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 __FBSDID("$FreeBSD$"); 19 20 #include <math.h> 21 22 #include "math_private.h" 23 24 /* 25 * Polynomial evaluator: 26 * P[0] x^n + P[1] x^(n-1) + ... + P[n] 27 */ 28 static inline long double 29 __polevll(long double x, long double *PP, int n) 30 { 31 long double y; 32 long double *P; 33 34 P = PP; 35 y = *P++; 36 do { 37 y = y * x + *P++; 38 } while (--n); 39 40 return (y); 41 } 42 43 /* 44 * Polynomial evaluator: 45 * x^n + P[0] x^(n-1) + P[1] x^(n-2) + ... + P[n] 46 */ 47 static inline long double 48 __p1evll(long double x, long double *PP, int n) 49 { 50 long double y; 51 long double *P; 52 53 P = PP; 54 n -= 1; 55 y = x + *P++; 56 do { 57 y = y * x + *P++; 58 } while (--n); 59 60 return (y); 61 } 62 63 /* powl.c 64 * 65 * Power function, long double precision 66 * 67 * 68 * 69 * SYNOPSIS: 70 * 71 * long double x, y, z, powl(); 72 * 73 * z = powl( x, y ); 74 * 75 * 76 * 77 * DESCRIPTION: 78 * 79 * Computes x raised to the yth power. Analytically, 80 * 81 * x**y = exp( y log(x) ). 82 * 83 * Following Cody and Waite, this program uses a lookup table 84 * of 2**-i/32 and pseudo extended precision arithmetic to 85 * obtain several extra bits of accuracy in both the logarithm 86 * and the exponential. 87 * 88 * 89 * 90 * ACCURACY: 91 * 92 * The relative error of pow(x,y) can be estimated 93 * by y dl ln(2), where dl is the absolute error of 94 * the internally computed base 2 logarithm. At the ends 95 * of the approximation interval the logarithm equal 1/32 96 * and its relative error is about 1 lsb = 1.1e-19. Hence 97 * the predicted relative error in the result is 2.3e-21 y . 98 * 99 * Relative error: 100 * arithmetic domain # trials peak rms 101 * 102 * IEEE +-1000 40000 2.8e-18 3.7e-19 103 * .001 < x < 1000, with log(x) uniformly distributed. 104 * -1000 < y < 1000, y uniformly distributed. 105 * 106 * IEEE 0,8700 60000 6.5e-18 1.0e-18 107 * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed. 108 * 109 * 110 * ERROR MESSAGES: 111 * 112 * message condition value returned 113 * pow overflow x**y > MAXNUM INFINITY 114 * pow underflow x**y < 1/MAXNUM 0.0 115 * pow domain x<0 and y noninteger 0.0 116 * 117 */ 118 119 #include <sys/cdefs.h> 120 __FBSDID("$FreeBSD$"); 121 122 #include <float.h> 123 #include <math.h> 124 125 #include "math_private.h" 126 127 /* Table size */ 128 #define NXT 32 129 /* log2(Table size) */ 130 #define LNXT 5 131 132 /* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z) 133 * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1 134 */ 135 static long double P[] = { 136 8.3319510773868690346226E-4L, 137 4.9000050881978028599627E-1L, 138 1.7500123722550302671919E0L, 139 1.4000100839971580279335E0L, 140 }; 141 static long double Q[] = { 142 /* 1.0000000000000000000000E0L,*/ 143 5.2500282295834889175431E0L, 144 8.4000598057587009834666E0L, 145 4.2000302519914740834728E0L, 146 }; 147 /* A[i] = 2^(-i/32), rounded to IEEE long double precision. 148 * If i is even, A[i] + B[i/2] gives additional accuracy. 