| /* |
| * Copyright (c) 2014 Advanced Micro Devices, Inc. |
| * |
| * Permission is hereby granted, free of charge, to any person obtaining a copy |
| * of this software and associated documentation files (the "Software"), to deal |
| * in the Software without restriction, including without limitation the rights |
| * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell |
| * copies of the Software, and to permit persons to whom the Software is |
| * furnished to do so, subject to the following conditions: |
| * |
| * The above copyright notice and this permission notice shall be included in |
| * all copies or substantial portions of the Software. |
| * |
| * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR |
| * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, |
| * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE |
| * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER |
| * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, |
| * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN |
| * THE SOFTWARE. |
| */ |
| |
| #include "math64.h" |
| |
| // ==================================================== |
| // Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. |
| // |
| // Developed at SunPro, 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, i) |
| // 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 |
| // |
| |
| #define pi 3.14159265358979311600e+00 /* 0x400921FB, 0x54442D18 */ |
| |
| #define a0 7.72156649015328655494e-02 /* 0x3FB3C467, 0xE37DB0C8 */ |
| #define a1 3.22467033424113591611e-01 /* 0x3FD4A34C, 0xC4A60FAD */ |
| #define a2 6.73523010531292681824e-02 /* 0x3FB13E00, 0x1A5562A7 */ |
| #define a3 2.05808084325167332806e-02 /* 0x3F951322, 0xAC92547B */ |
| #define a4 7.38555086081402883957e-03 /* 0x3F7E404F, 0xB68FEFE8 */ |
| #define a5 2.89051383673415629091e-03 /* 0x3F67ADD8, 0xCCB7926B */ |
| #define a6 1.19270763183362067845e-03 /* 0x3F538A94, 0x116F3F5D */ |
| #define a7 5.10069792153511336608e-04 /* 0x3F40B6C6, 0x89B99C00 */ |
| #define a8 2.20862790713908385557e-04 /* 0x3F2CF2EC, 0xED10E54D */ |
| #define a9 1.08011567247583939954e-04 /* 0x3F1C5088, 0x987DFB07 */ |
| #define a10 2.52144565451257326939e-05 /* 0x3EFA7074, 0x428CFA52 */ |
| #define a11 4.48640949618915160150e-05 /* 0x3F07858E, 0x90A45837 */ |
| |
| #define tc 1.46163214496836224576e+00 /* 0x3FF762D8, 0x6356BE3F */ |
| #define tf -1.21486290535849611461e-01 /* 0xBFBF19B9, 0xBCC38A42 */ |
| #define tt -3.63867699703950536541e-18 /* 0xBC50C7CA, 0xA48A971F */ |
| |
| #define t0 4.83836122723810047042e-01 /* 0x3FDEF72B, 0xC8EE38A2 */ |
| #define t1 -1.47587722994593911752e-01 /* 0xBFC2E427, 0x8DC6C509 */ |
| #define t2 6.46249402391333854778e-02 /* 0x3FB08B42, 0x94D5419B */ |
| #define t3 -3.27885410759859649565e-02 /* 0xBFA0C9A8, 0xDF35B713 */ |
| #define t4 1.79706750811820387126e-02 /* 0x3F9266E7, 0x970AF9EC */ |
| #define t5 -1.03142241298341437450e-02 /* 0xBF851F9F, 0xBA91EC6A */ |
| #define t6 6.10053870246291332635e-03 /* 0x3F78FCE0, 0xE370E344 */ |
