/* -------------------------------------------------------------- */ /* (C)Copyright 2007,2008, */ /* International Business Machines Corporation */ /* All Rights Reserved. */ /* */ /* Redistribution and use in source and binary forms, with or */ /* without modification, are permitted provided that the */ /* following conditions are met: */ /* */ /* - Redistributions of source code must retain the above copyright*/ /* notice, this list of conditions and the following disclaimer. */ /* */ /* - Redistributions in binary form must reproduce the above */ /* copyright notice, this list of conditions and the following */ /* disclaimer in the documentation and/or other materials */ /* provided with the distribution. */ /* */ /* - Neither the name of IBM Corporation nor the names of its */ /* contributors may be used to endorse or promote products */ /* derived from this software without specific prior written */ /* permission. */ /* */ /* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND */ /* CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, */ /* INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF */ /* MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE */ /* DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR */ /* CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, */ /* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT */ /* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; */ /* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) */ /* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN */ /* CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR */ /* OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, */ /* EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */ /* -------------------------------------------------------------- */ /* PROLOG END TAG zYx */ #ifdef __SPU__ #ifndef _TGAMMAD2_H_ #define _TGAMMAD2_H_ 1 #include #include "simdmath.h" #include "recipd2.h" #include "truncd2.h" #include "expd2.h" #include "logd2.h" #include "divd2.h" #include "sind2.h" #include "powd2.h" /* * FUNCTION * vector double _tgammad2(vector double x) * * DESCRIPTION * _tgammad2 * * This is an interesting function to approximate fast * and accurately. We take a fairly standard approach - break * the domain into 5 separate regions: * * 1. [-infinity, 0) - use * 2. [0, 1) - push x into [1,2), then adjust the * result. * 3. [1, 2) - use a rational approximation. * 4. [2, 10) - pull back into [1, 2), then adjust * the result. * 5. [10, +infinity] - use Stirling's Approximation. * * * Special Cases: * - tgamma(+/- 0) returns +/- infinity * - tgamma(negative integer) returns NaN * - tgamma(-infinity) returns NaN * - tgamma(infinity) returns infinity * */ /* * Coefficients for Stirling's Series for Gamma() */ /* 1/ 1 */ #define STIRLING_00 1.000000000000000000000000000000000000E0 /* 1/ 12 */ #define STIRLING_01 8.333333333333333333333333333333333333E-2 /* 1/ 288 */ #define STIRLING_02 3.472222222222222222222222222222222222E-3 /* -139/ 51840 */ #define STIRLING_03 -2.681327160493827160493827160493827160E-3 /* -571/ 2488320 */ #define STIRLING_04 -2.294720936213991769547325102880658436E-4 /* 163879/ 209018880 */ #define STIRLING_05 7.840392217200666274740348814422888497E-4 /* 5246819/ 75246796800 */ #define STIRLING_06 6.972813758365857774293988285757833083E-5 /* -534703531/ 902961561600 */ #define STIRLING_07 -5.921664373536938828648362256044011874E-4 /* -4483131259/ 86684309913600 */ #define STIRLING_08 -5.171790908260592193370578430020588228E-5 /* 432261921612371/ 514904800886784000 */ #define STIRLING_09 8.394987206720872799933575167649834452E-4 /* 6232523202521089/ 86504006548979712000 */ #define STIRLING_10 7.204895416020010559085719302250150521E-5 /* -25834629665134204969/ 13494625021640835072000 */ #define STIRLING_11 -1.914438498565477526500898858328522545E-3 /* -1579029138854919086429/ 9716130015581401251840000 */ #define STIRLING_12 -1.625162627839158168986351239802709981E-4 /* 746590869962651602203151/ 116593560186976815022080000 */ #define STIRLING_13 6.403362833808069794823638090265795830E-3 /* 1511513601028097903631961/ 2798245444487443560529920000 */ #define STIRLING_14 5.401647678926045151804675085702417355E-4 /* -8849272268392873147705987190261/ 299692087104605205332754432000000 */ #define STIRLING_15 -2.952788094569912050544065105469382445E-2 /* -142801712490607530608130701097701/ 57540880724084199423888850944000000 */ #define STIRLING_16 -2.481743600264997730915658368743464324E-3 /* * Rational Approximation Coefficients for the * domain [1, 2). */ #define TGD2_P00 -1.8211798563156931777484715e+05 #define TGD2_P01 -8.7136501560410004458390176e+04 #define TGD2_P02 -3.9304030489789496641606092e+04 #define TGD2_P03 -1.2078833505605729442322627e+04 #define TGD2_P04 -2.2149136023607729839568492e+03 #define TGD2_P05 -7.2672456596961114883015398e+02 #define TGD2_P06 -2.2126466212611862971471055e+01 #define TGD2_P07 -2.0162424149396112937893122e+01 #define TGD2_Q00 1.0000000000000000000000000 #define TGD2_Q01 -1.8212849094205905566923320e+05 #define TGD2_Q02 -1.9220660507239613798446953e+05 #define TGD2_Q03 2.9692670736656051303725690e+04 #define TGD2_Q04 3.0352658363629092491464689e+04 #define TGD2_Q05 -1.0555895821041505769244395e+04 #define TGD2_Q06 1.2786642579487202056043316e+03 #define TGD2_Q07 -5.5279768804094054246434098e+01 static __inline vector double _tgammad2(vector double x) { vector double signbit = spu_splats(-0.0); vector double zerod = spu_splats(0.0); vector double halfd = spu_splats(0.5); vector double oned = spu_splats(1.0); vector double ninep9d = (vec_double2)spu_splats(0x4023FFFFFFFFFFFFull); vector double twohd = spu_splats(200.0); vector double pi = spu_splats(SM_PI); vector double sqrt2pi = spu_splats(2.50662827463100050241576528481); vector double inf = (vector double)spu_splats(0x7FF0000000000000ull); vector double nan = (vector double)spu_splats(0x7FF8000000000000ull); vector double xabs; vector double xscaled; vector double xtrunc; vector double xinv; vector double nresult; vector double rresult; /* Rational Approx result */ vector double sresult; /* Stirling's result */ vector double result; vector double pr,qr; vector unsigned long long gt0 = spu_cmpgt(x, zerod); vector unsigned long long gt1 = spu_cmpgt(x, oned); vector unsigned long long gt9p9 = spu_cmpgt(x, ninep9d); vector unsigned long long gt200 = spu_cmpgt(x, twohd); xabs = spu_andc(x, signbit); /* * For x in [0, 1], add 1 to x, use rational * approximation, then use: * * gamma(x) = gamma(x+1)/x * */ xabs = spu_sel(spu_add(xabs, oned), xabs, gt1); xtrunc = _truncd2(xabs); /* * For x in [2, 10): */ xscaled = spu_add(oned, spu_sub(xabs, xtrunc)); /* * For x in [1,2), use a rational approximation. */ pr = spu_madd(xscaled, spu_splats(TGD2_P07), spu_splats(TGD2_P06)); pr = spu_madd(pr, xscaled, spu_splats(TGD2_P05)); pr = spu_madd(pr, xscaled, spu_splats(TGD2_P04)); pr = spu_madd(pr, xscaled, spu_splats(TGD2_P03)); pr = spu_madd(pr, xscaled, spu_splats(TGD2_P02)); pr = spu_madd(pr, xscaled, spu_splats(TGD2_P01)); pr = spu_madd(pr, xscaled, spu_splats(TGD2_P00)); qr = spu_madd(xscaled, spu_splats(TGD2_Q07), spu_splats(TGD2_Q06)); qr = spu_madd(qr, xscaled, spu_splats(TGD2_Q05)); qr = spu_madd(qr, xscaled, spu_splats(TGD2_Q04)); qr = spu_madd(qr, xscaled, spu_splats(TGD2_Q03)); qr = spu_madd(qr, xscaled, spu_splats(TGD2_Q02)); qr = spu_madd(qr, xscaled, spu_splats(TGD2_Q01)); qr = spu_madd(qr, xscaled, spu_splats(TGD2_Q00)); rresult = _divd2(pr, qr); rresult = spu_sel(_divd2(rresult, x), rresult, gt1); /* * If x was in [2,10) and we pulled it into [1,2), we need to push * it back out again. */ rresult = spu_sel(rresult, spu_mul(rresult, xscaled), spu_cmpgt(x, xscaled)); /* [2,3) */ xscaled = spu_add(xscaled, oned); rresult = spu_sel(rresult, spu_mul(rresult, xscaled), spu_cmpgt(x, xscaled)); /* [3,4) */ xscaled = spu_add(xscaled, oned); rresult = spu_sel(rresult, spu_mul(rresult, xscaled), spu_cmpgt(x, xscaled)); /* [4,5) */ xscaled = spu_add(xscaled, oned); rresult = spu_sel(rresult, spu_mul(rresult, xscaled), spu_cmpgt(x, xscaled)); /* [5,6) */ xscaled = spu_add(xscaled, oned); rresult = spu_sel(rresult, spu_mul(rresult, xscaled), spu_cmpgt(x, xscaled)); /* [6,7) */ xscaled = spu_add(xscaled, oned); rresult = spu_sel(rresult, spu_mul(rresult, xscaled), spu_cmpgt(x, xscaled)); /* [7,8) */ xscaled = spu_add(xscaled, oned); rresult = spu_sel(rresult, spu_mul(rresult, xscaled), spu_cmpgt(x, xscaled)); /* [8,9) */ xscaled = spu_add(xscaled, oned); rresult = spu_sel(rresult, spu_mul(rresult, xscaled), spu_cmpgt(x, xscaled)); /* [9,10) */ /* * For x >= 10, we use Stirling's Approximation */ vector double sum; xinv = _recipd2(xabs); sum = spu_madd(xinv, spu_splats(STIRLING_16), spu_splats(STIRLING_15)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_14)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_13)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_12)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_11)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_10)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_09)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_08)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_07)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_06)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_05)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_04)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_03)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_02)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_01)); sum = spu_madd(sum, xinv, spu_splats(STIRLING_00)); sum = spu_mul(sum, sqrt2pi); sum = spu_mul(sum, _powd2(x, spu_sub(x, halfd))); sresult = spu_mul(sum, _expd2(spu_or(x, signbit))); /* * Choose rational approximation or Stirling's result. */ result = spu_sel(rresult, sresult, gt9p9); result = spu_sel(result, inf, gt200); /* For x < 0, use: * * gamma(x) = pi/(x*gamma(-x)*sin(x*pi)) * or * gamma(x) = pi/(gamma(1 - x)*sin(x*pi)) */ nresult = _divd2(pi, spu_mul(x, spu_mul(result, _sind2(spu_mul(x, pi))))); result = spu_sel(nresult, result, gt0); /* * x = non-positive integer, return NaN. */ result = spu_sel(result, nan, spu_andc(spu_cmpeq(x, xtrunc), gt0)); return result; } #endif /* _TGAMMAD2_H_ */ #endif /* __SPU__ */