1/* 2 * Copyright (c) 2014 Advanced Micro Devices, Inc. 3 * 4 * Permission is hereby granted, free of charge, to any person obtaining a copy 5 * of this software and associated documentation files (the "Software"), to deal 6 * in the Software without restriction, including without limitation the rights 7 * to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 8 * copies of the Software, and to permit persons to whom the Software is 9 * furnished to do so, subject to the following conditions: 10 * 11 * The above copyright notice and this permission notice shall be included in 12 * all copies or substantial portions of the Software. 13 * 14 * THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR 15 * IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, 16 * FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE 17 * AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER 18 * LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, 19 * OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN 20 * THE SOFTWARE. 21 */ 22 23#include <clc/clc.h> 24 25#include "math.h" 26#include "../clcmacro.h" 27 28/* 29 * ==================================================== 30 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 31 * 32 * Developed at SunPro, a Sun Microsystems, Inc. business. 33 * Permission to use, copy, modify, and distribute this 34 * software is freely granted, provided that this notice 35 * is preserved. 36 * ==================================================== 37 */ 38 39#define erx_f 8.4506291151e-01f /* 0x3f58560b */ 40 41// Coefficients for approximation to erf on [00.84375] 42 43#define efx 1.2837916613e-01f /* 0x3e0375d4 */ 44#define efx8 1.0270333290e+00f /* 0x3f8375d4 */ 45 46#define pp0 1.2837916613e-01f /* 0x3e0375d4 */ 47#define pp1 -3.2504209876e-01f /* 0xbea66beb */ 48#define pp2 -2.8481749818e-02f /* 0xbce9528f */ 49#define pp3 -5.7702702470e-03f /* 0xbbbd1489 */ 50#define pp4 -2.3763017452e-05f /* 0xb7c756b1 */ 51#define qq1 3.9791721106e-01f /* 0x3ecbbbce */ 52#define qq2 6.5022252500e-02f /* 0x3d852a63 */ 53#define qq3 5.0813062117e-03f /* 0x3ba68116 */ 54#define qq4 1.3249473704e-04f /* 0x390aee49 */ 55#define qq5 -3.9602282413e-06f /* 0xb684e21a */ 56 57// Coefficients for approximation to erf in [0.843751.25] 58 59#define pa0 -2.3621185683e-03f /* 0xbb1acdc6 */ 60#define pa1 4.1485610604e-01f /* 0x3ed46805 */ 61#define pa2 -3.7220788002e-01f /* 0xbebe9208 */ 62#define pa3 3.1834661961e-01f /* 0x3ea2fe54 */ 63#define pa4 -1.1089469492e-01f /* 0xbde31cc2 */ 64#define pa5 3.5478305072e-02f /* 0x3d1151b3 */ 65#define pa6 -2.1663755178e-03f /* 0xbb0df9c0 */ 66#define qa1 1.0642088205e-01f /* 0x3dd9f331 */ 67#define qa2 5.4039794207e-01f /* 0x3f0a5785 */ 68#define qa3 7.1828655899e-02f /* 0x3d931ae7 */ 69#define qa4 1.2617121637e-01f /* 0x3e013307 */ 70#define qa5 1.3637083583e-02f /* 0x3c5f6e13 */ 71#define qa6 1.1984500103e-02f /* 0x3c445aa3 */ 72 73// Coefficients for approximation to erfc in [1.251/0.35] 74 75#define ra0 -9.8649440333e-03f /* 0xbc21a093 */ 76#define ra1 -6.9385856390e-01f /* 0xbf31a0b7 */ 77#define ra2 -1.0558626175e+01f /* 0xc128f022 */ 78#define ra3 -6.2375331879e+01f /* 0xc2798057 */ 79#define ra4 -1.6239666748e+02f /* 0xc322658c */ 80#define ra5 -1.8460508728e+02f /* 0xc3389ae7 */ 81#define ra6 -8.1287437439e+01f /* 0xc2a2932b */ 82#define ra7 -9.8143291473e+00f /* 0xc11d077e */ 83#define sa1 1.9651271820e+01f /* 0x419d35ce */ 84#define sa2 1.3765776062e+02f /* 0x4309a863 */ 85#define sa3 4.3456588745e+02f /* 0x43d9486f */ 86#define sa4 6.4538726807e+02f /* 0x442158c9 */ 87#define sa5 4.2900814819e+02f /* 0x43d6810b */ 88#define sa6 1.0863500214e+02f /* 0x42d9451f */ 89#define sa7 6.5702495575e+00f /* 0x40d23f7c */ 90#define