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/* -------------------------------------------------------------- */ /* (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 _ERF_UTILS_H_ #define _ERF_UTILS_H_ 1 #include <spu_intrinsics.h> /* * This file contains approximation methods for the erf and erfc functions. */ #define SQRT_PI 1.7724538509055160272981674833411451827975494561223871282138077898529113E0 #define INV_SQRT_PI 5.6418958354775628694807945156077258584405062932899885684408572171064247E-1 #define TWO_OVER_SQRT_PI 1.1283791670955125738961589031215451716881012586579977136881714434212849E0 /* * Coefficients of Taylor Series Expansion of Error Function */ #define TAYLOR_ERF_00 1.0000000000000000000000000000000000000000000000000000000000000000000000E0 #define TAYLOR_ERF_01 -3.3333333333333333333333333333333333333333333333333333333333333333333333E-1 #define TAYLOR_ERF_02 1.0000000000000000000000000000000000000000000000000000000000000000000000E-1 #define TAYLOR_ERF_03 -2.3809523809523809523809523809523809523809523809523809523809523809523810E-2 #define TAYLOR_ERF_04 4.6296296296296296296296296296296296296296296296296296296296296296296296E-3 #define TAYLOR_ERF_05 -7.5757575757575757575757575757575757575757575757575757575757575757575758E-4 #define TAYLOR_ERF_06 1.0683760683760683760683760683760683760683760683760683760683760683760684E-4 #define TAYLOR_ERF_07 -1.3227513227513227513227513227513227513227513227513227513227513227513228E-5 #define TAYLOR_ERF_08 1.4589169000933706816059757236227824463118580765639589169000933706816060E-6 #define TAYLOR_ERF_09 -1.4503852223150468764503852223150468764503852223150468764503852223150469E-7 #define TAYLOR_ERF_10 1.3122532963802805072646342487612328882170152011421852691693961535231377E-8 #define TAYLOR_ERF_11 -1.0892221037148573380457438428452921206544394950192051641327003645844226E-9 #define TAYLOR_ERF_12 8.3507027951472395916840361284805729250173694618139062583507027951472396E-11 #define TAYLOR_ERF_13 -5.9477940136376350368119915445018325676761890753660300985403866062302276E-12 #define TAYLOR_ERF_14 3.9554295164585257633971372340283122987009139171153402133150354277885750E-13 #define TAYLOR_ERF_15 -2.4668270102644569277100425760606678852113226579859111007771188689434124E-14 #define TAYLOR_ERF_16 1.4483264643598137264964265124598618265445265605599099265926266086599580E-15 #define TAYLOR_ERF_17 -8.0327350124157736091398445228866286178099792434415172399254921152569101E-17 #define TAYLOR_ERF_18 4.2214072888070882330314498243398198441944335363431396906515348954052831E-18 #define TAYLOR_ERF_19 -2.1078551914421358248605080094544309613386510235451574703658136454790212E-19 #define TAYLOR_ERF_20 1.0025164934907719167019489313258878962464315843690383090764235630936808E-20 #define TAYLOR_ERF_21 -4.5518467589282002862436219473268442686715055325725991884976042178118399E-22 #define TAYLOR_ERF_22 1.9770647538779051748330883205561040762916640191981996475292624380394860E-23 #define TAYLOR_ERF_23 -8.2301492992142213568444934713251326025092396728879726307878639881384709E-25 #define TAYLOR_ERF_24 3.2892603491757517327524761322472893904586246991984244357740612877764297E-26 #define TAYLOR_ERF_25 -1.2641078988989163521950692586675857265291969432213552733563059066748632E-27 #define TAYLOR_ERF_26 4.6784835155184857737263085770716162592880293254201102279514950101899871E-29 #define TAYLOR_ERF_27 -1.6697617934173720269864939702679842541566703989714871520634965356233624E-30 #define TAYLOR_ERF_28 5.7541916439821717721965644338808981189609568886862025916975131240153466E-32 #define TAYLOR_ERF_29 -1.9169428621097825307726719621929350834644917747230482041306735714136456E-33 #define TAYLOR_ERF_30 6.1803075882227961374638057797477142035193997108557291827163792739565622E-35 #define TAYLOR_ERF_31 -1.9303572088151078565555153741147494440075954038003045578376811864380455E-36 #define TAYLOR_ERF_32 5.8467550074688362962979552196744814890614668480489993819122074396921572E-38 #define TAYLOR_ERF_33 -1.7188560628017836239681912676564509126594090688520350964463748691994130E-39 #define TAYLOR_ERF_34 4.9089239645234229670020807729318930583197104694410209489303971115243253E-41 #define TAYLOR_ERF_35 -1.3630412617791395763506783635102640685072837923196396196225247512884444E-42 #define TAYLOR_ERF_36 3.6824935154611457351939940566677606112639706717920248475342183158858278E-44 #define TAYLOR_ERF_37 -9.6872802388707617538436600409638387251268417672366779772972229571050606E-46 #define TAYLOR_ERF_38 2.4830690974549115910398991902675594818336060579041382375163763560590552E-47 #define TAYLOR_ERF_39 -6.2056579196373967059419746072899084745598074150801247740591035188752759E-49 #define TAYLOR_ERF_40 1.5131079495412170980537530678268603996611876104670674603415715370097123E-50 #define TAYLOR_ERF_41 -3.6015793098101259166133998969725445892611283117200253978156713046660799E-52 #define TAYLOR_ERF_42 8.3734196838722815428266720293759440030440798283686864991232694198118944E-54 #define TAYLOR_ERF_43 -1.9025412272898795272394202686366085010926137006451172211319911806576077E-55 #define TAYLOR_ERF_44 4.2267897541935525758383443148974703675959497435169866761614717241371774E-57 #define TAYLOR_ERF_45 -9.1864295023986856959612367283485924961181813717463202485560679718732304E-59 /* * Taylor Series Expansion of Erf * * infinite * --------- * - n 2n * 2 * x - -1 * x * erf(x) = ---- * - ------------ * sqrt(pi) - (2n + 1) * n! * - * --------- * n = 0 * * 45 terms give us accurate results for 0 <= x < 2.5 */ #define TAYLOR_ERF(_xabs, _xsqu, _tresult) { \ _tresult = spu_madd(_xsqu, spu_splats(TAYLOR_ERF_45), spu_splats(TAYLOR_ERF_44)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_43)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_42)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_41)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_40)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_39)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_38)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_37)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_36)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_35)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_34)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_33)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_32)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_31)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_30)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_29)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_28)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_27)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_26)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_25)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_24)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_23)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_22)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_21)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_20)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_19)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_18)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_17)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_16)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_15)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_14)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_13)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_12)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_11)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_10)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_09)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_08)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_07)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_06)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_05)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_04)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_03)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_02)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_01)); \ _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_00)); \ _tresult = spu_mul(_tresult, _xabs); \ _tresult = spu_mul(_tresult, spu_splats(TWO_OVER_SQRT_PI)); \ } /* * Continued Fractions Approximation of Erfc() * ( ) * 1 ( 1 v 2v 3v ) * erfc(x) = ------------------------- * ( --- --- --- --- ... ) * sqrt(pi) * x * exp(x^2) ( 1+ 1+ 1+ 1+ ) * ( ) * Continued Fractions * 1 * v = ----- * 2*x^2 * * We are using a backward recurrence calculation to estimate the continued fraction. * * p = a p + b q * m,n m m+1,n m m+1,n * * q = p * m,n m+1,n * * With, * * p = a ; q = 1 * n,n n n,n * * * a = 0, b = 1, * 0 0 * * a = 1, b = n/2x^2 * n n * * * F = p / q * 0,n 0,n 0,n * * Ref: "Computing the Incomplete Gamma Function to Arbitrary Precision", * by Serge Winitzki, Department of Physics, Ludwig-Maximilians University, Munich, Germany. * */ #define CONTFRAC_ERFCF4(_xabs, _xsqu, _presult) { \ vec_float4 v; \ vec_float4 p, q, plast, qlast; \ vec_float4 factor; \ vec_float4 inv_xsqu; \ inv_xsqu = _recipf4(_xsqu); \ v = spu_mul(inv_xsqu, onehalff); \ p = spu_splats(1.945f); q = onef; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 4.0f)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 3.0f)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 2.0f)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 1.0f)), plast); q = plast; plast = p; qlast = q; \ p = qlast; q = plast; \ factor = spu_mul(spu_splats((float)SQRT_PI), spu_mul(_xabs, _expf4(_xsqu))); \ _presult = _divf4(p, spu_mul(factor, q)); \ } #define CONTFRAC_ERFC(_xabs, _xsqu, _presult) { \ vec_double2 v; \ vec_double2 p, q, plast, qlast; \ vec_double2 factor; \ vec_double2 inv_xsqu; \ inv_xsqu = _recipd2(_xsqu); \ v = spu_mul(inv_xsqu, onehalfd); \ p = spu_splats(3.025); q = oned; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(40.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(39.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(38.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(37.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(36.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(35.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(34.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(33.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(32.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(31.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(30.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(29.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(28.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(27.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(26.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(25.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(24.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(23.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(22.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(21.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(20.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(19.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(18.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(17.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(16.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(15.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(14.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(13.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(12.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(11.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats(10.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 9.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 8.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 7.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 6.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 5.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 4.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 3.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 2.0)), plast); q = plast; plast = p; qlast = q; \ p = spu_madd(qlast, spu_mul(v, spu_splats( 1.0)), plast); q = plast; plast = p; qlast = q; \ p = qlast; q = plast; \ factor = spu_mul(spu_splats(SQRT_PI), spu_mul(_xabs, _expd2(_xsqu))); \ _presult = _divd2(p, spu_mul(factor, q)); \ } #endif /* _ERF_UTILS_H_ */ #endif /* __SPU__ */