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1 /* Implementation of gamma function according to ISO C.
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145
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2 Copyright (C) 1997-2018 Free Software Foundation, Inc.
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3 This file is part of the GNU C Library.
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4 Contributed by Ulrich Drepper <drepper@cygnus.com>, 1997 and
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145
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5 Jakub Jelinek <jj@ultra.linux.cz, 1999.
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6
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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7 The GNU C Library is free software; you can redistribute it and/or
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8 modify it under the terms of the GNU Lesser General Public
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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9 License as published by the Free Software Foundation; either
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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10 version 2.1 of the License, or (at your option) any later version.
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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11
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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12 The GNU C Library is distributed in the hope that it will be useful,
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13 but WITHOUT ANY WARRANTY; without even the implied warranty of
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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14 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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15 Lesser General Public License for more details.
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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16
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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17 You should have received a copy of the GNU Lesser General Public
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18 License along with the GNU C Library; if not, see
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19 <http://www.gnu.org/licenses/>. */
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20
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Nobuyasu Oshiro <dimolto@cr.ie.u-ryukyu.ac.jp>
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21 #include "quadmath-imp.h"
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22 __float128
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23 tgammaq (__float128 x)
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24 {
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25 int sign;
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26 __float128 ret;
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27 ret = __quadmath_gammaq_r (x, &sign);
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28 return sign < 0 ? -ret : ret;
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29 }
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30
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31 /* Coefficients B_2k / 2k(2k-1) of x^-(2k-1) inside exp in Stirling's
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32 approximation to gamma function. */
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33
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34 static const __float128 gamma_coeff[] =
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35 {
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36 0x1.5555555555555555555555555555p-4Q,
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37 -0xb.60b60b60b60b60b60b60b60b60b8p-12Q,
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38 0x3.4034034034034034034034034034p-12Q,
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39 -0x2.7027027027027027027027027028p-12Q,
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40 0x3.72a3c5631fe46ae1d4e700dca8f2p-12Q,
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41 -0x7.daac36664f1f207daac36664f1f4p-12Q,
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42 0x1.a41a41a41a41a41a41a41a41a41ap-8Q,
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43 -0x7.90a1b2c3d4e5f708192a3b4c5d7p-8Q,
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44 0x2.dfd2c703c0cfff430edfd2c703cp-4Q,
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45 -0x1.6476701181f39edbdb9ce625987dp+0Q,
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46 0xd.672219167002d3a7a9c886459cp+0Q,
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47 -0x9.cd9292e6660d55b3f712eb9e07c8p+4Q,
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48 0x8.911a740da740da740da740da741p+8Q,
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49 -0x8.d0cc570e255bf59ff6eec24b49p+12Q,
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50 };
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51
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52 #define NCOEFF (sizeof (gamma_coeff) / sizeof (gamma_coeff[0]))
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53
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54 /* Return gamma (X), for positive X less than 1775, in the form R *
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55 2^(*EXP2_ADJ), where R is the return value and *EXP2_ADJ is set to
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56 avoid overflow or underflow in intermediate calculations. */
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57
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58 static __float128
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59 gammal_positive (__float128 x, int *exp2_adj)
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60 {
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61 int local_signgam;
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62 if (x < 0.5Q)
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63 {
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64 *exp2_adj = 0;
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65 return expq (__quadmath_lgammaq_r (x + 1, &local_signgam)) / x;
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66 }
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67 else if (x <= 1.5Q)
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68 {
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69 *exp2_adj = 0;
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70 return expq (__quadmath_lgammaq_r (x, &local_signgam));
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71 }
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72 else if (x < 12.5Q)
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73 {
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74 /* Adjust into the range for using exp (lgamma). */
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75 *exp2_adj = 0;
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76 __float128 n = ceilq (x - 1.5Q);
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77 __float128 x_adj = x - n;
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78 __float128 eps;
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79 __float128 prod = __quadmath_gamma_productq (x_adj, 0, n, &eps);
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80 return (expq (__quadmath_lgammaq_r (x_adj, &local_signgam))
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81 * prod * (1 + eps));
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82 }
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83 else
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84 {
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85 __float128 eps = 0;
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86 __float128 x_eps = 0;
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87 __float128 x_adj = x;
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88 __float128 prod = 1;
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89 if (x < 24)
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90 {
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91 /* Adjust into the range for applying Stirling's
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92 approximation. */
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93 __float128 n = ceilq (24 - x);
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94 x_adj = x + n;
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95 x_eps = (x - (x_adj - n));
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96 prod = __quadmath_gamma_productq (x_adj - n, x_eps, n, &eps);
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97 }
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98 /* The result is now gamma (X_ADJ + X_EPS) / (PROD * (1 + EPS)).
