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 #include <sys/cdefs.h> 18 #include <math.h> 19 20 #include "math_private.h" 21 22 /* 23 * Polynomial evaluator: 24 * P[0] x^n + P[1] x^(n-1) + ... + P[n] 25 */ 26 static inline long double 27 __polevll(long double x, long double *PP, int n) 28 { 29 long double y; 30 long double *P; 31 32 P = PP; 33 y = *P++; 34 do { 35 y = y * x + *P++; 36 } while (--n); 37 38 return (y); 39 } 40 41 /* 42 * Polynomial evaluator: 43 * x^n + P[0] x^(n-1) + P[1] x^(n-2) + ... + P[n] 44 */ 45 static inline long double 46 __p1evll(long double x, long double *PP, int n) 47 { 48 long double y; 49 long double *P; 50 51 P = PP; 52 n -= 1; 53 y = x + *P++; 54 do { 55 y = y * x + *P++; 56 } while (--n); 57 58 return (y); 59 } 60 61 /* powl.c 62 * 63 * Power function, long double precision 64 * 65 * 66 * 67 * SYNOPSIS: 68 * 69 * long double x, y, z, powl(); 70 * 71 * z = powl( x, y ); 72 * 73 * 74 * 75 * DESCRIPTION: 76 * 77 * Computes x raised to the yth power. Analytically, 78 * 79 * x**y = exp( y log(x) ). 80 * 81 * Following Cody and Waite, this program uses a lookup table 82 * of 2**-i/32 and pseudo extended precision arithmetic to 83 * obtain several extra bits of accuracy in both the logarithm 84 * and the exponential. 85 * 86 * 87 * 88 * ACCURACY: 89 * 90 * The relative error of pow(x,y) can be estimated 91 * by y dl ln(2), where dl is the absolute error of 92 * the internally computed base 2 logarithm. At the ends 93 * of the approximation interval the logarithm equal 1/32 94 * and its relative error is about 1 lsb = 1.1e-19. Hence 95 * the predicted relative error in the result is 2.3e-21 y . 96 * 97 * Relative error: 98 * arithmetic domain # trials peak rms 99 * 100 * IEEE +-1000 40000 2.8e-18 3.7e-19 101 * .001 < x < 1000, with log(x) uniformly distributed. 102 * -1000 < y < 1000, y uniformly distributed. 103 * 104 * IEEE 0,8700 60000 6.5e-18 1.0e-18 105 * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed. 106 * 107 * 108 * ERROR MESSAGES: 109 * 110 * message condition value returned 111 * pow overflow x**y > MAXNUM INFINITY 112 * pow underflow x**y < 1/MAXNUM 0.0 113 * pow domain x<0 and y noninteger 0.0 114 * 115 */ 116 117 #include <sys/cdefs.h> 118 #include <float.h> 119 #include <math.h> 120 121 #include "math_private.h" 122 123 /* Table size */ 124 #define NXT 32 125 /* log2(Table size) */ 126 #define LNXT 5 127 128 /* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z) 129 * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1 130 */ 131 static long double P[] = { 132 8.3319510773868690346226E-4L, 133 4.9000050881978028599627E-1L, 134 1.7500123722550302671919E0L, 135 1.4000100839971580279335E0L, 136 }; 137 static long double Q[] = { 138 /* 1.0000000000000000000000E0L,*/ 139 5.2500282295834889175431E0L, 140 8.4000598057587009834666E0L, 141 4.2000302519914740834728E0L, 142 }; 143 /* A[i] = 2^(-i/32), rounded to IEEE long double precision. 144 * If i is even, A[i] + B[i/2] gives additional accuracy. 145 */ 146 static long double A[33] = { 147 1.0000000000000000000000E0L, 148 9.7857206208770013448287E-1L, 149 9.5760328069857364691013E-1L, 150 9.3708381705514995065011E-1L, 151 9.1700404320467123175367E-1L, 152 8.9735453750155359320742E-1L, 153 8.7812608018664974155474E-1L, 154 8.5930964906123895780165E-1L, 155 8.4089641525371454301892E-1L, 156 8.2287773907698242225554E-1L, 157 8.0524516597462715409607E-1L, 158 7.8799042255394324325455E-1L, 159 7.7110541270397041179298E-1L, 160 7.5458221379671136985669E-1L, 161 7.3841307296974965571198E-1L, 162 7.2259040348852331001267E-1L, 