1 /* 2 * CDDL HEADER START 3 * 4 * The contents of this file are subject to the terms of the 5 * Common Development and Distribution License (the "License"). 6 * You may not use this file except in compliance with the License. 7 * 8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE 9 * or http://www.opensolaris.org/os/licensing. 10 * See the License for the specific language governing permissions 11 * and limitations under the License. 12 * 13 * When distributing Covered Code, include this CDDL HEADER in each 14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE. 15 * If applicable, add the following below this CDDL HEADER, with the 16 * fields enclosed by brackets "[]" replaced with your own identifying 17 * information: Portions Copyright [yyyy] [name of copyright owner] 18 * 19 * CDDL HEADER END 20 */ 21 22 /* 23 * Copyright 2011 Nexenta Systems, Inc. All rights reserved. 24 */ 25 /* 26 * Copyright 2006 Sun Microsystems, Inc. All rights reserved. 27 * Use is subject to license terms. 28 */ 29 30 #include <sys/isa_defs.h> 31 #include "libm_inlines.h" 32 33 #ifdef _LITTLE_ENDIAN 34 #define HI(x) *(1+(int*)x) 35 #define LO(x) *(unsigned*)x 36 #else 37 #define HI(x) *(int*)x 38 #define LO(x) *(1+(unsigned*)x) 39 #endif 40 41 #ifdef __RESTRICT 42 #define restrict _Restrict 43 #else 44 #define restrict 45 #endif 46 47 /* float rhypotf(float x, float y) 48 * 49 * Method : 50 * 1. Special cases: 51 * for x or y = Inf => 0; 52 * for x or y = NaN => QNaN; 53 * for x and y = 0 => +Inf + divide-by-zero; 54 * 2. Computes d = x * x + y * y; 55 * 3. Computes reciprocal square root from: 56 * d = m * 2**n 57 * Where: 58 * m = [0.5, 2), 59 * n = ((exponent + 1) & ~1). 60 * Then: 61 * rsqrtf(d) = 1/sqrt( m * 2**n ) = (2 ** (-n/2)) * (1/sqrt(m)) 62 * 4. Computes 1/sqrt(m) from: 63 * 1/sqrt(m) = (1/sqrt(m0)) * (1/sqrt(1 + (1/m0)*dm)) 64 * Where: 65 * m = m0 + dm, 66 * m0 = 0.5 * (1 + k/64) for m = [0.5, 0.5+127/256), k = [0, 63]; 67 * m0 = 1.0 * (0 + k/64) for m = [0.5+127/256, 1.0+127/128), k = [64, 127]; 68 * Then: 69 * 1/sqrt(m0), 1/m0 are looked up in a table, 70 * 1/sqrt(1 + (1/m0)*dm) is computed using approximation: 71 * 1/sqrt(1 + z) = ((a3 * z + a2) * z + a1) * z + a0 72 * where z = [-1/64, 1/64]. 73 * 74 * Accuracy: 75 * The maximum relative error for the approximating 76 * polynomial is 2**(-27.87). 77 * Maximum error observed: less than 0.535 ulp after 3.000.000.000 78 * results. 79 */ 80 81 #pragma align 32 (__vlibm_TBL_rhypotf) 82 83 static const double __vlibm_TBL_rhypotf[] = { 84 /* 85 i = [0,63] 86 TBL[2*i+0] = 1.0 / (*(double*)&(0x3ff0000000000000LL + (i << 46))); 87 TBL[2*i+1] = (double)(0.5/sqrtl(2) / sqrtl(*(double*)&(0x3ff0000000000000LL + (i << 46)))); 88 TBL[128+2*i+0] = 1.0 / (*(double*)&(0x3ff0000000000000LL + (i << 46))); 89 TBL[128+2*i+1] = (double)(0.25 / sqrtl(*(double*)&(0x3ff0000000000000LL + (i << 46)))); 90 */ 91 1.0000000000000000000e+00, 3.5355339059327378637e-01, 92 9.8461538461538467004e-01, 3.5082320772281166965e-01, 93 9.6969696969696972388e-01, 3.4815531191139570399e-01, 94 9.5522388059701490715e-01, 3.4554737023254405992e-01, 95 9.4117647058823528106e-01, 3.4299717028501769400e-01, 96 9.2753623188405798228e-01, 3.4050261230349943009e-01, 97 