| 1 | |
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| 2 | /* @(#)er_lgamma.c 5.1 93/09/24 */ |
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| 3 | /* |
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| 4 | * ==================================================== |
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| 5 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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| 6 | * |
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| 7 | * Developed at SunPro, a Sun Microsystems, Inc. business. |
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| 8 | * Permission to use, copy, modify, and distribute this |
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| 9 | * software is freely granted, provided that this notice |
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| 10 | * is preserved. |
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| 11 | * ==================================================== |
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| 12 | * |
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| 13 | */ |
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| 14 | |
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| 15 | /* |
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| 16 | FUNCTION |
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| 17 | <<gamma>>, <<gammaf>>, <<lgamma>>, <<lgammaf>>, <<gamma_r>>, |
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| 18 | <<gammaf_r>>, <<lgamma_r>>, <<lgammaf_r>>---logarithmic gamma |
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| 19 | function |
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| 20 | INDEX |
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| 21 | gamma |
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| 22 | INDEX |
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| 23 | gammaf |
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| 24 | INDEX |
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| 25 | lgamma |
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| 26 | INDEX |
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| 27 | lgammaf |
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| 28 | INDEX |
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| 29 | gamma_r |
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| 30 | INDEX |
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| 31 | gammaf_r |
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| 32 | INDEX |
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| 33 | lgamma_r |
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| 34 | INDEX |
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| 35 | lgammaf_r |
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| 36 | |
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| 37 | SYNOPSIS |
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| 38 | #include <math.h> |
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| 39 | double gamma(double <[x]>); |
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| 40 | float gammaf(float <[x]>); |
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| 41 | double lgamma(double <[x]>); |
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| 42 | float lgammaf(float <[x]>); |
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| 43 | double gamma_r(double <[x]>, int *<[signgamp]>); |
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| 44 | float gammaf_r(float <[x]>, int *<[signgamp]>); |
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| 45 | double lgamma_r(double <[x]>, int *<[signgamp]>); |
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| 46 | float lgammaf_r(float <[x]>, int *<[signgamp]>); |
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| 47 | |
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| 48 | DESCRIPTION |
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| 49 | <<gamma>> calculates |
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| 50 | @tex |
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| 51 | $\mit ln\bigl(\Gamma(x)\bigr)$, |
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| 52 | @end tex |
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| 53 | the natural logarithm of the gamma function of <[x]>. The gamma function |
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| 54 | (<<exp(gamma(<[x]>))>>) is a generalization of factorial, and retains |
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| 55 | the property that |
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| 56 | @ifnottex |
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| 57 | <<exp(gamma(N))>> is equivalent to <<N*exp(gamma(N-1))>>. |
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| 58 | @end ifnottex |
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| 59 | @tex |
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| 60 | $\mit \Gamma(N)\equiv N\times\Gamma(N-1)$. |
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| 61 | @end tex |
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| 62 | Accordingly, the results of the gamma function itself grow very |
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| 63 | quickly. <<gamma>> is defined as |
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| 64 | @tex |
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| 65 | $\mit ln\bigl(\Gamma(x)\bigr)$ rather than simply $\mit \Gamma(x)$ |
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| 66 | @end tex |
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| 67 | @ifnottex |
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| 68 | the natural log of the gamma function, rather than the gamma function |
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| 69 | itself, |
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| 70 | @end ifnottex |
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| 71 | to extend the useful range of results representable. |
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| 72 | |
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| 73 | The sign of the result is returned in the global variable <<signgam>>, |
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| 74 | which is declared in math.h. |
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| 75 | |
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| 76 | <<gammaf>> performs the same calculation as <<gamma>>, but uses and |
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| 77 | returns <<float>> values. |
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| 78 | |
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| 79 | <<lgamma>> and <<lgammaf>> are alternate names for <<gamma>> and |
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| 80 | <<gammaf>>. The use of <<lgamma>> instead of <<gamma>> is a reminder |
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| 81 | that these functions compute the log of the gamma function, rather |
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| 82 | than the gamma function itself. |
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| 83 | |
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| 84 | The functions <<gamma_r>>, <<gammaf_r>>, <<lgamma_r>>, and |
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| 85 | <<lgammaf_r>> are just like <<gamma>>, <<gammaf>>, <<lgamma>>, and |
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| 86 | <<lgammaf>>, respectively, but take an additional argument. This |
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| 87 | additional argument is a pointer to an integer. This additional |
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| 88 | argument is used to return the sign of the result, and the global |
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| 89 | variable <<signgam>> is not used. These functions may be used for |
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| 90 | reentrant calls (but they will still set the global variable <<errno>> |
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| 91 | if an error occurs). |
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| 92 | |
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| 93 | RETURNS |
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| 94 | Normally, the computed result is returned. |
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| 95 | |
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| 96 | When <[x]> is a nonpositive integer, <<gamma>> returns <<HUGE_VAL>> |
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| 97 | and <<errno>> is set to <<EDOM>>. If the result overflows, <<gamma>> |
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| 98 | returns <<HUGE_VAL>> and <<errno>> is set to <<ERANGE>>. |
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| 99 | |
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| 100 | You can modify this error treatment using <<matherr>>. |
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| 101 | |
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| 102 | PORTABILITY |
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| 103 | Neither <<gamma>> nor <<gammaf>> is ANSI C. */ |
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| 104 | |
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| 105 | /* lgamma_r(x, signgamp) |
