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 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 erf(float x) { 110 int hx = as_uint(x); 111 int ix = hx & 0x7fffffff; 112 float absx = as_float(ix); 113 114 float x2 = absx * absx; 115 float t = 1.0f / x2; 116 float tt = absx - 1.0f; 117 t = absx < 1.25f ? tt : t; 118 t = absx < 0.84375f ? x2 : t; 119 120 float u, v, tu, tv; 121 122 // |x| < 6 123 u = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, rb6, rb5), rb4), rb3), rb2), rb1), rb0); 124 v = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, sb7, sb6), sb5), sb4), sb3), sb2), sb1); 125 126 tu = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, ra7, ra6), ra5), ra4), ra3), ra2), ra1), ra0); 127 tv = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, sa8, sa7), sa6), sa5), sa4), sa3), sa2), sa1); 128 u = absx < 0x1.6db6dcp+1f ? tu : u; 129 v = absx < 0x1.6db6dcp+1f ? tv : v; 130 131 tu = mad(t, mad(t, mad(t, mad(t, mad(t, mad(t, pa6, pa5), pa4), pa3), pa2), pa1), pa0); 132 tv = mad(t, mad(t, mad(t, mad(t, mad(t, qa6, qa5), qa4), qa3), qa2), qa1); 133 u = absx < 1.25f ? tu : u; 134 v = absx < 1.25f ? tv : v; 135 136 tu = mad(t, mad(t, mad(t, mad(t, pp4, pp3), pp2), pp1), pp0); 137 tv = mad(t, mad(t, mad(t, mad(t, qq5, qq4), qq3), qq2), qq1); 138 u = absx < 0.84375f ? tu : u; 139 v = absx < 0.84375f ? tv : v; 140 141 v = mad(t, v, 1.0f); 142 float q = MATH_DIVIDE(u, v); 143 144 float ret = 1.0f; 145 146 // |x| < 6 147 float z = as_float(ix & 0xfffff000); 148 float r = exp(mad(-z, z, -0.5625f)) * exp(mad(z-absx, z+absx, q)); 149 r = 1.0f - MATH_DIVIDE(r, absx); 150 ret = absx < 6.0f ? r : ret; 151 152 r = erx + q; 153 ret = absx < 1.25f ? r : ret; 154 155 ret = as_float((hx & 0x80000000) | as_int(ret)); 156 157 r = mad(x, q, x); 158 ret = absx < 0.84375f ? r : ret; 159 160 // Prevent underflow 161 r = 0.125f * mad(8.0f, x, efx8 * x); 162 ret = absx < 0x1.0p-28f ? r : ret; 163 164 ret = isnan(x) ? x : ret; 165 166 return ret; 167} 168 169_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, float, erf, float); 170 171#ifdef cl_khr_fp64 172 173#pragma OPENCL EXTENSION cl_khr_fp64 : enable 174 175/* 176 * ==================================================== 177 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved. 178 * 179 * Developed at SunPro, a Sun Microsystems, Inc. business. 180 * Permission to use, copy, modify, and distribute this 181 * software is freely granted, provided that this notice 182 * is preserved. 183 * ==================================================== 184 */ 185 186/* double erf(double x) 187 * double erfc(double x) 188 * x 189 * 2 |\ 190 * erf(x) = --------- | exp(-t*t)dt 191 * sqrt(pi) \| 192 * 0 193 * 194 * erfc(x) = 1-erf(x) 195 * Note that 196 * erf(-x) = -erf(x) 197 * erfc(-x) = 2 - erfc(x) 198 * 199 * Method: 200 * 1. For |x| in [0, 0.84375] 201 * erf(x) = x + x*R(x^2) 202 * erfc(x) = 1 - erf(x) if x in [-.84375,0.25] 203 * = 0.5 + ((0.5-x)-x*R) if x in [0.25,0.84375] 204 * where R = P/Q where P is an odd poly of degree 8 and 205 * Q is an odd poly of degree 10. 206 * -57.90 207 * | R - (erf(x)-x)/x | <= 2 208 * 209 * 210 * Remark. The formula is derived by noting 211 * erf(x) = (2/sqrt(pi))*(x - x^3/3 + x^5/10 - x^7/42 + ....) 