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// Derived from SciPy's special/cephes/unity.c
// https://github.com/scipy/scipy/blob/master/scipy/special/cephes/unity.c
// Made freely available by Stephen L. Moshier without support or guarantee.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
// Copyright ©1984, ©1996 by Stephen L. Moshier
// Portions Copyright ©2016 The Gonum Authors. All rights reserved.
package cephes
import "math"
// Relative error approximations for function arguments near unity.
// log1p(x) = log(1+x)
// expm1(x) = exp(x) - 1
// cosm1(x) = cos(x) - 1
// lgam1p(x) = lgam(1+x)
const (
invSqrt2 = 1 / math.Sqrt2
pi4 = math.Pi / 4
euler = 0.577215664901532860606512090082402431 // Euler constant
)
// Coefficients for
//
// log(1+x) = x - \frac{x^2}{2} + \frac{x^3 lP(x)}{lQ(x)}
//
// for
//
// \frac{1}{\sqrt{2}} <= x < \sqrt{2}
//
// Theoretical peak relative error = 2.32e-20
var lP = [...]float64{
4.5270000862445199635215e-5,
4.9854102823193375972212e-1,
6.5787325942061044846969e0,
2.9911919328553073277375e1,
6.0949667980987787057556e1,
5.7112963590585538103336e1,
2.0039553499201281259648e1,
}
var lQ = [...]float64{
1.5062909083469192043167e1,
8.3047565967967209469434e1,
2.2176239823732856465394e2,
3.0909872225312059774938e2,
2.1642788614495947685003e2,
6.0118660497603843919306e1,
}
// log1p computes
//
// log(1 + x)
func log1p(x float64) float64 {
z := 1 + x
if z < invSqrt2 || z > math.Sqrt2 {
return math.Log(z)
}
z = x * x
z = -0.5*z + x*(z*polevl(x, lP[:], 6)/p1evl(x, lQ[:], 6))
return x + z
}
// log1pmx computes
//
// log(1 + x) - x
func log1pmx(x float64) float64 {
if math.Abs(x) < 0.5 {
xfac := x
res := 0.0
var term float64
for n := 2; n < maxIter; n++ {
xfac *= -x
term = xfac / float64(n)
res += term
if math.Abs(term) < machEp*math.Abs(res) {
break
}
}
return res
}
return log1p(x) - x
}
// Coefficients for
//
// e^x = 1 + \frac{2x eP(x^2)}{eQ(x^2) - eP(x^2)}
//
// for
//
// -0.5 <= x <= 0.5
var eP = [...]float64{
1.2617719307481059087798e-4,
3.0299440770744196129956e-2,
9.9999999999999999991025e-1,
}
var eQ = [...]float64{
3.0019850513866445504159e-6,
2.5244834034968410419224e-3,
2.2726554820815502876593e-1,
2.0000000000000000000897e0,
}
// expm1 computes
//
// expm1(x) = e^x - 1
func expm1(x float64) float64 {
if math.IsInf(x, 0) {
if math.IsNaN(x) || x > 0 {
return x
}
return -1
}
if x < -0.5 || x > 0.5 {
return math.Exp(x) - 1
}
xx := x * x
r := x * polevl(xx, eP[:], 2)
r = r / (polevl(xx, eQ[:], 3) - r)
return r + r
}
var coscof = [...]float64{
4.7377507964246204691685e-14,
-1.1470284843425359765671e-11,
2.0876754287081521758361e-9,
-2.7557319214999787979814e-7,
2.4801587301570552304991e-5,
-1.3888888888888872993737e-3,
4.1666666666666666609054e-2,
}
// cosm1 computes
//
// cosm1(x) = cos(x) - 1
func cosm1(x float64) float64 {
if x < -pi4 || x > pi4 {
return math.Cos(x) - 1
}
xx := x * x
xx = -0.5*xx + xx*xx*polevl(xx, coscof[:], 6)
return xx
}
// lgam1pTayler computes
//
// lgam(x + 1)
//
// around x = 0 using its Taylor series.
func lgam1pTaylor(x float64) float64 {
if x == 0 {
return 0
}
res := -euler * x
xfac := -x
for n := 2; n < 42; n++ {
nf := float64(n)
xfac *= -x
coeff := Zeta(nf, 1) * xfac / nf
res += coeff
if math.Abs(coeff) < machEp*math.Abs(res) {
break
}
}
return res
}
// lgam1p computes
//
// lgam(x + 1)
func lgam1p(x float64) float64 {
if math.Abs(x) <= 0.5 {
return lgam1pTaylor(x)
} else if math.Abs(x-1) < 0.5 {
return math.Log(x) + lgam1pTaylor(x-1)
}
return lgam(x + 1)
}
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