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 "config.h" 26#include "math.h" 27#include "tables.h" 28#include "../clcmacro.h" 29 30/* 31 compute pow using log and exp 32 x^y = exp(y * log(x)) 33 34 we take care not to lose precision in the intermediate steps 35 36 When computing log, calculate it in splits, 37 38 r = f * (p_invead + p_inv_tail) 39 r = rh + rt 40 41 calculate log polynomial using r, in end addition, do 42 poly = poly + ((rh-r) + rt) 43 44 lth = -r 45 ltt = ((xexp * log2_t) - poly) + logT 46 lt = lth + ltt 47 48 lh = (xexp * log2_h) + logH 49 l = lh + lt 50 51 Calculate final log answer as gh and gt, 52 gh = l & higher-half bits 53 gt = (((ltt - (lt - lth)) + ((lh - l) + lt)) + (l - gh)) 54 55 yh = y & higher-half bits 56 yt = y - yh 57 58 Before entering computation of exp, 59 vs = ((yt*gt + yt*gh) + yh*gt) 60 v = vs + yh*gh 61 vt = ((yh*gh - v) + vs) 62 63 In calculation of exp, add vt to r that is used for poly 64 At the end of exp, do 65 ((((expT * poly) + expT) + expH*poly) + expH) 66*/ 67 68_CLC_DEF _CLC_OVERLOAD float __clc_pow(float x, float y) 69{ 70 71 int ix = as_int(x); 72 int ax = ix & EXSIGNBIT_SP32; 73 int xpos = ix == ax; 74 75 int iy = as_int(y); 76 int ay = iy & EXSIGNBIT_SP32; 77 int ypos = iy == ay; 78 79 /* Extra precise log calculation 80 * First handle case that x is close to 1 81 */ 82 float r = 1.0f - as_float(ax); 83 int near1 = fabs(r) < 0x1.0p-4f; 84 float r2 = r*r; 85 86 /* Coefficients are just 1/3, 1/4, 1/5 and 1/6 */ 87 float poly = mad(r, 88 mad(r, 89 mad(r, 90 mad(r, 0x1.24924ap-3f, 0x1.555556p-3f), 91 0x1.99999ap-3f), 92 0x1.000000p-2f), 93 0x1.555556p-2f); 94 95 poly *= r2*r; 96 97 float lth_near1 = -r2 * 0.5f; 98 float ltt_near1 = -poly; 99 float lt_near1 = lth_near1 + ltt_near1; 100 float lh_near1 = -r; 101 float l_near1 = lh_near1 + lt_near1; 102 103 /* Computations for x not near 1 */ 104 int m = (int)(ax >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32; 105 float mf = (float)m; 106 int ixs = as_int(as_float(ax | 0x3f800000) - 1.0f); 107 float mfs = (float)((ixs >> EXPSHIFTBITS_SP32) - 253); 108 int c = m == -127; 109 int ixn = c ? ixs : ax; 110 float mfn = c ? mfs : mf; 111 112 int indx = (ixn & 0x007f0000) + ((ixn & 0x00008000) << 1); 113 114 /* F - Y */ 115 float f = as_float(0x3f000000 | indx) - as_float(0x3f000000 | (ixn & MANTBITS_SP32)); 116 117 indx = indx >> 16; 118 float2 tv = USE_TABLE(log_inv_tbl_ep, indx); 119 float rh = f * tv.s0; 120 float rt = f * tv.s1; 121 r = rh + rt; 122 123 poly = mad(r, mad(r, 0x1.0p-2f, 0x1.555556p-2f), 0x1.0p-1f) * (r*r); 124 poly += (rh - r) + rt; 125 126 const float LOG2_HEAD = 0x1.62e000p-1f; /* 0.693115234 */ 127 const float LOG2_TAIL = 0x1.0bfbe8p-15f; /* 0.0000319461833 */ 128 tv = USE_TABLE(loge_tbl, indx); 129 float lth = -r; 130 float ltt = mad(mfn, LOG2_TAIL, -poly) + tv.s1; 131 float lt = lth + ltt; 132 float lh = mad(mfn, LOG2_HEAD, tv.s0); 133 float l = lh + lt; 134 135 /* Select near 1 or not */ 136 lth = near1 ? lth_near1 : lth; 137 ltt = near1 ? ltt_near1 : ltt; 138 lt = near1 ? lt_near1 : lt; 139 lh = near1 ? lh_near1 : lh; 140 l = near1 ? l_near1 : l; 141 142 float gh = as_float(as_int(l) & 0xfffff000); 143 float gt = ((ltt - (lt - lth)) + ((lh - l) + lt)) + (l - gh); 144 145 float yh = as_float(iy & 0xfffff000); 146 147 float yt = y - yh; 148 149 float ylogx_s = mad(gt, yh, mad(gh, yt, yt*gt)); 150 float ylogx = mad(yh, gh, ylogx_s); 151 float ylogx_t = mad(yh, gh, -ylogx) + ylogx_s; 152 153 /* Extra precise exp of ylogx */ 154 const float R_64_BY_LOG2 = 0x1.715476p+6f; /* 64/log2 : 92.332482616893657 */ 155 int n = convert_int(ylogx * R_64_BY_LOG2); 156 float nf = (float) n; 157 158 int j = n & 0x3f; 159 m = n >> 6; 160 int m2 = m << EXPSHIFTBITS_SP32; 161 162 const float R_LOG2_BY_64_LD = 0x1.620000p-7f; /* log2/64 lead: 0.0108032227 */ 163 const float R_LOG2_BY_64_TL = 0x1.c85fdep-16f; /* log2/64 tail: 0.0000272020388 */ 164 r = mad(nf, -R_LOG2_BY_64_TL, mad(nf, -R_LOG2_BY_64_LD, ylogx)) + ylogx_t; 165 166 /* Truncated Taylor series for e^r */ 167 poly = mad(mad(mad(r, 0x1.555556p-5f, 0x1.555556p-3f), r, 0x1.000000p-1f), r*r, r); 168 169 tv = USE_TABLE(exp_tbl_ep, j); 170 171 float expylogx = mad(tv.s0, poly, mad(tv.s1, poly, tv.s1)) + tv.s0; 172 float sexpylogx = expylogx * as_float(0x1 << (m + 149)); 173 float texpylogx = as_float(as_int(expylogx) + m2); 174 expylogx = m < -125 ? sexpylogx : texpylogx; 175 176 /* Result is +-Inf if (ylogx + ylogx_t) > 128*log2 */ 177 expylogx = (ylogx > 0x1.62e430p+6f) | (ylogx == 0x1.62e430p+6f & ylogx_t > -0x1.05c610p-22f) ? as_float(PINFBITPATT_SP32) : expylogx; 178 179 /* Result is 0 if ylogx < -149*log2 */ 180 expylogx = ylogx < -0x1.9d1da0p+6f ? 0.0f : expylogx; 181 182 /* Classify y: 183 * inty = 0 means not an integer. 184 * inty = 1 means odd integer. 185 * inty = 2 means even integer. 186 */ 187 188 int yexp = (int)(ay >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32 + 1; 189 int mask = (1 << (24 - yexp)) - 1; 190 int yodd = ((iy >> (24 - yexp)) & 0x1) != 0; 191 int inty = yodd ? 1 : 2; 192 inty = (iy & mask) != 0 ? 0 : inty; 193 inty = yexp < 1 ? 0 : inty; 194 inty = yexp > 24 ? 2 : inty; 195 196 float signval = as_float((as_uint(expylogx) ^ SIGNBIT_SP32)); 197 expylogx = ((inty == 1) & !xpos) ? signval : expylogx; 198 int ret = as_int(expylogx); 199 200 /* Corner case handling */ 201 ret = (!xpos & (inty == 0)) ? QNANBITPATT_SP32 : ret; 202 ret = ax < 0x3f800000 & iy == NINFBITPATT_SP32 ? PINFBITPATT_SP32 : ret; 203 ret = ax > 0x3f800000 & iy == NINFBITPATT_SP32 ? 0 : ret; 204 ret = ax < 0x3f800000 & iy == PINFBITPATT_SP32 ? 0 : ret; 205 ret = ax > 0x3f800000 & iy == PINFBITPATT_SP32 ? PINFBITPATT_SP32 : ret; 206 int xinf = xpos ? PINFBITPATT_SP32 : NINFBITPATT_SP32; 207 ret = ((ax == 0) & !ypos & (inty == 1)) ? xinf : ret; 208 ret = ((ax == 0) & !ypos & (inty != 1)) ? PINFBITPATT_SP32 : ret; 209 int xzero = xpos ? 0 : 0x80000000; 210 ret = ((ax == 0) & ypos & (inty == 1)) ? xzero : ret; 211 ret = ((ax == 0) & ypos & (inty != 1)) ? 0 : ret; 212 ret = ((ax == 0) & (iy == NINFBITPATT_SP32)) ? PINFBITPATT_SP32 : ret; 213 ret = ((ix == 0xbf800000) & (ay == PINFBITPATT_SP32)) ? 0x3f800000 : ret; 214 ret = ((ix == NINFBITPATT_SP32) & !ypos & (inty == 1)) ? 0x80000000 : ret; 215 ret = ((ix == NINFBITPATT_SP32) & !ypos & (inty != 1)) ? 