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| 106 | * Reentrant version of the logarithm of the Gamma function |
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| 107 | * with user provide pointer for the sign of Gamma(x). |
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| 108 | * |
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| 109 | * Method: |
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| 110 | * 1. Argument Reduction for 0 < x <= 8 |
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| 111 | * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may |
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| 112 | * reduce x to a number in [1.5,2.5] by |
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| 113 | * lgamma(1+s) = log(s) + lgamma(s) |
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| 114 | * for example, |
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| 115 | * lgamma(7.3) = log(6.3) + lgamma(6.3) |
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| 116 | * = log(6.3*5.3) + lgamma(5.3) |
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| 117 | * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) |
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| 118 | * 2. Polynomial approximation of lgamma around its |
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| 119 | * minimun ymin=1.461632144968362245 to maintain monotonicity. |
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| 120 | * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use |
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| 121 | * Let z = x-ymin; |
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| 122 | * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) |
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| 123 | * where |
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| 124 | * poly(z) is a 14 degree polynomial. |
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| 125 | * 2. Rational approximation in the primary interval [2,3] |
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| 126 | * We use the following approximation: |
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| 127 | * s = x-2.0; |
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| 128 | * lgamma(x) = 0.5*s + s*P(s)/Q(s) |
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| 129 | * with accuracy |
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| 130 | * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 |
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| 131 | * Our algorithms are based on the following observation |
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| 132 | * |
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| 133 | * zeta(2)-1 2 zeta(3)-1 3 |
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| 134 | * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... |
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| 135 | * 2 3 |
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| 136 | * |
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| 137 | * where Euler = 0.5771... is the Euler constant, which is very |
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| 138 | * close to 0.5. |
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| 139 | * |
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| 140 | * 3. For x>=8, we have |
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| 141 | * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... |
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| 142 | * (better formula: |
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| 143 | * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) |
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| 144 | * Let z = 1/x, then we approximation |
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| 145 | * f(z) = lgamma(x) - (x-0.5)(log(x)-1) |
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| 146 | * by |
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| 147 | * 3 5 11 |
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| 148 | * w = w0 + w1*z + w2*z + w3*z + ... + w6*z |
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| 149 | * where |
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| 150 | * |w - f(z)| < 2**-58.74 |
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| 151 | * |
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| 152 | * 4. For negative x, since (G is gamma function) |
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| 153 | * -x*G(-x)*G(x) = pi/sin(pi*x), |
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| 154 | * we have |
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| 155 | * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) |
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| 156 | * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 |
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| 157 | * Hence, for x<0, signgam = sign(sin(pi*x)) and |
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| 158 | * lgamma(x) = log(|Gamma(x)|) |
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| 159 | * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); |
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| 160 | * Note: one should avoid compute pi*(-x) directly in the |
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| 161 | * computation of sin(pi*(-x)). |
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| 162 | * |
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| 163 | * 5. Special Cases |
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| 164 | * lgamma(2+s) ~ s*(1-Euler) for tiny s |
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| 165 | * lgamma(1)=lgamma(2)=0 |
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| 166 | * lgamma(x) ~ -log(x) for tiny x |
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| 167 | * lgamma(0) = lgamma(inf) = inf |
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| 168 | * lgamma(-integer) = +-inf |
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| 169 | * |
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| 170 | */ |
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| 171 | |
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| 172 | #include "fdlibm.h" |
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| 173 | |
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| 174 | #ifdef __STDC__ |
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| 175 | static const double |
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| 176 | #else |
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| 177 | static double |
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| 178 | #endif |
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| 179 | two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */ |
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| 180 | half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ |
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| 181 | one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ |
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| 182 | pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ |
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| 183 | a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ |
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| 184 | a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ |
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| 185 | a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ |
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| 186 | a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ |
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| 187 | a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ |
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| 188 | a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ |
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| 189 | a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ |
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| 190 | a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ |
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| 191 | a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ |
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| 192 | a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ |
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| 193 | a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ |
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| 194 | a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ |
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| 195 | tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ |
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| 196 | tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ |
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| 197 | /* tt = -(tail of tf) */ |
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| 198 | tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ |
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| 199 | t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ |
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| 200 | t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ |
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| 201 | t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ |
