1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_powl.c */ 2 /* 3 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 4 * 5 * Permission to use, copy, modify, and distribute this software for any 6 * purpose with or without fee is hereby granted, provided that the above 7 * copyright notice and this permission notice appear in all copies. 8 * 9 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 10 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 11 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 12 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 13 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 14 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 15 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 16 */ 17 /* powl.c 18 * 19 * Power function, long double precision 20 * 21 * 22 * SYNOPSIS: 23 * 24 * long double x, y, z, powl(); 25 * 26 * z = powl( x, y ); 27 * 28 * 29 * DESCRIPTION: 30 * 31 * Computes x raised to the yth power. Analytically, 32 * 33 * x**y = exp( y log(x) ). 34 * 35 * Following Cody and Waite, this program uses a lookup table 36 * of 2**-i/32 and pseudo extended precision arithmetic to 37 * obtain several extra bits of accuracy in both the logarithm 38 * and the exponential. 39 * 40 * 41 * ACCURACY: 42 * 43 * The relative error of pow(x,y) can be estimated 44 * by y dl ln(2), where dl is the absolute error of 45 * the internally computed base 2 logarithm. At the ends 46 * of the approximation interval the logarithm equal 1/32 47 * and its relative error is about 1 lsb = 1.1e-19. Hence 48 * the predicted relative error in the result is 2.3e-21 y . 49 * 50 * Relative error: 51 * arithmetic domain # trials peak rms 52 * 53 * IEEE +-1000 40000 2.8e-18 3.7e-19 54 * .001 < x < 1000, with log(x) uniformly distributed. 55 * -1000 < y < 1000, y uniformly distributed. 56 * 57 * IEEE 0,8700 60000 6.5e-18 1.0e-18 58 * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed. 59 * 60 * 61 * ERROR MESSAGES: 62 * 63 * message condition value returned 64 * pow overflow x**y > MAXNUM INFINITY 65 * pow underflow x**y < 1/MAXNUM 0.0 66 * pow domain x<0 and y noninteger 0.0 67 * 68 */ 69 70 #include "libm.h" 71 72 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024 73 long double powl(long double x, long double y) 74 { 75 return pow(x, y); 76 } 77 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384 78 79 /* Table size */ 80 #define NXT 32 81 82 /* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z) 83 * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1 84 */ 85 static const long double P[] = { 86 8.3319510773868690346226E-4L, 87 4.9000050881978028599627E-1L, 88 1.7500123722550302671919E0L, 89 1.4000100839971580279335E0L, 90 }; 91 static const long double Q[] = { 92 /* 1.0000000000000000000000E0L,*/ 93 5.2500282295834889175431E0L, 94 8.4000598057587009834666E0L, 95 4.2000302519914740834728E0L, 96 }; 97 /* A[i] = 2^(-i/32), rounded to IEEE long double precision. 98 * If i is even, A[i] + B[i/2] gives additional accuracy. 