| #define t7 -3.68452016781138256760e-03 /* 0xBF6E2EFF, 0xB3E914D7 */ |
| #define t8 2.25964780900612472250e-03 /* 0x3F6282D3, 0x2E15C915 */ |
| #define t9 -1.40346469989232843813e-03 /* 0xBF56FE8E, 0xBF2D1AF1 */ |
| #define t10 8.81081882437654011382e-04 /* 0x3F4CDF0C, 0xEF61A8E9 */ |
| #define t11 -5.38595305356740546715e-04 /* 0xBF41A610, 0x9C73E0EC */ |
| #define t12 3.15632070903625950361e-04 /* 0x3F34AF6D, 0x6C0EBBF7 */ |
| #define t13 -3.12754168375120860518e-04 /* 0xBF347F24, 0xECC38C38 */ |
| #define t14 3.35529192635519073543e-04 /* 0x3F35FD3E, 0xE8C2D3F4 */ |
| |
| #define u0 -7.72156649015328655494e-02 /* 0xBFB3C467, 0xE37DB0C8 */ |
| #define u1 6.32827064025093366517e-01 /* 0x3FE4401E, 0x8B005DFF */ |
| #define u2 1.45492250137234768737e+00 /* 0x3FF7475C, 0xD119BD6F */ |
| #define u3 9.77717527963372745603e-01 /* 0x3FEF4976, 0x44EA8450 */ |
| #define u4 2.28963728064692451092e-01 /* 0x3FCD4EAE, 0xF6010924 */ |
| #define u5 1.33810918536787660377e-02 /* 0x3F8B678B, 0xBF2BAB09 */ |
| |
| #define v1 2.45597793713041134822e+00 /* 0x4003A5D7, 0xC2BD619C */ |
| #define v2 2.12848976379893395361e+00 /* 0x40010725, 0xA42B18F5 */ |
| #define v3 7.69285150456672783825e-01 /* 0x3FE89DFB, 0xE45050AF */ |
| #define v4 1.04222645593369134254e-01 /* 0x3FBAAE55, 0xD6537C88 */ |
| #define v5 3.21709242282423911810e-03 /* 0x3F6A5ABB, 0x57D0CF61 */ |
| |
| #define s0 -7.72156649015328655494e-02 /* 0xBFB3C467, 0xE37DB0C8 */ |
| #define s1 2.14982415960608852501e-01 /* 0x3FCB848B, 0x36E20878 */ |
| #define s2 3.25778796408930981787e-01 /* 0x3FD4D98F, 0x4F139F59 */ |
| #define s3 1.46350472652464452805e-01 /* 0x3FC2BB9C, 0xBEE5F2F7 */ |
| #define s4 2.66422703033638609560e-02 /* 0x3F9B481C, 0x7E939961 */ |
| #define s5 1.84028451407337715652e-03 /* 0x3F5E26B6, 0x7368F239 */ |
| #define s6 3.19475326584100867617e-05 /* 0x3F00BFEC, 0xDD17E945 */ |
| |
| #define r1 1.39200533467621045958e+00 /* 0x3FF645A7, 0x62C4AB74 */ |
| #define r2 7.21935547567138069525e-01 /* 0x3FE71A18, 0x93D3DCDC */ |
| #define r3 1.71933865632803078993e-01 /* 0x3FC601ED, 0xCCFBDF27 */ |
| #define r4 1.86459191715652901344e-02 /* 0x3F9317EA, 0x742ED475 */ |
| #define r5 7.77942496381893596434e-04 /* 0x3F497DDA, 0xCA41A95B */ |
| #define r6 7.32668430744625636189e-06 /* 0x3EDEBAF7, 0xA5B38140 */ |
| |
| #define w0 4.18938533204672725052e-01 /* 0x3FDACFE3, 0x90C97D69 */ |
| #define w1 8.33333333333329678849e-02 /* 0x3FB55555, 0x5555553B */ |
| #define w2 -2.77777777728775536470e-03 /* 0xBF66C16C, 0x16B02E5C */ |
| #define w3 7.93650558643019558500e-04 /* 0x3F4A019F, 0x98CF38B6 */ |
| #define w4 -5.95187557450339963135e-04 /* 0xBF4380CB, 0x8C0FE741 */ |
| #define w5 8.36339918996282139126e-04 /* 0x3F4B67BA, 0x4CDAD5D1 */ |
| #define w6 -1.63092934096575273989e-03 /* 0xBF5AB89D, 0x0B9E43E4 */ |
| |
| __attribute__ ((overloadable, always_inline)) double |
| lgamma_r(double x, int *ip) |
| { |
| ulong ux = as_ulong(x); |
| ulong ax = ux & EXSIGNBIT_DP64; |
| double absx = as_double(ax); |
| |
| if (ax >= 0x7ff0000000000000UL) { |
| // +-Inf, NaN |
| *ip = 1; |