163 7.0710678118654752438189E-1L, 164 6.9195494098191597746178E-1L, 165 6.7712777346844636413344E-1L, 166 6.6261832157987064729696E-1L, 167 6.4841977732550483296079E-1L, 168 6.3452547859586661129850E-1L, 169 6.2092890603674202431705E-1L, 170 6.0762367999023443907803E-1L, 171 5.9460355750136053334378E-1L, 172 5.8186242938878875689693E-1L, 173 5.6939431737834582684856E-1L, 174 5.5719337129794626814472E-1L, 175 5.4525386633262882960438E-1L, 176 5.3357020033841180906486E-1L, 177 5.2213689121370692017331E-1L, 178 5.1094857432705833910408E-1L, 179 5.0000000000000000000000E-1L, 180 }; 181 static long double B[17] = { 182 0.0000000000000000000000E0L, 183 2.6176170809902549338711E-20L, 184 -1.0126791927256478897086E-20L, 185 1.3438228172316276937655E-21L, 186 1.2207982955417546912101E-20L, 187 -6.3084814358060867200133E-21L, 188 1.3164426894366316434230E-20L, 189 -1.8527916071632873716786E-20L, 190 1.8950325588932570796551E-20L, 191 1.5564775779538780478155E-20L, 192 6.0859793637556860974380E-21L, 193 -2.0208749253662532228949E-20L, 194 1.4966292219224761844552E-20L, 195 3.3540909728056476875639E-21L, 196 -8.6987564101742849540743E-22L, 197 -1.2327176863327626135542E-20L, 198 0.0000000000000000000000E0L, 199 }; 200 201 /* 2^x = 1 + x P(x), 202 * on the interval -1/32 <= x <= 0 203 */ 204 static long double R[] = { 205 1.5089970579127659901157E-5L, 206 1.5402715328927013076125E-4L, 207 1.3333556028915671091390E-3L, 208 9.6181291046036762031786E-3L, 209 5.5504108664798463044015E-2L, 210 2.4022650695910062854352E-1L, 211 6.9314718055994530931447E-1L, 212 }; 213 214 #define douba(k) A[k] 215 #define doubb(k) B[k] 216 #define MEXP (NXT*16384.0L) 217 /* The following if denormal numbers are supported, else -MEXP: */ 218 #define MNEXP (-NXT*(16384.0L+64.0L)) 219 /* log2(e) - 1 */ 220 #define LOG2EA 0.44269504088896340735992L 221 222 #define F W 223 #define Fa Wa 224 #define Fb Wb 225 #define G W 226 #define Ga Wa 227 #define Gb u 228 #define H W 229 #define Ha Wb 230 #define Hb Wb 231 232 static const long double MAXLOGL = 1.1356523406294143949492E4L; 233 static const long double MINLOGL = -1.13994985314888605586758E4L; 234 static const long double LOGE2L = 6.9314718055994530941723E-1L; 235 static volatile long double z; 236 static long double w, W, Wa, Wb, ya, yb, u; 237 static const long double huge = 0x1p10000L; 238 #if 0 /* XXX Prevent gcc from erroneously constant folding this. */ 239 static const long double twom10000 = 0x1p-10000L; 240 #else 241 static volatile long double twom10000 = 0x1p-10000L; 242 #endif 243 244 static long double reducl( long double ); 245 static long double powil ( long double, int ); 246 247 long double 248 powl(long double x, long double y) 249 { 250 /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */ 251 int i, nflg, iyflg, yoddint; 252 long e; 253 254 if( y == 0.0L ) 255 return( 1.0L ); 256 257 if( x == 1.0L ) 258 return( 1.0L ); 259 260 if( isnan(x) ) 261 return ( nan_mix(x, y) ); 262 if( isnan(y) ) 263 return ( nan_mix(x, y) ); 264 265 if( y == 1.0L ) 266 return( x ); 267 268 if( !isfinite(y) && x == -1.0L ) 269 return( 1.0L ); 270 271 if( y >= LDBL_MAX ) 272 { 273 if( x > 1.0L ) 274 return( INFINITY ); 275 if( x > 0.0L && x < 1.0L ) 276 return( 0.0L ); 277 if( x < -1.0L ) 278 return( INFINITY ); 279 if( x > -1.0L && x < 0.0L ) 280 return( 0.0L ); 281 } 282 if( y <= -LDBL_MAX ) 283 { 284 if( x > 1.0L ) 285 return( 0.0L ); 286 if( x > 0.0L && x < 1.0L ) 287 return( INFINITY ); 288 if( x < -1.0L ) 289 return( 0.0L ); 290 if( x > -1.0L && x < 0.0L ) 291 return( INFINITY ); 292 } 293 if( x >= LDBL_MAX ) 294 { 295 if( y > 0.0L ) 296 return( INFINITY ); 297 return( 0.0L ); 298 } 299 300 w = floorl(y); 301 /* Set iyflg to 1 if y is an integer. */ 302 iyflg = 0; 303 if( w == y ) 304 iyflg = 1; 305 306 /* Test for odd integer y. */ 307 yoddint = 0; 308 if( iyflg ) 309 { 310 ya = fabsl(y); 311 ya = floorl(0.5L * ya); 312 yb = 0.5L * fabsl(w); 313 if( ya != yb ) 314 yoddint = 1; 315 } 316 317 if( x <= -LDBL_MAX ) 318 { 319 if( y > 0.0L ) 320 { 321 if( yoddint ) 322 return( -INFINITY ); 323 return( INFINITY ); 324 } 325 if( y < 0.0L ) 326 { 327 if( yoddint ) 328 return( -0.0L ); 329 return( 0.0 ); 330 } 331 } 332 333 334 nflg = 0; /* flag = 1 if x<0 raised to integer power */ 335 if( x <= 0.0L ) 336 { 337 if( x == 0.0L ) 338 { 339 if( y < 0.0 ) 340 { 341 if( signbit(x) && yoddint ) 342 return( -INFINITY ); 343 return( INFINITY ); 344 } 345 if( y > 0.0 ) 346 { 347 if( signbit(x) && yoddint ) 348 return( -0.0L ); 349 return( 0.0 ); 350 } 351 if( y == 0.0L ) 352 return( 1.0L ); /* 0**0 */ 353 else 354 return( 0.0L ); /* 0**y */ 355 } 356 else 357 { 358 if( iyflg == 0 ) 359 return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */ 360 nflg = 1; 361 } 362 } 363 364 /* Integer power of an integer. */ 365 366 if( iyflg ) 367 { 368 i = w; 369 w = floorl(x); 370 if( (w == x) && (fabsl(y) < 32768.0) ) 371 { 372 w = powil( x, (int) y ); 373 return( w ); 374 } 375 } 376 377 378 if( nflg ) 379 x = fabsl(x); 380 381 /* separate significand from exponent */ 382 x = frexpl( x, &i ); 383 e = i; 384 385 /* find significand in antilog table A[] */ 386 i = 1; 387 if( x <= douba(17) ) 388 i = 17; 389 if( x <= douba(i+8) ) 390 i += 8; 391 if( x <= douba(i+4) ) 392 i += 4; 393 if( x <= douba(i+2) ) 394 i += 2; 395 if( x >= douba(1) ) 396 i = -1; 397 i += 1; 398 399 400 /* Find (x - A[i])/A[i] 401 * in order to compute log(x/A[i]): 402 * 403 * log(x) = log( a x/a ) = log(a) + log(x/a) 404 * 405 * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a 406 */ 407 x -= douba(i); 408 x -= doubb(i/2); 409 x /= douba(i); 410 411 412 /* rational approximation for log(1+v): 413 * 414 * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v) 415 */ 416 z = x*x; 417 w = x * ( z * __polevll( x, P, 3 ) / __p1evll( x, Q, 3 ) ); 418 w = w - ldexpl( z, -1 ); /* w - 0.5 * z */ 419 420 /* Convert to base 2 logarithm: 421 * multiply by log2(e) = 1 + LOG2EA 422 */ 423 z = LOG2EA * w; 424 z += w; 425 z += LOG2EA * x; 426 z += x; 427 428 /* Compute exponent term of the base 2 logarithm. */ 429 w = -i; 430 w = ldexpl( w, -LNXT ); /* divide by NXT */ 431 w += e; 432 /* Now base 2 log of x is w + z. */ 433 434 /* Multiply base 2 log by y, in extended precision. */ 435 436 /* separate y into large part ya 437 * and small part yb less than 1/NXT 438 */ 439 ya = reducl(y); 440 yb = y - ya; 441 442 /* (w+z)(ya+yb) 443 * = w*ya + w*yb + z*y 444 */ 445 F = z * y + w * yb; 446 Fa = reducl(F); 447 Fb = F - Fa; 448 449 G = Fa + w * ya; 450 Ga = reducl(G); 451 Gb = G - Ga; 452 453 H = Fb + Gb; 454 Ha = reducl(H); 455 w = ldexpl( Ga+Ha, LNXT ); 456 457 /* Test the power of 2 for overflow */ 458 if( w > MEXP ) 459 return (huge * huge); /* overflow */ 460 461 if( w < MNEXP ) 462 return (twom10000 * twom10000); /* underflow */ 463 464 e = w; 465 Hb = H - Ha; 466 467 if( Hb > 0.0L ) 468 { 469 e += 1; 470 Hb -= (1.0L/NXT); /*0.0625L;*/ 471 } 472 473 /* Now the product y * log2(x) = Hb + e/NXT. 474 * 475 * Compute base 2 exponential of Hb, 476 * where -0.0625 <= Hb <= 0. 