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Diffstat (limited to 'lib/libm/lgamma_r.c')
| -rw-r--r-- | lib/libm/lgamma_r.c | 314 |
1 files changed, 314 insertions, 0 deletions
diff --git a/lib/libm/lgamma_r.c b/lib/libm/lgamma_r.c new file mode 100644 index 00000000..00eeb27b --- /dev/null +++ b/lib/libm/lgamma_r.c @@ -0,0 +1,314 @@ +/* origin: FreeBSD /usr/src/lib/msun/src/e_lgamma_r.c */ +/* + * ==================================================== + * 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. + * ==================================================== + * + */ +/* 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(neg.integer) = inf and raise divide-by-zero + * lgamma(inf) = inf + * lgamma(-inf) = inf (bug for bug compatible with C99!?) + * + */ + +#include "libm.h" + +static const double 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 */ + +/* sin(pi*x) assuming x > 2^-100, if sin(pi*x)==0 the sign is arbitrary */ +static double sin_pi(double x) +{ + int n; + + /* spurious inexact if odd int */ + x = 2.0 * (x * 0.5 - floor(x * 0.5)); /* x mod 2.0 */ + + n = (int)(x * 4.0); + n = (n + 1) / 2; + x -= n * 0.5f; + x *= pi; + + switch (n) { + default: /* case 4: */ + case 0: + return __sin(x, 0.0, 0); + case 1: + return __cos(x, 0.0); + case 2: + return __sin(-x, 0.0, 0); + case 3: + return -__cos(x, 0.0); + } +} + +double __lgamma_r(double x, int *signgamp) +{ + union { + double f; + uint64_t i; + } u = { x }; + double_t t, y, z, nadj, p, p1, p2, p3, q, r, w; + uint32_t ix; + int sign, i; + + /* purge off +-inf, NaN, +-0, tiny and negative arguments */ + *signgamp = 1; + sign = u.i >> 63; + ix = u.i >> 32 & 0x7fffffff; + if (ix >= 0x7ff00000) + return x * x; + if (ix < (0x3ff - 70) << 20) { /* |x|<2**-70, return -log(|x|) */ + if (sign) { + x = -x; + *signgamp = -1; + } + return -log(x); + } + if (sign) { + x = -x; + t = sin_pi(x); + if (t == 0.0) /* -integer */ + return 1.0 / (x - x); + if (t > 0.0) + *signgamp = -1; + else + t = -t; + nadj = log(pi / (t * x)); + } + + /* purge off 1 and 2 */ + if ((ix == 0x3ff00000 || ix == 0x40000000) && (uint32_t)u.i == 0) + r = 0; + /* for x < 2.0 */ + else if (ix < 0x40000000) { + if (ix <= 0x3feccccc) { /* lgamma(x) = lgamma(x+1)-log(x) */ + r = -log(x); + if (ix >= 0x3FE76944) { + y = 1.0 - x; + i = 0; + } else if (ix >= 0x3FCDA661) { + y = x - (tc - 1.0); + i = 1; + } else { + y = x; + i = 2; + } + } else { + r = 0.0; + if (ix >= 0x3FFBB4C3) { /* [1.7316,2] */ + y = 2.0 - x; + i = 0; + } else if (ix >= 0x3FF3B4C4) { /* [1.23,1.73] */ + y = x - tc; + i = 1; + } else { + y = x - 1.0; + 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 = 1.0 + + 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; + y = x - (double)i; + p = y * + (s0 + + y * (s1 + + y * (s2 + y * (s3 + y * (s4 + y * (s5 + y * s6)))))); + q = 1.0 + + y * (r1 + + y * (r2 + y * (r3 + y * (r4 + y * (r5 + y * r6))))); + r = 0.5 * y + p / q; + z = 1.0; /* 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 += log(z); + break; + } + } else if (ix < 0x43900000) { /* 8.0 <= x < 2**58 */ + t = log(x); + z = 1.0 / x; + y = z * z; + w = w0 + + z * (w1 + + y * (w2 + y * (w3 + y * (w4 + y * (w5 + y * w6))))); + r = (x - 0.5) * (t - 1.0) + w; + } else /* 2**58 <= x <= inf */ + r = x * (log(x) - 1.0); + if (sign) + r = nadj - r; + return r; +} + +weak_alias(__lgamma_r, lgamma_r); |