sa8 -6.0424413532e-02f /* 0xbd777f97 */ 91 92// Coefficients for approximation to erfc in [1/.3528] 93 94#define rb0 -9.8649431020e-03f /* 0xbc21a092 */ 95#define rb1 -7.9928326607e-01f /* 0xbf4c9dd4 */ 96#define rb2 -1.7757955551e+01f /* 0xc18e104b */ 97#define rb3 -1.6063638306e+02f /* 0xc320a2ea */ 98#define rb4 -6.3756646729e+02f /* 0xc41f6441 */ 99#define rb5 -1.0250950928e+03f /* 0xc480230b */ 100#define rb6 -4.8351919556e+02f /* 0xc3f1c275 */ 101#define sb1 3.0338060379e+01f /* 0x41f2b459 */ 102#define sb2 3.2579251099e+02f /* 0x43a2e571 */ 103#define sb3 1.5367296143e+03f /* 0x44c01759 */ 104#define sb4 3.1998581543e+03f /* 0x4547fdbb */ 105#define sb5 2.5530502930e+03f /* 0x451f90ce */ 106#define sb6 4.7452853394e+02f /* 0x43ed43a7 */ 107#define sb7 -2.2440952301e+01f /* 0xc1b38712 */ 108 109_CLC_OVERLOAD _CLC_DEF float erfc(float x) { 110 int hx = as_int(x); 111 int ix = hx & 0x7fffffff; 112 float absx = as_float(ix); 113 114 // Argument for polys 115 float x2 = absx * absx; 116 float t = 1.0f / x2; 117 float tt = absx - 1.0f; 118 t = absx < 1.25f ? tt : t; 119 t = absx < 0.84375f ? x2 : t; 120 121 // Evaluate polys 122 float tu, tv, u, v; 123 124 u = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, rb6, rb5), rb4), rb3), rb2), rb1), rb0); 125 v = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, sb7, sb6), sb5), sb4), sb3), sb2), sb1); 126 127 tu = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, ra7, ra6), ra5), ra4), ra3), ra2), ra1), ra0); 128 tv = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, sa8, sa7), sa6), sa5), sa4), sa3), sa2), sa1); 129 u = absx < 0x1.6db6dap+1f ? tu : u; 130 v = absx < 0x1.6db6dap+1f ? tv : v; 131 132 tu = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, pa6, pa5), pa4), pa3), pa2), pa1), pa0); 133 tv = mad(t, mad(t, mad(t, mad(t, mad(t, qa6, qa5), qa4), qa3), qa2), qa1); 134 u = absx < 1.25f ? tu : u; 135 v = absx < 1.25f ? tv : v; 136 137 tu = mad(t, mad(t, mad(t, mad(t, pp4, pp3), pp2), pp1), pp0); 138 tv = mad(t, mad(t, mad(t, mad(t, qq5, qq4), qq3), qq2), qq1); 139 u = absx < 0.84375f ? tu : u; 140 v = absx < 0.84375f ? tv : v; 141 142 v = mad(t, v, 1.0f); 143 144 float q = MATH_DIVIDE(u, v); 145 146 float ret = 0.0f; 147 148 float z = as_float(ix & 0xfffff000); 149 float r = exp(mad(-z, z, -0.5625f)) * exp(mad(z - absx, z + absx, q)); 150 r = MATH_DIVIDE(r, absx); 151 t = 2.0f - r; 152 r = x < 0.0f ? t : r; 153 ret = absx < 28.0f ? r : ret; 154 155 r = 1.0f - erx_f - q; 156 t = erx_f + q + 1.0f; 157 r = x < 0.0f ? t : r; 158 ret = absx < 1.25f ? r : ret; 159 160 r = 0.5f - mad(x, q, x - 0.5f); 161 ret = absx < 0.84375f ? r : ret; 162 163 ret = x < -6.0f ? 2.0f : ret; 164 165 ret = isnan(x) ? x : ret; 166 167 return ret; 168} 169 170_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, float, erfc, float); 171 172#ifdef cl_khr_fp64 173 174#pragma OPENCL EXTENSION cl_khr_fp64 : enable 175 176/* 177 * ==================================================== 178 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 179 * 180 * Developed at SunPro, a Sun Microsystems, Inc. business. 181 * Permission to use, copy, modify, and distribute this 182 * software is freely granted, provided that this notice 183 * is preserved. 184 * ==================================================== 185 */ 186 187/* double erf(double x) 188 * double erfc(double x) 189 * x 190 * 2 |\ 191 * erf(x) = --------- | exp(-t*t)dt 192 * sqrt(pi) \| 193 * 0 194 * 195 * erfc(x) = 1-erf(x) 196 * Note that 197 * erf(-x) = -erf(x) 198 * erfc(-x) = 2 - erfc(x) 199 * 200 * Method: 201 * 1. For |x| in [0, 0.84375] 202 * erf(x) = x + x*R(x^2) 203 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 204 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 205 * where R = P/Q where P is an odd poly of degree 8 and 206 * Q is an odd poly of degree 10. 