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