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// Copyright 2010 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
// The original C code, the long comment, and the constants
// below are from http://netlib.sandia.gov/cephes/cprob/gamma.c.
// The go code is a simplified version of the original C.
//
// tgamma.c
//
// Gamma function
//
// SYNOPSIS:
//
// double x, y, tgamma();
// extern int signgam;
//
// y = tgamma( x );
//
// DESCRIPTION:
//
// Returns gamma function of the argument. The result is
// correctly signed, and the sign (+1 or -1) is also
// returned in a global (extern) variable named signgam.
// This variable is also filled in by the logarithmic gamma
// function lgamma().
//
// Arguments |x| <= 34 are reduced by recurrence and the function
// approximated by a rational function of degree 6/7 in the
// interval (2,3). Large arguments are handled by Stirling's
// formula. Large negative arguments are made positive using
// a reflection formula.
//
// ACCURACY:
//
// Relative error:
// arithmetic domain # trials peak rms
// DEC -34, 34 10000 1.3e-16 2.5e-17
// IEEE -170,-33 20000 2.3e-15 3.3e-16
// IEEE -33, 33 20000 9.4e-16 2.2e-16
// IEEE 33, 171.6 20000 2.3e-15 3.2e-16
//
// Error for arguments outside the test range will be larger
// owing to error amplification by the exponential function.
//
// Cephes Math Library Release 2.8: June, 2000
// Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier
//
// The readme file at http://netlib.sandia.gov/cephes/ says:
// Some software in this archive may be from the book _Methods and
// Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster
// International, 1989) or from the Cephes Mathematical Library, a
// commercial product. In either event, it is copyrighted by the author.
// What you see here may be used freely but it comes with no support or
// guarantee.
//
// The two known misprints in the book are repaired here in the
// source listings for the gamma function and the incomplete beta
// integral.
//
// Stephen L. Moshier
// moshier@na-net.ornl.gov
var _P = []float64{
1.60119522476751861407e-04,
1.19135147006586384913e-03,
1.04213797561761569935e-02,
4.76367800457137231464e-02,
2.07448227648435975150e-01,
4.94214826801497100753e-01,
9.99999999999999996796e-01,
}
var _Q = []float64{
-2.31581873324120129819e-05,
5.39605580493303397842e-04,
-4.45641913851797240494e-03,
1.18139785222060435552e-02,
3.58236398605498653373e-02,
-2.34591795718243348568e-01,
7.14304917030273074085e-02,
1.00000000000000000320e+00,
}
var _S = []float64{
7.87311395793093628397e-04,
-2.29549961613378126380e-04,
-2.68132617805781232825e-03,
3.47222221605458667310e-03,
8.33333333333482257126e-02,
}
// Gamma function computed by Stirling's formula.
// The polynomial is valid for 33 <= x <= 172.
func stirling(x float64) float64 {
const (
SqrtTwoPi = 2.506628274631000502417
MaxStirling = 143.01608
)
w := 1 / x
w = 1 + w*((((_S[0]*w+_S[1])*w+_S[2])*w+_S[3])*w+_S[4])
y := Exp(x)
if x > MaxStirling { // avoid Pow() overflow
v := Pow(x, 0.5*x-0.25)
y = v * (v / y)
} else {
y = Pow(x, x-0.5) / y
}
y = SqrtTwoPi * y * w
return y
}
// Gamma(x) returns the Gamma function of x.
//
// Special cases are:
// Gamma(Inf) = Inf
// Gamma(-Inf) = -Inf
// Gamma(NaN) = NaN
// Large values overflow to +Inf.
// Negative integer values equal ┬▒Inf.
func Gamma(x float64) float64 {
const Euler = 0.57721566490153286060651209008240243104215933593992 // A001620
// special cases
switch {
case x < -MaxFloat64 || x != x: // IsInf(x, -1) || IsNaN(x):
return x
case x < -170.5674972726612 || x > 171.61447887182298:
return Inf(1)
}
q := Fabs(x)
p := Floor(q)
if q > 33 {
if x >= 0 {
return stirling(x)
}
signgam := 1
if ip := int(p); ip&1 == 0 {
signgam = -1
}
z := q - p
if z > 0.5 {
p = p + 1
z = q - p
}
z = q * Sin(Pi*z)
if z == 0 {
return Inf(signgam)
}
z = Pi / (Fabs(z) * stirling(q))
return float64(signgam) * z
}
// Reduce argument
z := 1.0
for x >= 3 {
x = x - 1
z = z * x
}
for x < 0 {
if x > -1e-09 {
goto small
}
z = z / x
x = x + 1
}
for x < 2 {
if x < 1e-09 {
goto small
}
z = z / x
x = x + 1
}
if x == 2 {
return z
}
x = x - 2
p = (((((x*_P[0]+_P[1])*x+_P[2])*x+_P[3])*x+_P[4])*x+_P[5])*x + _P[6]
q = ((((((x*_Q[0]+_Q[1])*x+_Q[2])*x+_Q[3])*x+_Q[4])*x+_Q[5])*x+_Q[6])*x + _Q[7]
return z * p / q
small:
if x == 0 {
return Inf(1)
}
return z / ((1 + Euler*x) * x)
}