1 /* $OpenBSD: e_powl.c,v 1.7 2017/01/21 08:29:13 krw Exp $ */ 2 3 /* 4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net> 5 * 6 * Permission to use, copy, modify, and distribute this software for any 7 * purpose with or without fee is hereby granted, provided that the above 8 * copyright notice and this permission notice appear in all copies. 9 * 10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES 11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF 12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR 13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES 14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN 15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF 16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE. 17 */ 18 19 /* powl.c 20 * 21 * Power function, long double precision 22 * 23 * 24 * 25 * SYNOPSIS: 26 * 27 * long double x, y, z, powl(); 28 * 29 * z = powl( x, y ); 30 * 31 * 32 * 33 * DESCRIPTION: 34 * 35 * Computes x raised to the yth power. Analytically, 36 * 37 * x**y = exp( y log(x) ). 38 * 39 * Following Cody and Waite, this program uses a lookup table 40 * of 2**-i/32 and pseudo extended precision arithmetic to 41 * obtain several extra bits of accuracy in both the logarithm 42 * and the exponential. 43 * 44 * 45 * 46 * ACCURACY: 47 * 48 * The relative error of pow(x,y) can be estimated 49 * by y dl ln(2), where dl is the absolute error of 50 * the internally computed base 2 logarithm. At the ends 51 * of the approximation interval the logarithm equal 1/32 52 * and its relative error is about 1 lsb = 1.1e-19. Hence 53 * the predicted relative error in the result is 2.3e-21 y . 54 * 55 * Relative error: 56 * arithmetic domain # trials peak rms 57 * 58 * IEEE +-1000 40000 2.8e-18 3.7e-19 59 * .001 < x < 1000, with log(x) uniformly distributed. 60 * -1000 < y < 1000, y uniformly distributed. 61 * 62 * IEEE 0,8700 60000 6.5e-18 1.0e-18 63 * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed. 64 * 65 * 66 * ERROR MESSAGES: 67 * 68 * message condition value returned 69 * pow overflow x**y > MAXNUM INFINITY 70 * pow underflow x**y < 1/MAXNUM 0.0 71 * pow domain x<0 and y noninteger 0.0 72 * 73 */ 74 75 #include <float.h> 76 #include <math.h> 77 78 #include "math_private.h" 79 80 /* Table size */ 81 #define NXT 32 82 /* log2(Table size) */ 83 #define LNXT 5 84 85 /* log(1+x) = x - .5x^2 + x^3 * P(z)/Q(z) 86 * on the domain 2^(-1/32) - 1 <= x <= 2^(1/32) - 1 87 */ 88 static long double P[] = { 89 8.3319510773868690346226E-4L, 90 4.9000050881978028599627E-1L, 91 1.7500123722550302671919E0L, 92 1.4000100839971580279335E0L, 93 }; 94 static long double Q[] = { 95 /* 1.0000000000000000000000E0L,*/ 96 5.2500282295834889175431E0L, 97 8.4000598057587009834666E0L, 98 4.2000302519914740834728E0L, 99 }; 100 /* A[i] = 2^(-i/32), rounded to IEEE long double precision. 101 * If i is even, A[i] + B[i/2] gives additional accuracy. 102 */ 103 static long double A[33] = { 104 1.0000000000000000000000E0L, 105 9.7857206208770013448287E-1L, 106 9.5760328069857364691013E-1L, 107 9.3708381705514995065011E-1L, 108 9.1700404320467123175367E-1L, 109 8.9735453750155359320742E-1L, 110 8.7812608018664974155474E-1L, 111 8.5930964906123895780165E-1L, 112 8.4089641525371454301892E-1L, 113 8.2287773907698242225554E-1L, 114 8.0524516597462715409607E-1L, 115 7.8799042255394324325455E-1L, 116 7.7110541270397041179298E-1L, 117 7.5458221379671136985669E-1L, 118 7.3841307296974965571198E-1L, 119 7.2259040348852331001267E-1L, 120 7.0710678118654752438189E-1L, 121 6.9195494098191597746178E-1L, 122 6.7712777346844636413344E-1L, 123 6.6261832157987064729696E-1L, 124 6.4841977732550483296079E-1L, 125 6.3452547859586661129850E-1L, 126 6.2092890603674202431705E-1L, 127 6.0762367999023443907803E-1L, 