src/math/powl.c

   1 /* origin: OpenBSD /usr/src/lib/libm/src/ld80/e_powl.c */
   2 /*
   3  * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
   4  *
   5  * Permission to use, copy, modify, and distribute this software for any
   6  * purpose with or without fee is hereby granted, provided that the above
   7  * copyright notice and this permission notice appear in all copies.
   8  *
   9  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
  10  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
  11  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
  12  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
  13  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
  14  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
  15  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
  16  */
  17 /*                                                      powl.c
  18  *
  19  *      Power function, long double precision
  20  *
  21  *
  22  * SYNOPSIS:
  23  *
  24  * long double x, y, z, powl();
  25  *
  26  * z = powl( x, y );
  27  *
  28  *
  29  * DESCRIPTION:
  30  *
  31  * Computes x raised to the yth power.  Analytically,
  32  *
  33  *      x**y  =  exp( y log(x) ).
  34  *
  35  * Following Cody and Waite, this program uses a lookup table
  36  * of 2**-i/32 and pseudo extended precision arithmetic to
  37  * obtain several extra bits of accuracy in both the logarithm
  38  * and the exponential.
  39  *
  40  *
  41  * ACCURACY:
  42  *
  43  * The relative error of pow(x,y) can be estimated
  44  * by   y dl ln(2),   where dl is the absolute error of
  45  * the internally computed base 2 logarithm.  At the ends
  46  * of the approximation interval the logarithm equal 1/32
  47  * and its relative error is about 1 lsb = 1.1e-19.  Hence
  48  * the predicted relative error in the result is 2.3e-21 y .
  49  *
  50  *                      Relative error:
  51  * arithmetic   domain     # trials      peak         rms
  52  *
  53  *    IEEE     +-1000       40000      2.8e-18      3.7e-19
  54  * .001 < x < 1000, with log(x) uniformly distributed.
  55  * -1000 < y < 1000, y uniformly distributed.
  56  *
  57  *    IEEE     0,8700       60000      6.5e-18      1.0e-18
  58  * 0.99 < x < 1.01, 0 < y < 8700, uniformly distributed.
  59  *
  60  *
  61  * ERROR MESSAGES:
  62  *
  63  *   message         condition      value returned
  64  * pow overflow     x**y > MAXNUM      INFINITY
  65  * pow underflow   x**y < 1/MAXNUM       0.0
  66  * pow domain      x<0 and y noninteger  0.0
  67  *
  68  */
  69
  70 #include "libm.h"
  71
  72 #if LDBL_MANT_DIG == 53 && LDBL_MAX_EXP == 1024
  73 long double powl(long double x, long double y)
  74 {
  75         return pow(x, y);
  76 }
  77 #elif LDBL_MANT_DIG == 64 && LDBL_MAX_EXP == 16384
  78
  79 /* Table size */
  80 #define NXT 32
  81
  82 /* log(1+x) =  x - .5x^2 + x^3 *  P(z)/Q(z)
  83  * on the domain  2^(-1/32) - 1  <=  x  <=  2^(1/32) - 1
  84  */
  85 static const long double P[] = {
  86  8.3319510773868690346226E-4L,
  87  4.9000050881978028599627E-1L,
  88  1.7500123722550302671919E0L,
  89  1.4000100839971580279335E0L,
  90 };
  91 static const long double Q[] = {
  92 /* 1.0000000000000000000000E0L,*/
  93  5.2500282295834889175431E0L,
  94  8.4000598057587009834666E0L,
  95  4.2000302519914740834728E0L,
  96 };
  97 /* A[i] = 2^(-i/32), rounded to IEEE long double precision.
  98  * If i is even, A[i] + B[i/2] gives additional accuracy.