207 * -57.90 208 * | R - (erf(x)-x)/x | <= 2 209 * 210 * 211 * Remark. The formula is derived by noting 212 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 213 * and that 214 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 215 * is close to one. The interval is chosen because the fix 216 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 217 * near 0.6174), and by some experiment, 0.84375 is chosen to 218 * guarantee the error is less than one ulp for erf. 219 * 220 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 221 * c = 0.84506291151 rounded to single (24 bits) 222 * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 223 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 224 * 1+(c+P1(s)/Q1(s)) if x < 0 225 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 226 * Remark: here we use the taylor series expansion at x=1. 227 * erf(1+s) = erf(1) + s*Poly(s) 228 * = 0.845.. + P1(s)/Q1(s) 229 * That is, we use rational approximation to approximate 230 * erf(1+s) - (c = (single)0.84506291151) 231 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 232 * where 233 * P1(s) = degree 6 poly in s 234 * Q1(s) = degree 6 poly in s 235 * 236 * 3. For x in [1.25,1/0.35(~2.857143)], 237 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 238 * erf(x) = 1 - erfc(x) 239 * where 240 * R1(z) = degree 7 poly in z, (z=1/x^2) 241 * S1(z) = degree 8 poly in z 242 * 243 * 4. For x in [1/0.35,28] 244 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 245 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 246 * = 2.0 - tiny (if x <= -6) 247 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 248 * erf(x) = sign(x)*(1.0 - tiny) 249 * where 250 * R2(z) = degree 6 poly in z, (z=1/x^2) 251 * S2(z) = degree 7 poly in z 252 * 253 * Note1: 254 * To compute exp(-x*x-0.5625+R/S), let s be a single 255 * precision number and s := x; then 256 * -x*x = -s*s + (s-x)*(s+x) 257 * exp(-x*x-0.5626+R/S) = 258 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 259 * Note2: 260 * Here 4 and 5 make use of the asymptotic series 261 * exp(-x*x) 262 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 263 * x*sqrt(pi) 264 * We use rational approximation to approximate 265 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 266 * Here is the error bound for R1/S1 and R2/S2 267 * |R1/S1 - f(x)| < 2**(-62.57) 268 * |R2/S2 - f(x)| < 2**(-61.52) 269 * 270 * 5. For inf > x >= 28 271 * erf(x) = sign(x) *(1 - tiny) (raise inexact) 272 * erfc(x) = tiny*tiny (raise underflow) if x > 0 273 * = 2 - tiny if x<0 274 * 275 * 7. Special case: 276 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 277 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 278 * erfc/erf(NaN) is NaN 279 */ 280 281#define AU0 -9.86494292470009928597e-03 282#define AU1 -7.99283237680523006574e-01 283#define AU2 -1.77579549177547519889e+01 284#define AU3 -1.60636384855821916062e+02 285#define AU4 -6.37566443368389627722e+02 286#define AU5 -1.02509513161107724954e+03 287#define AU6 -4.83519191608651397019e+02 288 289#define AV0 3.03380607434824582924e+01 290#define AV1 3.25792512996573918826e+02 291#define AV2 1.53672958608443695994e+03 292#define AV3 3.19985821950859553908e+03 293#define AV4 2.55305040643316442583e+03 294#define AV5 4.74528541206955367215e+02 295#define AV6 -2.24409524465858183362e+01 296 297#define BU0 -9.86494403484714822705e-03 298#define BU1 -6.93858572707181764372e-01 299#define BU2 -1.05586262253232909814e+01 300#define BU3 -6.23753324503260060396e+01 301#define BU4 -1.62396669462573470355e+02 302#define BU5 -1.84605092906711035994e+02 303#define BU6 -8.12874355063065934246e+01 