0 : ret; 216 ret = ((ix == NINFBITPATT_SP32) & ypos & (inty == 1)) ? NINFBITPATT_SP32 : ret; 217 ret = ((ix == NINFBITPATT_SP32) & ypos & (inty != 1)) ? PINFBITPATT_SP32 : ret; 218 ret = ((ix == PINFBITPATT_SP32) & !ypos) ? 0 : ret; 219 ret = ((ix == PINFBITPATT_SP32) & ypos) ? PINFBITPATT_SP32 : ret; 220 ret = (ax > PINFBITPATT_SP32) ? ix : ret; 221 ret = (ay > PINFBITPATT_SP32) ? iy : ret; 222 ret = ay == 0 ? 0x3f800000 : ret; 223 ret = ix == 0x3f800000 ? 0x3f800000 : ret; 224 225 return as_float(ret); 226} 227_CLC_BINARY_VECTORIZE(_CLC_DEF _CLC_OVERLOAD, float, __clc_pow, float, float) 228 229#ifdef cl_khr_fp64 230_CLC_DEF _CLC_OVERLOAD double __clc_pow(double x, double y) 231{ 232 const double real_log2_tail = 5.76999904754328540596e-08; 233 const double real_log2_lead = 6.93147122859954833984e-01; 234 235 long ux = as_long(x); 236 long ax = ux & (~SIGNBIT_DP64); 237 int xpos = ax == ux; 238 239 long uy = as_long(y); 240 long ay = uy & (~SIGNBIT_DP64); 241 int ypos = ay == uy; 242 243 // Extended precision log 244 double v, vt; 245 { 246 int exp = (int)(ax >> 52) - 1023; 247 int mask_exp_1023 = exp == -1023; 248 double xexp = (double) exp; 249 long mantissa = ax & 0x000FFFFFFFFFFFFFL; 250 251 long temp_ux = as_long(as_double(0x3ff0000000000000L | mantissa) - 1.0); 252 exp = ((temp_ux & 0x7FF0000000000000L) >> 52) - 2045; 253 double xexp1 = (double) exp; 254 long mantissa1 = temp_ux & 0x000FFFFFFFFFFFFFL; 255 256 xexp = mask_exp_1023 ? xexp1 : xexp; 257 mantissa = mask_exp_1023 ? mantissa1 : mantissa; 258 259 long rax = (mantissa & 0x000ff00000000000) + ((mantissa & 0x0000080000000000) << 1); 260 int index = rax >> 44; 261 262 double F = as_double(rax | 0x3FE0000000000000L); 263 double Y = as_double(mantissa | 0x3FE0000000000000L); 264 double f = F - Y; 265 double2 tv = USE_TABLE(log_f_inv_tbl, index); 266 double log_h = tv.s0; 267 double log_t = tv.s1; 268 double f_inv = (log_h + log_t) * f; 269 double r1 = as_double(as_long(f_inv) & 0xfffffffff8000000L); 270 double r2 = fma(-F, r1, f) * (log_h + log_t); 271 double r = r1 + r2; 272 273 double poly = fma(r, 274 fma(r, 275 fma(r, 276 fma(r, 1.0/7.0, 1.0/6.0), 277 1.0/5.0), 278 1.0/4.0), 279 1.0/3.0); 280 poly = poly * r * r * r; 281 282 double hr1r1 = 0.5*r1*r1; 283 double poly0h = r1 + hr1r1; 284 double poly0t = r1 - poly0h + hr1r1; 285 poly = fma(r1, r2, fma(0.5*r2, r2, poly)) + r2 + poly0t; 286 287 tv = USE_TABLE(powlog_tbl, index); 288 log_h = tv.s0; 289 log_t = tv.s1; 290 291 double resT_t = fma(xexp, real_log2_tail, + log_t) - poly; 292 double resT = resT_t - poly0h; 293 double resH = fma(xexp, real_log2_lead, log_h); 294 double resT_h = poly0h; 295 296 double H = resT + resH; 297 double H_h = as_double(as_long(H) & 0xfffffffff8000000L); 298 double T = (resH - H + resT) + (resT_t - (resT + resT_h)) + (H - H_h); 299 H = H_h; 300 301 double y_head = as_double(uy & 0xfffffffff8000000L); 302 double y_tail = y - y_head; 303 304 double temp = fma(y_tail, H, fma(y_head, T, y_tail*T)); 305 v = fma(y_head, H, temp); 306 vt = fma(y_head, H, -v) + temp; 307 } 308 309 // Now calculate exp of (v,vt) 310 311 double expv; 312 { 313 const double max_exp_arg = 709.782712893384; 314 const double min_exp_arg = -745.1332191019411; 315 const double sixtyfour_by_lnof2 = 92.33248261689366; 316 const double lnof2_by_64_head = 0.010830424260348081; 317 const double lnof2_by_64_tail = -4.359010638708991e-10; 318 319 double temp = v * sixtyfour_by_lnof2; 320 int n = (int)temp; 321 double dn = (double)n; 322 int j = n & 0x0000003f; 323 int m = n >> 6; 324 325 double2 tv = USE_TABLE(two_to_jby64_ep_tbl, j); 326 double f1 = tv.s0; 327 double f2 = tv.s1; 328 double f = f1 + f2; 329 330 double r1 = fma(dn, -lnof2_by_64_head, v); 331 double r2 = dn * lnof2_by_64_tail; 332 double r = (r1 + r2) + vt; 333 334 double q = fma(r, 335 fma(r, 336 fma(r, 337 fma(r, 1.38889490863777199667e-03, 8.33336798434219616221e-03), 338 4.16666666662260795726e-02), 339 1.66666666665260878863e-01), 340 5.00000000000000008883e-01); 341 q = fma(r*r, q, r); 342 343 expv = fma(f, q, f2) + f1; 344 expv = ldexp(expv, m); 345 346 expv = v > max_exp_arg ? as_double(0x7FF0000000000000L) : expv; 347 expv = v < min_exp_arg ? 0.0 : expv; 348 } 349 350 // See whether y is an integer. 351 // inty = 0 means not an integer. 352 // inty = 1 means odd integer. 353 // inty = 2 means even integer. 354 355 int inty; 356 { 357 int yexp = (int)(ay >> EXPSHIFTBITS_DP64) - EXPBIAS_DP64 + 1; 358 inty = yexp < 1 ? 0 : 2; 359 inty = yexp > 53 ? 2 : inty; 360 long mask = (1L << (53 - yexp)) - 1L; 361 int inty1 = (((ay & ~mask) >> (53 - yexp)) & 1L) == 1L ? 1 : 2; 362 inty1 = (ay & mask) != 0 ? 0 : inty1; 363 inty = !(yexp < 1) & !(yexp > 53) ? inty1 : inty; 364 } 365 366 expv *= (inty == 1) & !xpos ? -1.0 : 1.0; 367 368 long ret = as_long(expv); 369 370 // Now all the edge cases 371 ret = !xpos & (inty == 0) ? QNANBITPATT_DP64 : ret; 372 ret = ax < 0x3ff0000000000000L & uy == NINFBITPATT_DP64 ? PINFBITPATT_DP64 : ret; 373 ret = ax > 0x3ff0000000000000L & uy == NINFBITPATT_DP64 ? 0L : ret; 374 ret = ax < 0x3ff0000000000000L & uy == PINFBITPATT_DP64 ? 0L : ret; 375 ret = ax > 0x3ff0000000000000L & uy == PINFBITPATT_DP64 ? PINFBITPATT_DP64 : ret; 376 long xinf = xpos ? PINFBITPATT_DP64 : NINFBITPATT_DP64; 377 ret = ((ax == 0L) & !ypos & (inty == 1)) ? xinf : ret; 378 ret = ((ax == 0L) & !ypos & (inty != 1)) ? PINFBITPATT_DP64 : ret; 379 long xzero = xpos ? 0L : 0x8000000000000000L; 380 ret = ((ax == 0L) & ypos & (inty == 1)) ? xzero : ret; 381 ret = ((ax == 0L) & ypos & (inty != 1)) ? 0L : ret; 382 ret = ((ax == 0L) & (uy == NINFBITPATT_DP64)) ? PINFBITPATT_DP64 : ret; 383 ret = ((ux == 0xbff0000000000000L) & (ay == PINFBITPATT_DP64)) ? 0x3ff0000000000000L : ret; 384 ret = ((ux == NINFBITPATT_DP64) & !ypos & (inty == 1)) ? 0x8000000000000000L : ret; 385 ret = ((ux == NINFBITPATT_DP64) & !ypos & (inty != 1)) ? 0L : ret; 386 ret = ((ux == NINFBITPATT_DP64) & ypos & (inty == 1)) ? NINFBITPATT_DP64 : ret; 387 ret = ((ux == NINFBITPATT_DP64) & ypos & (inty != 1)) ? PINFBITPATT_DP64 : ret; 388 ret = (ux == PINFBITPATT_DP64) & !ypos ? 0L : ret; 389 ret = (ux == PINFBITPATT_DP64) & ypos ? PINFBITPATT_DP64 : ret; 390 ret = ax > PINFBITPATT_DP64 ? ux : ret; 391 ret = ay > PINFBITPATT_DP64 ? uy : ret; 392 ret = ay == 0L ? 0x3ff0000000000000L : ret; 393 ret = ux == 0x3ff0000000000000L ? 0x3ff0000000000000L : ret; 394 395 return as_double(ret); 396} 397_CLC_BINARY_VECTORIZE(_CLC_DEF _CLC_OVERLOAD, double, __clc_pow, double, double) 398#endif 399