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| 202 | t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ |
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| 203 | t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ |
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| 204 | t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ |
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| 205 | t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ |
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| 206 | t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ |
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| 207 | t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ |
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| 208 | t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ |
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| 209 | t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ |
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| 210 | t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ |
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| 211 | t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ |
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| 212 | t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ |
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| 213 | t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ |
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| 214 | u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ |
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| 215 | u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ |
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| 216 | u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ |
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| 217 | u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ |
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| 218 | u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ |
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| 219 | u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ |
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| 220 | v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ |
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| 221 | v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ |
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| 222 | v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ |
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| 223 | v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ |
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| 224 | v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ |
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| 225 | s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ |
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| 226 | s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ |
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| 227 | s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ |
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| 228 | s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ |
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| 229 | s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ |
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| 230 | s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ |
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| 231 | s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ |
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| 232 | r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ |
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| 233 | r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ |
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| 234 | r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ |
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| 235 | r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ |
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| 236 | r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ |
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| 237 | r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ |
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| 238 | w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ |
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| 239 | w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ |
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| 240 | w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ |
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| 241 | w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ |
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| 242 | w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ |
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| 243 | w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ |
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| 244 | w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ |
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| 245 | |
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| 246 | #ifdef __STDC__ |
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| 247 | static const double zero= 0.00000000000000000000e+00; |
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| 248 | #else |
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| 249 | static double zero= 0.00000000000000000000e+00; |
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| 250 | #endif |
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| 251 | |
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| 252 | #ifdef __STDC__ |
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| 253 | static double sin_pi(double x) |
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| 254 | #else |
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| 255 | static double sin_pi(x) |
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| 256 | double x; |
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| 257 | #endif |
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| 258 | { |
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| 259 | double y,z; |
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| 260 | __int32_t n,ix; |
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| 261 | |
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| 262 | GET_HIGH_WORD(ix,x); |
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| 263 | ix &= 0x7fffffff; |
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| 264 | |
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| 265 | if(ix<0x3fd00000) return sin(pi*x); |
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| 266 | y = -x; /* x is assume negative */ |
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| 267 | |
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| 268 | /* |
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| 269 | * argument reduction, make sure inexact flag not raised if input |
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| 270 | * is an integer |
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| 271 | */ |
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| 272 | z = floor(y); |
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| 273 | if(z!=y) { /* inexact anyway */ |
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| 274 | y *= 0.5; |
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| 275 | y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */ |
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| 276 | n = (__int32_t) (y*4.0); |
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| 277 | } else { |
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| 278 | if(ix>=0x43400000) { |
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| 279 | y = zero; n = 0; /* y must be even */ |
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| 280 | } else { |
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| 281 | if(ix<0x43300000) z = y+two52; /* exact */ |
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| 282 | GET_LOW_WORD(n,z); |
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| 283 | n &= 1; |
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| 284 | y = n; |
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| 285 | n<<= 2; |
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| 286 | } |
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| 287 | } |
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| 288 | switch (n) { |
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| 289 | case 0: y = sin(pi*y); break; |
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| 290 | case 1: |
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| 291 | case 2: y = cos(pi*(0.5-y)); break; |
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| 292 | case 3: |
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| 293 | case 4: y = sin(pi*(one-y)); break; |
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| 294 | case 5: |
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| 295 | case 6: y = -cos(pi*(y-1.5)); break; |
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| 296 | default: y = sin(pi*(y-2.0)); break; |
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| 297 | } |
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| 298 | return -y; |