304#define BU7 -9.81432934416914548592e+00 305 306#define BV0 1.96512716674392571292e+01 307#define BV1 1.37657754143519042600e+02 308#define BV2 4.34565877475229228821e+02 309#define BV3 6.45387271733267880336e+02 310#define BV4 4.29008140027567833386e+02 311#define BV5 1.08635005541779435134e+02 312#define BV6 6.57024977031928170135e+00 313#define BV7 -6.04244152148580987438e-02 314 315#define CU0 -2.36211856075265944077e-03 316#define CU1 4.14856118683748331666e-01 317#define CU2 -3.72207876035701323847e-01 318#define CU3 3.18346619901161753674e-01 319#define CU4 -1.10894694282396677476e-01 320#define CU5 3.54783043256182359371e-02 321#define CU6 -2.16637559486879084300e-03 322 323#define CV0 1.06420880400844228286e-01 324#define CV1 5.40397917702171048937e-01 325#define CV2 7.18286544141962662868e-02 326#define CV3 1.26171219808761642112e-01 327#define CV4 1.36370839120290507362e-02 328#define CV5 1.19844998467991074170e-02 329 330#define DU0 1.28379167095512558561e-01 331#define DU1 -3.25042107247001499370e-01 332#define DU2 -2.84817495755985104766e-02 333#define DU3 -5.77027029648944159157e-03 334#define DU4 -2.37630166566501626084e-05 335 336#define DV0 3.97917223959155352819e-01 337#define DV1 6.50222499887672944485e-02 338#define DV2 5.08130628187576562776e-03 339#define DV3 1.32494738004321644526e-04 340#define DV4 -3.96022827877536812320e-06 341 342_CLC_OVERLOAD _CLC_DEF double erfc(double x) { 343 long lx = as_long(x); 344 long ax = lx & 0x7fffffffffffffffL; 345 double absx = as_double(ax); 346 int xneg = lx != ax; 347 348 // Poly arg 349 double x2 = x * x; 350 double xm1 = absx - 1.0; 351 double t = 1.0 / x2; 352 t = absx < 1.25 ? xm1 : t; 353 t = absx < 0.84375 ? x2 : t; 354 355 356 // Evaluate rational poly 357 // XXX Need to evaluate if we can grab the 14 coefficients from a 358 // table faster than evaluating 3 pairs of polys 359 double tu, tv, u, v; 360 361 // |x| < 28 362 u = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, AU6, AU5), AU4), AU3), AU2), AU1), AU0); 363 v = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, AV6, AV5), AV4), AV3), AV2), AV1), AV0); 364 365 tu = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BU7, BU6), BU5), BU4), BU3), BU2), BU1), BU0); 366 tv = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BV7, BV6), BV5), BV4), BV3), BV2), BV1), BV0); 367 u = absx < 0x1.6db6dp+1 ? tu : u; 368 v = absx < 0x1.6db6dp+1 ? tv : v; 369 370 tu = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, CU6, CU5), CU4), CU3), CU2), CU1), CU0); 371 tv = fma(t, fma(t, fma(t, fma(t, fma(t, CV5, CV4), CV3), CV2), CV1), CV0); 372 u = absx < 1.25 ? tu : u; 373 v = absx < 1.25 ? tv : v; 374 375 tu = fma(t, fma(t, fma(t, fma(t, DU4, DU3), DU2), DU1), DU0); 376 tv = fma(t, fma(t, fma(t, fma(t, DV4, DV3), DV2), DV1), DV0); 377 u = absx < 0.84375 ? tu : u; 378 v = absx < 0.84375 ? tv : v; 379 380 v = fma(t, v, 1.0); 381 double q = u / v; 382 383 384 // Evaluate return value 385 386 // |x| < 28 387 double z = as_double(ax & 0xffffffff00000000UL); 388 double ret = exp(-z * z - 0.5625) * exp((z - absx) * (z + absx) + q) / absx; 389 t = 2.0 - ret; 390 ret = xneg ? t : ret; 391 392 const double erx = 8.45062911510467529297e-01; 393 z = erx + q + 1.0; 394 t = 1.0 - erx - q; 395 t = xneg ? z : t; 396 ret = absx < 1.25 ? t : ret; 397 398 // z = 1.0 - fma(x, q, x); 399 // t = 0.5 - fma(x, q, x - 0.5); 400 // t = xneg == 1 | absx < 0.25 ? z : t; 401 t = fma(-x, q, 1.0 - x); 402 ret = absx < 0.84375 ? t : ret; 403 404 ret = x >= 28.0 ? 0.0 : ret; 405 ret = x <= -6.0 ? 2.0 : ret; 406 ret = ax > 0x7ff0000000000000UL ? x : ret; 407 408 return ret; 409} 410 411_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, double, erfc, double); 412 413#endif 414