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#include <clc/clcmacro.h> 25#include <clc/math/clc_fabs.h> 26#include <clc/math/clc_mad.h> 27#include <clc/math/clc_subnormal_config.h> 28#include <clc/math/math.h> 29#include <clc/math/tables.h> 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_CLC_DEF _CLC_OVERLOAD float __clc_rootn(float x, int ny) { 68 float y = MATH_RECIP((float)ny); 69 70 int ix = as_int(x); 71 int ax = ix & EXSIGNBIT_SP32; 72 int xpos = ix == ax; 73 74 int iy = as_int(y); 75 int ay = iy & EXSIGNBIT_SP32; 76 int ypos = iy == ay; 77 78 // Extra precise log calculation 79 // First handle case that x is close to 1 80 float r = 1.0f - as_float(ax); 81 int near1 = __clc_fabs(r) < 0x1.0p-4f; 82 float r2 = r * r; 83 84 // Coefficients are just 1/3, 1/4, 1/5 and 1/6 85 float poly = __clc_mad( 86 r, 87 __clc_mad(r, 88 __clc_mad(r, __clc_mad(r, 0x1.24924ap-3f, 0x1.555556p-3f), 89 0x1.99999ap-3f), 90 0x1.000000p-2f), 91 0x1.555556p-2f); 92 93 poly *= r2 * r; 94 95 float lth_near1 = -r2 * 0.5f; 96 float ltt_near1 = -poly; 97 float lt_near1 = lth_near1 + ltt_near1; 98 float lh_near1 = -r; 99 float l_near1 = lh_near1 + lt_near1; 100 101 // Computations for x not near 1 102 int m = (int)(ax >> EXPSHIFTBITS_SP32) - EXPBIAS_SP32; 103 float mf = (float)m; 104 int ixs = as_int(as_float(ax | 0x3f800000) - 1.0f); 105 float mfs = (float)((ixs >> EXPSHIFTBITS_SP32) - 253); 106 int c = m == -127; 107 int ixn = c ? ixs : ax; 108 float mfn = c ? mfs : mf; 109 110 int indx = (ixn & 0x007f0000) + ((ixn & 0x00008000) << 1); 111 112 // F - Y 113 float f = as_float(0x3f000000 | indx) - 114 as_float(0x3f000000 | (ixn & MANTBITS_SP32)); 115 116 indx = indx >> 16; 117 float2 tv = USE_TABLE(log_inv_tbl_ep, indx); 118 float rh = f * tv.s0; 119 float rt = f * tv.s1; 120 r = rh + rt; 121 122 poly = __clc_mad(r, __clc_mad(r, 0x1.0p-2f, 0x1.555556p-2f), 0x1.0p-1f) * 123 (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 = __clc_mad(mfn, LOG2_TAIL, -poly) + tv.s1; 131 float lt = lth + ltt; 132 float lh = __clc_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 fny = (float)ny; 148 float fnyh = as_float(as_int(fny) & 0xfffff000); 149 float fnyt = (float)(ny - (int)fnyh); 150 float yt = MATH_DIVIDE(__clc_mad(-fnyt, yh, __clc_mad(-fnyh, yh, 1.0f)), fny); 151 152 float ylogx_s = __clc_mad(gt, yh, __clc_mad(gh, yt, yt * gt)); 153 float ylogx = __clc_mad(yh, gh, ylogx_s); 154 float ylogx_t = __clc_mad(yh, gh, -ylogx) + ylogx_s; 155 156 // Extra precise exp of ylogx 157 const float R_64_BY_LOG2 = 0x1.715476p+6f; // 64/log2 : 92.332482616893657 158 int n = convert_int(ylogx * R_64_BY_LOG2); 159 float nf = (float)n; 160 161 int j = n & 0x3f; 162 m = n >> 6; 163 int m2 = m << EXPSHIFTBITS_SP32; 164 165 // log2/64 lead: 0.0108032227 166 const float R_LOG2_BY_64_LD = 0x1.620000p-7f; 167 // log2/64 tail: 0.0000272020388 168 const float R_LOG2_BY_64_TL = 0x1.c85fdep-16f; 169 r = __clc_mad(nf, -R_LOG2_BY_64_TL, __clc_mad(nf, -R_LOG2_BY_64_LD, ylogx)) + 170 ylogx_t; 171 172 // Truncated Taylor series for e^r 173 poly = __clc_mad(__clc_mad(__clc_mad(r, 0x1.555556p-5f, 0x1.555556p-3f), r, 174 0x1.000000p-1f), 175 r * r, r); 176 177 tv = USE_TABLE(exp_tbl_ep, j); 178 179 float expylogx = 180 __clc_mad(tv.s0, poly, __clc_mad(tv.s1, poly, tv.s1)) + tv.s0; 181 float sexpylogx = __clc_fp32_subnormals_supported() 182 ? expylogx * as_float(0x1 << (m + 149)) 183 : 0.0f; 184 185 float texpylogx = as_float(as_int(expylogx) + m2); 186 expylogx = m < -125 ? sexpylogx : texpylogx; 187 188 // Result is +-Inf if (ylogx + ylogx_t) > 128*log2 189 expylogx = ((ylogx > 0x1.62e430p+6f) | 190 (ylogx == 0x1.62e430p+6f & ylogx_t > -0x1.05c610p-22f)) 191 ? as_float(PINFBITPATT_SP32) 192 : expylogx; 193 194 // Result is 0 if ylogx < -149*log2 195 expylogx = ylogx < -0x1.9d1da0p+6f ? 