1// Copyright 2017 The Go Authors. All rights reserved.
2// Use of this source code is governed by a BSD-style
3// license that can be found in the LICENSE file.
4
5package math
6
7/*
8	Inverse of the floating-point error function.
9*/
10
11// This implementation is based on the rational approximation
12// of percentage points of normal distribution available from
13// https://www.jstor.org/stable/2347330.
14
15const (
16	// Coefficients for approximation to erf in |x| <= 0.85
17	a0 = 1.1975323115670912564578e0
18	a1 = 4.7072688112383978012285e1
19	a2 = 6.9706266534389598238465e2
20	a3 = 4.8548868893843886794648e3
21	a4 = 1.6235862515167575384252e4
22	a5 = 2.3782041382114385731252e4
23	a6 = 1.1819493347062294404278e4
24	a7 = 8.8709406962545514830200e2
25	b0 = 1.0000000000000000000e0
26	b1 = 4.2313330701600911252e1
27	b2 = 6.8718700749205790830e2
28	b3 = 5.3941960214247511077e3
29	b4 = 2.1213794301586595867e4
30	b5 = 3.9307895800092710610e4
31	b6 = 2.8729085735721942674e4
32	b7 = 5.2264952788528545610e3
33	// Coefficients for approximation to erf in 0.85 < |x| <= 1-2*exp(-25)
34	c0 = 1.42343711074968357734e0
35	c1 = 4.63033784615654529590e0
36	c2 = 5.76949722146069140550e0
37	c3 = 3.64784832476320460504e0
38	c4 = 1.27045825245236838258e0
39	c5 = 2.41780725177450611770e-1
40	c6 = 2.27238449892691845833e-2
41	c7 = 7.74545014278341407640e-4
42	d0 = 1.4142135623730950488016887e0
43	d1 = 2.9036514445419946173133295e0
44	d2 = 2.3707661626024532365971225e0
45	d3 = 9.7547832001787427186894837e-1
46	d4 = 2.0945065210512749128288442e-1
47	d5 = 2.1494160384252876777097297e-2
48	d6 = 7.7441459065157709165577218e-4
49	d7 = 1.4859850019840355905497876e-9
50	// Coefficients for approximation to erf in 1-2*exp(-25) < |x| < 1
51	e0 = 6.65790464350110377720e0
52	e1 = 5.46378491116411436990e0
53	e2 = 1.78482653991729133580e0
54	e3 = 2.96560571828504891230e-1
55	e4 = 2.65321895265761230930e-2
56	e5 = 1.24266094738807843860e-3
57	e6 = 2.71155556874348757815e-5
58	e7 = 2.01033439929228813265e-7
59	f0 = 1.414213562373095048801689e0
60	f1 = 8.482908416595164588112026e-1
61	f2 = 1.936480946950659106176712e-1
62	f3 = 2.103693768272068968719679e-2
63	f4 = 1.112800997078859844711555e-3
64	f5 = 2.611088405080593625138020e-5
65	f6 = 2.010321207683943062279931e-7
66	f7 = 2.891024605872965461538222e-15
67)
68
69// Erfinv returns the inverse error function of x.
70//
71// Special cases are:
72//	Erfinv(1) = +Inf
73//	Erfinv(-1) = -Inf
74//	Erfinv(x) = NaN if x < -1 or x > 1
75//	Erfinv(NaN) = NaN
76func Erfinv(x float64) float64 {
77	// special cases
78	if IsNaN(x) || x <= -1 || x >= 1 {
79		if x == -1 || x == 1 {
80			return Inf(int(x))
81		}
82		return NaN()
83	}
84
85	sign := false
86	if x < 0 {
87		x = -x
88		sign = true
89	}
90
91	var ans float64
92	if x <= 0.85 { // |x| <= 0.85
93		r := 0.180625 - 0.25*x*x
94		z1 := ((((((a7*r+a6)*r+a5)*r+a4)*r+a3)*r+a2)*r+a1)*r + a0
95		z2 := ((((((b7*r+b6)*r+b5)*r+b4)*r+b3)*r+b2)*r+b1)*r + b0
96		ans = (x * z1) / z2
97	} else {
98		var z1, z2 float64
99		r := Sqrt(Ln2 - Log(1.0-x))
100		if r <= 5.0 {
101			r -= 1.6
102			z1 = ((((((c7*r+c6)*r+c5)*r+c4)*r+c3)*r+c2)*r+c1)*r + c0
103			z2 = ((((((d7*r+d6)*r+d5)*r+d4)*r+d3)*r+d2)*r+d1)*r + d0
104		} else {
105			r -= 5.0
106			z1 = ((((((e7*r+e6)*r+e5)*r+e4)*r+e3)*r+e2)*r+e1)*r + e0
107			z2 = ((((((f7*r+f6)*r+f5)*r+f4)*r+f3)*r+f2)*r+f1)*r + f0
108		}
109		ans = z1 / z2
110	}
111
112	if sign {
113		return -ans
114	}
115	return ans
116}
117
118// Erfcinv returns the inverse of Erfc(x).
119//
120// Special cases are:
121//	Erfcinv(0) = +Inf
122//	Erfcinv(2) = -Inf
123//	Erfcinv(x) = NaN if x < 0 or x > 2
124//	Erfcinv(NaN) = NaN
125func Erfcinv(x float64) float64 {
126	return Erfinv(1 - x)
127}
128