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 /* @(#)e_lgamma_r.c 1.3 95/01/18 */ /* * ==================================================== * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. * * Developed at SunSoft, a Sun Microsystems, Inc. business. * Permission to use, copy, modify, and distribute this * software is freely granted, provided that this notice * is preserved. * ==================================================== * */ #ifndef lint static char rcsid[] = "\$FreeBSD: src/lib/msun/src/e_lgamma_r.c,v 1.8 2005/02/04 18:26:06 das Exp \$"; #endif /* __ieee754_lgamma_r(x, signgamp) * Reentrant version of the logarithm of the Gamma function * with user provide pointer for the sign of Gamma(x). * * Method: * 1. Argument Reduction for 0 < x <= 8 * Since gamma(1+s)=s*gamma(s), for x in [0,8], we may * reduce x to a number in [1.5,2.5] by * lgamma(1+s) = log(s) + lgamma(s) * for example, * lgamma(7.3) = log(6.3) + lgamma(6.3) * = log(6.3*5.3) + lgamma(5.3) * = log(6.3*5.3*4.3*3.3*2.3) + lgamma(2.3) * 2. Polynomial approximation of lgamma around its * minimun ymin=1.461632144968362245 to maintain monotonicity. * On [ymin-0.23, ymin+0.27] (i.e., [1.23164,1.73163]), use * Let z = x-ymin; * lgamma(x) = -1.214862905358496078218 + z^2*poly(z) * where * poly(z) is a 14 degree polynomial. * 2. Rational approximation in the primary interval [2,3] * We use the following approximation: * s = x-2.0; * lgamma(x) = 0.5*s + s*P(s)/Q(s) * with accuracy * |P/Q - (lgamma(x)-0.5s)| < 2**-61.71 * Our algorithms are based on the following observation * * zeta(2)-1 2 zeta(3)-1 3 * lgamma(2+s) = s*(1-Euler) + --------- * s - --------- * s + ... * 2 3 * * where Euler = 0.5771... is the Euler constant, which is very * close to 0.5. * * 3. For x>=8, we have * lgamma(x)~(x-0.5)log(x)-x+0.5*log(2pi)+1/(12x)-1/(360x**3)+.... * (better formula: * lgamma(x)~(x-0.5)*(log(x)-1)-.5*(log(2pi)-1) + ...) * Let z = 1/x, then we approximation * f(z) = lgamma(x) - (x-0.5)(log(x)-1) * by * 3 5 11 * w = w0 + w1*z + w2*z + w3*z + ... + w6*z * where * |w - f(z)| < 2**-58.74 * * 4. For negative x, since (G is gamma function) * -x*G(-x)*G(x) = pi/sin(pi*x), * we have * G(x) = pi/(sin(pi*x)*(-x)*G(-x)) * since G(-x) is positive, sign(G(x)) = sign(sin(pi*x)) for x<0 * Hence, for x<0, signgam = sign(sin(pi*x)) and * lgamma(x) = log(|Gamma(x)|) * = log(pi/(|x*sin(pi*x)|)) - lgamma(-x); * Note: one should avoid compute pi*(-x) directly in the * computation of sin(pi*(-x)). * * 5. Special Cases * lgamma(2+s) ~ s*(1-Euler) for tiny s * lgamma(1)=lgamma(2)=0 * lgamma(x) ~ -log(x) for tiny x * lgamma(0) = lgamma(inf) = inf * lgamma(-integer) = +-inf * */ #include "math.h" #include "math_private.h" static const double two52= 4.50359962737049600000e+15, /* 0x43300000, 0x00000000 */ half= 5.00000000000000000000e-01, /* 0x3FE00000, 0x00000000 */ one = 1.00000000000000000000e+00, /* 0x3FF00000, 0x00000000 */ pi = 3.14159265358979311600e+00, /* 0x400921FB, 0x54442D18 */ a0 = 7.72156649015328655494e-02, /* 0x3FB3C467, 0xE37DB0C8 */ a1 = 3.22467033424113591611e-01, /* 0x3FD4A34C, 0xC4A60FAD */ a2 = 6.73523010531292681824e-02, /* 0x3FB13E00, 0x1A5562A7 */ a3 = 2.05808084325167332806e-02, /* 0x3F951322, 0xAC92547B */ a4 = 7.38555086081402883957e-03, /* 