0.0f : expylogx; 196 197 // Classify y: 198 // inty = 0 means not an integer. 199 // inty = 1 means odd integer. 200 // inty = 2 means even integer. 201 202 int inty = 2 - (ny & 1); 203 204 float signval = as_float((as_uint(expylogx) ^ SIGNBIT_SP32)); 205 expylogx = ((inty == 1) & !xpos) ? signval : expylogx; 206 int ret = as_int(expylogx); 207 208 // Corner case handling 209 ret = (!xpos & (inty == 2)) ? QNANBITPATT_SP32 : ret; 210 int xinf = xpos ? PINFBITPATT_SP32 : NINFBITPATT_SP32; 211 ret = ((ax == 0) & !ypos & (inty == 1)) ? xinf : ret; 212 ret = ((ax == 0) & !ypos & (inty == 2)) ? PINFBITPATT_SP32 : ret; 213 ret = ((ax == 0) & ypos & (inty == 2)) ? 0 : ret; 214 int xzero = xpos ? 0 : 0x80000000; 215 ret = ((ax == 0) & ypos & (inty == 1)) ? xzero : ret; 216 ret = 217 ((ix == NINFBITPATT_SP32) & ypos & (inty == 1)) ? NINFBITPATT_SP32 : ret; 218 ret = ((ix == NINFBITPATT_SP32) & !ypos & (inty == 1)) ? 0x80000000 : ret; 219 ret = ((ix == PINFBITPATT_SP32) & !ypos) ? 0 : ret; 220 ret = ((ix == PINFBITPATT_SP32) & ypos) ? PINFBITPATT_SP32 : ret; 221 ret = ax > PINFBITPATT_SP32 ? ix : ret; 222 ret = ny == 0 ? QNANBITPATT_SP32 : ret; 223 224 return as_float(ret); 225} 226_CLC_BINARY_VECTORIZE(_CLC_DEF _CLC_OVERLOAD, float, __clc_rootn, float, int) 227 228#ifdef cl_khr_fp64 229_CLC_DEF _CLC_OVERLOAD double __clc_rootn(double x, int ny) { 230 const double real_log2_tail = 5.76999904754328540596e-08; 231 const double real_log2_lead = 6.93147122859954833984e-01; 232 233 double dny = (double)ny; 234 double y = 1.0 / dny; 235 236 long ux = as_long(x); 237 long ax = ux & (~SIGNBIT_DP64); 238 int xpos = ax == ux; 239 240 long uy = as_long(y); 241 long ay = uy & (~SIGNBIT_DP64); 242 int ypos = ay == uy; 243 244 // Extended precision log 245 double v, vt; 246 { 247 int exp = (int)(ax >> 52) - 1023; 248 int mask_exp_1023 = exp == -1023; 249 double xexp = (double)exp; 250 long mantissa = ax & 0x000FFFFFFFFFFFFFL; 251 252 long temp_ux = as_long(as_double(0x3ff0000000000000L | mantissa) - 1.0); 253 exp = ((temp_ux & 0x7FF0000000000000L) >> 52) - 2045; 254 double xexp1 = (double)exp; 255 long mantissa1 = temp_ux & 0x000FFFFFFFFFFFFFL; 256 257 xexp = mask_exp_1023 ? xexp1 : xexp; 258 mantissa = mask_exp_1023 ? mantissa1 : mantissa; 259 260 long rax = (mantissa & 0x000ff00000000000) + 261 ((mantissa & 0x0000080000000000) << 1); 262 int index = rax >> 44; 263 264 double F = as_double(rax | 0x3FE0000000000000L); 265 double Y = as_double(mantissa | 0x3FE0000000000000L); 266 double f = F - Y; 267 double2 tv = USE_TABLE(log_f_inv_tbl, index); 268 double log_h = tv.s0; 269 double log_t = tv.s1; 270 double f_inv = (log_h + log_t) * f; 271 double r1 = as_double(as_long(f_inv) & 0xfffffffff8000000L); 272 double r2 = fma(-F, r1, f) * (log_h + log_t); 273 double r = r1 + r2; 274 275 double poly = fma( 276 r, fma(r, fma(r, fma(r, 1.0 / 7.0, 1.0 / 6.0), 1.0 / 5.0), 1.0 / 4.0), 277 1.0 / 3.0); 278 poly = poly * r * r * r; 279 280 double hr1r1 = 0.5 * r1 * r1; 281 double poly0h = r1 + hr1r1; 282 double poly0t = r1 - poly0h + hr1r1; 283 poly = fma(r1, r2, fma(0.5 * r2, r2, poly)) + r2 + poly0t; 284 285 tv = USE_TABLE(powlog_tbl, index); 286 