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// Copyright 2017 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
/*
Inverse of the floating-point error function.
*/
// This implementation is based on the rational approximation
// of percentage points of normal distribution available from
// https://www.jstor.org/stable/2347330.
const (
// Coefficients for approximation to erf in |x| <= 0.85
a0 = 1.1975323115670912564578e0
a1 = 4.7072688112383978012285e1
a2 = 6.9706266534389598238465e2
a3 = 4.8548868893843886794648e3
a4 = 1.6235862515167575384252e4
a5 = 2.3782041382114385731252e4
a6 = 1.1819493347062294404278e4
a7 = 8.8709406962545514830200e2
b0 = 1.0000000000000000000e0
b1 = 4.2313330701600911252e1
b2 = 6.8718700749205790830e2
b3 = 5.3941960214247511077e3
b4 = 2.1213794301586595867e4
b5 = 3.9307895800092710610e4
b6 = 2.8729085735721942674e4
b7 = 5.2264952788528545610e3
// Coefficients for approximation to erf in 0.85 < |x| <= 1-2*exp(-25)
c0 = 1.42343711074968357734e0
c1 = 4.63033784615654529590e0
c2 = 5.76949722146069140550e0
c3 = 3.64784832476320460504e0
c4 = 1.27045825245236838258e0
c5 = 2.41780725177450611770e-1
c6 = 2.27238449892691845833e-2
c7 = 7.74545014278341407640e-4
d0 = 1.4142135623730950488016887e0
d1 = 2.9036514445419946173133295e0
d2 = 2.3707661626024532365971225e0
d3 = 9.7547832001787427186894837e-1
d4 = 2.0945065210512749128288442e-1
d5 = 2.1494160384252876777097297e-2
d6 = 7.7441459065157709165577218e-4
d7 = 1.4859850019840355905497876e-9
// Coefficients for approximation to erf in 1-2*exp(-25) < |x| < 1
e0 = 6.65790464350110377720e0
e1 = 5.46378491116411436990e0
e2 = 1.78482653991729133580e0
e3 = 2.96560571828504891230e-1
e4 = 2.65321895265761230930e-2
e5 = 1.24266094738807843860e-3
e6 = 2.71155556874348757815e-5
e7 = 2.01033439929228813265e-7
f0 = 1.414213562373095048801689e0
f1 = 8.482908416595164588112026e-1
f2 = 1.936480946950659106176712e-1
f3 = 2.103693768272068968719679e-2
f4 = 1.112800997078859844711555e-3
f5 = 2.611088405080593625138020e-5
f6 = 2.010321207683943062279931e-7
f7 = 2.891024605872965461538222e-15
)
// Erfinv returns the inverse error function of x.
//
// Special cases are:
// Erfinv(1) = +Inf
// Erfinv(-1) = -Inf
// Erfinv(x) = NaN if x < -1 or x > 1
// Erfinv(NaN) = NaN
func Erfinv(x float64) float64 {
// special cases
if IsNaN(x) || x <= -1 || x >= 1 {
if x == -1 || x == 1 {
return Inf(int(x))
}
return NaN()
}
sign := false
if x < 0 {
x = -x
sign = true
}
var ans float64
if x <= 0.85 { // |x| <= 0.85
r := 0.180625 - 0.25*x*x
z1 := ((((((a7*r+a6)*r+a5)*r+a4)*r+a3)*r+a2)*r+a1)*r + a0
z2 := ((((((b7*r+b6)*r+b5)*r+b4)*r+b3)*r+b2)*r+b1)*r + b0
ans = (x * z1) / z2
} else {
var z1, z2 float64
r := Sqrt(Ln2 - Log(1.0-x))
if r <= 5.0 {
r -= 1.6
z1 = ((((((c7*r+c6)*r+c5)*r+c4)*r+c3)*r+c2)*r+c1)*r + c0
z2 = ((((((d7*r+d6)*r+d5)*r+d4)*r+d3)*r+d2)*r+d1)*r + d0
} else {
r -= 5.0
z1 = ((((((e7*r+e6)*r+e5)*r+e4)*r+e3)*r+e2)*r+e1)*r + e0
z2 = ((((((f7*r+f6)*r+f5)*r+f4)*r+f3)*r+f2)*r+f1)*r + f0
}
ans = z1 / z2
}
if sign {
return -ans
}
return ans
}
// Erfcinv returns the inverse of Erfc(x).
//
// Special cases are:
// Erfcinv(0) = +Inf
// Erfcinv(2) = -Inf
// Erfcinv(x) = NaN if x < 0 or x > 2
// Erfcinv(NaN) = NaN
func Erfcinv(x float64) float64 {
return Erfinv(1 - x)
}
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