Source file src/pkg/math/gamma.go
1 // Copyright 2010 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 5 package math 6 7 // The original C code, the long comment, and the constants 8 // below are from http://netlib.sandia.gov/cephes/cprob/gamma.c. 9 // The go code is a simplified version of the original C. 10 // 11 // tgamma.c 12 // 13 // Gamma function 14 // 15 // SYNOPSIS: 16 // 17 // double x, y, tgamma(); 18 // extern int signgam; 19 // 20 // y = tgamma( x ); 21 // 22 // DESCRIPTION: 23 // 24 // Returns gamma function of the argument. The result is 25 // correctly signed, and the sign (+1 or -1) is also 26 // returned in a global (extern) variable named signgam. 27 // This variable is also filled in by the logarithmic gamma 28 // function lgamma(). 29 // 30 // Arguments |x| <= 34 are reduced by recurrence and the function 31 // approximated by a rational function of degree 6/7 in the 32 // interval (2,3). Large arguments are handled by Stirling's 33 // formula. Large negative arguments are made positive using 34 // a reflection formula. 35 // 36 // ACCURACY: 37 // 38 // Relative error: 39 // arithmetic domain # trials peak rms 40 // DEC -34, 34 10000 1.3e-16 2.5e-17 41 // IEEE -170,-33 20000 2.3e-15 3.3e-16 42 // IEEE -33, 33 20000 9.4e-16 2.2e-16 43 // IEEE 33, 171.6 20000 2.3e-15 3.2e-16 44 // 45 // Error for arguments outside the test range will be larger 46 // owing to error amplification by the exponential function. 47 // 48 // Cephes Math Library Release 2.8: June, 2000 49 // Copyright 1984, 1987, 1989, 1992, 2000 by Stephen L. Moshier 50 // 51 // The readme file at http://netlib.sandia.gov/cephes/ says: 52 // Some software in this archive may be from the book _Methods and 53 // Programs for Mathematical Functions_ (Prentice-Hall or Simon & Schuster 54 // International, 1989) or from the Cephes Mathematical Library, a 55 // commercial product. In either event, it is copyrighted by the author. 56 // What you see here may be used freely but it comes with no support or 57 // guarantee. 58 // 59 // The two known misprints in the book are repaired here in the 60 // source listings for the gamma function and the incomplete beta 61 // integral. 62 // 63 // Stephen L. Moshier 64 // [email protected] 65 66 var _gamP = [...]float64{ 67 1.60119522476751861407e-04, 68 1.19135147006586384913e-03, 69 1.04213797561761569935e-02, 70 4.76367800457137231464e-02, 71 2.07448227648435975150e-01, 72 4.94214826801497100753e-01, 73 9.99999999999999996796e-01, 74 } 75 var _gamQ = [...]float64{ 76 -2.31581873324120129819e-05, 77 5.39605580493303397842e-04, 78 -4.45641913851797240494e-03, 79 1.18139785222060435552e-02, 80 3.58236398605498653373e-02, 81 -2.34591795718243348568e-01, 82 7.14304917030273074085e-02, 83 1.00000000000000000320e+00, 84 } 85 var _gamS = [...]float64{ 86 7.87311395793093628397e-04, 87 -2.29549961613378126380e-04, 88 -2.68132617805781232825e-03, 89 3.47222221605458667310e-03, 90 8.33333333333482257126e-02, 91 } 92 93 // Gamma function computed by Stirling's formula. 94 // The polynomial is valid for 33 <= x <= 172. 95 func stirling(x float64) float64 { 96 const ( 97 SqrtTwoPi = 2.506628274631000502417 98 MaxStirling = 143.01608 99 ) 100 w := 1 / x 101 w = 1 + w*((((_gamS[0]*w+_gamS[1])*w+_gamS[2])*w+_gamS[3])*w+_gamS[4]) 102 y := Exp(x) 103 if x > MaxStirling { // avoid Pow() overflow 104 v := Pow(x, 0.5*x-0.25) 105 y = v * (v / y) 106 } else { 107 y = Pow(x, x-0.5) / y 108 } 109 y = SqrtTwoPi * y * w 110 return y 111 } 112 113 // Gamma(x) returns the Gamma function of x. 114 // 115 // Special cases are: 116 // Gamma(±Inf) = ±Inf 117 // Gamma(NaN) = NaN 118 // Large values overflow to +Inf. 119 // Zero and negative integer arguments return ±Inf. 120 func Gamma(x float64) float64 { 121 const Euler = 0.57721566490153286060651209008240243104215933593992 // A001620 122 // special cases 123 switch { 124 case IsInf(x, -1) || IsNaN(x): 125 return x 126 case x < -170.5674972726612 || x > 171.61447887182298: 127 return Inf(1) 128 } 129 q := Abs(x) 130 p := Floor(q) 131 if q > 33 { 132 if x >= 0 { 133 return stirling(x) 134 } 135 signgam := 1 136 if ip := int(p); ip&1 == 0 { 137 signgam = -1 138 } 139 z := q - p 140 if z > 0.5 { 141 p = p + 1 142 z = q - p 143 } 144 z = q * Sin(Pi*z) 145 if z == 0 { 146 return Inf(signgam) 147 } 148 z = Pi / (Abs(z) * stirling(q)) 149 return float64(signgam) * z 150 } 151 152 // Reduce argument 153 z := 1.0 154 for x >= 3 { 155 x = x - 1 156 z = z * x 157 } 158 for x < 0 { 159 if x > -1e-09 { 160 goto small 161 } 162 z = z / x 163 x = x + 1 164 } 165 for x < 2 { 166 if x < 1e-09 { 167 goto small 168 } 169 z = z / x 170 x = x + 1 171 } 172 173 if x == 2 { 174 return z 175 } 176 177 x = x - 2 178 p = (((((x*_gamP[0]+_gamP[1])*x+_gamP[2])*x+_gamP[3])*x+_gamP[4])*x+_gamP[5])*x + _gamP[6] 179 q = ((((((x*_gamQ[0]+_gamQ[1])*x+_gamQ[2])*x+_gamQ[3])*x+_gamQ[4])*x+_gamQ[5])*x+_gamQ[6])*x + _gamQ[7] 180 return z * p / q 181 182 small: 183 if x == 0 { 184 return Inf(1) 185 } 186 return z / ((1 + Euler*x) * x) 187 }