log_h = tv.s0; 287 log_t = tv.s1; 288 289 double resT_t = fma(xexp, real_log2_tail, +log_t) - poly; 290 double resT = resT_t - poly0h; 291 double resH = fma(xexp, real_log2_lead, log_h); 292 double resT_h = poly0h; 293 294 double H = resT + resH; 295 double H_h = as_double(as_long(H) & 0xfffffffff8000000L); 296 double T = (resH - H + resT) + (resT_t - (resT + resT_h)) + (H - H_h); 297 H = H_h; 298 299 double y_head = as_double(uy & 0xfffffffff8000000L); 300 double y_tail = y - y_head; 301 302 double fnyh = as_double(as_long(dny) & 0xfffffffffff00000); 303 double fnyt = (double)(ny - (int)fnyh); 304 y_tail = fma(-fnyt, y_head, fma(-fnyh, y_head, 1.0)) / dny; 305 306 double temp = fma(y_tail, H, fma(y_head, T, y_tail * T)); 307 v = fma(y_head, H, temp); 308 vt = fma(y_head, H, -v) + temp; 309 } 310 311 // Now calculate exp of (v,vt) 312 313 double expv; 314 { 315 const double max_exp_arg = 709.782712893384; 316 const double min_exp_arg = -745.1332191019411; 317 const double sixtyfour_by_lnof2 = 92.33248261689366; 318 const double lnof2_by_64_head = 0.010830424260348081; 319 const double lnof2_by_64_tail = -4.359010638708991e-10; 320 321 double temp = v * sixtyfour_by_lnof2; 322 int n = (int)temp; 323 double dn = (double)n; 324 int j = n & 0x0000003f; 325 int m = n >> 6; 326 327 double2 tv = USE_TABLE(two_to_jby64_ep_tbl, j); 328 double f1 = tv.s0; 329 double f2 = tv.s1; 330 double f = f1 + f2; 331 332 double r1 = fma(dn, -lnof2_by_64_head, v); 333 double r2 = dn * lnof2_by_64_tail; 334 double r = (r1 + r2) + vt; 335 336 double q = fma( 337 r, 338 fma(r, 339 fma(r, 340 fma(r, 1.38889490863777199667e-03, 8.33336798434219616221e-03), 341 4.16666666662260795726e-02), 342 1.66666666665260878863e-01), 343 5.00000000000000008883e-01); 344 q = fma(r * r, q, r); 345 346 expv = fma(f, q, f2) + f1; 347 expv = ldexp(expv, m); 348 349 expv = v > max_exp_arg ? as_double(0x7FF0000000000000L) : expv; 350 expv = v < min_exp_arg ? 0.0 : expv; 351 } 352 353 // See whether y is an integer. 354 // inty = 0 means not an integer. 355 // inty = 1 means odd integer. 356 // inty = 2 means even integer. 357 358 int inty = 2 - (ny & 1); 359 360 expv *= ((inty == 1) & !xpos) ? -1.0 : 1.0; 361 362 long ret = as_long(expv); 363 364 // Now all the edge cases 365 ret = (!xpos & (inty == 2)) ? QNANBITPATT_DP64 : ret; 366 long xinf = xpos ? PINFBITPATT_DP64 : NINFBITPATT_DP64; 367 ret = ((ax == 0L) & !ypos & (inty == 1)) ? xinf : ret; 368 ret = ((ax == 0L) & !ypos & (inty == 2)) ? PINFBITPATT_DP64 : ret; 369 ret = ((ax == 0L) & ypos & (inty == 2)) ? 0L : ret; 370 long xzero = xpos ? 0L : 0x8000000000000000L; 371 ret = ((ax == 0L) & ypos & (inty == 1)) ? xzero : ret; 372 ret = 373 ((ux == NINFBITPATT_DP64) & ypos & (inty == 1)) ? NINFBITPATT_DP64 : ret; 374 ret = ((ux == NINFBITPATT_DP64) & !ypos & (inty == 1)) ? 0x8000000000000000L 375 : ret; 376 ret = ((ux == PINFBITPATT_DP64) & !ypos) ? 0L : ret; 377 ret = ((ux == PINFBITPATT_DP64) & ypos) ? PINFBITPATT_DP64 : ret; 378 ret = ax > PINFBITPATT_DP64 ? ux : ret; 379 ret = ny == 0 ? QNANBITPATT_DP64 : ret; 380 return as_double(ret); 381} 382_CLC_BINARY_VECTORIZE(_CLC_DEF _CLC_OVERLOAD, double, __clc_rootn, double, int) 383#endif 384 385#ifdef cl_khr_fp16 386 387#pragma OPENCL EXTENSION cl_khr_fp16 : enable 388 389_CLC_OVERLOAD _CLC_DEF half __clc_rootn(half x, int y) { 390 return (half)__clc_rootn((float)x, y); 391} 392 393_CLC_BINARY_VECTORIZE(_CLC_OVERLOAD _CLC_DEF, half, __clc_rootn, half, int); 394 395#endif 396