1 /*- 2 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 3 * 4 * Permission to use, copy, modify, and distribute this software for any 5 * purpose with or without fee is hereby granted, provided that the above 6 * copyright notice and this permission notice appear in all copies. 7 * 8 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 9 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 10 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 11 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 12 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 13 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 14 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 15 */ 16 17 /* powl.c 18 * 19 * Power function, long double precision 20 * 21 * 22 * 23 * SYNOPSIS: 24 * 25 * long double x, y, z, powl(); 26 * 27 * z = powl( x, y ); 28 * 29 * 30 * 31 * DESCRIPTION: 32 * 33 * Computes x raised to the yth power. Analytically, 34 * 35 * x**y = exp( y log(x) ). 36 * 37 * Following Cody and Waite, this program uses a lookup table 38 * of 2**-i/32 and pseudo extended precision arithmetic to 39 * obtain several extra bits of accuracy in both the logarithm 40 * and the exponential. 41 * 42 * 43 * 44 * ACCURACY: 45 * 46 * The relative error of pow(x,y) can be estimated 47 * by y dl ln(2), where dl is the absolute error of 48 * the internally computed base 2 logarithm. At the ends 49 * of the approximation interval the logarithm equal 1/32 50 * and its relative error is about 1 lsb = 1.1e-19. Hence 51 * the predicted relative error in the result is 2.3e-21 y . 52 * 53 * Relative error: 54 * arithmetic domain # trials peak rms 55 * 56 * IEEE +-1000 40000 2.8e-18 3.7e-19 57 * .001 < x < 1000, with log(x) uniformly distributed. 58 * -1000 < y < 1000, y uniformly distributed. 59 * 60 * IEEE 0,8700 60000 6.5e-18 1.0e-18 61 * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed. 62 * 63 * 64 * ERROR MESSAGES: 65 * 66 * message condition value returned 67 * pow overflow x**y > MAXNUM INFINITY 68 * pow underflow x**y < 1/MAXNUM 0.0 69 * pow domain x<0 and y noninteger 0.0 70 * 71 */ 72 73 #include <sys/cdefs.h> 74 __FBSDID("$FreeBSD$"); 75 76 #include <float.h> 77 #include <math.h> 78 79 #include "math_private.h" 80 81 /* Table size */ 82 #define NXT 32 83 /* log2(Table size) */ 84 #define LNXT 5 85 86 /* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z) 87 * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1 88 */ 89 static long double P[] = { 90 8.3319510773868690346226E-4L, 91 4.9000050881978028599627E-1L, 92 1.7500123722550302671919E0L, 93 1.4000100839971580279335E0L, 94 }; 95 static long double Q[] = { 96 /* 1.0000000000000000000000E0L,*/ 97 5.2500282295834889175431E0L, 98 8.4000598057587009834666E0L, 99 4.2000302519914740834728E0L, 100 }; 101 /* A[i] = 2^(-i/32), rounded to IEEE long double precision. 102 * If i is even, A[i] + B[i/2] gives additional accuracy. 103 */ 104 static long double A[33] = { 105 1.0000000000000000000000E0L, 106 9.7857206208770013448287E-1L, 107 9.5760328069857364691013E-1L, 108 9.3708381705514995065011E-1L, 109 9.1700404320467123175367E-1L, 110 8.9735453750155359320742E-1L, 111 8.7812608018664974155474E-1L, 112 8.5930964906123895780165E-1L, 113 8.4089641525371454301892E-1L, 114 8.2287773907698242225554E-1L, 115 8.0524516597462715409607E-1L, 116 7.8799042255394324325455E-1L, 117 7.7110541270397041179298E-1L, 118 7.5458221379671136985669E-1L, 119 7.3841307296974965571198E-1L, 120 7.2259040348852331001267E-1L, 121 7.0710678118654752438189E-1L, 122 