341 lines
12 KiB
Go
341 lines
12 KiB
Go
|
// 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(x) 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
|
||
|
|
// TODO(rsc): Remove manual inlining of IsNaN, IsInf
|
||
|
|
// when compiler does it for us
|
||
|
|
switch {
|
||
|
|
case x != x: // IsNaN(x):
|
||
|
|
return NaN()
|
||
|
|
case x > MaxFloat64: // IsInf(x, 1):
|
||
|
|
return 1
|
||
|
|
case x < -MaxFloat64: // 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) precison 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(x) 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
|
||
|
|
// TODO(rsc): Remove manual inlining of IsNaN, IsInf
|
||
|
|
// when compiler does it for us
|
||
|
|
switch {
|
||
|
|
case x != x: // IsNaN(x):
|
||
|
|
return NaN()
|
||
|
|
case x > MaxFloat64: // IsInf(x, 1):
|
||
|
|
return 0
|
||
|
|
case x < -MaxFloat64: // 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) precison 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
|
||
|
|
}
|