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99 Compute gamma (X_ADJ + X_EPS) using Stirling's approximation,
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100 starting by computing pow (X_ADJ, X_ADJ) with a power of 2
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101 factored out. */
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102 __float128 exp_adj = -eps;
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103 __float128 x_adj_int = roundq (x_adj);
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104 __float128 x_adj_frac = x_adj - x_adj_int;
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105 int x_adj_log2;
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106 __float128 x_adj_mant = frexpq (x_adj, &x_adj_log2);
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107 if (x_adj_mant < M_SQRT1_2q)
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108 {
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109 x_adj_log2--;
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110 x_adj_mant *= 2;
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111 }
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112 *exp2_adj = x_adj_log2 * (int) x_adj_int;
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113 __float128 ret = (powq (x_adj_mant, x_adj)
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114 * exp2q (x_adj_log2 * x_adj_frac)
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115 * expq (-x_adj)
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116 * sqrtq (2 * M_PIq / x_adj)
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117 / prod);
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118 exp_adj += x_eps * logq (x_adj);
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119 __float128 bsum = gamma_coeff[NCOEFF - 1];
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120 __float128 x_adj2 = x_adj * x_adj;
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121 for (size_t i = 1; i <= NCOEFF - 1; i++)
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122 bsum = bsum / x_adj2 + gamma_coeff[NCOEFF - 1 - i];
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123 exp_adj += bsum / x_adj;
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124 return ret + ret * expm1q (exp_adj);
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125 }
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126 }
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127
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128 __float128
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129 __quadmath_gammaq_r (__float128 x, int *signgamp)
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130 {
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131 int64_t hx;
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132 uint64_t lx;
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133 __float128 ret;
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134
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135 GET_FLT128_WORDS64 (hx, lx, x);
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136
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137 if (((hx & 0x7fffffffffffffffLL) | lx) == 0)
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138 {
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139 /* Return value for x == 0 is Inf with divide by zero exception. */
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140 *signgamp = 0;
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141 return 1.0 / x;
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142 }
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143 if (hx < 0 && (uint64_t) hx < 0xffff000000000000ULL && rintq (x) == x)
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145
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144 {
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145 /* Return value for integer x < 0 is NaN with invalid exception. */
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146 *signgamp = 0;
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147 return (x - x) / (x - x);
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148 }
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149 if (hx == 0xffff000000000000ULL && lx == 0)
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145
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150 {
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151 /* x == -Inf. According to ISO this is NaN. */
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152 *signgamp = 0;
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153 return x - x;
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154 }
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155 if ((hx & 0x7fff000000000000ULL) == 0x7fff000000000000ULL)
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156 {
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157 /* Positive infinity (return positive infinity) or NaN (return
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158 NaN). */
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159 *signgamp = 0;
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160 return x + x;
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161 }
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162
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163 if (x >= 1756)
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164 {
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165 /* Overflow. */
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166 *signgamp = 0;
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167 return FLT128_MAX * FLT128_MAX;
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168 }
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169 else
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170 {
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171 SET_RESTORE_ROUNDF128 (FE_TONEAREST);
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172 if (x > 0)
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173 {
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174 *signgamp = 0;
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175 int exp2_adj;
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176 ret = gammal_positive (x, &exp2_adj);
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177 ret = scalbnq (ret, exp2_adj);
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178 }
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179 else if (x >= -FLT128_EPSILON / 4)
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180 {
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181 *signgamp = 0;
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182 ret = 1 / x;
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183 }
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184 else
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185 {
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186 __float128 tx = truncq (x);
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187 *signgamp = (tx == 2 * truncq (tx / 2)) ? -1 : 1;
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188 if (x <= -1775)
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189 /* Underflow. */
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190 ret = FLT128_MIN * FLT128_MIN;
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191 else
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192 {
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193 __float128 frac = tx - x;
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194 if (frac > 0.5Q)
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195 frac = 1 - frac;
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196 __float128 sinpix = (frac <= 0.25Q
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197 ? sinq (M_PIq * frac)
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198 : cosq (M_PIq * (0.5Q - frac)));
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199 int exp2_adj;
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200 ret = M_PIq / (-x * sinpix
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201 * gammal_positive (-x, &exp2_adj));
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202 ret = scalbnq (ret, -exp2_adj);
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203 math_check_force_underflow_nonneg (ret);
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204 }
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205 }
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206 }
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207 if (isinfq (ret) && x != 0)
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208 {
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209 if (*signgamp < 0)
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210 return -(-copysignq (FLT128_MAX, ret) * FLT128_MAX);
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211 else
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212 return copysignq (FLT128_MAX, ret) * FLT128_MAX;
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213 }
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214 else if (ret == 0)
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215 {
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216 if (*signgamp < 0)
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217 return -(-copysignq (FLT128_MIN, ret) * FLT128_MIN);
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218 else
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219 return copysignq (FLT128_MIN, ret) * FLT128_MIN;
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220 }
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221 else
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222 return ret;
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223 }
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