99 */ 100 static const long double A[33] = { 101 1.0000000000000000000000E0L, 102 9.7857206208770013448287E-1L, 103 9.5760328069857364691013E-1L, 104 9.3708381705514995065011E-1L, 105 9.1700404320467123175367E-1L, 106 8.9735453750155359320742E-1L, 107 8.7812608018664974155474E-1L, 108 8.5930964906123895780165E-1L, 109 8.4089641525371454301892E-1L, 110 8.2287773907698242225554E-1L, 111 8.0524516597462715409607E-1L, 112 7.8799042255394324325455E-1L, 113 7.7110541270397041179298E-1L, 114 7.5458221379671136985669E-1L, 115 7.3841307296974965571198E-1L, 116 7.2259040348852331001267E-1L, 117 7.0710678118654752438189E-1L, 118 6.9195494098191597746178E-1L, 119 6.7712777346844636413344E-1L, 120 6.6261832157987064729696E-1L, 121 6.4841977732550483296079E-1L, 122 6.3452547859586661129850E-1L, 123 6.2092890603674202431705E-1L, 124 6.0762367999023443907803E-1L, 125 5.9460355750136053334378E-1L, 126 5.8186242938878875689693E-1L, 127 5.6939431737834582684856E-1L, 128 5.5719337129794626814472E-1L, 129 5.4525386633262882960438E-1L, 130 5.3357020033841180906486E-1L, 131 5.2213689121370692017331E-1L, 132 5.1094857432705833910408E-1L, 133 5.0000000000000000000000E-1L, 134 }; 135 static const long double B[17] = { 136 0.0000000000000000000000E0L, 137 2.6176170809902549338711E-20L, 138 -1.0126791927256478897086E-20L, 139 1.3438228172316276937655E-21L, 140 1.2207982955417546912101E-20L, 141 -6.3084814358060867200133E-21L, 142 1.3164426894366316434230E-20L, 143 -1.8527916071632873716786E-20L, 144 1.8950325588932570796551E-20L, 145 1.5564775779538780478155E-20L, 146 6.0859793637556860974380E-21L, 147 -2.0208749253662532228949E-20L, 148 1.4966292219224761844552E-20L, 149 3.3540909728056476875639E-21L, 150 -8.6987564101742849540743E-22L, 151 -1.2327176863327626135542E-20L, 152 0.0000000000000000000000E0L, 153 }; 154 155 /* 2^x = 1 + x P(x), 156 * on the interval -1/32 <= x <= 0 157 */ 158 static const long double R[] = { 159 1.5089970579127659901157E-5L, 160 1.5402715328927013076125E-4L, 161 1.3333556028915671091390E-3L, 162 9.6181291046036762031786E-3L, 163 5.5504108664798463044015E-2L, 164 2.4022650695910062854352E-1L, 165 6.9314718055994530931447E-1L, 166 }; 167 168 #define MEXP (NXT*16384.0L) 169 /* The following if denormal numbers are supported, else -MEXP: */ 170 #define MNEXP (-NXT*(16384.0L+64.0L)) 171 /* log2(e) - 1 */ 172 #define LOG2EA 0.44269504088896340735992L 173 174 #define F W 175 #define Fa Wa 176 #define Fb Wb 177 #define G W 178 #define Ga Wa 179 #define Gb u 180 #define H W 181 #define Ha Wb 182 #define Hb Wb 183 184 static const long double MAXLOGL = 1.1356523406294143949492E4L; 185 static const long double MINLOGL = -1.13994985314888605586758E4L; 186 static const long double LOGE2L = 6.9314718055994530941723E-1L; 187 static const long double huge = 0x1p10000L; 188 /* XXX Prevent gcc from erroneously constant folding this. */ 189 static const volatile long double twom10000 = 0x1p-10000L; 190 191 static long double reducl(long double); 192 static long double powil(long double, int); 193 194 long double powl(long double x, long double y) 195 { 196 /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */ 197 int i, nflg, iyflg, yoddint; 198 long e; 199 volatile long double z=0; 200 long double w=0, W=0, Wa=0, Wb=0, ya=0, yb=0, u=0; 201 202 /* make sure no invalid exception is raised by nan comparision */ 203 if (isnan(x)) { 204 if (!isnan(y) && y == 0.0) 205 return 1.0; 206 return x; 207 } 208 if (isnan(y)) { 209 if (x == 1.0) 210 return 1.0; 211 return y; 212 } 213 if (x == 1.0) 214 return 1.0; /* 1**y = 1, even if y is nan */ 215 if (x == -1.0 && !isfinite(y)) 216 return 1.0; /* -1**inf = 1 */ 