1 /* -------------------------------------------------------------- */ 2 /* (C)Copyright 2007,2008, */ 3 /* International Business Machines Corporation */ 4 /* All Rights Reserved. */ 5 /* */ 6 /* Redistribution and use in source and binary forms, with or */ 7 /* without modification, are permitted provided that the */ 8 /* following conditions are met: */ 9 /* */ 10 /* - Redistributions of source code must retain the above copyright*/ 11 /* notice, this list of conditions and the following disclaimer. */ 12 /* */ 13 /* - Redistributions in binary form must reproduce the above */ 14 /* copyright notice, this list of conditions and the following */ 15 /* disclaimer in the documentation and/or other materials */ 16 /* provided with the distribution. */ 17 /* */ 18 /* - Neither the name of IBM Corporation nor the names of its */ 19 /* contributors may be used to endorse or promote products */ 20 /* derived from this software without specific prior written */ 21 /* permission. */ 22 /* */ 23 /* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND */ 24 /* CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, */ 25 /* INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF */ 26 /* MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE */ 27 /* DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR */ 28 /* CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, */ 29 /* SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT */ 30 /* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; */ 31 /* LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) */ 32 /* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN */ 33 /* CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR */ 34 /* OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, */ 35 /* EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. */ 36 /* -------------------------------------------------------------- */ 37 /* PROLOG END TAG zYx */ 38 #ifdef __SPU__ 39 #ifndef _ERF_UTILS_H_ 40 #define _ERF_UTILS_H_ 1 41 42 #include <spu_intrinsics.h> 43 44 45 /* 46 * This file contains approximation methods for the erf and erfc functions. 47 */ 48 49 50 #define SQRT_PI 1.7724538509055160272981674833411451827975494561223871282138077898529113E0 51 #define INV_SQRT_PI 5.6418958354775628694807945156077258584405062932899885684408572171064247E-1 52 #define TWO_OVER_SQRT_PI 1.1283791670955125738961589031215451716881012586579977136881714434212849E0 53 54 /* 55 * Coefficients of Taylor Series Expansion of Error Function 56 */ 57 #define TAYLOR_ERF_00 1.0000000000000000000000000000000000000000000000000000000000000000000000E0 58 #define TAYLOR_ERF_01 -3.3333333333333333333333333333333333333333333333333333333333333333333333E-1 59 #define TAYLOR_ERF_02 1.0000000000000000000000000000000000000000000000000000000000000000000000E-1 60 #define TAYLOR_ERF_03 -2.3809523809523809523809523809523809523809523809523809523809523809523810E-2 61 #define TAYLOR_ERF_04 4.6296296296296296296296296296296296296296296296296296296296296296296296E-3 62 #define TAYLOR_ERF_05 -7.5757575757575757575757575757575757575757575757575757575757575757575758E-4 63 #define TAYLOR_ERF_06 1.0683760683760683760683760683760683760683760683760683760683760683760684E-4 64 #define TAYLOR_ERF_07 -1.3227513227513227513227513227513227513227513227513227513227513227513228E-5 65 #define TAYLOR_ERF_08 1.4589169000933706816059757236227824463118580765639589169000933706816060E-6 66 #define TAYLOR_ERF_09 -1.4503852223150468764503852223150468764503852223150468764503852223150469E-7 67 #define TAYLOR_ERF_10 1.3122532963802805072646342487612328882170152011421852691693961535231377E-8 68 #define TAYLOR_ERF_11 -1.0892221037148573380457438428452921206544394950192051641327003645844226E-9 69 #define TAYLOR_ERF_12 8.3507027951472395916840361284805729250173694618139062583507027951472396E-11 70 #define TAYLOR_ERF_13 -5.9477940136376350368119915445018325676761890753660300985403866062302276E-12 71 #define TAYLOR_ERF_14 3.9554295164585257633971372340283122987009139171153402133150354277885750E-13 72 #define TAYLOR_ERF_15 -2.4668270102644569277100425760606678852113226579859111007771188689434124E-14 73 #define TAYLOR_ERF_16 