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// Copyright ©2016 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
/*
* Cephes Math Library Release 2.1: January, 1989
* Copyright 1984, 1987, 1989 by Stephen L. Moshier
* Direct inquiries to 30 Frost Street, Cambridge, MA 02140
*/
package cephes
import "math"
// TODO(btracey): There is currently an implementation of this functionality
// in gonum/stat/distuv. Find out which implementation is better, and rectify
// by having distuv call this, or moving this implementation into
// gonum/mathext/internal/gonum.
// math.Sqrt(2*pi)
const s2pi = 2.50662827463100050242e0
// approximation for 0 <= |y - 0.5| <= 3/8
var P0 = [5]float64{
-5.99633501014107895267e1,
9.80010754185999661536e1,
-5.66762857469070293439e1,
1.39312609387279679503e1,
-1.23916583867381258016e0,
}
var Q0 = [8]float64{
/* 1.00000000000000000000E0, */
1.95448858338141759834e0,
4.67627912898881538453e0,
8.63602421390890590575e1,
-2.25462687854119370527e2,
2.00260212380060660359e2,
-8.20372256168333339912e1,
1.59056225126211695515e1,
-1.18331621121330003142e0,
}
// Approximation for interval z = math.Sqrt(-2 log y ) between 2 and 8
// i.e., y between exp(-2) = .135 and exp(-32) = 1.27e-14.
var P1 = [9]float64{
4.05544892305962419923e0,
3.15251094599893866154e1,
5.71628192246421288162e1,
4.40805073893200834700e1,
1.46849561928858024014e1,
2.18663306850790267539e0,
-1.40256079171354495875e-1,
-3.50424626827848203418e-2,
-8.57456785154685413611e-4,
}
var Q1 = [8]float64{
/* 1.00000000000000000000E0, */
1.57799883256466749731e1,
4.53907635128879210584e1,
4.13172038254672030440e1,
1.50425385692907503408e1,
2.50464946208309415979e0,
-1.42182922854787788574e-1,
-3.80806407691578277194e-2,
-9.33259480895457427372e-4,
}
// Approximation for interval z = math.Sqrt(-2 log y ) between 8 and 64
// i.e., y between exp(-32) = 1.27e-14 and exp(-2048) = 3.67e-890.
var P2 = [9]float64{
3.23774891776946035970e0,
6.91522889068984211695e0,
3.93881025292474443415e0,
1.33303460815807542389e0,
2.01485389549179081538e-1,
1.23716634817820021358e-2,
3.01581553508235416007e-4,
2.65806974686737550832e-6,
6.23974539184983293730e-9,
}
var Q2 = [8]float64{
/* 1.00000000000000000000E0, */
6.02427039364742014255e0,
3.67983563856160859403e0,
1.37702099489081330271e0,
2.16236993594496635890e-1,
1.34204006088543189037e-2,
3.28014464682127739104e-4,
2.89247864745380683936e-6,
6.79019408009981274425e-9,
}
// Ndtri returns the argument, x, for which the area under the
// Gaussian probability density function (integrated from
// minus infinity to x) is equal to y.
func Ndtri(y0 float64) float64 {
// For small arguments 0 < y < exp(-2), the program computes
// z = math.Sqrt( -2.0 * math.Log(y) ); then the approximation is
// x = z - math.Log(z)/z - (1/z) P(1/z) / Q(1/z).
// There are two rational functions P/Q, one for 0 < y < exp(-32)
// and the other for y up to exp(-2). For larger arguments,
// w = y - 0.5, and x/math.Sqrt(2pi) = w + w**3 R(w**2)/S(w**2)).
var x, y, z, y2, x0, x1 float64
var code int
if y0 <= 0.0 {
if y0 < 0 {
panic(paramOutOfBounds)
}
return math.Inf(-1)
}
if y0 >= 1.0 {
if y0 > 1 {
panic(paramOutOfBounds)
}
return math.Inf(1)
}
code = 1
y = y0
if y > (1.0 - 0.13533528323661269189) { /* 0.135... = exp(-2) */
y = 1.0 - y
code = 0
}
if y > 0.13533528323661269189 {
y = y - 0.5
y2 = y * y
x = y + y*(y2*polevl(y2, P0[:], 4)/p1evl(y2, Q0[:], 8))
x = x * s2pi
return (x)
}
x = math.Sqrt(-2.0 * math.Log(y))
x0 = x - math.Log(x)/x
z = 1.0 / x
if x < 8.0 { /* y > exp(-32) = 1.2664165549e-14 */
x1 = z * polevl(z, P1[:], 8) / p1evl(z, Q1[:], 8)
} else {
x1 = z * polevl(z, P2[:], 8) / p1evl(z, Q2[:], 8)
}
x = x0 - x1
if code != 0 {
x = -x
}
return (x)
}
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