1 |
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2 /* |
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3 * Copyright (c) 1998, 2001, Oracle and/or its affiliates. All rights reserved. |
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4 * DO NOT ALTER OR REMOVE COPYRIGHT NOTICES OR THIS FILE HEADER. |
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5 * |
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6 * This code is free software; you can redistribute it and/or modify it |
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7 * under the terms of the GNU General Public License version 2 only, as |
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8 * published by the Free Software Foundation. Oracle designates this |
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9 * particular file as subject to the "Classpath" exception as provided |
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10 * by Oracle in the LICENSE file that accompanied this code. |
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11 * |
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12 * This code is distributed in the hope that it will be useful, but WITHOUT |
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13 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or |
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14 * FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License |
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15 * version 2 for more details (a copy is included in the LICENSE file that |
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16 * accompanied this code). |
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17 * |
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18 * You should have received a copy of the GNU General Public License version |
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19 * 2 along with this work; if not, write to the Free Software Foundation, |
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20 * Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301 USA. |
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21 * |
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22 * Please contact Oracle, 500 Oracle Parkway, Redwood Shores, CA 94065 USA |
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23 * or visit www.oracle.com if you need additional information or have any |
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24 * questions. |
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25 */ |
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26 |
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27 /* __ieee754_lgamma_r(x, signgamp) |
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28 * Reentrant version of the logarithm of the Gamma function |
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29 * with user provide pointer for the sign of Gamma(x). |
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30 * |
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31 * Method: |
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32 * 1. Argument Reduction for 0 < x <= 8 |
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33 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may |
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34 * reduce x to a number in [1.5,2.5] by |
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35 * lgamma(1+s) = log(s) + lgamma(s) |
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36 * for example, |
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37 * lgamma(7.3) = log(6.3) + lgamma(6.3) |
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38 * = log(6.3*5.3) + lgamma(5.3) |
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39 * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) |
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40 * 2. Polynomial approximation of lgamma around its |
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41 * minimun ymin=1.461632144968362245 to maintain monotonicity. |
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42 * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use |
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43 * Let z = x-ymin; |
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44 * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) |
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45 * where |
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46 * poly(z) is a 14 degree polynomial. |
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47 * 2. Rational approximation in the primary interval [2,3] |
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48 * We use the following approximation: |
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49 * s = x-2.0; |
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50 * lgamma(x) = 0.5*s + s*P(s)/Q(s) |
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51 * with accuracy |
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52 * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 |
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53 * Our algorithms are based on the following observation |
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54 * |
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55 * zeta(2)-1 2 zeta(3)-1 3 |
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56 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... |
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57 * 2 3 |
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58 * |
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59 * where Euler = 0.5771... is the Euler constant, which is very |
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60 * close to 0.5. |
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61 * |
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62 * 3. For x>=8, we have |
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63 * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... |
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64 * (better formula: |
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65 * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) |
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66 * Let z = 1/x, then we approximation |
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67 * f(z) = lgamma(x) - (x-0.5)(log(x)-1) |
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68 * by |
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69 * 3 5 11 |
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70 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z |
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71 * where |
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72 * |w - f(z)| < 2**-58.74 |
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73 * |
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74 * 4. For negative x, since (G is gamma function) |
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75 * -x*G(-x)*G(x) = pi/sin(pi*x), |
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76 * we have |
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77 * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) |
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78 * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 |
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79 * Hence, for x<0, signgam = sign(sin(pi*x)) and |
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80 * lgamma(x) = log(|Gamma(x)|) |
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81 * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); |
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82 * Note: one should avoid compute pi*(-x) directly in the |
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83 * computation of sin(pi*(-x)). |
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84 * |
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85 * 5. Special Cases |
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86 * lgamma(2+s) ~ s*(1-Euler) for tiny s |
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87 * lgamma(1)=lgamma(2)=0 |
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88 * lgamma(x) ~ -log(x) for tiny x |
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89 * lgamma(0) = lgamma(inf) = inf |
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90 * lgamma(-integer) = +-inf |
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91 * |
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92 */ |
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93 |
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94 #include "fdlibm.h" |
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95 |
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96 #ifdef __STDC__ |
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97 static const double |
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98 #else |
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99 static double |
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100 #endif |
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101 two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */ |
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102 half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ |
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103 one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ |
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104 pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ |
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105 a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ |
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106 a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ |
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107 a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ |
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108 a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ |
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109 a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ |
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110 a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ |
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111 a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ |
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112 a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ |
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113 a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ |
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114 a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ |
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115 a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ |
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116 a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ |
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117 tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ |
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118 tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ |
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119 /* tt = -(tail of tf) */ |
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120 tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ |
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121 t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ |
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122 t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ |
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123 t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ |
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124 t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ |
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125 t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ |
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126 t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ |
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127 t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ |
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128 t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ |
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129 t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ |
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130 t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ |
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131 t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ |
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132 t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ |
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133 t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ |
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134 t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ |
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135 t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ |
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136 u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ |
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137 u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ |
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138 u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ |
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139 u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ |
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140 u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ |
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141 u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ |
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142 v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ |
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143 v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ |
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144 v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ |
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145 v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ |
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146 v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ |
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147 s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ |
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148 s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ |
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149 s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ |
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150 s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ |
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151 s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ |
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152 s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ |
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153 s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ |
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154 r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ |
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155 r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ |
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156 r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ |
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157 r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ |
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158 r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ |
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159 r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ |
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160 w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ |
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161 w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ |
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162 w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ |
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163 w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ |
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164 w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ |
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165 w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ |
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166 w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ |
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167 |
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168 static double zero= 0.00000000000000000000e+00; |
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169 |
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170 #ifdef __STDC__ |
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171 static double sin_pi(double x) |
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172 #else |
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173 static double sin_pi(x) |
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174 double x; |
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175 #endif |
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176 { |
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177 double y,z; |
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178 int n,ix; |
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179 |
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180 ix = 0x7fffffff&__HI(x); |
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181 |
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182 if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0); |
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183 y = -x; /* x is assume negative */ |
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184 |
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185 /* |
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186 * argument reduction, make sure inexact flag not raised if input |
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187 * is an integer |
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188 */ |
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189 z = floor(y); |
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190 if(z!=y) { /* inexact anyway */ |
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191 y *= 0.5; |
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192 y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */ |
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193 n = (int) (y*4.0); |
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194 } else { |
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195 if(ix>=0x43400000) { |
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196 y = zero; n = 0; /* y must be even */ |
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197 } else { |
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198 if(ix<0x43300000) z = y+two52; /* exact */ |
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199 n = __LO(z)&1; /* lower word of z */ |
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200 y = n; |
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201 n<<= 2; |
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202 } |
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203 } |
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204 switch (n) { |
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205 case 0: y = __kernel_sin(pi*y,zero,0); break; |
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206 case 1: |
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207 case 2: y = __kernel_cos(pi*(0.5-y),zero); break; |
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208 case 3: |
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209 case 4: y = __kernel_sin(pi*(one-y),zero,0); break; |
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210 case 5: |
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211 case 6: y = -__kernel_cos(pi*(y-1.5),zero); break; |
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212 default: y = __kernel_sin(pi*(y-2.0),zero,0); break; |
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213 } |
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214 return -y; |
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215 } |
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216 |
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217 |
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218 #ifdef __STDC__ |
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219 double __ieee754_lgamma_r(double x, int *signgamp) |
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220 #else |
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221 double __ieee754_lgamma_r(x,signgamp) |
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222 double x; int *signgamp; |
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223 #endif |
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224 { |
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225 double t,y,z,nadj=0,p,p1,p2,p3,q,r,w; |
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226 int i,hx,lx,ix; |
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227 |
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228 hx = __HI(x); |
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229 lx = __LO(x); |
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230 |
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231 /* purge off +-inf, NaN, +-0, and negative arguments */ |
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232 *signgamp = 1; |
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233 ix = hx&0x7fffffff; |
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234 if(ix>=0x7ff00000) return x*x; |
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235 if((ix|lx)==0) return one/zero; |
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236 if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */ |
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237 if(hx<0) { |
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238 *signgamp = -1; |
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239 return -__ieee754_log(-x); |
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240 } else return -__ieee754_log(x); |
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241 } |
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242 if(hx<0) { |
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243 if(ix>=0x43300000) /* |x|>=2**52, must be -integer */ |
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244 return one/zero; |
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245 t = sin_pi(x); |
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246 if(t==zero) return one/zero; /* -integer */ |
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247 nadj = __ieee754_log(pi/fabs(t*x)); |
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248 if(t<zero) *signgamp = -1; |
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249 x = -x; |
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250 } |
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251 |
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252 /* purge off 1 and 2 */ |
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253 if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0; |
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254 /* for x < 2.0 */ |
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255 else if(ix<0x40000000) { |
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256 if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ |
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257 r = -__ieee754_log(x); |
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258 if(ix>=0x3FE76944) {y = one-x; i= 0;} |
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259 else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;} |
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260 else {y = x; i=2;} |
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261 } else { |
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262 r = zero; |
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263 if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */ |
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264 else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */ |
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265 else {y=x-one;i=2;} |
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266 } |
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267 switch(i) { |
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268 case 0: |
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269 z = y*y; |
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270 p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); |
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271 p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); |
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272 p = y*p1+p2; |
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273 r += (p-0.5*y); break; |
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274 case 1: |
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275 z = y*y; |
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276 w = z*y; |
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277 p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ |
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278 p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); |
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279 p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); |
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280 p = z*p1-(tt-w*(p2+y*p3)); |
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281 r += (tf + p); break; |
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282 case 2: |
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283 p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); |
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284 p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); |
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285 r += (-0.5*y + p1/p2); |
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286 } |
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287 } |
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288 else if(ix<0x40200000) { /* x < 8.0 */ |
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289 i = (int)x; |
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290 t = zero; |
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291 y = x-(double)i; |
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292 p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); |
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293 q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); |
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294 r = half*y+p/q; |
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295 z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ |
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296 switch(i) { |
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297 case 7: z *= (y+6.0); /* FALLTHRU */ |
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298 case 6: z *= (y+5.0); /* FALLTHRU */ |
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299 case 5: z *= (y+4.0); /* FALLTHRU */ |
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300 case 4: z *= (y+3.0); /* FALLTHRU */ |
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301 case 3: z *= (y+2.0); /* FALLTHRU */ |
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302 r += __ieee754_log(z); break; |
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303 } |
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304 /* 8.0 <= x < 2**58 */ |
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305 } else if (ix < 0x43900000) { |
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306 t = __ieee754_log(x); |
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307 z = one/x; |
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308 y = z*z; |
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309 w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); |
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310 r = (x-half)*(t-one)+w; |
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311 } else |
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312 /* 2**58 <= x <= inf */ |
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313 r = x*(__ieee754_log(x)-one); |
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314 if(hx<0) r = nadj - r; |
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315 return r; |
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316 } |
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