newlib/libm/ld80/e_powl.c

   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 "../ld/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 "../ld/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 }