9.1428571428571425717e-01, 3.3806170189140660742e-01, 98 9.0140845070422537244e-01, 3.3567254331867563133e-01, 99 8.8888888888888883955e-01, 3.3333333333333331483e-01, 100 8.7671232876712323900e-01, 3.3104235544094717802e-01, 101 8.6486486486486491287e-01, 3.2879797461071458287e-01, 102 8.5333333333333338810e-01, 3.2659863237109043599e-01, 103 8.4210526315789469010e-01, 3.2444284226152508843e-01, 104 8.3116883116883122362e-01, 3.2232918561015211356e-01, 105 8.2051282051282048435e-01, 3.2025630761017426229e-01, 106 8.1012658227848100001e-01, 3.1822291367029204023e-01, 107 8.0000000000000004441e-01, 3.1622776601683794118e-01, 108 7.9012345679012341293e-01, 3.1426968052735443360e-01, 109 7.8048780487804880757e-01, 3.1234752377721214378e-01, 110 7.7108433734939763049e-01, 3.1046021028253312224e-01, 111 7.6190476190476186247e-01, 3.0860669992418382490e-01, 112 7.5294117647058822484e-01, 3.0678599553894819740e-01, 113 7.4418604651162789665e-01, 3.0499714066520933198e-01, 114 7.3563218390804596680e-01, 3.0323921743156134756e-01, 115 7.2727272727272729291e-01, 3.0151134457776362918e-01, 116 7.1910112359550559802e-01, 2.9981267559834456904e-01, 117 7.1111111111111113825e-01, 2.9814239699997197031e-01, 118 7.0329670329670335160e-01, 2.9649972666444046610e-01, 119 6.9565217391304345895e-01, 2.9488391230979427160e-01, 120 6.8817204301075274309e-01, 2.9329423004270660513e-01, 121 6.8085106382978721751e-01, 2.9172998299578911663e-01, 122 6.7368421052631577428e-01, 2.9019050004400465115e-01, 123 6.6666666666666662966e-01, 2.8867513459481286553e-01, 124 6.5979381443298967813e-01, 2.8718326344709527165e-01, 125 6.5306122448979586625e-01, 2.8571428571428569843e-01, 126 6.4646464646464651960e-01, 2.8426762180748055275e-01, 127 6.4000000000000001332e-01, 2.8284271247461900689e-01, 128 6.3366336633663367106e-01, 2.8143901789211672737e-01, 129 6.2745098039215685404e-01, 2.8005601680560193723e-01, 130 6.2135922330097081989e-01, 2.7869320571664707442e-01, 131 6.1538461538461541878e-01, 2.7735009811261457369e-01, 132 6.0952380952380957879e-01, 2.7602622373694168934e-01, 133 6.0377358490566035432e-01, 2.7472112789737807015e-01, 134 5.9813084112149528249e-01, 2.7343437080986532361e-01, 135 5.9259259259259255970e-01, 2.7216552697590867815e-01, 136 5.8715596330275232617e-01, 2.7091418459143856712e-01, 137 5.8181818181818178992e-01, 2.6967994498529684888e-01, 138 5.7657657657657657158e-01, 2.6846242208560971987e-01, 139 5.7142857142857139685e-01, 2.6726124191242439654e-01, 140 5.6637168141592919568e-01, 2.6607604209509572168e-01, 141 5.6140350877192979340e-01, 2.6490647141300877054e-01, 142 5.5652173913043478937e-01, 2.6375218935831479250e-01, 143 5.5172413793103447510e-01, 2.6261286571944508772e-01, 144 5.4700854700854706358e-01, 2.6148818018424535570e-01, 145 5.4237288135593220151e-01, 2.6037782196164771520e-01, 146 5.3781512605042014474e-01, 2.5928148942086576278e-01, 147 5.3333333333333332593e-01, 2.5819888974716115326e-01, 148 5.2892561983471075848e-01, 2.5712973861329002645e-01, 149 5.2459016393442625681e-01, 2.5607375986579195004e-01, 150 5.2032520325203257539e-01, 2.5503068522533534068e-01, 151 5.1612903225806450180e-01, 2.5400025400038100942e-01, 152 5.1200000000000001066e-01, 2.5298221281347033074e-01, 153 