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| 299 | } |
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| 300 | |
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| 301 | |
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| 302 | #ifdef __STDC__ |
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| 303 | double lgamma_r(double x, int *signgamp) |
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| 304 | #else |
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| 305 | double lgamma_r(x,signgamp) |
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| 306 | double x; int *signgamp; |
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| 307 | #endif |
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| 308 | { |
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| 309 | double t,y,z,nadj,p,p1,p2,p3,q,r,w; |
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| 310 | __int32_t i,hx,lx,ix; |
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| 311 | |
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| 312 | nadj = 0; |
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| 313 | |
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| 314 | EXTRACT_WORDS(hx,lx,x); |
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| 315 | |
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| 316 | /* purge off +-inf, NaN, +-0, and negative arguments */ |
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| 317 | *signgamp = 1; |
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| 318 | ix = hx&0x7fffffff; |
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| 319 | if(ix>=0x7ff00000) return x*x; |
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| 320 | if((ix|lx)==0) return one/zero; |
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| 321 | if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */ |
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| 322 | if(hx<0) { |
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| 323 | *signgamp = -1; |
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| 324 | return -log(-x); |
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| 325 | } else return -log(x); |
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| 326 | } |
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| 327 | if(hx<0) { |
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| 328 | if(ix>=0x43300000) /* |x|>=2**52, must be -integer */ |
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| 329 | return one/zero; |
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| 330 | t = sin_pi(x); |
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| 331 | if(t==zero) return one/zero; /* -integer */ |
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| 332 | nadj = log(pi/fabs(t*x)); |
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| 333 | if(t<zero) *signgamp = -1; |
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| 334 | x = -x; |
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| 335 | } |
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| 336 | |
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| 337 | /* purge off 1 and 2 */ |
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| 338 | if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0; |
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| 339 | /* for x < 2.0 */ |
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| 340 | else if(ix<0x40000000) { |
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| 341 | if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ |
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| 342 | r = -log(x); |
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| 343 | if(ix>=0x3FE76944) {y = one-x; i= 0;} |
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| 344 | else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;} |
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| 345 | else {y = x; i=2;} |
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| 346 | } else { |
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| 347 | r = zero; |
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| 348 | if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */ |
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| 349 | else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */ |
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| 350 | else {y=x-one;i=2;} |
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| 351 | } |
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| 352 | switch(i) { |
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| 353 | case 0: |
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| 354 | z = y*y; |
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| 355 | p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); |
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| 356 | p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); |
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| 357 | p = y*p1+p2; |
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| 358 | r += (p-0.5*y); break; |
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| 359 | case 1: |
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| 360 | z = y*y; |
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| 361 | w = z*y; |
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| 362 | p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ |
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| 363 | p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); |
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| 364 | p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); |
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| 365 | p = z*p1-(tt-w*(p2+y*p3)); |
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| 366 | r += (tf + p); break; |
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| 367 | case 2: |
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| 368 | p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); |
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| 369 | p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); |
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| 370 | r += (-0.5*y + p1/p2); |
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| 371 | } |
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| 372 | } |
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| 373 | else if(ix<0x40200000) { /* x < 8.0 */ |
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| 374 | i = (__int32_t)x; |
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| 375 | t = zero; |
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| 376 | y = x-(double)i; |
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| 377 | p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); |
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| 378 | q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); |
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| 379 | r = half*y+p/q; |
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| 380 | z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ |
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| 381 | switch(i) { |
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| 382 | case 7: z *= (y+6.0); /* FALLTHRU */ |
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| 383 | case 6: z *= (y+5.0); /* FALLTHRU */ |
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| 384 | case 5: z *= (y+4.0); /* FALLTHRU */ |
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| 385 | case 4: z *= (y+3.0); /* FALLTHRU */ |
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| 386 | case 3: z *= (y+2.0); /* FALLTHRU */ |
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| 387 | r += log(z); break; |
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| 388 | } |
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| 389 | /* 8.0 <= x < 2**58 */ |
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| 390 | } else if (ix < 0x43900000) { |
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| 391 | t = log(x); |
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| 392 | z = one/x; |
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| 393 | y = z*z; |
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| 394 | w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); |
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| 395 | r = (x-half)*(t-one)+w; |
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| 396 | } else |
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| 397 | /* 2**58 <= x <= inf */ |
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| 398 | r = x*(log(x)-one); |
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| 399 | if(hx<0) r = nadj - r; |
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| 400 | return r; |
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| 401 | } |
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| 402 | |
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| 403 | double |
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| 404 | lgamma(double x) |
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| 405 | { |
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| 406 | return lgamma_r(x, &(_REENT_SIGNGAM(_REENT))); |
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| 407 | } |
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