1.4483264643598137264964265124598618265445265605599099265926266086599580E-15 74 #define TAYLOR_ERF_17 -8.0327350124157736091398445228866286178099792434415172399254921152569101E-17 75 #define TAYLOR_ERF_18 4.2214072888070882330314498243398198441944335363431396906515348954052831E-18 76 #define TAYLOR_ERF_19 -2.1078551914421358248605080094544309613386510235451574703658136454790212E-19 77 #define TAYLOR_ERF_20 1.0025164934907719167019489313258878962464315843690383090764235630936808E-20 78 #define TAYLOR_ERF_21 -4.5518467589282002862436219473268442686715055325725991884976042178118399E-22 79 #define TAYLOR_ERF_22 1.9770647538779051748330883205561040762916640191981996475292624380394860E-23 80 #define TAYLOR_ERF_23 -8.2301492992142213568444934713251326025092396728879726307878639881384709E-25 81 #define TAYLOR_ERF_24 3.2892603491757517327524761322472893904586246991984244357740612877764297E-26 82 #define TAYLOR_ERF_25 -1.2641078988989163521950692586675857265291969432213552733563059066748632E-27 83 #define TAYLOR_ERF_26 4.6784835155184857737263085770716162592880293254201102279514950101899871E-29 84 #define TAYLOR_ERF_27 -1.6697617934173720269864939702679842541566703989714871520634965356233624E-30 85 #define TAYLOR_ERF_28 5.7541916439821717721965644338808981189609568886862025916975131240153466E-32 86 #define TAYLOR_ERF_29 -1.9169428621097825307726719621929350834644917747230482041306735714136456E-33 87 #define TAYLOR_ERF_30 6.1803075882227961374638057797477142035193997108557291827163792739565622E-35 88 #define TAYLOR_ERF_31 -1.9303572088151078565555153741147494440075954038003045578376811864380455E-36 89 #define TAYLOR_ERF_32 5.8467550074688362962979552196744814890614668480489993819122074396921572E-38 90 #define TAYLOR_ERF_33 -1.7188560628017836239681912676564509126594090688520350964463748691994130E-39 91 #define TAYLOR_ERF_34 4.9089239645234229670020807729318930583197104694410209489303971115243253E-41 92 #define TAYLOR_ERF_35 -1.3630412617791395763506783635102640685072837923196396196225247512884444E-42 93 #define TAYLOR_ERF_36 3.6824935154611457351939940566677606112639706717920248475342183158858278E-44 94 #define TAYLOR_ERF_37 -9.6872802388707617538436600409638387251268417672366779772972229571050606E-46 95 #define TAYLOR_ERF_38 2.4830690974549115910398991902675594818336060579041382375163763560590552E-47 96 #define TAYLOR_ERF_39 -6.2056579196373967059419746072899084745598074150801247740591035188752759E-49 97 #define TAYLOR_ERF_40 1.5131079495412170980537530678268603996611876104670674603415715370097123E-50 98 #define TAYLOR_ERF_41 -3.6015793098101259166133998969725445892611283117200253978156713046660799E-52 99 #define TAYLOR_ERF_42 8.3734196838722815428266720293759440030440798283686864991232694198118944E-54 100 #define TAYLOR_ERF_43 -1.9025412272898795272394202686366085010926137006451172211319911806576077E-55 101 #define TAYLOR_ERF_44 4.2267897541935525758383443148974703675959497435169866761614717241371774E-57 102 #define TAYLOR_ERF_45 -9.1864295023986856959612367283485924961181813717463202485560679718732304E-59 103 104 /* 105 * Taylor Series Expansion of Erf 106 * 107 * infinite 108 * --------- 109 * - n 2n 110 * 2 * x - -1 * x 111 * erf(x) = ---- * - ------------ 112 * sqrt(pi) - (2n + 1) * n! 113 * - 114 * --------- 115 * n = 0 116 * 117 * 45 terms give us accurate results for 0 <= x < 2.5 118 */ 119 #define TAYLOR_ERF(_xabs, _xsqu, _tresult) { \ 120 _tresult = spu_madd(_xsqu, spu_splats(TAYLOR_ERF_45), spu_splats(TAYLOR_ERF_44)); \ 121 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_43)); \ 122 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_42)); \ 123 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_41)); \ 124 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_40)); \ 125 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_39)); \ 126 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_38)); \ 127 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_37)); \ 128 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_36)); \ 129 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_35)); \ 130 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_34)); \ 131 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_33)); \ 132 