qq1+z*(qq2+z*(qq3+z*(qq4+z*qq5))))
|
| 289 |
|
|
y := r / s
|
| 290 |
|
|
if x < 0.25 { // |x| < 1/4
|
| 291 |
|
|
temp = x + x*y
|
| 292 |
|
|
} else {
|
| 293 |
|
|
temp = 0.5 + (x*y + (x - 0.5))
|
| 294 |
|
|
}
|
| 295 |
|
|
}
|
| 296 |
|
|
if sign {
|
| 297 |
|
|
return 1 + temp
|
| 298 |
|
|
}
|
| 299 |
|
|
return 1 - temp
|
| 300 |
|
|
}
|
| 301 |
|
|
if x < 1.25 { // 0.84375 <= |x| < 1.25
|
| 302 |
|
|
s := x - 1
|
| 303 |
|
|
P := pa0 + s*(pa1+s*(pa2+s*(pa3+s*(pa4+s*(pa5+s*pa6)))))
|
| 304 |
|
|
Q := 1 + s*(qa1+s*(qa2+s*(qa3+s*(qa4+s*(qa5+s*qa6)))))
|
| 305 |
|
|
if sign {
|
| 306 |
|
|
return 1 + erx + P/Q
|
| 307 |
|
|
}
|
| 308 |
|
|
return 1 - erx - P/Q
|
| 309 |
|
|
|
| 310 |
|
|
}
|
| 311 |
|
|
if x < 28 { // |x| < 28
|
| 312 |
|
|
s := 1 / (x * x)
|
| 313 |
|
|
var R, S float64
|
| 314 |
|
|
if x < 1/0.35 { // |x| < 1 / 0.35 ~ 2.857143
|
| 315 |
|
|
R = ra0 + s*(ra1+s*(ra2+s*(ra3+s*(ra4+s*(ra5+s*(ra6+s*ra7))))))
|
| 316 |
|
|
S = 1 + s*(sa1+s*(sa2+s*(sa3+s*(sa4+s*(sa5+s*(sa6+s*(sa7+s*sa8)))))))
|
| 317 |
|
|
} else { // |x| >= 1 / 0.35 ~ 2.857143
|
| 318 |
|
|
if sign && x > 6 {
|
| 319 |
|
|
return 2 // x < -6
|
| 320 |
|
|
}
|
| 321 |
|
|
R = rb0 + s*(rb1+s*(rb2+s*(rb3+s*(rb4+s*(rb5+s*rb6)))))
|
| 322 |
|
|
S = 1 + s*(sb1+s*(sb2+s*(sb3+s*(sb4+s*(sb5+s*(sb6+s*sb7))))))
|
| 323 |
|
|
}
|
| 324 |
|
|
z := Float64frombits(Float64bits(x) & 0xffffffff00000000) // pseudo-single (20-bit) precison x
|
| 325 |
|
|
r := Exp(-z*z-0.5625) * Exp((z-x)*(z+x)+R/S)
|
| 326 |
|
|
if sign {
|
| 327 |
|
|
return 2 - r/x
|
| 328 |
|
|
}
|
| 329 |
|
|
return r / x
|
| 330 |
|
|
}
|
| 331 |
|
|
if sign {
|
| 332 |
|
|
return 2
|
| 333 |
|
|
}
|
| 334 |
|
|
return 0
|
| 335 |
|
|
}
|