212 * and that 213 * 2/sqrt(pi) = 1.128379167095512573896158903121545171688 214 * is close to one. The interval is chosen because the fix 215 * point of erf(x) is near 0.6174 (i.e., erf(x)=x when x is 216 * near 0.6174), and by some experiment, 0.84375 is chosen to 217 * guarantee the error is less than one ulp for erf. 218 * 219 * 2. For |x| in [0.84375,1.25], let s = |x| - 1, and 220 * c = 0.84506291151 rounded to single (24 bits) 221 * erf(x) = sign(x) * (c + P1(s)/Q1(s)) 222 * erfc(x) = (1-c) - P1(s)/Q1(s) if x > 0 223 * 1+(c+P1(s)/Q1(s)) if x < 0 224 * |P1/Q1 - (erf(|x|)-c)| <= 2**-59.06 225 * Remark: here we use the taylor series expansion at x=1. 226 * erf(1+s) = erf(1) + s*Poly(s) 227 * = 0.845.. + P1(s)/Q1(s) 228 * That is, we use rational approximation to approximate 229 * erf(1+s) - (c = (single)0.84506291151) 230 * Note that |P1/Q1|< 0.078 for x in [0.84375,1.25] 231 * where 232 * P1(s) = degree 6 poly in s 233 * Q1(s) = degree 6 poly in s 234 * 235 * 3. For x in [1.25,1/0.35(~2.857143)], 236 * erfc(x) = (1/x)*exp(-x*x-0.5625+R1/S1) 237 * erf(x) = 1 - erfc(x) 238 * where 239 * R1(z) = degree 7 poly in z, (z=1/x^2) 240 * S1(z) = degree 8 poly in z 241 * 242 * 4. For x in [1/0.35,28] 243 * erfc(x) = (1/x)*exp(-x*x-0.5625+R2/S2) if x > 0 244 * = 2.0 - (1/x)*exp(-x*x-0.5625+R2/S2) if -6<x<0 245 * = 2.0 - tiny (if x <= -6) 246 * erf(x) = sign(x)*(1.0 - erfc(x)) if x < 6, else 247 * erf(x) = sign(x)*(1.0 - tiny) 248 * where 249 * R2(z) = degree 6 poly in z, (z=1/x^2) 250 * S2(z) = degree 7 poly in z 251 * 252 * Note1: 253 * To compute exp(-x*x-0.5625+R/S), let s be a single 254 * precision number and s := x; then 255 * -x*x = -s*s + (s-x)*(s+x) 256 * exp(-x*x-0.5626+R/S) = 257 * exp(-s*s-0.5625)*exp((s-x)*(s+x)+R/S); 258 * Note2: 259 * Here 4 and 5 make use of the asymptotic series 260 * exp(-x*x) 261 * erfc(x) ~ ---------- * ( 1 + Poly(1/x^2) ) 262 * x*sqrt(pi) 263 * We use rational approximation to approximate 264 * g(s)=f(1/x^2) = log(erfc(x)*x) - x*x + 0.5625 265 * Here is the error bound for R1/S1 and R2/S2 266 * |R1/S1 - f(x)| < 2**(-62.57) 267 * |R2/S2 - f(x)| < 2**(-61.52) 268 * 269 * 5. For inf > x >= 28 270 * erf(x) = sign(x) *(1 - tiny) (raise inexact) 271 * erfc(x) = tiny*tiny (raise underflow) if x > 0 272 * = 2 - tiny if x<0 273 * 274 * 7. Special case: 275 * erf(0) = 0, erf(inf) = 1, erf(-inf) = -1, 276 * erfc(0) = 1, erfc(inf) = 0, erfc(-inf) = 2, 277 * erfc/erf(NaN) is NaN 278 */ 279 280#define AU0 -9.86494292470009928597e-03 281#define AU1 -7.99283237680523006574e-01 282#define AU2 -1.77579549177547519889e+01 283#define AU3 -1.60636384855821916062e+02 284#define AU4 -6.37566443368389627722e+02 285#define AU5 -1.02509513161107724954e+03 286#define AU6 -4.83519191608651397019e+02 287 288#define AV1 3.03380607434824582924e+01 289#define AV2 3.25792512996573918826e+02 290#define AV3 1.53672958608443695994e+03 291#define AV4 3.19985821950859553908e+03 292#define AV5 2.55305040643316442583e+03 293#define AV6 4.74528541206955367215e+02 294#define AV7 -2.24409524465858183362e+01 295 296#define BU0 -9.86494403484714822705e-03 297#define BU1 -6.93858572707181764372e-01 298#define BU2 -1.05586262253232909814e+01 299#define BU3 -6.23753324503260060396e+01 300#define BU4 -1.62396669462573470355e+02 301#define BU5 -1.84605092906711035994e+02 302#define BU6 -8.12874355063065934246e+01 303#define BU7 -9.81432934416914548592e+00 304 305#define BV1 1.96512716674392571292e+01 