0x3F7E404F, 0xB68FEFE8 */ a5 = 2.89051383673415629091e-03, /* 0x3F67ADD8, 0xCCB7926B */ a6 = 1.19270763183362067845e-03, /* 0x3F538A94, 0x116F3F5D */ a7 = 5.10069792153511336608e-04, /* 0x3F40B6C6, 0x89B99C00 */ a8 = 2.20862790713908385557e-04, /* 0x3F2CF2EC, 0xED10E54D */ a9 = 1.08011567247583939954e-04, /* 0x3F1C5088, 0x987DFB07 */ a10 = 2.52144565451257326939e-05, /* 0x3EFA7074, 0x428CFA52 */ a11 = 4.48640949618915160150e-05, /* 0x3F07858E, 0x90A45837 */ tc = 1.46163214496836224576e+00, /* 0x3FF762D8, 0x6356BE3F */ tf = -1.21486290535849611461e-01, /* 0xBFBF19B9, 0xBCC38A42 */ /* tt = -(tail of tf) */ tt = -3.63867699703950536541e-18, /* 0xBC50C7CA, 0xA48A971F */ t0 = 4.83836122723810047042e-01, /* 0x3FDEF72B, 0xC8EE38A2 */ t1 = -1.47587722994593911752e-01, /* 0xBFC2E427, 0x8DC6C509 */ t2 = 6.46249402391333854778e-02, /* 0x3FB08B42, 0x94D5419B */ t3 = -3.27885410759859649565e-02, /* 0xBFA0C9A8, 0xDF35B713 */ t4 = 1.79706750811820387126e-02, /* 0x3F9266E7, 0x970AF9EC */ t5 = -1.03142241298341437450e-02, /* 0xBF851F9F, 0xBA91EC6A */ t6 = 6.10053870246291332635e-03, /* 0x3F78FCE0, 0xE370E344 */ t7 = -3.68452016781138256760e-03, /* 0xBF6E2EFF, 0xB3E914D7 */ t8 = 2.25964780900612472250e-03, /* 0x3F6282D3, 0x2E15C915 */ t9 = -1.40346469989232843813e-03, /* 0xBF56FE8E, 0xBF2D1AF1 */ t10 = 8.81081882437654011382e-04, /* 0x3F4CDF0C, 0xEF61A8E9 */ t11 = -5.38595305356740546715e-04, /* 0xBF41A610, 0x9C73E0EC */ t12 = 3.15632070903625950361e-04, /* 0x3F34AF6D, 0x6C0EBBF7 */ t13 = -3.12754168375120860518e-04, /* 0xBF347F24, 0xECC38C38 */ t14 = 3.35529192635519073543e-04, /* 0x3F35FD3E, 0xE8C2D3F4 */ u0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ u1 = 6.32827064025093366517e-01, /* 0x3FE4401E, 0x8B005DFF */ u2 = 1.45492250137234768737e+00, /* 0x3FF7475C, 0xD119BD6F */ u3 = 9.77717527963372745603e-01, /* 0x3FEF4976, 0x44EA8450 */ u4 = 2.28963728064692451092e-01, /* 0x3FCD4EAE, 0xF6010924 */ u5 = 1.33810918536787660377e-02, /* 0x3F8B678B, 0xBF2BAB09 */ v1 = 2.45597793713041134822e+00, /* 0x4003A5D7, 0xC2BD619C */ v2 = 2.12848976379893395361e+00, /* 0x40010725, 0xA42B18F5 */ v3 = 7.69285150456672783825e-01, /* 0x3FE89DFB, 0xE45050AF */ v4 = 1.04222645593369134254e-01, /* 0x3FBAAE55, 0xD6537C88 */ v5 = 3.21709242282423911810e-03, /* 0x3F6A5ABB, 0x57D0CF61 */ s0 = -7.72156649015328655494e-02, /* 0xBFB3C467, 0xE37DB0C8 */ s1 = 2.14982415960608852501e-01, /* 0x3FCB848B, 0x36E20878 */ s2 = 3.25778796408930981787e-01, /* 0x3FD4D98F, 0x4F139F59 */ s3 = 1.46350472652464452805e-01, /* 0x3FC2BB9C, 0xBEE5F2F7 */ s4 = 2.66422703033638609560e-02, /* 0x3F9B481C, 0x7E939961 */ s5 = 1.84028451407337715652e-03, /* 0x3F5E26B6, 0x7368F239 */ s6 = 3.19475326584100867617e-05, /* 0x3F00BFEC, 0xDD17E945 */ r1 = 1.39200533467621045958e+00, /* 0x3FF645A7, 0x62C4AB74 */ r2 = 7.21935547567138069525e-01, /* 0x3FE71A18, 0x93D3DCDC */ r3 = 1.71933865632803078993e-01, /* 0x3FC601ED, 0xCCFBDF27 */ r4 = 1.86459191715652901344e-02, /* 0x3F9317EA, 0x742ED475 */ r5 = 7.77942496381893596434e-04, /* 0x3F497DDA, 0xCA41A95B */ r6 = 7.32668430744625636189e-06, /* 0x3EDEBAF7, 0xA5B38140 */ w0 = 4.18938533204672725052e-01, /* 0x3FDACFE3, 0x90C97D69 */ w1 = 8.33333333333329678849e-02, /* 0x3FB55555, 0x5555553B */ w2 = -2.77777777728775536470e-03, /* 