// 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
}