1 | /*
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2 | "A Precision Approximation of the Gamma Function" - Cornelius Lanczos (1964)
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3 | "Lanczos Implementation of the Gamma Function" - Paul Godfrey (2001)
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4 | "An Analysis of the Lanczos Gamma Approximation" - Glendon Ralph Pugh (2004)
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5 |
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6 | approximation method:
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7 |
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8 | (x - 0.5) S(x)
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9 | Gamma(x) = (x + g - 0.5) * ----------------
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10 | exp(x + g - 0.5)
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11 |
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12 | with
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13 | a1 a2 a3 aN
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14 | S(x) ~= [ a0 + ----- + ----- + ----- + ... + ----- ]
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15 | x + 1 x + 2 x + 3 x + N
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16 |
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17 | with a0, a1, a2, a3,.. aN constants which depend on g.
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18 |
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19 | for x < 0 the following reflection formula is used:
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20 |
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21 | Gamma(x)*Gamma(-x) = -pi/(x sin(pi x))
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22 |
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23 | most ideas and constants are from boost and python
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24 | */
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25 | #include "libm.h"
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26 |
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27 | static const double pi = 3.141592653589793238462643383279502884;
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28 |
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29 | /* sin(pi x) with x > 0x1p-100, if sin(pi*x)==0 the sign is arbitrary */
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30 | static double sinpi(double x)
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31 | {
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32 | int n;
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33 |
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34 | /* argument reduction: x = |x| mod 2 */
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35 | /* spurious inexact when x is odd int */
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36 | x = x * 0.5;
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37 | x = 2 * (x - floor(x));
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38 |
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39 | /* reduce x into [-.25,.25] */
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40 | n = 4 * x;
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41 | n = (n+1)/2;
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42 | x -= n * 0.5;
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43 |
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44 | x *= pi;
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45 | switch (n) {
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46 | default: /* case 4 */
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47 | case 0:
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48 | return __sin(x, 0, 0);
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49 | case 1:
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50 | return __cos(x, 0);
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51 | case 2:
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52 | return __sin(-x, 0, 0);
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53 | case 3:
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54 | return -__cos(x, 0);
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55 | }
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56 | }
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57 |
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58 | #define N 12
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59 | //static const double g = 6.024680040776729583740234375;
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60 | static const double gmhalf = 5.524680040776729583740234375;
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61 | static const double Snum[N+1] = {
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62 | 23531376880.410759688572007674451636754734846804940,
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63 | 42919803642.649098768957899047001988850926355848959,
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64 | 35711959237.355668049440185451547166705960488635843,
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65 | 17921034426.037209699919755754458931112671403265390,
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66 | 6039542586.3520280050642916443072979210699388420708,
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67 | 1439720407.3117216736632230727949123939715485786772,
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68 | 248874557.86205415651146038641322942321632125127801,
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69 | 31426415.585400194380614231628318205362874684987640,
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70 | 2876370.6289353724412254090516208496135991145378768,
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71 | 186056.26539522349504029498971604569928220784236328,
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72 | 8071.6720023658162106380029022722506138218516325024,
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73 | 210.82427775157934587250973392071336271166969580291,
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74 | 2.5066282746310002701649081771338373386264310793408,
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75 | };
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76 | static const double Sden[N+1] = {
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77 | 0, 39916800, 120543840, 150917976, 105258076, 45995730, 13339535,
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78 | 2637558, 357423, 32670, 1925, 66, 1,
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79 | };
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80 | /* n! for small integer n */
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81 | static const double fact[] = {
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82 | 1, 1, 2, 6, 24, 120, 720, 5040.0, 40320.0, 362880.0, 3628800.0, 39916800.0,
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83 | 479001600.0, 6227020800.0, 87178291200.0, 1307674368000.0, 20922789888000.0,
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84 | 355687428096000.0, 6402373705728000.0, 121645100408832000.0,
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85 | 2432902008176640000.0, 51090942171709440000.0, 1124000727777607680000.0,
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86 | };
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87 |
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88 | /* S(x) rational function for positive x */
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89 | static double S(double x)
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90 | {
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91 | double_t num = 0, den = 0;
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92 | int i;
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93 |
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94 | /* to avoid overflow handle large x differently */
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95 | if (x < 8)
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96 | for (i = N; i >= 0; i--) {
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97 | num = num * x + Snum[i];
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98 | den = den * x + Sden[i];
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99 | }
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100 | else
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101 | for (i = 0; i <= N; i++) {
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102 | num = num / x + Snum[i];
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103 | den = den / x + Sden[i];
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104 | }
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105 | return num/den;
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106 | }
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107 |