149 */ 150 static long double A[33] = { 151 1.0000000000000000000000E0L, 152 9.7857206208770013448287E-1L, 153 9.5760328069857364691013E-1L, 154 9.3708381705514995065011E-1L, 155 9.1700404320467123175367E-1L, 156 8.9735453750155359320742E-1L, 157 8.7812608018664974155474E-1L, 158 8.5930964906123895780165E-1L, 159 8.4089641525371454301892E-1L, 160 8.2287773907698242225554E-1L, 161 8.0524516597462715409607E-1L, 162 7.8799042255394324325455E-1L, 163 7.7110541270397041179298E-1L, 164 7.5458221379671136985669E-1L, 165 7.3841307296974965571198E-1L, 166 7.2259040348852331001267E-1L, 167 7.0710678118654752438189E-1L, 168 6.9195494098191597746178E-1L, 169 6.7712777346844636413344E-1L, 170 6.6261832157987064729696E-1L, 171 6.4841977732550483296079E-1L, 172 6.3452547859586661129850E-1L, 173 6.2092890603674202431705E-1L, 174 6.0762367999023443907803E-1L, 175 5.9460355750136053334378E-1L, 176 5.8186242938878875689693E-1L, 177 5.6939431737834582684856E-1L, 178 5.5719337129794626814472E-1L, 179 5.4525386633262882960438E-1L, 180 5.3357020033841180906486E-1L, 181 5.2213689121370692017331E-1L, 182 5.1094857432705833910408E-1L, 183 5.0000000000000000000000E-1L, 184 }; 185 static long double B[17] = { 186 0.0000000000000000000000E0L, 187 2.6176170809902549338711E-20L, 188 -1.0126791927256478897086E-20L, 189 1.3438228172316276937655E-21L, 190 1.2207982955417546912101E-20L, 191 -6.3084814358060867200133E-21L, 192 1.3164426894366316434230E-20L, 193 -1.8527916071632873716786E-20L, 194 1.8950325588932570796551E-20L, 195 1.5564775779538780478155E-20L, 196 6.0859793637556860974380E-21L, 197 -2.0208749253662532228949E-20L, 198 1.4966292219224761844552E-20L, 199 3.3540909728056476875639E-21L, 200 -8.6987564101742849540743E-22L, 201 -1.2327176863327626135542E-20L, 202 0.0000000000000000000000E0L, 203 }; 204 205 /* 2^x = 1 + x P(x), 206 * on the interval -1/32 <= x <= 0 207 */ 208 static long double R[] = { 209 1.5089970579127659901157E-5L, 210 1.5402715328927013076125E-4L, 211 1.3333556028915671091390E-3L, 212 9.6181291046036762031786E-3L, 213 5.5504108664798463044015E-2L, 214 2.4022650695910062854352E-1L, 215 6.9314718055994530931447E-1L, 216 }; 217 218 #define douba(k) A[k] 219 #define doubb(k) B[k] 220 #define MEXP (NXT*16384.0L) 221 /* The following if denormal numbers are supported, else -MEXP: */ 222 #define MNEXP (-NXT*(16384.0L+64.0L)) 223 /* log2(e) - 1 */ 224 #define LOG2EA 0.44269504088896340735992L 225 226 #define F W 227 #define Fa Wa 228 #define Fb Wb 229 #define G W 230 #define Ga Wa 231 #define Gb u 232 #define H W 233 #define Ha Wb 234 #define Hb Wb 235 236 static const long double MAXLOGL = 1.1356523406294143949492E4L; 237 static const long double MINLOGL = -1.13994985314888605586758E4L; 238 static const long double LOGE2L = 6.9314718055994530941723E-1L; 239 static volatile long double z; 240 static long double w, W, Wa, Wb, ya, yb, u; 241 static const long double huge = 0x1p10000L; 242 #if 0 /* XXX Prevent gcc from erroneously constant folding this. */ 243 static const long double twom10000 = 0x1p-10000L; 244 #else 245 static volatile long double twom10000 = 0x1p-10000L; 246 #endif 247 248 static long double reducl( long double ); 249 static long double powil ( long double, int ); 250 251 long double 252 powl(long double x, long double y) 253 { 254 /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */ 255 int i, nflg, iyflg, yoddint; 256 long e; 257 258 if( y == 0.0L ) 259 return( 1.0L ); 260 261 if( x == 1.0L ) 262 return( 1.0L ); 263 264 if( isnan(x) ) 265 return ( nan_mix(x, y) ); 266 if( isnan(y) ) 267 return ( nan_mix(x, y) ); 268 269 if( y == 1.0L ) 270 return( x ); 271 272 if( !isfinite(y) && x == -1.0L ) 273 return( 1.0L ); 274 275 if( y >= LDBL_MAX ) 276 { 277 if( x > 1.0L ) 278 return( INFINITY ); 279 if( x > 0.0L && x < 1.0L ) 280 return( 0.0L ); 281 if( x < -1.0L ) 282 return( INFINITY ); 283 if( x > -1.0L && x < 0.0L ) 284 return( 0.0L ); 285 } 286 if( y <= -LDBL_MAX ) 287 { 288 if( x > 1.0L ) 289 return( 0.0L ); 290 if( x > 0.0L && x < 1.0L ) 291 return( INFINITY ); 292 if( x < -1.0L ) 293 return( 0.0L ); 294 if( x > -1.0L && x < 0.0L ) 295 return( INFINITY ); 296 } 297 if( x >= LDBL_MAX ) 298 { 299 if( y > 0.0L ) 300 return( INFINITY ); 301 return( 0.0L ); 302 } 303 304 w = floorl(y); 305 /* Set iyflg to 1 if y is an integer. */ 306 iyflg = 0; 307 if( w == y ) 308 iyflg = 1; 309 310 /* Test for odd integer y. */ 311 yoddint = 0; 312 if( iyflg ) 313 { 314 ya = fabsl(y); 315 ya = floorl(0.5L * ya); 316 yb = 0.5L * fabsl(w); 317 if( ya != yb ) 318 yoddint = 1; 319 } 320 321 if( x <= -LDBL_MAX ) 322 { 323 if( y > 0.0L ) 324 { 325 if( yoddint ) 326 return( -INFINITY ); 327 return( INFINITY ); 328 } 329 if( y < 0.0L ) 330 { 331 if( yoddint ) 332 return( -0.0L ); 333 return( 0.0 ); 334 } 335 } 336 337 338 nflg = 0; /* flag = 1 if x<0 raised to integer power */ 339 if( x <= 0.0L ) 340 { 341 if( x == 0.0L ) 342 { 343 if( y < 0.0 ) 344 { 345 if( signbit(x) && yoddint ) 346 return( -INFINITY ); 347 return( INFINITY ); 348 } 349 if( y > 0.0 ) 350 { 351 if( signbit(x) && yoddint ) 352 return( -0.0L ); 353 return( 0.0 ); 354 } 355 if( y == 0.0L ) 356 return( 1.0L ); /* 0**0 */ 357 else 358 return( 0.0L ); /* 0**y */ 359 } 360 else 361 { 362 if( iyflg == 0 ) 363 return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */ 364 nflg = 1; 365 } 366 } 367 368 /* Integer power of an integer. */ 369 370 if( iyflg ) 371 { 372 i = w; 373 w = floorl(x); 374 if( (w == x) && (fabsl(y) < 32768.0) ) 375 { 376 w = powil( x, (int) y ); 377 return( w ); 378 } 379 } 380 381 382 if( nflg ) 383 x = fabsl(x); 384 385 /* separate significand from exponent */ 386 x = frexpl( x, &i ); 387 e = i; 388 389 /* find significand in antilog table A[] */ 390 i = 1; 391 if( x <= douba(17) ) 392 i = 17; 393 if( x <= douba(i+8) ) 394 i += 8; 395 if( x <= douba(i+4) ) 396 i += 4; 397 if( x <= douba(i+2) ) 398 i += 2; 399 if( x >= douba(1) ) 400 i = -1; 401 i += 1; 402 403 404 /* Find (x - A[i])/A[i] 405 * in order to compute log(x/A[i]): 406 * 407 * log(x) = log( a x/a ) = log(a) + log(x/a) 408 * 409 * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a 410 */ 411 x -= douba(i); 412 x -= doubb(i/2); 413 x /= douba(i); 414 415 416 /* rational approximation for log(1+v): 417 * 418 * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v) 419 */ 420 z = x*x; 421 w = x * ( z * __polevll( x, P, 3 ) / __p1evll( x, Q, 3 ) ); 422 w = w - ldexpl( z, -1 ); /* w - 0.5 * z */ 423 424 /* Convert to base 2 logarithm: 425 * multiply by log2(e) = 1 + LOG2EA 426 */ 427 z = LOG2EA * w; 428 z += w; 429 z += LOG2EA * x; 430 z += x; 431 432 /* Compute exponent term of the base 2 logarithm. */ 433 w = -i; 434 w = ldexpl( w, -LNXT ); /* divide by NXT */ 435 w += e; 436 /* Now base 2 log of x is w + z. */ 437 438 /* Multiply base 2 log by y, in extended precision. */ 439 440 /* separate y into large part ya 441 * and small part yb less than 1/NXT 442 */ 443 ya = reducl(y); 444 yb = y - ya; 445 446 /* (w+z)(ya+yb) 447 * = w*ya + w*yb + z*y 448 */ 449 F = z * y + w * yb; 450 Fa = reducl(F); 451 Fb = F - Fa; 452 453 G = Fa + w * ya; 454 Ga = reducl(G); 455 Gb = G - Ga; 456 457 H = Fb + Gb; 458 Ha = reducl(H); 459 w = ldexpl( Ga+Ha, LNXT ); 460 461 /* Test the power of 2 for overflow */ 462 if( w > MEXP ) 463 return (huge * huge); /* overflow */ 464 465 if( w < MNEXP ) 466 return (twom10000 * twom10000); /* underflow */ 467 468 e = w; 469 Hb = H - Ha; 470 471 if( Hb > 0.0L ) 472 { 473 e += 1; 474 Hb -= (1.0L/NXT); /*0.0625L;*/ 475 } 476 477 /* Now the product y * log2(x) = Hb + e/NXT. 478 * 479 * Compute base 2 exponential of Hb, 480 * where -0.0625 <= Hb <= 0. 