306#define BV2 1.37657754143519042600e+02 307#define BV3 4.34565877475229228821e+02 308#define BV4 6.45387271733267880336e+02 309#define BV5 4.29008140027567833386e+02 310#define BV6 1.08635005541779435134e+02 311#define BV7 6.57024977031928170135e+00 312#define BV8 -6.04244152148580987438e-02 313 314#define CU0 -2.36211856075265944077e-03 315#define CU1 4.14856118683748331666e-01 316#define CU2 -3.72207876035701323847e-01 317#define CU3 3.18346619901161753674e-01 318#define CU4 -1.10894694282396677476e-01 319#define CU5 3.54783043256182359371e-02 320#define CU6 -2.16637559486879084300e-03 321 322#define CV1 1.06420880400844228286e-01 323#define CV2 5.40397917702171048937e-01 324#define CV3 7.18286544141962662868e-02 325#define CV4 1.26171219808761642112e-01 326#define CV5 1.36370839120290507362e-02 327#define CV6 1.19844998467991074170e-02 328 329#define DU0 1.28379167095512558561e-01 330#define DU1 -3.25042107247001499370e-01 331#define DU2 -2.84817495755985104766e-02 332#define DU3 -5.77027029648944159157e-03 333#define DU4 -2.37630166566501626084e-05 334 335#define DV1 3.97917223959155352819e-01 336#define DV2 6.50222499887672944485e-02 337#define DV3 5.08130628187576562776e-03 338#define DV4 1.32494738004321644526e-04 339#define DV5 -3.96022827877536812320e-06 340 341_CLC_OVERLOAD _CLC_DEF double erf(double y) { 342 double x = fabs(y); 343 double x2 = x * x; 344 double xm1 = x - 1.0; 345 346 // Poly variable 347 double t = 1.0 / x2; 348 t = x < 1.25 ? xm1 : t; 349 t = x < 0.84375 ? x2 : t; 350 351 double u, ut, v, vt; 352 353 // Evaluate rational poly 354 // XXX We need to see of we can grab 16 coefficents from a table 355 // faster than evaluating 3 of the poly pairs 356 // if (x < 6.0) 357 u = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, AU6, AU5), AU4), AU3), AU2), AU1), AU0); 358 v = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, AV7, AV6), AV5), AV4), AV3), AV2), AV1); 359 360 ut = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BU7, BU6), BU5), BU4), BU3), BU2), BU1), BU0); 361 vt = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, BV8, BV7), BV6), BV5), BV4), BV3), BV2), BV1); 362 u = x < 0x1.6db6ep+1 ? ut : u; 363 v = x < 0x1.6db6ep+1 ? vt : v; 364 365 ut = fma(t, fma(t, fma(t, fma(t, fma(t, fma(t, CU6, CU5), CU4), CU3), CU2), CU1), CU0); 366 vt = fma(t, fma(t, fma(t, fma(t, fma(t, CV6, CV5), CV4), CV3), CV2), CV1); 367 u = x < 1.25 ? ut : u; 368 v = x < 1.25 ? vt : v; 369 370 ut = fma(t, fma(t, fma(t, fma(t, DU4, DU3), DU2), DU1), DU0); 371 vt = fma(t, fma(t, fma(t, fma(t, DV5, DV4), DV3), DV2), DV1); 372 u = x < 0.84375 ? ut : u; 373 v = x < 0.84375 ? vt : v; 374 375 v = fma(t, v, 1.0); 376 377 // Compute rational approximation 378 double q = u / v; 379 380 // Compute results 381 double z = as_double(as_long(x) & 0xffffffff00000000L); 382 double r = exp(-z * z - 0.5625) * exp((z - x) * (z + x) + q); 383 r = 1.0 - r / x; 384 385 double ret = x < 6.0 ? r : 1.0; 386 387 r = 8.45062911510467529297e-01 + q; 388 ret = x < 1.25 ? r : ret; 389 390 q = x < 0x1.0p-28 ? 1.28379167095512586316e-01 : q; 391 392 r = fma(x, q, x); 393 ret = x < 0.84375 ? r : ret; 394 395 ret = isnan(x) ? x : ret; 396 397 return y < 0.0 ? -ret : ret; 398} 399 400_CLC_UNARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, double, erf, double); 401 402#endif 403