1/* Copyright JS Foundation and other contributors, http://js.foundation 2 * 3 * Licensed under the Apache License, Version 2.0 (the "License"); 4 * you may not use this file except in compliance with the License. 5 * You may obtain a copy of the License at 6 * 7 * http://www.apache.org/licenses/LICENSE-2.0 8 * 9 * Unless required by applicable law or agreed to in writing, software 10 * distributed under the License is distributed on an "AS IS" BASIS 11 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. 12 * See the License for the specific language governing permissions and 13 * limitations under the License. 14 * 15 * This file is based on work under the following copyright and permission 16 * notice: 17 * 18 * Copyright (C) 2004 by Sun Microsystems, Inc. All rights reserved. 19 * 20 * Permission to use, copy, modify, and distribute this 21 * software is freely granted, provided that this notice 22 * is preserved. 23 * 24 * @(#)e_pow.c 1.5 04/04/22 25 */ 26 27#include "jerry-libm-internal.h" 28 29/* pow(x,y) return x**y 30 * 31 * n 32 * Method: Let x = 2 * (1+f) 33 * 1. Compute and return log2(x) in two pieces: 34 * log2(x) = w1 + w2, 35 * where w1 has 53-24 = 29 bit trailing zeros. 36 * 2. Perform y*log2(x) = n+y' by simulating muti-precision 37 * arithmetic, where |y'|<=0.5. 38 * 3. Return x**y = 2**n*exp(y'*log2) 39 * 40 * Special cases: 41 * 0. +1 ** (anything) is 1 42 * 1. (anything) ** 0 is 1 43 * 2. (anything) ** 1 is itself 44 * 3. (anything) ** NAN is NAN 45 * 4. NAN ** (anything except 0) is NAN 46 * 5. +-(|x| > 1) ** +INF is +INF 47 * 6. +-(|x| > 1) ** -INF is +0 48 * 7. +-(|x| < 1) ** +INF is +0 49 * 8. +-(|x| < 1) ** -INF is +INF 50 * 9. -1 ** +-INF is 1 51 * 10. +0 ** (+anything except 0, NAN) is +0 52 * 11. -0 ** (+anything except 0, NAN, odd integer) is +0 53 * 12. +0 ** (-anything except 0, NAN) is +INF 54 * 13. -0 ** (-anything except 0, NAN, odd integer) is +INF 55 * 14. -0 ** (odd integer) = -( +0 ** (odd integer) ) 56 * 15. +INF ** (+anything except 0,NAN) is +INF 57 * 16. +INF ** (-anything except 0,NAN) is +0 58 * 17. -INF ** (anything) = -0 ** (-anything) 59 * 18. (-anything) ** (integer) is (-1)**(integer)*(+anything**integer) 60 * 19. (-anything except 0 and inf) ** (non-integer) is NAN 61 * 62 * Accuracy: 63 * pow(x,y) returns x**y nearly rounded. In particular 64 * pow(integer,integer) 65 * always returns the correct integer provided it is 66 * representable. 67 * 68 * Constants: 69 * The hexadecimal values are the intended ones for the following 70 * constants. The decimal values may be used, provided that the 71 * compiler will convert from decimal to binary accurately enough 72 * to produce the hexadecimal values shown. 