| return absx; |
| } |
| |
| if (absx < 0x1.0p-70) { |
| *ip = ax == ux ? 1 : -1; |
| return -log(absx); |
| } |
| |
| // Handle rest of range |
| double r; |
| |
| if (absx < 2.0) { |
| int i = 0; |
| double y = 2.0 - absx; |
| |
| int c = absx < 0x1.bb4c3p+0; |
| double t = absx - tc; |
| i = c ? 1 : i; |
| y = c ? t : y; |
| |
| c = absx < 0x1.3b4c4p+0; |
| t = absx - 1.0; |
| i = c ? 2 : i; |
| y = c ? t : y; |
| |
| c = absx <= 0x1.cccccp-1; |
| t = -log(absx); |
| r = c ? t : 0.0; |
| t = 1.0 - absx; |
| i = c ? 0 : i; |
| y = c ? t : y; |
| |
| c = absx < 0x1.76944p-1; |
| t = absx - (tc - 1.0); |
| i = c ? 1 : i; |
| y = c ? t : y; |
| |
| c = absx < 0x1.da661p-3; |
| i = c ? 2 : i; |
| y = c ? absx : y; |
| |
| double p, q; |
| |
| switch (i) { |
| case 0: |
| p = fma(y, fma(y, fma(y, fma(y, a11, a10), a9), a8), a7); |
| p = fma(y, fma(y, fma(y, fma(y, p, a6), a5), a4), a3); |
| p = fma(y, fma(y, fma(y, p, a2), a1), a0); |
| r = fma(y, p - 0.5, r); |
| break; |
| case 1: |
| p = fma(y, fma(y, fma(y, fma(y, t14, t13), t12), t11), t10); |
| p = fma(y, fma(y, fma(y, fma(y, fma(y, p, t9), t8), t7), t6), t5); |
| p = fma(y, fma(y, fma(y, fma(y, fma(y, p, t4), t3), t2), t1), t0); |
| p = fma(y*y, p, -tt); |
| r += (tf + p); |
| break; |
| case 2: |
| p = y*fma(y, fma(y, fma(y, fma(y, fma(y, u5, u4), u3), u2), u1), u0); |
| q = fma(y, fma(y, fma(y, fma(y, fma(y, v5, v4), v3), v2), v1), 1.0); |
| r += fma(-0.5, y, p/q); |
| } |
| } else if (absx < 8.0) { |
| int i = absx; |
| double y = absx - (double)i; |
| double p = y*fma(y, fma(y, fma(y, fma(y, fma(y, fma(y, s6, s5), s4), s3), s2), s1), s0); |
| double q = fma(y, fma(y, fma(y, fma(y, fma(y, fma(y, r6, r5), r4), r3), r2), r1), 1.0); |
| r = fma(0.5, y, p/q); |
| double z = 1.0; |
| // lgamma(1+s) = log(s) + lgamma(s) |
| double y6 = y + 6.0; |
| double y5 = y + 5.0; |
| double y4 = y + 4.0; |
| double y3 = y + 3.0; |
| double y2 = y + 2.0; |
| z *= i > 6 ? y6 : 1.0; |
| z *= i > 5 ? y5 : 1.0; |
| z *= i > 4 ? y4 : 1.0; |
| z *= i > 3 ? y3 : 1.0; |
| z *= i > 2 ? y2 : 1.0; |
| r += log(z); |
| } else { |
| double z = 1.0 / absx; |
| double z2 = z * z; |
| double w = fma(z, fma(z2, fma(z2, fma(z2, fma(z2, fma(z2, w6, w5), w4), w3), w2), w1), w0); |
| r = (absx - 0.5) * (log(absx) - 1.0) + w; |
| } |
| |
| if (x < 0.0) { |
| double t = sinpi(x); |
| r = log(pi / fabs(t * x)) - r; |
| r = t == 0.0 ? as_double(PINFBITPATT_DP64) : r; |
| *ip = t < 0.0 ? -1 : 1; |
| } else |
| *ip = 1; |
| |
| return r; |
| } |
| |
| #if __OPENCL_C_VERSION__ < 200 |
| __attribute__ ((overloadable, always_inline)) double |
| lgamma_r(double x, __local int *ip) |
| { |
| int i; |
| double ret = lgamma_r(x, &i); |
| *ip = i; |
| return ret; |
| } |
| |
| __attribute__ ((overloadable, always_inline)) double |
| lgamma_r(double x, __global int *ip) |
| { |
| int i; |
| double ret = lgamma_r(x, &i); |
| *ip = i; |
| return ret; |
| } |
| #endif |
| |
| __attribute__ ((overloadable, always_inline)) double |
| lgamma(double x) |
| { |
| int i; |
| return lgamma_r(x, &i); |
| } |
| |