128 5.9460355750136053334378E-1L, 129 5.8186242938878875689693E-1L, 130 5.6939431737834582684856E-1L, 131 5.5719337129794626814472E-1L, 132 5.4525386633262882960438E-1L, 133 5.3357020033841180906486E-1L, 134 5.2213689121370692017331E-1L, 135 5.1094857432705833910408E-1L, 136 5.0000000000000000000000E-1L, 137 }; 138 static long double B[17] = { 139 0.0000000000000000000000E0L, 140 2.6176170809902549338711E-20L, 141 -1.0126791927256478897086E-20L, 142 1.3438228172316276937655E-21L, 143 1.2207982955417546912101E-20L, 144 -6.3084814358060867200133E-21L, 145 1.3164426894366316434230E-20L, 146 -1.8527916071632873716786E-20L, 147 1.8950325588932570796551E-20L, 148 1.5564775779538780478155E-20L, 149 6.0859793637556860974380E-21L, 150 -2.0208749253662532228949E-20L, 151 1.4966292219224761844552E-20L, 152 3.3540909728056476875639E-21L, 153 -8.6987564101742849540743E-22L, 154 -1.2327176863327626135542E-20L, 155 0.0000000000000000000000E0L, 156 }; 157 158 /* 2^x = 1 + x P(x), 159 * on the interval -1/32 <= x <= 0 160 */ 161 static long double R[] = { 162 1.5089970579127659901157E-5L, 163 1.5402715328927013076125E-4L, 164 1.3333556028915671091390E-3L, 165 9.6181291046036762031786E-3L, 166 5.5504108664798463044015E-2L, 167 2.4022650695910062854352E-1L, 168 6.9314718055994530931447E-1L, 169 }; 170 171 #define douba(k) A[k] 172 #define doubb(k) B[k] 173 #define MEXP (NXT*16384.0L) 174 /* The following if denormal numbers are supported, else -MEXP: */ 175 #define MNEXP (-NXT*(16384.0L+64.0L)) 176 /* log2(e) - 1 */ 177 #define LOG2EA 0.44269504088896340735992L 178 179 #define F W 180 #define Fa Wa 181 #define Fb Wb 182 #define G W 183 #define Ga Wa 184 #define Gb u 185 #define H W 186 #define Ha Wb 187 #define Hb Wb 188 189 static const long double MAXLOGL = 1.1356523406294143949492E4L; 190 static const long double MINLOGL = -1.13994985314888605586758E4L; 191 static const long double LOGE2L = 6.9314718055994530941723E-1L; 192 static volatile long double z; 193 static long double w, W, Wa, Wb, ya, yb, u; 194 static const long double huge = 0x1p10000L; 195 #if 0 /* XXX Prevent gcc from erroneously constant folding this. */ 196 static const long double twom10000 = 0x1p-10000L; 197 #else 198 static volatile long double twom10000 = 0x1p-10000L; 199 #endif 200 201 static long double reducl( long double ); 202 static long double powil ( long double, int ); 203 204 long double 205 powl(long double x, long double y) 206 { 207 /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */ 208 int i, nflg, iyflg, yoddint; 209 long e; 210 211 if( y == 0.0L ) 212 return( 1.0L ); 213 214 if( x == 1.0L ) 215 return( 1.0L ); 216 217 if( isnan(x) ) 218 return( x ); 219 if( isnan(y) ) 220 return( y ); 221 222 if( y == 1.0L ) 223 return( x ); 224 225 if( !isfinite(y) && x == -1.0L ) 226 return( 1.0L ); 227 228 if( y >= LDBL_MAX ) 229 { 230 if( x > 1.0L ) 231 return( INFINITY ); 232 if( x > 0.0L && x < 1.0L ) 233 return( 0.0L ); 234 if( x < -1.0L ) 235 return( INFINITY ); 236 if( x > -1.0L && x < 0.0L ) 237 return( 0.0L ); 238 } 239 if( y <= -LDBL_MAX ) 240 { 241 if( x > 1.0L ) 242 return( 0.0L ); 243 if( x > 0.0L && x < 1.0L ) 244 return( INFINITY ); 245 if( x < -1.0L ) 246 return( 0.0L ); 247 if( x > -1.0L && x < 0.0L ) 248 return( INFINITY ); 249 } 250 if( x >= LDBL_MAX ) 251 { 252 if( y > 0.0L ) 253 return( INFINITY ); 254 return( 0.0L ); 255 } 256 257 w = floorl(y); 258 /* Set iyflg to 1 if y is an integer. */ 259 iyflg = 0; 260 if( w == y ) 261 iyflg = 1; 262 263 /* Test for odd integer y. */ 264 yoddint = 0; 265 if( iyflg ) 266 { 267 ya = fabsl(y); 268 ya = floorl(0.5L * ya); 269 yb = 0.5L * fabsl(w); 270 if( ya != yb ) 271 yoddint = 1; 272 } 273 274 if( x <= -LDBL_MAX ) 275 { 276 if( y > 0.0L ) 277 { 278 if( yoddint ) 279 return( -INFINITY ); 280 return( INFINITY ); 281 } 282 if( y < 0.0L ) 283 { 284 if( yoddint ) 285 return( -0.0L ); 286 return( 0.0 ); 287 } 288 } 289 290 291 nflg = 0; /* flag = 1 if x<0 raised to integer power */ 292 if( x <= 0.0L ) 293 { 294 if( x == 0.0L ) 295 { 296 if( y < 0.0 ) 297 { 298 if( signbit(x) && yoddint ) 299 return( -INFINITY ); 300 return( INFINITY ); 301 } 302 if( y > 0.0 ) 303 { 304 if( signbit(x) && yoddint ) 305 return( -0.0L ); 306 return( 0.0 ); 307 } 308 if( y == 0.0L ) 309 return( 1.0L ); /* 0**0 */ 310 else 311 return( 0.0L ); /* 0**y */ 312 } 313 else 314 { 315 if( iyflg == 0 ) 316 return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */ 317 nflg = 1; 318 } 319 } 320 321 /* Integer power of an integer. */ 322 323 if( iyflg ) 324 { 325 i = w; 326 w = floorl(x); 327 if( (w == x) && (fabsl(y) < 32768.0) ) 328 { 329 w = powil( x, (int) y ); 330 return( w ); 331 } 332 } 333 334 335 if( nflg ) 336 x = fabsl(x); 337 338 /* separate significand from exponent */ 339 x = frexpl( x, &i ); 340 e = i; 341 342 /* find significand in antilog table A[] */ 343 i = 1; 344 if( x <= douba(17) ) 345 i = 17; 346 if( x <= douba(i+8) ) 347 i += 8; 348 if( x <= douba(i+4) ) 349 i += 4; 350 if( x <= douba(i+2) ) 351 i += 2; 352 if( x >= douba(1) ) 353 i = -1; 354 i += 1; 355 356 357 /* Find (x - A[i])/A[i] 358 * in order to compute log(x/A[i]): 359 * 360 * log(x) = log( a x/a ) = log(a) + log(x/a) 361 * 362 * log(x/a) = log(1+v), v = x/a - 1 = (x-a)/a 363 */ 364 x -= douba(i); 365 x -= doubb(i/2); 366 x /= douba(i); 367 368 369 /* rational approximation for log(1+v): 370 * 371 * log(1+v) = v - v**2/2 + v**3 P(v) / Q(v) 372 */ 373 z = x*x; 374 w = x * ( z * __polevll( x, P, 3 ) / __p1evll( x, Q, 3 ) ); 375 w = w - ldexpl( z, -1 ); /* w - 0.5 * z */ 376 377 /* Convert to base 2 logarithm: 378 * multiply by log2(e) = 1 + LOG2EA 379 */ 380 z = LOG2EA * w; 381 z += w; 382 z += LOG2EA * x; 383 z += x; 384 385 /* Compute exponent term of the base 2 logarithm. */ 386 w = -i; 387 w = ldexpl( w, -LNXT ); /* divide by NXT */ 388 w += e; 389 /* Now base 2 log of x is w + z. */ 390 391 /* Multiply base 2 log by y, in extended precision. */ 392 393 /* separate y into large part ya 394 * and small part yb less than 1/NXT 395 */ 396 ya = reducl(y); 397 yb = y - ya; 398 399 /* (w+z)(ya+yb) 400 * = w*ya + w*yb + z*y 401 */ 402 F = z * y + w * yb; 403 Fa = reducl(F); 404 Fb = F - Fa; 405 406 G = Fa + w * ya; 407 Ga = reducl(G); 408 Gb = G - Ga; 409 410 H = Fb + Gb; 411 Ha = reducl(H); 412 w = ldexpl( Ga+Ha, LNXT ); 413 414 /* Test the power of 2 for overflow */ 415 if( w > MEXP ) 416 return (huge * huge); /* overflow */ 417 418 if( w < MNEXP ) 419 return (twom10000 * twom10000); /* underflow */ 420 421 e = w; 422 Hb = H - Ha; 423 424 if( Hb > 0.0L ) 425 { 426 e += 1; 427 Hb -= (1.0L/NXT); /*0.0625L;*/ 428 } 429 430 /* Now the product y * log2(x) = Hb + e/NXT. 431 * 432 * Compute base 2 exponential of Hb, 433 * where -0.0625 <= Hb <= 0. 434 */ 435 z = Hb * __polevll( Hb, R, 6 ); /* z = 2**Hb - 1 */ 436 437 /* Express e/NXT as an integer plus a negative number of (1/NXT)ths. 438 * Find lookup table entry for the fractional power of 2. 439 */ 440 if( e < 0 ) 441 i = 0; 442 else 443 i = 1; 444 i = e/NXT + i; 445 e = NXT*i - e; 446 w = douba( e ); 447 z = w * z; /* 2**-e * ( 1 + (2**Hb-1) ) */ 448 z = z + w; 449 z = ldexpl( z, i ); /* multiply by integer power of 2 */ 450 451 if( nflg ) 452 { 453 /* For negative x, 454 * find out if the integer exponent 455 * is odd or even. 456 */ 457 w = ldexpl( y, -1 ); 458 w = floorl(w); 459 w = ldexpl( w, 1 ); 460 if( w != y ) 461 z = -z; /* odd exponent */ 462 } 463 464 return( z ); 465 } 466 DEF_STD(powl); 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