73 */ 74 75static const double bp[] = 76{ 77 1.0, 78 1.5, 79}; 80static const double dp_h[] = 81{ 82 0.0, 83 5.84962487220764160156e-01, /* 0x3FE2B803, 0x40000000 */ 84}; 85static const double dp_l[] = 86{ 87 0.0, 88 1.35003920212974897128e-08, /* 0x3E4CFDEB, 0x43CFD006 */ 89}; 90 91#define zero 0.0 92#define one 1.0 93#define two 2.0 94#define two53 9007199254740992.0 /* 0x43400000, 0x00000000 */ 95#define huge 1.0e300 96#define tiny 1.0e-300 97/* poly coefs for (3/2) * (log(x) - 2s - 2/3 * s**3 */ 98#define L1 5.99999999999994648725e-01 /* 0x3FE33333, 0x33333303 */ 99#define L2 4.28571428578550184252e-01 /* 0x3FDB6DB6, 0xDB6FABFF */ 100#define L3 3.33333329818377432918e-01 /* 0x3FD55555, 0x518F264D */ 101#define L4 2.72728123808534006489e-01 /* 0x3FD17460, 0xA91D4101 */ 102#define L5 2.30660745775561754067e-01 /* 0x3FCD864A, 0x93C9DB65 */ 103#define L6 2.06975017800338417784e-01 /* 0x3FCA7E28, 0x4A454EEF */ 104#define P1 1.66666666666666019037e-01 /* 0x3FC55555, 0x5555553E */ 105#define P2 -2.77777777770155933842e-03 /* 0xBF66C16C, 0x16BEBD93 */ 106#define P3 6.61375632143793436117e-05 /* 0x3F11566A, 0xAF25DE2C */ 107#define P4 -1.65339022054652515390e-06 /* 0xBEBBBD41, 0xC5D26BF1 */ 108#define P5 4.13813679705723846039e-08 /* 0x3E663769, 0x72BEA4D0 */ 109#define lg2 6.93147180559945286227e-01 /* 0x3FE62E42, 0xFEFA39EF */ 110#define lg2_h 6.93147182464599609375e-01 /* 0x3FE62E43, 0x00000000 */ 111#define lg2_l -1.90465429995776804525e-09 /* 0xBE205C61, 0x0CA86C39 */ 112#define ovt 8.0085662595372944372e-0017 /* -(1024-log2(ovfl+.5ulp)) */ 113#define cp 9.61796693925975554329e-01 /* 0x3FEEC709, 0xDC3A03FD = 2 / (3 ln2) */ 114#define cp_h 9.61796700954437255859e-01 /* 0x3FEEC709, 0xE0000000 = (float) cp */ 115#define cp_l -7.02846165095275826516e-09 /* 0xBE3E2FE0, 0x145B01F5 = tail of cp_h */ 116#define ivln2 1.44269504088896338700e+00 /* 0x3FF71547, 0x652B82FE = 1 / ln2 */ 117#define ivln2_h 1.44269502162933349609e+00 /* 0x3FF71547, 0x60000000 = 24b 1 / ln2 */ 118#define ivln2_l 1.92596299112661746887e-08 /* 0x3E54AE0B, 0xF85DDF44 = 1 / ln2 tail */ 119 120double 121pow (double x, double y) 122{ 123 double_accessor t1, ax, p_h, y1, t, z; 124 double z_h, z_l, p_l; 125 double t2, r, s, u, v, w; 126 int i, j, k, yisint, n; 127 int hx, hy, ix, iy; 128 unsigned lx, ly; 129 130 hx = __HI (x); 131 lx = __LO (x); 132 hy = __HI (y); 133 ly = __LO (y); 134 ix = hx & 0x7fffffff; 135 iy = hy & 0x7fffffff; 136 137 /* x == one: 1**y = 1 */ 138 if (((hx - 0x3ff00000) | lx) == 0) 139 { 140 return one; 141 } 142 143 /* y == zero: x**0 = 1 */ 144 if ((iy | ly) == 0) 145 { 146 return one; 147 } 148 149 /* +-NaN return x + y */ 150 if (ix > 0x7ff00000 || ((ix == 0x7ff00000) && (lx != 0)) || iy > 0x7ff00000 || ((iy == 0x7ff00000) && (ly != 0))) 151 { 152 return x + y; 153 } 154 155 /* determine if y is an odd int when x < 0 156 * yisint = 0 ... y is not an integer 157 * yisint = 1 ... y is an odd int 158 * yisint = 2 ... y is an even int 159 */ 160 yisint = 0; 161 if (hx < 0) 162 { 163 if (iy >= 0x43400000) /* even integer y */ 164 { 165 yisint = 2; 166 } 167 else if (iy >= 0x3ff00000) 168 { 169 k = (iy >> 20) - 0x3ff; /* exponent */ 170 if (k > 20) 171 { 172 j = ly >> (52 - k); 173 if ((j << (52 - k)) == ly) 174 { 175 yisint = 2 - (j & 1); 176 } 177 } 178 else if (ly == 0) 179 { 180 j = iy >> (20 - k); 181 if ((j << (20 - k)) == iy) 182 { 183 yisint = 2 - (j & 1); 184 } 185 } 186 } 187 } 188 189 /* special value of y */ 190 if (ly == 0) 191 { 192 if (iy == 0x7ff00000) /* y is +-inf */ 193 { 194 if (((ix - 0x3ff00000) | lx) == 0) /* +-1**+-inf is 1 */ 195 { 196 return one; 197 } 198 else if (ix >= 