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_32)); \ 133 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_31)); \ 134 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_30)); \ 135 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_29)); \ 136 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_28)); \ 137 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_27)); \ 138 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_26)); \ 139 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_25)); \ 140 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_24)); \ 141 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_23)); \ 142 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_22)); \ 143 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_21)); \ 144 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_20)); \ 145 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_19)); \ 146 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_18)); \ 147 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_17)); \ 148 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_16)); \ 149 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_15)); \ 150 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_14)); \ 151 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_13)); \ 152 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_12)); \ 153 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_11)); \ 154 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_10)); \ 155 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_09)); \ 156 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_08)); \ 157 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_07)); \ 158 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_06)); \ 159 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_05)); \ 160 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_04)); \ 161 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_03)); \ 162 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_02)); \ 163 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_01)); \ 164 _tresult = spu_madd(_tresult, _xsqu, spu_splats(TAYLOR_ERF_00)); \ 165 _tresult = spu_mul(_tresult, _xabs); \ 166 _tresult = spu_mul(_tresult, spu_splats(TWO_OVER_SQRT_PI)); \ 167 } 168 169 170 /* 171 * Continued Fractions Approximation of Erfc() 172 * ( ) 173 * 1 ( 1 v 2v 3v ) 174 * erfc(x) = ------------------------- * ( --- --- --- --- ... ) 175 * sqrt(pi) * x * exp(x^2) ( 1+ 1+ 1+ 1+ ) 176 * ( ) 177 * Continued Fractions 178 * 1 179 * v = ----- 180 * 2*x^2 181 * 182 * We are using a backward recurrence calculation to estimate the continued fraction. 183 * 184 * p = a p + b q 185 * m,n m m+1,n m m+1,n 186 * 187 * q = p 188 * m,n m+1,n 189 * 190 * With, 191 * 192 * p = a ; q = 1 193 * n,n n n,n 194 * 195 * 196 * a = 0, b = 1, 197 * 0 0 198 * 199 * a = 1, b = n/2x^2 200 * n n 201 * 202 * 203 * F = p / q 204 * 0,n 0,n 0,n 205 * 206 * Ref: "Computing the Incomplete Gamma Function to Arbitrary Precision", 207 * by Serge Winitzki, Department of Physics, Ludwig-Maximilians University, Munich, Germany. 208 * 209 */ 210 211 #define CONTFRAC_ERFCF4(_xabs, _xsqu, _presult) { \ 212 vec_float4 v; \ 213 vec_float4 p, q, plast, qlast; \ 214 vec_float4 factor; \ 215 vec_float4 inv_xsqu; \ 216 inv_xsqu = _recipf4(_xsqu); \ 217 v = spu_mul(inv_xsqu, onehalff); \ 218 p = spu_splats(1.945f); q = onef; plast = p; qlast = q; \ 219 p = spu_madd(qlast, spu_mul(v, spu_splats( 4.0f)), plast); q = plast; plast = p; qlast = q; \ 220 p = spu_madd(qlast, spu_mul(v, spu_splats( 3.0f)), plast); q = plast; plast = p; qlast = q; \ 221 p = spu_madd(qlast, spu_mul(v, spu_splats( 2.0f)), plast); q = plast; plast = p; qlast = q; \ 222 p = spu_madd(qlast, spu_mul(v, spu_splats( 1.0f)), plast); q = plast; plast = p; qlast = q; \ 223 p = qlast; q = plast; \ 224 factor = spu_mul(spu_splats((float)SQRT_PI), spu_mul(_xabs, _expf4(_xsqu))); \ 225 _presult = _divf4(p, spu_mul(factor, q)); \ 226 } 227 228 #define CONTFRAC_ERFC(_xabs, _xsqu, _presult) { \ 229 vec_double2 v; \ 230 vec_double2 p, q, plast, qlast; \ 231 vec_double2 factor; \ 232 vec_double2 inv_xsqu; \ 233 inv_xsqu = _recipd2(_xsqu); \ 234 