477 */ 478 z = Hb * __polevll( Hb, R, 6 ); /* z = 2**Hb - 1 */ 479 480 /* Express e/NXT as an integer plus a negative number of (1/NXT)ths. 481 * Find lookup table entry for the fractional power of 2. 482 */ 483 if( e < 0 ) 484 i = 0; 485 else 486 i = 1; 487 i = e/NXT + i; 488 e = NXT*i - e; 489 w = douba( e ); 490 z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */ 491 z = z + w; 492 z = ldexpl( z, i ); /* multiply by integer power of 2 */ 493 494 if( nflg ) 495 { 496 /* For negative x, 497 * find out if the integer exponent 498 * is odd or even. 499 */ 500 w = ldexpl( y, -1 ); 501 w = floorl(w); 502 w = ldexpl( w, 1 ); 503 if( w != y ) 504 z = -z; /* odd exponent */ 505 } 506 507 return( z ); 508 } 509 510 511 /* Find a multiple of 1/NXT that is within 1/NXT of x. */ 512 static inline long double 513 reducl(long double x) 514 { 515 long double t; 516 517 t = ldexpl( x, LNXT ); 518 t = floorl( t ); 519 t = ldexpl( t, -LNXT ); 520 return(t); 521 } 522 523 /* powil.c 524 * 525 * Real raised to integer power, long double precision 526 * 527 * 528 * 529 * SYNOPSIS: 530 * 531 * long double x, y, powil(); 532 * int n; 533 * 534 * y = powil( x, n ); 535 * 536 * 537 * 538 * DESCRIPTION: 539 * 540 * Returns argument x raised to the nth power. 541 * The routine efficiently decomposes n as a sum of powers of 542 * two. The desired power is a product of two-to-the-kth 543 * powers of x. Thus to compute the 32767 power of x requires 544 * 28 multiplications instead of 32767 multiplications. 545 * 546 * 547 * 548 * ACCURACY: 549 * 550 * 551 * Relative error: 552 * arithmetic x domain n domain # trials peak rms 553 * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18 554 * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18 555 * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17 556 * 557 * Returns MAXNUM on overflow, zero on underflow. 558 * 559 */ 560 561 static long double 562 powil(long double x, int nn) 563 { 564 long double ww, y; 565 long double s; 566 int n, e, sign, asign, lx; 567 568 if( x == 0.0L ) 569 { 570 if( nn == 0 ) 571 return( 1.0L ); 572 else if( nn < 0 ) 573 return( LDBL_MAX ); 574 else 575 return( 0.0L ); 576 } 577 578 if( nn == 0 ) 579 return( 1.0L ); 580 581 582 if( x < 0.0L ) 583 { 584 asign = -1; 585 x = -x; 586 } 587 else 588 asign = 0; 589 590 591 if( nn < 0 ) 592 { 593 sign = -1; 594 n = -nn; 595 } 596 else 597 { 598 sign = 1; 599 n = nn; 600 } 601 602 /* Overflow detection */ 603 604 /* Calculate approximate logarithm of answer */ 605 s = x; 606 s = frexpl( s, &lx ); 607 e = (lx - 1)*n; 608 if( (e == 0) || (e > 64) || (e < -64) ) 609 { 610 s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L); 611 s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L; 612 } 613 else 614 { 615 s = LOGE2L * e; 616 } 617 618 if( s > MAXLOGL ) 619 return (huge * huge); /* overflow */ 620 621 if( s < MINLOGL ) 622 return (twom10000 * twom10000); /* underflow */ 623 /* Handle tiny denormal answer, but with less accuracy 624 * since roundoff error in 1.0/x will be amplified. 625 * The precise demarcation should be the gradual underflow threshold. 626 */ 627 if( s < (-MAXLOGL+2.0L) ) 628 { 629 x = 1.0L/x; 630 sign = -sign; 631 } 632 633 /* First bit of the power */ 634 if( n & 1 ) 635 y = x; 636 637 else 638 { 639 y = 1.0L; 640 asign = 0; 641 } 642 643 ww = x; 644 n >>= 1; 645 while( n ) 646 { 647 ww = ww * ww; /* arg to the 2-to-the-kth power */ 648 if( n & 1 ) /* if that bit is set, then include in product */ 649 y *= ww; 650 n >>= 1; 651 } 652 653 if( asign ) 654 y = -y; /* odd power of negative number */ 655 if( sign < 0 ) 656 y = 1.0L/y; 657 return(y); 658 } 659