0x3ff00000) /* (|x|>1)**+-inf = inf,0 */ 199 { 200 return (hy >= 0) ? y : zero; 201 } 202 else /* (|x|<1)**-,+inf = inf,0 */ 203 { 204 return (hy < 0) ? -y : zero; 205 } 206 } 207 if (iy == 0x3ff00000) /* y is +-1 */ 208 { 209 if (hy < 0) 210 { 211 return one / x; 212 } 213 else 214 { 215 return x; 216 } 217 } 218 if (hy == 0x40000000) /* y is 2 */ 219 { 220 return x * x; 221 } 222 if (hy == 0x3fe00000) /* y is 0.5 */ 223 { 224 if (hx >= 0) /* x >= +0 */ 225 { 226 return sqrt (x); 227 } 228 } 229 } 230 231 ax.dbl = fabs (x); 232 /* special value of x */ 233 if (lx == 0) 234 { 235 if (ix == 0x7ff00000 || ix == 0 || ix == 0x3ff00000) 236 { 237 z.dbl = ax.dbl; /* x is +-0,+-inf,+-1 */ 238 if (hy < 0) 239 { 240 z.dbl = one / z.dbl; /* z = (1 / |x|) */ 241 } 242 if (hx < 0) 243 { 244 if (((ix - 0x3ff00000) | yisint) == 0) 245 { 246 z.dbl = NAN; /* (-1)**non-int is NaN */ 247 } 248 else if (yisint == 1) 249 { 250 z.dbl = -z.dbl; /* (x<0)**odd = -(|x|**odd) */ 251 } 252 } 253 return z.dbl; 254 } 255 } 256 257 n = (hx < 0) ? 0 : 1; 258 259 /* (x<0)**(non-int) is NaN */ 260 if ((n | yisint) == 0) 261 { 262 return NAN; 263 } 264 265 s = one; /* s (sign of result -ve**odd) = -1 else = 1 */ 266 if ((n | (yisint - 1)) == 0) 267 { 268 s = -one; /* (-ve)**(odd int) */ 269 } 270 271 /* |y| is huge */ 272 if (iy > 0x41e00000) /* if |y| > 2**31 */ 273 { 274 if (iy > 0x43f00000) /* if |y| > 2**64, must o/uflow */ 275 { 276 if (ix <= 0x3fefffff) 277 { 278 return (hy < 0) ? huge * huge : tiny * tiny; 279 } 280 if (ix >= 0x3ff00000) 281 { 282 return (hy > 0) ? huge * huge : tiny * tiny; 283 } 284 } 285 /* over/underflow if x is not close to one */ 286 if (ix < 0x3fefffff) 287 { 288 return (hy < 0) ? s * huge * huge : s * tiny * tiny; 289 } 290 if (ix > 0x3ff00000) 291 { 292 return (hy > 0) ? s * huge * huge : s * tiny * tiny; 293 } 294 /* now |1 - x| is tiny <= 2**-20, suffice to compute 295 log(x) by x - x^2 / 2 + x^3 / 3 - x^4 / 4 */ 296 t.dbl = ax.dbl - one; /* t has 20 trailing zeros */ 297 w = (t.dbl * t.dbl) * (0.5 - t.dbl * (0.3333333333333333333333 - t.dbl * 0.25)); 298 u = ivln2_h * t.dbl; /* ivln2_h has 21 sig. bits */ 299 v = t.dbl * ivln2_l - w * ivln2; 300 t1.dbl = u + v; 301 t1.as_int.lo = 0; 302 t2 = v - (t1.dbl - u); 303 } 304 else 305 { 306 double_accessor s_h, t_h; 307 double ss, s2, s_l, t_l; 308 309 n = 0; 310 /* take care subnormal number */ 311 if (ix < 0x00100000) 312 { 313 ax.dbl *= two53; 314 n -= 53; 315 ix = ax.as_int.hi; 316 } 317 n += ((ix) >> 20) - 0x3ff; 318 j = ix & 0x000fffff; 319 /* determine interval */ 320 ix = j | 0x3ff00000; /* normalize ix */ 321 if (j <= 0x3988E) /* |x| < sqrt(3/2) */ 322 { 323 k = 0; 324 } 325 else if (j < 0xBB67A) /* |x| < sqrt(3) */ 326 { 327 k = 1; 328 } 329 else 330 { 331 k = 0; 332 n += 1; 333 ix -= 0x00100000; 334 } 335 ax.as_int.hi = ix; 336 337 /* compute ss = s_h + s_l = (x - 1) / (x + 1) or (x - 1.5) / (x + 1.5) */ 338 u = ax.dbl - bp[k]; /* bp[0] = 1.0, bp[1] = 1.5 */ 339 v = one / (ax.dbl + bp[k]); 340 ss = u * v; 341 s_h.dbl = ss; 342 s_h.as_int.lo = 0; 343 /* t_h = ax + bp[k] High */ 344 t_h.dbl = zero; 345 t_h.as_int.hi = ((ix >> 1) | 0x20000000) + 0x00080000 + (k << 18); 