v = spu_mul(inv_xsqu, onehalfd); \ 235 p = spu_splats(3.025); q = oned; plast = p; qlast = q; \ 236 p = spu_madd(qlast, spu_mul(v, spu_splats(40.0)), plast); q = plast; plast = p; qlast = q; \ 237 p = spu_madd(qlast, spu_mul(v, spu_splats(39.0)), plast); q = plast; plast = p; qlast = q; \ 238 p = spu_madd(qlast, spu_mul(v, spu_splats(38.0)), plast); q = plast; plast = p; qlast = q; \ 239 p = spu_madd(qlast, spu_mul(v, spu_splats(37.0)), plast); q = plast; plast = p; qlast = q; \ 240 p = spu_madd(qlast, spu_mul(v, spu_splats(36.0)), plast); q = plast; plast = p; qlast = q; \ 241 p = spu_madd(qlast, spu_mul(v, spu_splats(35.0)), plast); q = plast; plast = p; qlast = q; \ 242 p = spu_madd(qlast, spu_mul(v, spu_splats(34.0)), plast); q = plast; plast = p; qlast = q; \ 243 p = spu_madd(qlast, spu_mul(v, spu_splats(33.0)), plast); q = plast; plast = p; qlast = q; \ 244 p = spu_madd(qlast, spu_mul(v, spu_splats(32.0)), plast); q = plast; plast = p; qlast = q; \ 245 p = spu_madd(qlast, spu_mul(v, spu_splats(31.0)), plast); q = plast; plast = p; qlast = q; \ 246 p = spu_madd(qlast, spu_mul(v, spu_splats(30.0)), plast); q = plast; plast = p; qlast = q; \ 247 p = spu_madd(qlast, spu_mul(v, spu_splats(29.0)), plast); q = plast; plast = p; qlast = q; \ 248 p = spu_madd(qlast, spu_mul(v, spu_splats(28.0)), plast); q = plast; plast = p; qlast = q; \ 249 p = spu_madd(qlast, spu_mul(v, spu_splats(27.0)), plast); q = plast; plast = p; qlast = q; \ 250 p = spu_madd(qlast, spu_mul(v, spu_splats(26.0)), plast); q = plast; plast = p; qlast = q; \ 251 p = spu_madd(qlast, spu_mul(v, spu_splats(25.0)), plast); q = plast; plast = p; qlast = q; \ 252 p = spu_madd(qlast, spu_mul(v, spu_splats(24.0)), plast); q = plast; plast = p; qlast = q; \ 253 p = spu_madd(qlast, spu_mul(v, spu_splats(23.0)), plast); q = plast; plast = p; qlast = q; \ 254 p = spu_madd(qlast, spu_mul(v, spu_splats(22.0)), plast); q = plast; plast = p; qlast = q; \ 255 p = spu_madd(qlast, spu_mul(v, spu_splats(21.0)), plast); q = plast; plast = p; qlast = q; \ 256 p = spu_madd(qlast, spu_mul(v, spu_splats(20.0)), plast); q = plast; plast = p; qlast = q; \ 257 p = spu_madd(qlast, spu_mul(v, spu_splats(19.0)), plast); q = plast; plast = p; qlast = q; \ 258 p = spu_madd(qlast, spu_mul(v, spu_splats(18.0)), plast); q = plast; plast = p; qlast = q; \ 259 p = spu_madd(qlast, spu_mul(v, spu_splats(17.0)), plast); q = plast; plast = p; qlast = q; \ 260 p = spu_madd(qlast, spu_mul(v, spu_splats(16.0)), plast); q = plast; plast = p; qlast = q; \ 261 p = spu_madd(qlast, spu_mul(v, spu_splats(15.0)), plast); q = plast; plast = p; qlast = q; \ 262 p = spu_madd(qlast, spu_mul(v, spu_splats(14.0)), plast); q = plast; plast = p; qlast = q; \ 263 p = spu_madd(qlast, spu_mul(v, spu_splats(13.0)), plast); q = plast; plast = p; qlast = q; \ 264 p = spu_madd(qlast, spu_mul(v, spu_splats(12.0)), plast); q = plast; plast = p; qlast = q; \ 265 p = spu_madd(qlast, spu_mul(v, spu_splats(11.0)), plast); q = plast; plast = p; qlast = q; \ 266 p = spu_madd(qlast, spu_mul(v, spu_splats(10.0)), plast); q = plast; plast = p; qlast = q; \ 267 p = spu_madd(qlast, spu_mul(v, spu_splats( 9.0)), plast); q = plast; plast = p; qlast = q; \ 268 p = spu_madd(qlast, spu_mul(v, spu_splats( 8.0)), plast); q = plast; plast = p; qlast = q; \ 269 p = spu_madd(qlast, spu_mul(v, spu_splats( 7.0)), plast); q = plast; plast = p; qlast = q; \ 270 p = spu_madd(qlast, spu_mul(v, spu_splats( 6.0)), plast); q = plast; plast = p; qlast = q; \ 271 p = spu_madd(qlast, spu_mul(v, spu_splats( 5.0)), plast); q = plast; plast = p; qlast = q; \ 272 p = spu_madd(qlast, spu_mul(v, spu_splats( 4.0)), plast); q = plast; plast = p; qlast = q; \ 273 p = spu_madd(qlast, spu_mul(v, spu_splats( 3.0)), plast); q = plast; plast = p; qlast = q; \ 274 p = spu_madd(qlast, spu_mul(v, spu_splats( 2.0)), plast); q = plast; plast = p; qlast = q; \ 275 p = spu_madd(qlast, spu_mul(v, spu_splats( 1.0)), plast); q = plast; plast = p; qlast = q; \ 276 p = qlast; q = plast; \ 277 factor = spu_mul(spu_splats(SQRT_PI), spu_mul(_xabs, _expd2(_xsqu))); \ 278 _presult = _divd2(p, spu_mul(factor, q)); \ 279 } 280 281 #endif /* _ERF_UTILS_H_ */ 282 #endif /* __SPU__ */ 283