5.0793650793650790831e-01, 2.5197631533948483540e-01, 154 5.0393700787401574104e-01, 2.5098232205526344041e-01, 155 1.0000000000000000000e+00, 2.5000000000000000000e-01, 156 9.8461538461538467004e-01, 2.4806946917841690703e-01, 157 9.6969696969696972388e-01, 2.4618298195866547551e-01, 158 9.5522388059701490715e-01, 2.4433888871261044695e-01, 159 9.4117647058823528106e-01, 2.4253562503633296910e-01, 160 9.2753623188405798228e-01, 2.4077170617153839660e-01, 161 9.1428571428571425717e-01, 2.3904572186687872426e-01, 162 9.0140845070422537244e-01, 2.3735633163877067897e-01, 163 8.8888888888888883955e-01, 2.3570226039551583908e-01, 164 8.7671232876712323900e-01, 2.3408229439226113655e-01, 165 8.6486486486486491287e-01, 2.3249527748763856860e-01, 166 8.5333333333333338810e-01, 2.3094010767585029797e-01, 167 8.4210526315789469010e-01, 2.2941573387056177213e-01, 168 8.3116883116883122362e-01, 2.2792115291927589338e-01, 169 8.2051282051282048435e-01, 2.2645540682891915352e-01, 170 8.1012658227848100001e-01, 2.2501758018520479077e-01, 171 8.0000000000000004441e-01, 2.2360679774997896385e-01, 172 7.9012345679012341293e-01, 2.2222222222222220989e-01, 173 7.8048780487804880757e-01, 2.2086305214969309541e-01, 174 7.7108433734939763049e-01, 2.1952851997938069295e-01, 175 7.6190476190476186247e-01, 2.1821789023599238999e-01, 176 7.5294117647058822484e-01, 2.1693045781865616384e-01, 177 7.4418604651162789665e-01, 2.1566554640687682354e-01, 178 7.3563218390804596680e-01, 2.1442250696755896233e-01, 179 7.2727272727272729291e-01, 2.1320071635561044232e-01, 180 7.1910112359550559802e-01, 2.1199957600127200541e-01, 181 7.1111111111111113825e-01, 2.1081851067789195153e-01, 182 7.0329670329670335160e-01, 2.0965696734438366011e-01, 183 6.9565217391304345895e-01, 2.0851441405707477061e-01, 184 6.8817204301075274309e-01, 2.0739033894608505104e-01, 185 6.8085106382978721751e-01, 2.0628424925175867233e-01, 186 6.7368421052631577428e-01, 2.0519567041703082322e-01, 187 6.6666666666666662966e-01, 2.0412414523193150862e-01, 188 6.5979381443298967813e-01, 2.0306923302672380549e-01, 189 6.5306122448979586625e-01, 2.0203050891044216364e-01, 190 6.4646464646464651960e-01, 2.0100756305184241945e-01, 191 6.4000000000000001332e-01, 2.0000000000000001110e-01, 192 6.3366336633663367106e-01, 1.9900743804199783060e-01, 193 6.2745098039215685404e-01, 1.9802950859533485772e-01, 194 6.2135922330097081989e-01, 1.9706585563285863860e-01, 195 6.1538461538461541878e-01, 1.9611613513818404453e-01, 196 6.0952380952380957879e-01, 1.9518001458970662965e-01, 197 6.0377358490566035432e-01, 1.9425717247145282696e-01, 198 5.9813084112149528249e-01, 1.9334729780913270658e-01, 199 5.9259259259259255970e-01, 1.9245008972987526219e-01, 200 5.8715596330275232617e-01, 1.9156525704423027490e-01, 201 5.8181818181818178992e-01, 1.9069251784911847580e-01, 202 5.7657657657657657158e-01, 1.8983159915049979682e-01, 203 5.7142857142857139685e-01, 1.8898223650461362655e-01, 204 5.6637168141592919568e-01, 1.8814417367671945613e-01, 205 5.6140350877192979340e-01, 1.8731716231633879777e-01, 206 5.5652173913043478937e-01, 1.8650096164806276300e-01, 207 5.5172413793103447510e-01, 1.8569533817705186074e-01, 208 5.4700854700854706358e-01, 1.8490006540840969729e-01, 