compiler/stdc/math/ld80/e_powl.c

   1 /*      $OpenBSD: e_powl.c,v 1.5 2013/11/12 20:35:19 martynas 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 const long double P[] = {
  89  8.3319510773868690346226E-4L,
  90  4.9000050881978028599627E-1L,
  91  1.7500123722550302671919E0L,
  92  1.4000100839971580279335E0L,
  93 };
  94 static const 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 const 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 const 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 const 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
 193 static const long double huge = 0x1p10000L;
 194 #if 0 /* XXX Prevent gcc from erroneously constant folding this. */
 195 static const long double twom10000 = 0x1p-10000L;
 196 #else
 197 static const volatile long double twom10000 __attribute__ ((__section__(".rodata"))) = 0x1p-10000L;
 198 #endif
 199
 200 static long double reducl( long double );
 201 static long double powil ( long double, int );
 202
 203 long double
 204 powl(long double x, long double y)
 205 {
 206 volatile long double z;
 207 long double w, W, Wa, Wb, ya, yb, u;
 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( x );
 220 if( isnan(y) )
 221         return( 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, (void *)P, 3 ) / __p1evll( x, (void *)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, (void *)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 }