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bd21366 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 | // 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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