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108 | double tgamma(double x)
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109 | {
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110 | union {double f; uint64_t i;} u = {x};
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111 | double absx, y;
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112 | double_t dy, z, r;
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113 | uint32_t ix = u.i>>32 & 0x7fffffff;
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114 | int sign = u.i>>63;
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115 |
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116 | /* special cases */
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117 | if (ix >= 0x7ff00000)
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118 | /* tgamma(nan)=nan, tgamma(inf)=inf, tgamma(-inf)=nan with invalid */
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119 | return x + INFINITY;
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120 | if (ix < (0x3ff-54)<<20)
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121 | /* |x| < 2^-54: tgamma(x) ~ 1/x, +-0 raises div-by-zero */
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122 | return 1/x;
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123 |
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124 | /* integer arguments */
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125 | /* raise inexact when non-integer */
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126 | if (x == floor(x)) {
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127 | if (sign)
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128 | return 0/0.0;
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129 | if (x <= sizeof fact/sizeof *fact)
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130 | return fact[(int)x - 1];
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131 | }
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132 |
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133 | /* x >= 172: tgamma(x)=inf with overflow */
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134 | /* x =< -184: tgamma(x)=+-0 with underflow */
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135 | if (ix >= 0x40670000) { /* |x| >= 184 */
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136 | if (sign) {
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137 | FORCE_EVAL((float)(0x1p-126/x));
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138 | if (floor(x) * 0.5 == floor(x * 0.5))
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139 | return 0;
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140 | return -0.0;
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141 | }
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142 | x *= 0x1p1023;
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143 | return x;
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144 | }
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145 |
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146 | absx = sign ? -x : x;
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147 |
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148 | /* handle the error of x + g - 0.5 */
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149 | y = absx + gmhalf;
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150 | if (absx > gmhalf) {
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151 | dy = y - absx;
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152 | dy -= gmhalf;
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153 | } else {
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154 | dy = y - gmhalf;
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155 | dy -= absx;
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156 | }
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157 |
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158 | z = absx - 0.5;
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159 | r = S(absx) * exp(-y);
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160 | if (x < 0) {
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161 | /* reflection formula for negative x */
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162 | /* sinpi(absx) is not 0, integers are already handled */
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163 | r = -pi / (sinpi(absx) * absx * r);
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164 | dy = -dy;
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165 | z = -z;
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166 | }
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167 | r += dy * (gmhalf+0.5) * r / y;
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168 | z = pow(y, 0.5*z);
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169 | y = r * z * z;
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170 | return y;
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171 | }
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172 |
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173 | #if 0
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174 | double __lgamma_r(double x, int *sign)
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175 | {
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176 | double r, absx;
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177 |
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178 | *sign = 1;
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179 |
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180 | /* special cases */
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181 | if (!isfinite(x))
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182 | /* lgamma(nan)=nan, lgamma(+-inf)=inf */
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183 | return x*x;
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184 |
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185 | /* integer arguments */
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186 | if (x == floor(x) && x <= 2) {
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187 | /* n <= 0: lgamma(n)=inf with divbyzero */
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188 | /* n == 1,2: lgamma(n)=0 */
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189 | if (x <= 0)
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190 | return 1/0.0;
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191 | return 0;
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192 | }
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193 |
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194 | absx = fabs(x);
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195 |
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196 | /* lgamma(x) ~ -log(|x|) for tiny |x| */
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197 | if (absx < 0x1p-54) {
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198 | *sign = 1 - 2*!!signbit(x);
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199 | return -log(absx);
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200 | }
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201 |
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202 | /* use tgamma for smaller |x| */
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203 | if (absx < 128) {
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204 | x = tgamma(x);
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205 | *sign = 1 - 2*!!signbit(x);
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206 | return log(fabs(x));
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207 | }
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208 |
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209 | /* second term (log(S)-g) could be more precise here.. */
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210 | /* or with stirling: (|x|-0.5)*(log(|x|)-1) + poly(1/|x|) */
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211 | r = (absx-0.5)*(log(absx+gmhalf)-1) + (log(S(absx)) - (gmhalf+0.5));
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212 | if (x < 0) {
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213 | /* reflection formula for negative x */
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214 | x = sinpi(absx);
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215 | *sign = 2*!!signbit(x) - 1;
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216 | r = log(pi/(fabs(x)*absx)) - r;
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217 | }
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218 | return r;
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219 | }
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220 |
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221 | weak_alias(__lgamma_r, lgamma_r);
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222 | #endif
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