217 if (y == 0.0) 218 return 1.0; /* x**0 = 1, even if x is nan */ 219 if (y == 1.0) 220 return x; 221 if (y >= LDBL_MAX) { 222 if (x > 1.0 || x < -1.0) 223 return INFINITY; 224 if (x != 0.0) 225 return 0.0; 226 } 227 if (y <= -LDBL_MAX) { 228 if (x > 1.0 || x < -1.0) 229 return 0.0; 230 if (x != 0.0 || y == -INFINITY) 231 return INFINITY; 232 } 233 if (x >= LDBL_MAX) { 234 if (y > 0.0) 235 return INFINITY; 236 return 0.0; 237 } 238 239 w = floorl(y); 240 241 /* Set iyflg to 1 if y is an integer. */ 242 iyflg = 0; 243 if (w == y) 244 iyflg = 1; 245 246 /* Test for odd integer y. */ 247 yoddint = 0; 248 if (iyflg) { 249 ya = fabsl(y); 250 ya = floorl(0.5 * ya); 251 yb = 0.5 * fabsl(w); 252 if( ya != yb ) 253 yoddint = 1; 254 } 255 256 if (x <= -LDBL_MAX) { 257 if (y > 0.0) { 258 if (yoddint) 259 return -INFINITY; 260 return INFINITY; 261 } 262 if (y < 0.0) { 263 if (yoddint) 264 return -0.0; 265 return 0.0; 266 } 267 } 268 nflg = 0; /* (x<0)**(odd int) */ 269 if (x <= 0.0) { 270 if (x == 0.0) { 271 if (y < 0.0) { 272 if (signbit(x) && yoddint) 273 /* (-0.0)**(-odd int) = -inf, divbyzero */ 274 return -1.0/0.0; 275 /* (+-0.0)**(negative) = inf, divbyzero */ 276 return 1.0/0.0; 277 } 278 if (signbit(x) && yoddint) 279 return -0.0; 280 return 0.0; 281 } 282 if (iyflg == 0) 283 return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */ 284 /* (x<0)**(integer) */ 285 if (yoddint) 286 nflg = 1; /* negate result */ 287 x = -x; 288 } 289 /* (+integer)**(integer) */ 290 if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) { 291 w = powil(x, (int)y); 292 return nflg ? -w : w; 293 } 294 295 /* separate significand from exponent */ 296 x = frexpl(x, &i); 297 e = i; 298 299 /* find significand in antilog table A[] */ 300 i = 1; 301 if (x <= A[17]) 302 i = 17; 303 if (x <= A[i+8]) 304 i += 8; 305 if (x <= A[i+4]) 306 i += 4; 307 if (x <= A[i+2]) 308 i += 2; 309 if (x >= A[1]) 310 i = -1; 311 i += 1; 312 313 /* Find (x - A[i])/A[i] 314 * in order to compute log(x/A[i]): 315 * 316 * log(x) = log( a x/a ) = log(a) + log(x/a) 317 * 318 * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a 319 */ 320 x -= A[i]; 321 x -= B[i/2]; 322 x /= A[i]; 323 324 /* rational approximation for log(1+v): 325 * 326 * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v) 327 */ 328 z = x*x; 329 w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3)); 330 w = w - 0.5*z; 331 332 /* Convert to base 2 logarithm: 333 * multiply by log2(e) = 1 + LOG2EA 334 */ 335 z = LOG2EA * w; 336 z += w; 337 z += LOG2EA * x; 338 z += x; 339 340 /* Compute exponent term of the base 2 logarithm. */ 341 w = -i; 342 w /= NXT; 343 w += e; 344 /* Now base 2 log of x is w + z. */ 345 346 /* Multiply base 2 log by y, in extended precision. */ 347 348 /* separate y into large part ya 349 * and small part yb less than 1/NXT 350 */ 351 ya = reducl(y); 352 yb = y - ya; 353 354 /* (w+z)(ya+yb) 355 * = w*ya + w*yb + z*y 356 */ 357 F = z * y + w * yb; 358 Fa = reducl(F); 359 Fb = F - Fa; 360 361 G = Fa + w * ya; 362 Ga = reducl(G); 363 Gb = G - Ga; 364 365 H = Fb + Gb; 366 Ha = reducl(H); 367 w = (Ga + Ha) * NXT; 368 369 /* Test the power of 2 for overflow */ 370 if (w > MEXP) 371 return huge * huge; /* overflow */ 372 if (w < MNEXP) 373 return twom10000 * twom10000; /* underflow */ 374 375 e = w; 376 Hb = H - Ha; 377 378 if (Hb > 0.0) { 379 e += 1; 380 Hb -= 1.0/NXT; /*0.0625L;*/ 381 } 382 383 /* Now the product y * log2(x) = Hb + e/NXT. 384 * 385 * Compute base 2 exponential of Hb, 386 * where -0.0625 <= Hb <= 0. 