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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 _gamP = [...]float64{
        1.60119522476751861407e-04,
        1.19135147006586384913e-03,
        1.04213797561761569935e-02,
        4.76367800457137231464e-02,
        2.07448227648435975150e-01,
        4.94214826801497100753e-01,
        9.99999999999999996796e-01,
}
var _gamQ = [...]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 _gamS = [...]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*((((_gamS[0]*w+_gamS[1])*w+_gamS[2])*w+_gamS[3])*w+_gamS[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(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 IsInf(x, -1) || IsNaN(x):
                return x
        case x < -170.5674972726612 || x > 171.61447887182298:
                return Inf(1)
        }
        q := Abs(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 / (Abs(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*_gamP[0]+_gamP[1])*x+_gamP[2])*x+_gamP[3])*x+_gamP[4])*x+_gamP[5])*x + _gamP[6]
        q = ((((((x*_gamQ[0]+_gamQ[1])*x+_gamQ[2])*x+_gamQ[3])*x+_gamQ[4])*x+_gamQ[5])*x+_gamQ[6])*x + _gamQ[7]
        return z * p / q

small:
        if x == 0 {
                return Inf(1)
        }
        return z / ((1 + Euler*x) * x)
}