6.9195494098191597746178E-1L, 123 6.7712777346844636413344E-1L, 124 6.6261832157987064729696E-1L, 125 6.4841977732550483296079E-1L, 126 6.3452547859586661129850E-1L, 127 6.2092890603674202431705E-1L, 128 6.0762367999023443907803E-1L, 129 5.9460355750136053334378E-1L, 130 5.8186242938878875689693E-1L, 131 5.6939431737834582684856E-1L, 132 5.5719337129794626814472E-1L, 133 5.4525386633262882960438E-1L, 134 5.3357020033841180906486E-1L, 135 5.2213689121370692017331E-1L, 136 5.1094857432705833910408E-1L, 137 5.0000000000000000000000E-1L, 138 }; 139 static long double B[17] = { 140 0.0000000000000000000000E0L, 141 2.6176170809902549338711E-20L, 142 -1.0126791927256478897086E-20L, 143 1.3438228172316276937655E-21L, 144 1.2207982955417546912101E-20L, 145 -6.3084814358060867200133E-21L, 146 1.3164426894366316434230E-20L, 147 -1.8527916071632873716786E-20L, 148 1.8950325588932570796551E-20L, 149 1.5564775779538780478155E-20L, 150 6.0859793637556860974380E-21L, 151 -2.0208749253662532228949E-20L, 152 1.4966292219224761844552E-20L, 153 3.3540909728056476875639E-21L, 154 -8.6987564101742849540743E-22L, 155 -1.2327176863327626135542E-20L, 156 0.0000000000000000000000E0L, 157 }; 158 159 /* 2^x = 1 + x P(x), 160 * on the interval -1/32 <= x <= 0 161 */ 162 static long double R[] = { 163 1.5089970579127659901157E-5L, 164 1.5402715328927013076125E-4L, 165 1.3333556028915671091390E-3L, 166 9.6181291046036762031786E-3L, 167 5.5504108664798463044015E-2L, 168 2.4022650695910062854352E-1L, 169 6.9314718055994530931447E-1L, 170 }; 171 172 #define douba(k) A[k] 173 #define doubb(k) B[k] 174 #define MEXP (NXT*16384.0L) 175 /* The following if denormal numbers are supported, else -MEXP: */ 176 #define MNEXP (-NXT*(16384.0L+64.0L)) 177 /* log2(e) - 1 */ 178 #define LOG2EA 0.44269504088896340735992L 179 180 #define F W 181 #define Fa Wa 182 #define Fb Wb 183 #define G W 184 #define Ga Wa 185 #define Gb u 186 #define H W 187 #define Ha Wb 188 #define Hb Wb 189 190 static const long double MAXLOGL = 1.1356523406294143949492E4L; 191 static const long double MINLOGL = -1.13994985314888605586758E4L; 192 static const long double LOGE2L = 6.9314718055994530941723E-1L; 193 static volatile long double z; 194 static long double w, W, Wa, Wb, ya, yb, u; 195 static const long double huge = 0x1p10000L; 196 #if 0 /* XXX Prevent gcc from erroneously constant folding this. */ 197 static const long double twom10000 = 0x1p-10000L; 198 #else 199 static volatile long double twom10000 = 0x1p-10000L; 200 #endif 201 202 static long double reducl( long double ); 203 static long double powil ( long double, int ); 204 205 long double 206 powl(long double x, long double y) 207 { 208 /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */ 209 int i, nflg, iyflg, yoddint; 210 long e; 211 212 if( y == 0.0L ) 213 return( 1.0L ); 214 215 if( x == 1.0L ) 216 return( 1.0L ); 217 218 if( isnan(x) ) 219 return ( nan_mix(x, y) ); 220 if( isnan(y) ) 221 return ( nan_mix(x, y) ); 222 223 if( y == 1.0L ) 224 return( x ); 225 226 if( !isfinite(y) && x == -1.0L ) 227 return( 1.0L ); 228 229 if( y >= LDBL_MAX ) 230 { 231 if( x > 1.0L ) 232 return( INFINITY ); 233 if( x > 0.0L && x < 1.0L ) 234 return( 0.0L ); 235 if( x < -1.0L ) 236 return( INFINITY ); 237 if( x > -1.0L && x < 0.0L ) 238 return( 0.0L ); 239 } 240 if( y <= -LDBL_MAX ) 241 { 242 if( x > 1.0L ) 243 return( 0.0L ); 244 if( x > 0.0L && x < 1.0L ) 245 return( INFINITY ); 246 if( x < -1.0L ) 247 return( 0.0L ); 248 if( x > -1.0L && x < 0.0L ) 249 return( INFINITY ); 250 } 251 if( x >= LDBL_MAX ) 252 { 