387 */ 388 z = Hb * __polevll(Hb, R, 6); /* z = 2**Hb - 1 */ 389 390 /* Express e/NXT as an integer plus a negative number of (1/NXT)ths. 391 * Find lookup table entry for the fractional power of 2. 392 */ 393 if (e < 0) 394 i = 0; 395 else 396 i = 1; 397 i = e/NXT + i; 398 e = NXT*i - e; 399 w = A[e]; 400 z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */ 401 z = z + w; 402 z = scalbnl(z, i); /* multiply by integer power of 2 */ 403 404 if (nflg) 405 z = -z; 406 return z; 407 } 408 409 410 /* Find a multiple of 1/NXT that is within 1/NXT of x. */ 411 static long double reducl(long double x) 412 { 413 long double t; 414 415 t = x * NXT; 416 t = floorl(t); 417 t = t / NXT; 418 return t; 419 } 420 421 /* 422 * Positive real raised to integer power, long double precision 423 * 424 * 425 * SYNOPSIS: 426 * 427 * long double x, y, powil(); 428 * int n; 429 * 430 * y = powil( x, n ); 431 * 432 * 433 * DESCRIPTION: 434 * 435 * Returns argument x>0 raised to the nth power. 436 * The routine efficiently decomposes n as a sum of powers of 437 * two. The desired power is a product of two-to-the-kth 438 * powers of x. Thus to compute the 32767 power of x requires 439 * 28 multiplications instead of 32767 multiplications. 440 * 441 * 442 * ACCURACY: 443 * 444 * Relative error: 445 * arithmetic x domain n domain # trials peak rms 446 * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18 447 * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18 448 * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17 449 * 450 * Returns MAXNUM on overflow, zero on underflow. 451 */ 452 453 static long double powil(long double x, int nn) 454 { 455 long double ww, y; 456 long double s; 457 int n, e, sign, lx; 458 459 if (nn == 0) 460 return 1.0; 461 462 if (nn < 0) { 463 sign = -1; 464 n = -nn; 465 } else { 466 sign = 1; 467 n = nn; 468 } 469 470 /* Overflow detection */ 471 472 /* Calculate approximate logarithm of answer */ 473 s = x; 474 s = frexpl( s, &lx); 475 e = (lx - 1)*n; 476 if ((e == 0) || (e > 64) || (e < -64)) { 477 s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L); 478 s = (2.9142135623730950L * s - 0.5 + lx) * nn * LOGE2L; 479 } else { 480 s = LOGE2L * e; 481 } 482 483 if (s > MAXLOGL) 484 return huge * huge; /* overflow */ 485 486 if (s < MINLOGL) 487 return twom10000 * twom10000; /* underflow */ 488 /* Handle tiny denormal answer, but with less accuracy 489 * since roundoff error in 1.0/x will be amplified. 490 * The precise demarcation should be the gradual underflow threshold. 491 */ 492 if (s < -MAXLOGL+2.0) { 493 x = 1.0/x; 494 sign = -sign; 495 } 496 497 /* First bit of the power */ 498 if (n & 1) 499 y = x; 500 else 501 y = 1.0; 502 503 ww = x; 504 n >>= 1; 505 while (n) { 506 ww = ww * ww; /* arg to the 2-to-the-kth power */ 507 if (n & 1) /* if that bit is set, then include in product */ 508 y *= ww; 509 n >>= 1; 510 } 511 512 if (sign < 0) 513 y = 1.0/y; 514 return y; 515 } 516 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384 517 // TODO: broken implementation to make things compile 518 long double powl(long double x, long double y) 519 { 520 return pow(x, y); 521 } 522 #endif 523