481 */ 482 z = Hb * __polevll( Hb, R, 6 ); /* z = 2**Hb - 1 */ 483 484 /* Express e/NXT as an integer plus a negative number of (1/NXT)ths. 485 * Find lookup table entry for the fractional power of 2. 486 */ 487 if( e < 0 ) 488 i = 0; 489 else 490 i = 1; 491 i = e/NXT + i; 492 e = NXT*i - e; 493 w = douba( e ); 494 z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */ 495 z = z + w; 496 z = ldexpl( z, i ); /* multiply by integer power of 2 */ 497 498 if( nflg ) 499 { 500 /* For negative x, 501 * find out if the integer exponent 502 * is odd or even. 503 */ 504 w = ldexpl( y, -1 ); 505 w = floorl(w); 506 w = ldexpl( w, 1 ); 507 if( w != y ) 508 z = -z; /* odd exponent */ 509 } 510 511 return( z ); 512 } 513 514 515 /* Find a multiple of 1/NXT that is within 1/NXT of x. */ 516 static inline long double 517 reducl(long double x) 518 { 519 long double t; 520 521 t = ldexpl( x, LNXT ); 522 t = floorl( t ); 523 t = ldexpl( t, -LNXT ); 524 return(t); 525 } 526 527 /* powil.c 528 * 529 * Real raised to integer power, long double precision 530 * 531 * 532 * 533 * SYNOPSIS: 534 * 535 * long double x, y, powil(); 536 * int n; 537 * 538 * y = powil( x, n ); 539 * 540 * 541 * 542 * DESCRIPTION: 543 * 544 * Returns argument x raised to the nth power. 545 * The routine efficiently decomposes n as a sum of powers of 546 * two. The desired power is a product of two-to-the-kth 547 * powers of x. Thus to compute the 32767 power of x requires 548 * 28 multiplications instead of 32767 multiplications. 549 * 550 * 551 * 552 * ACCURACY: 553 * 554 * 555 * Relative error: 556 * arithmetic x domain n domain # trials peak rms 557 * IEEE .001,1000 -1022,1023 50000 4.3e-17 7.8e-18 558 * IEEE 1,2 -1022,1023 20000 3.9e-17 7.6e-18 559 * IEEE .99,1.01 0,8700 10000 3.6e-16 7.2e-17 560 * 561 * Returns MAXNUM on overflow, zero on underflow. 562 * 563 */ 564 565 static long double 566 powil(long double x, int nn) 567 { 568 long double ww, y; 569 long double s; 570 int n, e, sign, asign, lx; 571 572 if( x == 0.0L ) 573 { 574 if( nn == 0 ) 575 return( 1.0L ); 576 else if( nn < 0 ) 577 return( LDBL_MAX ); 578 else 579 return( 0.0L ); 580 } 581 582 if( nn == 0 ) 583 return( 1.0L ); 584 585 586 if( x < 0.0L ) 587 { 588 asign = -1; 589 x = -x; 590 } 591 else 592 asign = 0; 593 594 595 if( nn < 0 ) 596 { 597 sign = -1; 598 n = -nn; 599 } 600 else 601 { 602 sign = 1; 603 n = nn; 604 } 605 606 /* Overflow detection */ 607 608 /* Calculate approximate logarithm of answer */ 609 s = x; 610 s = frexpl( s, &lx ); 611 e = (lx - 1)*n; 612 if( (e == 0) || (e > 64) || (e < -64) ) 613 { 614 s = (s - 7.0710678118654752e-1L) / (s + 7.0710678118654752e-1L); 615 s = (2.9142135623730950L * s - 0.5L + lx) * nn * LOGE2L; 616 } 617 else 618 { 619 s = LOGE2L * e; 620 } 621 622 if( s > MAXLOGL ) 623 return (huge * huge); /* overflow */ 624 625 if( s < MINLOGL ) 626 return (twom10000 * twom10000); /* underflow */ 627 /* Handle tiny denormal answer, but with less accuracy 628 * since roundoff error in 1.0/x will be amplified. 629 * The precise demarcation should be the gradual underflow threshold. 630 */ 631 if( s < (-MAXLOGL+2.0L) ) 632 { 633 x = 1.0L/x; 634 sign = -sign; 635 } 636 637 /* First bit of the power */ 638 if( n & 1 ) 639 y = x; 640 641 else 642 { 643 y = 1.0L; 644 asign = 0; 645 } 646 647 ww = x; 648 n >>= 1; 649 while( n ) 650 { 651 ww = ww * ww; /* arg to the 2-to-the-kth power */ 652 if( n & 1 ) /* if that bit is set, then include in product */ 653 y *= ww; 654 n >>= 1; 655 } 656 657 if( asign ) 658 y = -y; /* odd power of negative number */ 659 if( sign < 0 ) 660 y = 1.0L/y; 661 return(y); 662 } 663