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/* @(#)er_lgamma.c 5.1 93/09/24 */ |
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/* |
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* ==================================================== |
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
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* |
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* Developed at SunPro, a Sun Microsystems, Inc. business. |
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* Permission to use, copy, modify, and distribute this |
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* software is freely granted, provided that this notice |
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* is preserved. |
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* ==================================================== |
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*/ |
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/* lgamma_r(x, signgamp) |
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* Reentrant version of the logarithm of the Gamma function |
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* with user provide pointer for the sign of Gamma(x). |
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* |
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* Method: |
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* 1. Argument Reduction for 0 < x <= 8 |
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* Since gamma(1+s)=s*gamma(s), for x in [0,8], we may |
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* reduce x to a number in [1.5,2.5] by |
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* lgamma(1+s) = log(s) + lgamma(s) |
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* for example, |
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* lgamma(7.3) = log(6.3) + lgamma(6.3) |
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* = log(6.3*5.3) + lgamma(5.3) |
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* = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) |
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* 2. Polynomial approximation of lgamma around its |
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* minimun ymin=1.461632144968362245 to maintain monotonicity. |
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* On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use |
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* Let z = x-ymin; |
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* lgamma(x) = -1.214862905358496078218 + z^2*poly(z) |
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* where |
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* poly(z) is a 14 degree polynomial. |
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* 2. Rational approximation in the primary interval [2,3] |
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* We use the following approximation: |
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* s = x-2.0; |
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* lgamma(x) = 0.5*s + s*P(s)/Q(s) |
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* with accuracy |
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* |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 |
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* Our algorithms are based on the following observation |
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* |
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* zeta(2)-1 2 zeta(3)-1 3 |
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* lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... |
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* 2 3 |
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* |
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* where Euler = 0.5771... is the Euler constant, which is very |
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* close to 0.5. |
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* |
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* 3. For x>=8, we have |
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* lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... |
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* (better formula: |
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* lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) |
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* Let z = 1/x, then we approximation |
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* f(z) = lgamma(x) - (x-0.5)(log(x)-1) |
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* by |
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* 3 5 11 |
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* w = w0 + w1*z + w2*z + w3*z + ... + w6*z |
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* where |
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* |w - f(z)| < 2**-58.74 |
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* |
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* 4. For negative x, since (G is gamma function) |
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* -x*G(-x)*G(x) = pi/sin(pi*x), |
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* we have |
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* G(x) = pi/(sin(pi*x)*(-x)*G(-x)) |
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* since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 |
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* Hence, for x<0, signgam = sign(sin(pi*x)) and |
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* lgamma(x) = log(|Gamma(x)|) |
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* = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); |
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* Note: one should avoid compute pi*(-x) directly in the |
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* computation of sin(pi*(-x)). |
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* |
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* 5. Special Cases |
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* lgamma(2+s) ~ s*(1-Euler) for tiny s |
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* lgamma(1)=lgamma(2)=0 |
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* lgamma(x) ~ -log(x) for tiny x |
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* lgamma(0) = lgamma(inf) = inf |
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* lgamma(-integer) = +-inf |
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* |
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*/ |
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#include "math.h" |
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#include "math_private.h" |
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static const double |
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two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */ |
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half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ |
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one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ |
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pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ |
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a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ |
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a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ |
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a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ |
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a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ |
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a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ |
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a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ |
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a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ |
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a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ |
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a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ |
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a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ |
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a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ |
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a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ |
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tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ |
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tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ |
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/* tt = -(tail of tf) */ |
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tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ |
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t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ |
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t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ |
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t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ |
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t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ |
108 |
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t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ |
109 |
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t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ |
110 |
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t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ |
111 |
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t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ |
112 |
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t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ |
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t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ |
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t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ |
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t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ |
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t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ |
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t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ |
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t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ |
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u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ |
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u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ |
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u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ |
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u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ |
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u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ |
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u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ |
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v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ |
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v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ |
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v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ |
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v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ |
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v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ |
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s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ |
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s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ |
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s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ |
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s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ |
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s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ |
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s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ |
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s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ |
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r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ |
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r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ |
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r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ |
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r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ |
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r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ |
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r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ |
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w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ |
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w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ |
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w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ |
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w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ |
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w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ |
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w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ |
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w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ |
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static const double zero= 0.00000000000000000000e+00; |
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static double |
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sin_pi(double x) |
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{ |
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double y,z; |
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int n,ix; |
158 |
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159 |
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11580 |
GET_HIGH_WORD(ix,x); |
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ix &= 0x7fffffff; |
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✓✓ |
7540 |
if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0); |
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y = -x; /* x is assume negative */ |
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/* |
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* argument reduction, make sure inexact flag not raised if input |
167 |
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* is an integer |
168 |
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*/ |
169 |
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4040 |
z = floor(y); |
170 |
✓✗ |
4040 |
if(z!=y) { /* inexact anyway */ |
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4040 |
y *= 0.5; |
172 |
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y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */ |
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4040 |
n = (int) (y*4.0); |
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4040 |
} else { |
175 |
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if(ix>=0x43400000) { |
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y = zero; n = 0; /* y must be even */ |
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} else { |
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if(ix<0x43300000) z = y+two52; /* exact */ |
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GET_LOW_WORD(n,z); |
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n &= 1; |
181 |
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y = n; |
182 |
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n<<= 2; |
183 |
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} |
184 |
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} |
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✓✗✓✗
✓✗✓✓
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4040 |
switch (n) { |
186 |
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case 0: y = __kernel_sin(pi*y,zero,0); break; |
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case 1: |
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case 2: y = __kernel_cos(pi*(0.5-y),zero); break; |
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case 3: |
190 |
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1620 |
case 4: y = __kernel_sin(pi*(one-y),zero,0); break; |
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case 5: |
192 |
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1050 |
case 6: y = -__kernel_cos(pi*(y-1.5),zero); break; |
193 |
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780 |
default: y = __kernel_sin(pi*(y-2.0),zero,0); break; |
194 |
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} |
195 |
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4040 |
return -y; |
196 |
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5790 |
} |
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double |
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lgamma_r(double x, int *signgamp) |
201 |
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{ |
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double t,y,z,nadj,p,p1,p2,p3,q,r,w; |
203 |
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int i,hx,lx,ix; |
204 |
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205 |
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59730 |
EXTRACT_WORDS(hx,lx,x); |
206 |
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207 |
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/* purge off +-inf, NaN, +-0, and negative arguments */ |
208 |
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29865 |
*signgamp = 1; |
209 |
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29865 |
ix = hx&0x7fffffff; |
210 |
✓✓ |
29880 |
if(ix>=0x7ff00000) return x*x; |
211 |
✓✓ |
29850 |
if((ix|lx)==0) { |
212 |
✓✓ |
10 |
if(hx<0) |
213 |
|
5 |
*signgamp = -1; |
214 |
|
10 |
return one/zero; |
215 |
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} |
216 |
✓✓ |
29840 |
if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */ |
217 |
✓✓ |
80 |
if(hx<0) { |
218 |
|
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*signgamp = -1; |
219 |
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return - log(-x); |
220 |
|
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} else return - log(x); |
221 |
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} |
222 |
✓✓ |
29760 |
if(hx<0) { |
223 |
✗✓ |
5790 |
if(ix>=0x43300000) /* |x|>=2**52, must be -integer */ |
224 |
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return one/zero; |
225 |
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5790 |
t = sin_pi(x); |
226 |
✗✓ |
5790 |
if(t==zero) return one/zero; /* -integer */ |
227 |
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5790 |
nadj = log(pi/fabs(t*x)); |
228 |
✓✓ |
8610 |
if(t<zero) *signgamp = -1; |
229 |
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5790 |
x = -x; |
230 |
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5790 |
} |
231 |
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232 |
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/* purge off 1 and 2 */ |
233 |
✓✓✓✓
|
59530 |
if((((ix-0x3ff00000)|lx)==0)||(((ix-0x40000000)|lx)==0)) r = 0; |
234 |
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/* for x < 2.0 */ |
235 |
✓✓ |
29735 |
else if(ix<0x40000000) { |
236 |
✓✓ |
18490 |
if(ix<=0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ |
237 |
|
7140 |
r = - log(x); |
238 |
✓✓ |
7580 |
if(ix>=0x3FE76944) {y = one-x; i= 0;} |
239 |
✓✓ |
9020 |
else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;} |
240 |
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else {y = x; i=2;} |
241 |
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} else { |
242 |
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r = zero; |
243 |
✓✓ |
14280 |
if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */ |
244 |
✓✓ |
10980 |
else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */ |
245 |
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5860 |
else {y=x-one;i=2;} |
246 |
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} |
247 |
✓✓✓✓
|
37000 |
switch(i) { |
248 |
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case 0: |
249 |
|
3370 |
z = y*y; |
250 |
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3370 |
p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); |
251 |
|
3370 |
p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); |
252 |
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3370 |
p = y*p1+p2; |
253 |
|
3370 |
r += (p-0.5*y); break; |
254 |
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case 1: |
255 |
|
4880 |
z = y*y; |
256 |
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4880 |
w = z*y; |
257 |
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4880 |
p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ |
258 |
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4880 |
p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); |
259 |
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4880 |
p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); |
260 |
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4880 |
p = z*p1-(tt-w*(p2+y*p3)); |
261 |
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4880 |
r += (tf + p); break; |
262 |
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case 2: |
263 |
|
10240 |
p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); |
264 |
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10240 |
p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); |
265 |
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10240 |
r += (-0.5*y + p1/p2); |
266 |
|
10240 |
} |
267 |
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} |
268 |
✓✓ |
11245 |
else if(ix<0x40200000) { /* x < 8.0 */ |
269 |
|
16510 |
i = (int)x; |
270 |
|
|
t = zero; |
271 |
|
16510 |
y = x-(double)i; |
272 |
|
16510 |
p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); |
273 |
|
16510 |
q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); |
274 |
|
16510 |
r = half*y+p/q; |
275 |
|
|
z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ |
276 |
✓✓✓✓
✓✓ |
16510 |
switch(i) { |
277 |
|
320 |
case 7: z *= (y+6.0); /* FALLTHRU */ |
278 |
|
430 |
case 6: z *= (y+5.0); /* FALLTHRU */ |
279 |
|
690 |
case 5: z *= (y+4.0); /* FALLTHRU */ |
280 |
|
970 |
case 4: z *= (y+3.0); /* FALLTHRU */ |
281 |
|
2855 |
case 3: z *= (y+2.0); /* FALLTHRU */ |
282 |
|
2855 |
r += log(z); break; |
283 |
|
|
} |
284 |
|
|
/* 8.0 <= x < 2**58 */ |
285 |
✓✗ |
5750 |
} else if (ix < 0x43900000) { |
286 |
|
5750 |
t = log(x); |
287 |
|
5750 |
z = one/x; |
288 |
|
5750 |
y = z*z; |
289 |
|
5750 |
w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); |
290 |
|
5750 |
r = (x-half)*(t-one)+w; |
291 |
|
5750 |
} else |
292 |
|
|
/* 2**58 <= x <= inf */ |
293 |
|
|
r = x*(log(x)-one); |
294 |
✓✓ |
35550 |
if(hx<0) r = nadj - r; |
295 |
|
29760 |
return r; |
296 |
|
29865 |
} |
297 |
|
|
DEF_NONSTD(lgamma_r); |