346 t_l = ax.dbl - (t_h.dbl - bp[k]); 347 s_l = v * ((u - s_h.dbl * t_h.dbl) - s_h.dbl * t_l); 348 /* compute log(ax) */ 349 s2 = ss * ss; 350 r = s2 * s2 * (L1 + s2 * (L2 + s2 * (L3 + s2 * (L4 + s2 * (L5 + s2 * L6))))); 351 r += s_l * (s_h.dbl + ss); 352 s2 = s_h.dbl * s_h.dbl; 353 t_h.dbl = 3.0 + s2 + r; 354 t_h.as_int.lo = 0; 355 t_l = r - ((t_h.dbl - 3.0) - s2); 356 /* u + v = ss * (1 + ...) */ 357 u = s_h.dbl * t_h.dbl; 358 v = s_l * t_h.dbl + t_l * ss; 359 /* 2 / (3 * log2) * (ss + ...) */ 360 p_h.dbl = u + v; 361 p_h.as_int.lo = 0; 362 p_l = v - (p_h.dbl - u); 363 z_h = cp_h * p_h.dbl; /* cp_h + cp_l = 2 / (3 * log2) */ 364 z_l = cp_l * p_h.dbl + p_l * cp + dp_l[k]; 365 /* log2(ax) = (ss + ...) * 2 / (3 * log2) = n + dp_h + z_h + z_l */ 366 t.dbl = (double) n; 367 t1.dbl = (((z_h + z_l) + dp_h[k]) + t.dbl); 368 t1.as_int.lo = 0; 369 t2 = z_l - (((t1.dbl - t.dbl) - dp_h[k]) - z_h); 370 } 371 372 /* split up y into y1 + y2 and compute (y1 + y2) * (t1 + t2) */ 373 y1.dbl = y; 374 y1.as_int.lo = 0; 375 p_l = (y - y1.dbl) * t1.dbl + y * t2; 376 p_h.dbl = y1.dbl * t1.dbl; 377 z.dbl = p_l + p_h.dbl; 378 j = z.as_int.hi; 379 i = z.as_int.lo; 380 if (j >= 0x40900000) /* z >= 1024 */ 381 { 382 if (((j - 0x40900000) | i) != 0) /* if z > 1024 */ 383 { 384 return s * huge * huge; /* overflow */ 385 } 386 else 387 { 388 if (p_l + ovt > z.dbl - p_h.dbl) 389 { 390 return s * huge * huge; /* overflow */ 391 } 392 } 393 } 394 else if ((j & 0x7fffffff) >= 0x4090cc00) /* z <= -1075 */ 395 { 396 if (((j - 0xc090cc00) | i) != 0) /* z < -1075 */ 397 { 398 return s * tiny * tiny; /* underflow */ 399 } 400 else 401 { 402 if (p_l <= z.dbl - p_h.dbl) 403 { 404 return s * tiny * tiny; /* underflow */ 405 } 406 } 407 } 408 /* 409 * compute 2**(p_h + p_l) 410 */ 411 i = j & 0x7fffffff; 412 k = (i >> 20) - 0x3ff; 413 n = 0; 414 if (i > 0x3fe00000) /* if |z| > 0.5, set n = [z + 0.5] */ 415 { 416 n = j + (0x00100000 >> (k + 1)); 417 k = ((n & 0x7fffffff) >> 20) - 0x3ff; /* new k for n */ 418 t.dbl = zero; 419 t.as_int.hi = (n & ~(0x000fffff >> k)); 420 n = ((n & 0x000fffff) | 0x00100000) >> (20 - k); 421 if (j < 0) 422 { 423 n = -n; 424 } 425 p_h.dbl -= t.dbl; 426 } 427 t.dbl = p_l + p_h.dbl; 428 t.as_int.lo = 0; 429 u = t.dbl * lg2_h; 430 v = (p_l - (t.dbl - p_h.dbl)) * lg2 + t.dbl * lg2_l; 431 z.dbl = u + v; 432 w = v - (z.dbl - u); 433 t.dbl = z.dbl * z.dbl; 434 t1.dbl = z.dbl - t.dbl * (P1 + t.dbl * (P2 + t.dbl * (P3 + t.dbl * (P4 + t.dbl * P5)))); 435 r = (z.dbl * t1.dbl) / (t1.dbl - two) - (w + z.dbl * w); 436 z.dbl = one - (r - z.dbl); 437 j = z.as_int.hi; 438 j += (n << 20); 439 if ((j >> 20) <= 0) /* subnormal output */ 440 { 441 z.dbl = scalbn (z.dbl, n); 442 } 443 else 444 { 445 z.as_int.hi += (n << 20); 446 } 447 return s * z.dbl; 448} /* pow */ 449 450#undef zero 451#undef one 452#undef two 453#undef two53 454#undef huge 455#undef tiny 456#undef L1 457#undef L2 458#undef L3 459#undef L4 460#undef L5 461#undef L6 462#undef P1 463#undef P2 464#undef P3 465#undef P4 466#undef P5 467#undef lg2 468#undef lg2_h 469#undef lg2_l 470#undef ovt 471#undef cp 472#undef cp_h 473#undef cp_l 474#undef ivln2 475#undef ivln2_h 476#undef ivln2_l 477