209 5.4237288135593220151e-01, 1.8411492357966466327e-01, 210 5.3781512605042014474e-01, 1.8333969940564226464e-01, 211 5.3333333333333332593e-01, 1.8257418583505535814e-01, 212 5.2892561983471075848e-01, 1.8181818181818182323e-01, 213 5.2459016393442625681e-01, 1.8107149208503706128e-01, 214 5.2032520325203257539e-01, 1.8033392693348646030e-01, 215 5.1612903225806450180e-01, 1.7960530202677491007e-01, 216 5.1200000000000001066e-01, 1.7888543819998317663e-01, 217 5.0793650793650790831e-01, 1.7817416127494958844e-01, 218 5.0393700787401574104e-01, 1.7747130188322274291e-01, 219 }; 220 221 extern float fabsf(float); 222 223 static const double 224 A0 = 9.99999997962321453275e-01, 225 A1 =-4.99999998166077580600e-01, 226 A2 = 3.75066768969515586277e-01, 227 A3 =-3.12560092408808548438e-01; 228 229 static void 230 __vrhypotf_n(int n, float * restrict px, int stridex, float * restrict py, 231 int stridey, float * restrict pz, int stridez); 232 233 #pragma no_inline(__vrhypotf_n) 234 235 #define RETURN(ret) \ 236 { \ 237 *pz = (ret); \ 238 pz += stridez; \ 239 if (n_n == 0) \ 240 { \ 241 spx = px; spy = py; spz = pz; \ 242 ay0 = *(int*)py; \ 243 continue; \ 244 } \ 245 n--; \ 246 break; \ 247 } 248 249 250 void 251 __vrhypotf(int n, float * restrict px, int stridex, float * restrict py, 252 int stridey, float * restrict pz, int stridez) 253 { 254 float *spx, *spy, *spz; 255 int ax0, ay0, n_n; 256 float res, x0, y0; 257 258 while (n > 1) 259 { 260 n_n = 0; 261 spx = px; 262 spy = py; 263 spz = pz; 264 ax0 = *(int*)px; 265 ay0 = *(int*)py; 266 for (; n > 1 ; n--) 267 { 268 ax0 &= 0x7fffffff; 269 ay0 &= 0x7fffffff; 270 271 px += stridex; 272 273 if (ax0 >= 0x7f800000 || ay0 >= 0x7f800000) /* X or Y = NaN or Inf */ 274 { 275 x0 = *(px - stridex); 276 y0 = *py; 277 res = fabsf(x0) + fabsf(y0); 278 if (ax0 == 0x7f800000) res = 0.0f; 279 else if (ay0 == 0x7f800000) res = 0.0f; 280 ax0 = *(int*)px; 281 py += stridey; 282 RETURN (res) 283 } 284 ax0 = *(int*)px; 285 py += stridey; 286 if (ay0 == 0) /* Y = 0 */ 287 { 288 int tx = *(int*)(px - stridex) & 0x7fffffff; 289 if (tx == 0) /* X = 0 */ 290 { 291 RETURN (1.0f / 0.0f) 292 } 293 } 294 pz += stridez; 295 n_n++; 296 ay0 = *(int*)py; 297 } 298 if (n_n > 0) 299 __vrhypotf_n(n_n, spx, stridex, spy, stridey, spz, stridez); 300 } 301 if (n > 0) 302 { 303 ax0 = *(int*)px; 304 ay0 = *(int*)py; 305 x0 = *px; 306 y0 = *py; 307 308 ax0 &= 0x7fffffff; 309 ay0 &= 0x7fffffff; 310 311 if (ax0 >= 0x7f800000 || ay0 >= 0x7f800000) /* X or Y = NaN or Inf */ 312 { 313 res = fabsf(x0) + fabsf(y0); 314 if (ax0 == 0x7f800000) res = 0.0f; 315 else if (ay0 == 0x7f800000) res = 0.0f; 316 *pz = res; 317 } 318 else if (ax0 == 0 && ay0 == 0) /* X and Y = 0 */ 319 { 320 *pz = 1.0f / 0.0f; 321 } 322 else 323 { 324 double xx0, res0, hyp0, h_hi0 = 0, dbase0 = 0; 325 int ibase0, si0, hyp0h; 326 327 hyp0 = x0 * (double)x0 + y0 * (double)y0; 328 329 ibase0 = HI(&hyp0); 330 331 HI(&dbase0) = (0x60000000 - ((ibase0 & 0x7fe00000) >> 1)); 332 333 hyp0h = (ibase0 & 0x000fffff) | 0x3ff00000; 334 HI(&hyp0) = hyp0h; 335 HI(&h_hi0) = hyp0h & 0x7fffc000; 336 337 ibase0 >>= 10; 338 si0 = ibase0 & 0x7f0; 339 xx0 = ((double*)((char*)__vlibm_TBL_rhypotf + si0))[0]; 340 341 xx0 = (hyp0 - h_hi0) * xx0; 342 res0 = ((double*)((char*)__vlibm_TBL_rhypotf + si0))[1]; 343 res0 *= (((A3 * xx0 + A2) * xx0 + A1) * xx0 + A0); 344 res0 *= dbase0; 345 *pz = res0; 346 } 347 } 348 } 349 350 static void 351 __vrhypotf_n(int n, float * restrict px, int stridex, float * restrict py, 352 int stridey, float * restrict pz, int stridez) 353 { 354 double xx0, res0, hyp0, h_hi0 = 0, dbase0 = 0; 355 double xx1, res1, hyp1, h_hi1 = 0, dbase1 = 0; 356 double xx2, res2, hyp2, h_hi2 = 0, dbase2 = 0; 357 float x0, y0; 358 float x1, y1; 359 float x2, y2; 360 int ibase0, si0, hyp0h; 361 int ibase1, si1, hyp1h; 362 int ibase2, si2, hyp2h; 363 364 for (; n > 2 ; n -= 3) 365 { 366 x0 = *px; 367 px += stridex; 368 x1 = *px; 369 px += stridex; 370 x2 = *px; 371 px += stridex; 372 373 y0 = *py; 374 py += stridey; 375 y1 = *py; 376 py += stridey; 377 y2 = *py; 378 py += stridey; 379 380 hyp0 = x0 * (double)x0 + y0 * (double)y0; 381 hyp1 = x1 * (double)x1 + y1 * (double)y1; 382 hyp2 = x2 * (double)x2 + y2 * (double)y2; 383 384 ibase0 = HI(&hyp0); 385 ibase1 = HI(&hyp1); 386 ibase2 = HI(&hyp2); 387 388 HI(&dbase0) = (0x60000000 - ((ibase0 & 0x7fe00000) >> 1)); 389 HI(&dbase1) = (0x60000000 - ((ibase1 & 0x7fe00000) >> 1)); 390 HI(&dbase2) = (0x60000000 - ((ibase2 & 0x7fe00000) >> 1)); 391 392 hyp0h = (ibase0 & 0x000fffff) | 0x3ff00000; 393 hyp1h = (ibase1 & 0x000fffff) | 0x3ff00000; 394 hyp2h = (ibase2 & 0x000fffff) | 0x3ff00000; 395 HI(&hyp0) = hyp0h; 396 HI(&hyp1) = hyp1h; 397 HI(&hyp2) = hyp2h; 398 HI(&h_hi0) = hyp0h & 0x7fffc000; 399 HI(&h_hi1) = hyp1h & 0x7fffc000; 400 HI(&h_hi2) = hyp2h & 0x7fffc000; 401 402 ibase0 >>= 10; 403 ibase1 >>= 10; 404 ibase2 >>= 10; 405 si0 = ibase0 & 0x7f0; 406 si1 = ibase1 & 0x7f0; 407 si2 = ibase2 & 0x7f0; 408 xx0 = ((double*)((char*)__vlibm_TBL_rhypotf + si0))[0]; 409 xx1 = ((double*)((char*)__vlibm_TBL_rhypotf + si1))[0]; 410 xx2 = ((double*)((char*)__vlibm_TBL_rhypotf + si2))[0]; 411 412 xx0 = (hyp0 - h_hi0) * xx0; 413 xx1 = (hyp1 - h_hi1) * xx1; 414 xx2 = (hyp2 - h_hi2) * xx2; 415 res0 = ((double*)((char*)__vlibm_TBL_rhypotf + si0))[1]; 416 res1 = ((double*)((char*)__vlibm_TBL_rhypotf + si1))[1]; 417 res2 = ((double*)((char*)__vlibm_TBL_rhypotf + si2))[1]; 418 res0 *= (((A3 * xx0 + A2) * xx0 + A1) * xx0 + A0); 419 res1 *= (((A3 * xx1 + A2) * xx1 + A1) * xx1 + A0); 420 res2 *= (((A3 * xx2 + A2) * xx2 + A1) * xx2 + A0); 421 res0 *= dbase0; 422 res1 *= dbase1; 423 res2 *= dbase2; 424 *pz = res0; 425 pz += stridez; 426 *pz = res1; 427 pz += stridez; 428 *pz = res2; 429 pz += stridez; 430 } 431 432 for (; n > 0 ; n--) 433 { 434 x0 = *px; 435 px += stridex; 436 437 y0 = *py; 438 py += stridey; 439 440 hyp0 = x0 * (double)x0 + y0 * (double)y0; 441 442 ibase0 = HI(&hyp0); 443 444 HI(&dbase0) = (0x60000000 - ((ibase0 & 0x7fe00000) >> 1)); 445 446 hyp0h = (ibase0 & 0x000fffff) | 0x3ff00000; 447 HI(&hyp0) = hyp0h; 448 HI(&h_hi0) = hyp0h & 0x7fffc000; 449 450 ibase0 >>= 10; 451 si0 = ibase0 & 0x7f0; 452 xx0 = ((double*)((char*)__vlibm_TBL_rhypotf + si0))[0]; 453 454 xx0 = (hyp0 - h_hi0) * xx0; 455 res0 = ((double*)((char*)__vlibm_TBL_rhypotf + si0))[1]; 456 res0 *= (((A3 * xx0 + A2) * xx0 + A1) * xx0 + A0); 457 res0 *= dbase0; 458 *pz = res0; 459 pz += stridez; 460 } 461 } 462