0xBF66C16C, 0x16B02E5C */ w3 = 7.93650558643019558500e-04, /* 0x3F4A019F, 0x98CF38B6 */ w4 = -5.95187557450339963135e-04, /* 0xBF4380CB, 0x8C0FE741 */ w5 = 8.36339918996282139126e-04, /* 0x3F4B67BA, 0x4CDAD5D1 */ w6 = -1.63092934096575273989e-03; /* 0xBF5AB89D, 0x0B9E43E4 */ static const double zero= 0.00000000000000000000e+00; static double sin_pi(double x) { double y,z; int n,ix; GET_HIGH_WORD(ix,x); ix &= 0x7fffffff; if(ix<0x3fd00000) return __kernel_sin(pi*x,zero,0); y = -x; /* x is assume negative */ /* * argument reduction, make sure inexact flag not raised if input * is an integer */ z = floor(y); if(z!=y) { /* inexact anyway */ y *= 0.5; y = 2.0*(y - floor(y)); /* y = |x| mod 2.0 */ n = (int) (y*4.0); } else { if(ix>=0x43400000) { y = zero; n = 0; /* y must be even */ } else { if(ix<0x43300000) z = y+two52; /* exact */ GET_LOW_WORD(n,z); n &= 1; y = n; n<<= 2; } } switch (n) { case 0: y = __kernel_sin(pi*y,zero,0); break; case 1: case 2: y = __kernel_cos(pi*(0.5-y),zero); break; case 3: case 4: y = __kernel_sin(pi*(one-y),zero,0); break; case 5: case 6: y = -__kernel_cos(pi*(y-1.5),zero); break; default: y = __kernel_sin(pi*(y-2.0),zero,0); break; } return -y; } double __ieee754_lgamma_r(double x, int *signgamp) { double t,y,z,nadj,p,p1,p2,p3,q,r,w; int i,hx,lx,ix; EXTRACT_WORDS(hx,lx,x); /* purge off +-inf, NaN, +-0, and negative arguments */ *signgamp = 1; ix = hx&0x7fffffff; if(ix>=0x7ff00000) return x*x; if((ix|lx)==0) return one/zero; if(ix<0x3b900000) { /* |x|<2**-70, return -log(|x|) */ if(hx<0) { *signgamp = -1; return -__ieee754_log(-x); } else return -__ieee754_log(x); } if(hx<0) { if(ix>=0x43300000) /* |x|>=2**52, must be -integer */ return one/zero; t = sin_pi(x); if(t==zero) return one/zero; /* -integer */ nadj = __ieee754_log(pi/fabs(t*x)); if(t=0x3FE76944) {y = one-x; i= 0;} else if(ix>=0x3FCDA661) {y= x-(tc-one); i=1;} else {y = x; i=2;} } else { r = zero; if(ix>=0x3FFBB4C3) {y=2.0-x;i=0;} /* [1.7316,2] */ else if(ix>=0x3FF3B4C4) {y=x-tc;i=1;} /* [1.23,1.73] */ else {y=x-one;i=2;} } switch(i) { case 0: z = y*y; p1 = a0+z*(a2+z*(a4+z*(a6+z*(a8+z*a10)))); p2 = z*(a1+z*(a3+z*(a5+z*(a7+z*(a9+z*a11))))); p = y*p1+p2; r += (p-0.5*y); break; case 1: z = y*y; w = z*y; p1 = t0+w*(t3+w*(t6+w*(t9 +w*t12))); /* parallel comp */ p2 = t1+w*(t4+w*(t7+w*(t10+w*t13))); p3 = t2+w*(t5+w*(t8+w*(t11+w*t14))); p = z*p1-(tt-w*(p2+y*p3)); r += (tf + p); break; case 2: p1 = y*(u0+y*(u1+y*(u2+y*(u3+y*(u4+y*u5))))); p2 = one+y*(v1+y*(v2+y*(v3+y*(v4+y*v5)))); r += (-0.5*y + p1/p2); } } else if(ix<0x40200000) { /* x < 8.0 */ i = (int)x; t = zero; y = x-(double)i; p = y*(s0+y*(s1+y*(s2+y*(s3+y*(s4+y*(s5+y*s6)))))); q = one+y*(r1+y*(r2+y*(r3+y*(r4+y*(r5+y*r6))))); r = half*y+p/q; z = one; /* lgamma(1+s) = log(s) + lgamma(s) */ switch(i) { case 7: z *= (y+6.0); /* FALLTHRU */ case 6: z *= (y+5.0); /* FALLTHRU */ case 5: z *= (y+4.0); /* FALLTHRU */ case 4: z *= (y+3.0); /* FALLTHRU */ case 3: z *= (y+2.0); /* FALLTHRU */ r += __ieee754_log(z); break; } /* 8.0 <= x < 2**58 */ } else if (ix < 0x43900000) { t = __ieee754_log(x); z = one/x; y = z*z; w = w0+z*(w1+y*(w2+y*(w3+y*(w4+y*(w5+y*w6))))); r = (x-half)*(t-one)+w; } else /* 2**58 <= x <= inf */ r = x*(__ieee754_log(x)-one); if(hx<0) r = nadj - r; return r; }