253 if( y > 0.0L ) 254 return( INFINITY ); 255 return( 0.0L ); 256 } 257 258 w = floorl(y); 259 /* Set iyflg to 1 if y is an integer. */ 260 iyflg = 0; 261 if( w == y ) 262 iyflg = 1; 263 264 /* Test for odd integer y. */ 265 yoddint = 0; 266 if( iyflg ) 267 { 268 ya = fabsl(y); 269 ya = floorl(0.5L * ya); 270 yb = 0.5L * fabsl(w); 271 if( ya != yb ) 272 yoddint = 1; 273 } 274 275 if( x <= -LDBL_MAX ) 276 { 277 if( y > 0.0L ) 278 { 279 if( yoddint ) 280 return( -INFINITY ); 281 return( INFINITY ); 282 } 283 if( y < 0.0L ) 284 { 285 if( yoddint ) 286 return( -0.0L ); 287 return( 0.0 ); 288 } 289 } 290 291 292 nflg = 0; /* flag = 1 if x<0 raised to integer power */ 293 if( x <= 0.0L ) 294 { 295 if( x == 0.0L ) 296 { 297 if( y < 0.0 ) 298 { 299 if( signbit(x) && yoddint ) 300 return( -INFINITY ); 301 return( INFINITY ); 302 } 303 if( y > 0.0 ) 304 { 305 if( signbit(x) && yoddint ) 306 return( -0.0L ); 307 return( 0.0 ); 308 } 309 if( y == 0.0L ) 310 return( 1.0L ); /* 0**0 */ 311 else 312 return( 0.0L ); /* 0**y */ 313 } 314 else 315 { 316 if( iyflg == 0 ) 317 return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */ 318 nflg = 1; 319 } 320 } 321 322 /* Integer power of an integer. */ 323 324 if( iyflg ) 325 { 326 i = w; 327 w = floorl(x); 328 if( (w == x) && (fabsl(y) < 32768.0) ) 329 { 330 w = powil( x, (int) y ); 331 return( w ); 332 } 333 } 334 335 336 if( nflg ) 337 x = fabsl(x); 338 339 /* separate significand from exponent */ 340 x = frexpl( x, &i ); 341 e = i; 342 343 /* find significand in antilog table A[] */ 344 i = 1; 345 if( x <= douba(17) ) 346 i = 17; 347 if( x <= douba(i+8) ) 348 i += 8; 349 if( x <= douba(i+4) ) 350 i += 4; 351 if( x <= douba(i+2) ) 352 i += 2; 353 if( x >= douba(1) ) 354 i = -1; 355 i += 1; 356 357 358 /* Find (x - A[i])/A[i] 359 * in order to compute log(x/A[i]): 360 * 361 * log(x) = log( a x/a ) = log(a) + log(x/a) 362 * 363 * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a 364 */ 365 x -= douba(i); 366 x -= doubb(i/2); 367 x /= douba(i); 368 369 370 /* rational approximation for log(1+v): 371 * 372 * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v) 373 */ 374 z = x*x; 375 w = x * ( z * __polevll( x, P, 3 ) / __p1evll( x, Q, 3 ) ); 376 w = w - ldexpl( z, -1 ); /* w - 0.5 * z */ 377 378 /* Convert to base 2 logarithm: 379 * multiply by log2(e) = 1 + LOG2EA 380 */ 381 z = LOG2EA * w; 382 z += w; 383 z += LOG2EA * x; 384 z += x; 385 386 /* Compute exponent term of the base 2 logarithm. */ 387 w = -i; 388 w = ldexpl( w, -LNXT ); /* divide by NXT */ 389 w += e; 390 /* Now base 2 log of x is w + z. */ 391 392 /* Multiply base 2 log by y, in extended precision. */ 393 394 /* separate y into large part ya 395 * and small part yb less than 1/NXT 396 */ 397 ya = reducl(y); 398 yb = y - ya; 399 400 /* (w+z)(ya+yb) 401 * = w*ya + w*yb + z*y 402 */ 403 F = z * y + w * yb; 404 Fa = reducl(F); 405 Fb = F - Fa; 406 407 G = Fa + w * ya; 408 Ga = reducl(G); 409 Gb = G - Ga; 410 411 H = Fb + Gb; 412 Ha = reducl(H); 413 w = ldexpl( Ga+Ha, LNXT ); 414 415 /* Test the power of 2 for overflow */ 416 if( w > MEXP ) 417 return (huge * huge); /* overflow */ 418 419 if( w < MNEXP ) 420 return (twom10000 * twom10000); /* underflow */ 421 422 e = w; 423 Hb = H - Ha; 424 425 if( Hb > 0.0L ) 426 { 427 e += 1; 428 Hb -= (1.0L/NXT); /*0.0625L;*/ 429 } 430 431 /* Now the product y * log2(x) = Hb + e/NXT. 432 * 433 * Compute base 2 exponential of Hb, 434 * where -0.0625 <= Hb <= 0. 