| 1 | /* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */
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| 2 | /*
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| 3 | * ====================================================
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| 4 | * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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| 5 | *
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| 6 | * Developed at SunSoft, a Sun Microsystems, Inc. business.
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| 7 | * Permission to use, copy, modify, and distribute this
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| 8 | * software is freely granted, provided that this notice
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| 9 | * is preserved.
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| 10 | * ====================================================
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| 11 | *
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| 12 | */
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| 13 | /* lgamma_r(x, signgamp)
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| 14 | * Reentrant version of the logarithm of the Gamma function
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| 15 | * with user provide pointer for the sign of Gamma(x).
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| 16 | *
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| 17 | * Method:
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| 18 | * 1. Argument Reduction for 0 < x <= 8
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| 19 | * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may
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| 20 | * reduce x to a number in [1.5,2.5] by
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| 21 | * lgamma(1+s) = log(s) + lgamma(s)
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| 22 | * for example,
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| 23 | * lgamma(7.3) = log(6.3) + lgamma(6.3)
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| 24 | * = log(6.3*5.3) + lgamma(5.3)
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| 25 | * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3)
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| 26 | * 2. Polynomial approximation of lgamma around its
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| 27 | * minimun ymin=1.461632144968362245 to maintain monotonicity.
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| 28 | * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use
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| 29 | * Let z = x-ymin;
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| 30 | * lgamma(x) = -1.214862905358496078218 + z^2*poly(z)
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| 31 | * where
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| 32 | * poly(z) is a 14 degree polynomial.
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| 33 | * 2. Rational approximation in the primary interval [2,3]
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| 34 | * We use the following approximation:
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| 35 | * s = x-2.0;
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| 36 | * lgamma(x) = 0.5*s + s*P(s)/Q(s)
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| 37 | * with accuracy
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| 38 | * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71
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| 39 | * Our algorithms are based on the following observation
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| 40 | *
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| 41 | * zeta(2)-1 2 zeta(3)-1 3
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| 42 | * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ...
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| 43 | * 2 3
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| 44 | *
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| 45 | * where Euler = 0.5771... is the Euler constant, which is very
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| 46 | * close to 0.5.
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| 47 | *
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| 48 | * 3. For x>=8, we have
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| 49 | * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+....
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| 50 | * (better formula:
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| 51 | * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...)
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| 52 | * Let z = 1/x, then we approximation
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| 53 | * f(z) = lgamma(x) - (x-0.5)(log(x)-1)
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| 54 | * by
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| 55 | * 3 5 11
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| 56 | * w = w0 + w1*z + w2*z + w3*z + ... + w6*z
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| 57 | * where
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| 58 | * |w - f(z)| < 2**-58.74
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| 59 | *
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| 60 | * 4. For negative x, since (G is gamma function)
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| 61 | * -x*G(-x)*G(x) = pi/sin(pi*x),
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| 62 | * we have
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| 63 | * G(x) = pi/(sin(pi*x)*(-x)*G(-x))
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| 64 | * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0
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| 65 | * Hence, for x<0, signgam = sign(sin(pi*x)) and
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| 66 | * lgamma(x) = log(|Gamma(x)|)
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| 67 | * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x);
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| 68 | * Note: one should avoid compute pi*(-x) directly in the
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| 69 | * computation of sin(pi*(-x)).
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| 70 | *
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| 71 | * 5. Special Cases
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| 72 | * lgamma(2+s) ~ s*(1-Euler) for tiny s
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| 73 | * lgamma(1) = lgamma(2) = 0
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| 74 | * lgamma(x) ~ -log(|x|) for tiny x
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| 75 | * lgamma(0) = lgamma(neg.integer) = inf and raise divide-by-zero
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| 76 | * lgamma(inf) = inf
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| 77 | * lgamma(-inf) = inf (bug for bug compatible with C99!?)
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| 78 | *
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| 79 | */
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| 80 |
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| 81 | #include "libm.h"
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| 82 | #include "libc.h"
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| 83 |
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| 84 | static const double
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| 85 | pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */
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| 86 | a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */
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| 87 | a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */
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| 88 | a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */
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| 89 | a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */
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| 90 | a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */
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| 91 | a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */
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| 92 | a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */
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| 93 | a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */
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| 94 | a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */
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| 95 | a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */
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| 96 | a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */
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| 97 | a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */
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| 98 | tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */
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| 99 | tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */
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| 100 | /* tt = -(tail of tf) */
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| 101 | tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */
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| 102 | t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */
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| 103 | t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */
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| 104 | t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */
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| 105 | t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */
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| 106 | t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */
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| 107 | t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */
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| 108 | t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */
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| 109 | t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */
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| 110 | t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */
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| 111 | t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */
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| 112 | t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */
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| 113 | t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */
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| 114 | t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */
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| 115 | t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */
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| 116 | t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */
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| 117 | u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
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| 118 | u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */
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| 119 | u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */
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| 120 | u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */
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| 121 | u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */
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| 122 | u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */
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| 123 | v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */
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| 124 | v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */
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| 125 | v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */
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| 126 | v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */
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| 127 | v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */
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| 128 | s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */
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| 129 | s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */
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| 130 | s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */
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| 131 | s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */
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| 132 | s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */
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| 133 | s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */
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| 134 | s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */
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| 135 | r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */
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| 136 | r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */
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| 137 | r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */
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| 138 | r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */
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| 139 | r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */
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| 140 | r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */
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| 141 | w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */
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| 142 | w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */
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| 143 | w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */
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| 144 | w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */
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| 145 | w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */
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| 146 | w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */
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| 147 | w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */
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| 148 |
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| 149 | /* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */
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| 150 | static double sin_pi(double x)
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| 151 | {
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| 152 | int n;
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| 153 |
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| 154 | /* spurious inexact if odd int */
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| 155 | x = 2.0*(x*0.5 - floor(x*0.5)); /* x mod 2.0 */
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| 156 |
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| 157 | n = (int)(x*4.0);
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| 158 | n = (n+1)/2;
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| 159 | x -= n*0.5f;
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| 160 | x *= pi;
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| 161 |
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| 162 | switch (n) {
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| 163 | default: /* case 4: */
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| 164 | case 0: return __sin(x, 0.0, 0);
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| 165 | case 1: return __cos(x, 0.0);
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| 166 | case 2: return __sin(-x, 0.0, 0);
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| 167 | case 3: return -__cos(x, 0.0);
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| 168 | }
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| 169 | }
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| 170 |
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| 171 | double __lgamma_r(double x, int *signgamp)
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| 172 | {
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| 173 | union {double f; uint64_t i;} u = {x};
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| 174 | double_t t,y,z,nadj,p,p1,p2,p3,q,r,w;
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| 175 | uint32_t ix;
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| 176 | int sign,i;
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| 177 |
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| 178 | /* purge off +-inf, NaN, +-0, tiny and negative arguments */
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| 179 | *signgamp = 1;
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| 180 | sign = u.i>>63;
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| 181 | ix = u.i>>32 & 0x7fffffff;
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| 182 | if (ix >= 0x7ff00000)
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| 183 | return x*x;
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| 184 | if (ix < (0x3ff-70)<<20) { /* |x|<2**-70, return -log(|x|) */
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| 185 | if(sign) {
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| 186 | x = -x;
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| 187 | *signgamp = -1;
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| 188 | }
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| 189 | return -log(x);
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| 190 | }
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| 191 | if (sign) {
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| 192 | x = -x;
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| 193 | t = sin_pi(x);
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| 194 | if (t == 0.0) /* -integer */
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| 195 | return 1.0/(x-x);
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| 196 | if (t > 0.0)
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| 197 | *signgamp = -1;
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| 198 | else
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| 199 | t = -t;
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| 200 | nadj = log(pi/(t*x));
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| 201 | }
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| 202 |
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| 203 | /* purge off 1 and 2 */
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| 204 | if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t)u.i == 0)
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| 205 | r = 0;
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| 206 | /* for x < 2.0 */
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| 207 | else if (ix < 0x40000000) {
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| 208 | if (ix <= 0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */
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| 209 | r = -log(x);
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| 210 | if (ix >= 0x3FE76944) {
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| 211 | y = 1.0 - x;
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| 212 | i = 0;
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| 213 | } else if (ix >= 0x3FCDA661) {
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| 214 | y = x - (tc-1.0);
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| 215 | i = 1;
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| 216 | } else {
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| 217 | y = x;
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| 218 | i = 2;
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| 219 | }
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| 220 | } else {
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| 221 | r = 0.0;
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| 222 | if (ix >= 0x3FFBB4C3) { /* [1.7316,2] */
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| 223 | y = 2.0 - x;
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| 224 | i = 0;
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| 225 | } else if(ix >= 0x3FF3B4C4) { /* [1.23,1.73] */
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| 226 | y = x - tc;
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| 227 | i = 1;
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| 228 | } else {
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| 229 | y = x - 1.0;
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| 230 | i = 2;
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| 231 | }
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| 232 | }
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| 233 | switch (i) {
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| 234 | case 0:
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| 235 | z = y*y;
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| 236 | p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10))));
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| 237 | p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11)))));
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| 238 | p = y*p1+p2;
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| 239 | r += (p-0.5*y);
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| 240 | break;
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| 241 | case 1:
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| 242 | z = y*y;
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| 243 | w = z*y;
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| 244 | p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */
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| 245 | p2 = t1+w*(t4+w*(t7+w*(t10+w*t13)));
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| 246 | p3 = t2+w*(t5+w*(t8+w*(t11+w*t14)));
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| 247 | p = z*p1-(tt-w*(p2+y*p3));
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| 248 | r += tf + p;
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| 249 | break;
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| 250 | case 2:
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| 251 | p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5)))));
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| 252 | p2 = 1.0+y*(v1+y*(v2+y*(v3+y*(v4+y*v5))));
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| 253 | r += -0.5*y + p1/p2;
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| 254 | }
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| 255 | } else if (ix < 0x40200000) { /* x < 8.0 */
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| 256 | i = (int)x;
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| 257 | y = x - (double)i;
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| 258 | p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6))))));
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| 259 | q = 1.0+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6)))));
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| 260 | r = 0.5*y+p/q;
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| 261 | z = 1.0; /* lgamma(1+s) = log(s) + lgamma(s) */
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| 262 | switch (i) {
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| 263 | case 7: z *= y + 6.0; /* FALLTHRU */
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| 264 | case 6: z *= y + 5.0; /* FALLTHRU */
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| 265 | case 5: z *= y + 4.0; /* FALLTHRU */
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| 266 | case 4: z *= y + 3.0; /* FALLTHRU */
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| 267 | case 3: z *= y + 2.0; /* FALLTHRU */
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| 268 | r += log(z);
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| 269 | break;
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| 270 | }
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| 271 | } else if (ix < 0x43900000) { /* 8.0 <= x < 2**58 */
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| 272 | t = log(x);
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| 273 | z = 1.0/x;
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| 274 | y = z*z;
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| 275 | w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6)))));
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| 276 | r = (x-0.5)*(t-1.0)+w;
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| 277 | } else /* 2**58 <= x <= inf */
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| 278 | r = x*(log(x)-1.0);
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| 279 | if (sign)
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| 280 | r = nadj - r;
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| 281 | return r;
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| 282 | }
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| 283 |
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| 284 | weak_alias(__lgamma_r, lgamma_r);
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