435 */ 436 z = Hb * __polevll( Hb, R, 6 ); /* z = 2**Hb - 1 */ 437 438 /* Express e/NXT as an integer plus a negative number of (1/NXT)ths. 439 * Find lookup table entry for the fractional power of 2. 440 */ 441 if( e < 0 ) 442 i = 0; 443 else 444 i = 1; 445 i = e/NXT + i; 446 e = NXT*i - e; 447 w = douba( e ); 448 z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */ 449 z = z + w; 450 z = ldexpl( z, i ); /* multiply by integer power of 2 */ 451 452 if( nflg ) 453 { 454 /* For negative x, 455 * find out if the integer exponent 456 * is odd or even. 457 */ 458 w = ldexpl( y, -1 ); 459 w = floorl(w); 460 w = ldexpl( w, 1 ); 461 if( w != y ) 462 z = -z; /* odd exponent */ 463 } 464 465 return( z ); 466 } 467 468 469 /* Find a multiple of 1/NXT that is within 1/NXT of x. */ 470 static long double 471 reducl(long double x) 472 { 473 long double t; 474 475 t = ldexpl( x, LNXT ); 476 t = floorl( t ); 477 t = ldexpl( t, -LNXT ); 478 return(t); 479 } 480 481 /* powil.c 482 * 483 * Real raised to integer power, long double precision 484 * 485 * 486 * 487 * SYNOPSIS: 488 * 489 * long double x, y, powil(); 490 * int n; 491 * 492 * y = powil( x, n ); 493 * 494 * 495 * 496 * DESCRIPTION: 497 * 498 * Returns argument x raised to the nth power. 499 * The routine efficiently decomposes n as a sum of powers of 500 * two. The desired power is a product of two-to-the-kth 501 * powers of x. Thus to compute the 32767 power of x requires 502 * 28 multiplications instead of 32767 multiplications. 503 * 504 * 505 * 506 * ACCURACY: 507 * 508 * 509 * Relative error: 510 * arithmetic x domain n domain # trials peak rms 511 * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18 512 * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18 513 * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17 514 * 515 * Returns MAXNUM on overflow, zero on underflow. 516 * 517 */ 518 519 static long double 520 powil(long double x, int nn) 521 { 522 long double ww, y; 523 long double s; 524 int n, e, sign, asign, lx; 525 526 if( x == 0.0L ) 527 { 528 if( nn == 0 ) 529 return( 1.0L ); 530 else if( nn < 0 ) 531 return( LDBL_MAX ); 532 else 533 return( 0.0L ); 534 } 535 536 if( nn == 0 ) 537 return( 1.0L ); 538 539 540 if( x < 0.0L ) 541 { 542 asign = -1; 543 x = -x; 544 } 545 else 546 asign = 0; 547 548 549 if( nn < 0 ) 550 { 551 sign = -1; 552 n = -nn; 553 } 554 else 555 { 556 sign = 1; 557 n = nn; 558 } 559 560 /* Overflow detection */ 561 562 /* Calculate approximate logarithm of answer */ 563 s = x; 564 s = frexpl( s, &lx ); 565 e = (lx - 1)*n; 566 if( (e == 0) || (e > 64) || (e < -64) ) 567 { 568 s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L); 569 s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L; 570 } 571 else 572 { 573 s = LOGE2L * e; 574 } 575 576 if( s > MAXLOGL ) 577 return (huge * huge); /* overflow */ 578 579 if( s < MINLOGL ) 580 return (twom10000 * twom10000); /* underflow */ 581 /* Handle tiny denormal answer, but with less accuracy 582 * since roundoff error in 1.0/x will be amplified. 583 * The precise demarcation should be the gradual underflow threshold. 584 */ 585 if( s < (-MAXLOGL+2.0L) ) 586 { 587 x = 1.0L/x; 588 sign = -sign; 589 } 590 591 /* First bit of the power */ 592 if( n & 1 ) 593 y = x; 594 595 else 596 { 597 y = 1.0L; 598 asign = 0; 599 } 600 601 ww = x; 602 n >>= 1; 603 while( n ) 604 { 605 ww = ww * ww; /* arg to the 2-to-the-kth power */ 606 if( n & 1 ) /* if that bit is set, then include in product */ 607 y *= ww; 608 n >>= 1; 609 } 610 611 if( asign ) 612 y = -y; /* odd power of negative number */ 613 if( sign < 0 ) 614 y = 1.0L/y; 615 return(y); 616 } 617