  99  */
 100 static const long double A[33] = {
 101  1.0000000000000000000000E0L,
 102  9.7857206208770013448287E-1L,
 103  9.5760328069857364691013E-1L,
 104  9.3708381705514995065011E-1L,
 105  9.1700404320467123175367E-1L,
 106  8.9735453750155359320742E-1L,
 107  8.7812608018664974155474E-1L,
 108  8.5930964906123895780165E-1L,
 109  8.4089641525371454301892E-1L,
 110  8.2287773907698242225554E-1L,
 111  8.0524516597462715409607E-1L,
 112  7.8799042255394324325455E-1L,
 113  7.7110541270397041179298E-1L,
 114  7.5458221379671136985669E-1L,
 115  7.3841307296974965571198E-1L,
 116  7.2259040348852331001267E-1L,
 117  7.0710678118654752438189E-1L,
 118  6.9195494098191597746178E-1L,
 119  6.7712777346844636413344E-1L,
 120  6.6261832157987064729696E-1L,
 121  6.4841977732550483296079E-1L,
 122  6.3452547859586661129850E-1L,
 123  6.2092890603674202431705E-1L,
 124  6.0762367999023443907803E-1L,
 125  5.9460355750136053334378E-1L,
 126  5.8186242938878875689693E-1L,
 127  5.6939431737834582684856E-1L,
 128  5.5719337129794626814472E-1L,
 129  5.4525386633262882960438E-1L,
 130  5.3357020033841180906486E-1L,
 131  5.2213689121370692017331E-1L,
 132  5.1094857432705833910408E-1L,
 133  5.0000000000000000000000E-1L,
 134 };
 135 static const long double B[17] = {
 136  0.0000000000000000000000E0L,
 137  2.6176170809902549338711E-20L,
 138 -1.0126791927256478897086E-20L,
 139  1.3438228172316276937655E-21L,
 140  1.2207982955417546912101E-20L,
 141 -6.3084814358060867200133E-21L,
 142  1.3164426894366316434230E-20L,
 143 -1.8527916071632873716786E-20L,
 144  1.8950325588932570796551E-20L,
 145  1.5564775779538780478155E-20L,
 146  6.0859793637556860974380E-21L,
 147 -2.0208749253662532228949E-20L,
 148  1.4966292219224761844552E-20L,
 149  3.3540909728056476875639E-21L,
 150 -8.6987564101742849540743E-22L,
 151 -1.2327176863327626135542E-20L,
 152  0.0000000000000000000000E0L,
 153 };
 154
 155 /* 2^x = 1 + x P(x),
 156  * on the interval -1/32 <= x <= 0
 157  */
 158 static const long double R[] = {
 159  1.5089970579127659901157E-5L,
 160  1.5402715328927013076125E-4L,
 161  1.3333556028915671091390E-3L,
 162  9.6181291046036762031786E-3L,
 163  5.5504108664798463044015E-2L,
 164  2.4022650695910062854352E-1L,
 165  6.9314718055994530931447E-1L,
 166 };
 167
 168 #define MEXP (NXT*16384.0L)
 169 /* The following if denormal numbers are supported, else -MEXP: */
 170 #define MNEXP (-NXT*(16384.0L+64.0L))
 171 /* log2(e) - 1 */
 172 #define LOG2EA 0.44269504088896340735992L
 173
 174 #define F W
 175 #define Fa Wa
 176 #define Fb Wb
 177 #define G W
 178 #define Ga Wa
 179 #define Gb u
 180 #define H W
 181 #define Ha Wb
 182 #define Hb Wb
 183
 184 static const long double MAXLOGL = 1.1356523406294143949492E4L;
 185 static const long double MINLOGL = -1.13994985314888605586758E4L;
 186 static const long double LOGE2L = 6.9314718055994530941723E-1L;
 187 static const long double huge = 0x1p10000L;
 188 /* XXX Prevent gcc from erroneously constant folding this. */
 189 static const volatile long double twom10000 = 0x1p-10000L;
 190
 191 static long double reducl(long double);
 192 static long double powil(long double, int);
 193
 194 long double powl(long double x, long double y)
 195 {
 196         /* double F, Fa, Fb, G, Ga, Gb, H, Ha, Hb */
 197         int i, nflg, iyflg, yoddint;
 198         long e;
 199         volatile long double z=0;
 200         long double w=0, W=0, Wa=0, Wb=0, ya=0, yb=0, u=0;
 201
 202         /* make sure no invalid exception is raised by nan comparision */
 203         if (isnan(x)) {
 204                 if (!isnan(y) && y == 0.0)
 205                         return 1.0;
 206                 return x;
 207         }
 208         if (isnan(y)) {
 209                 if (x == 1.0)
 210                         return 1.0;
 211                 return y;
 212         }
 213         if (x == 1.0)
 214                 return 1.0; /* 1**y = 1, even if y is nan */
 215         if (y == 0.0)
 216                 return 1.0; /* x**0 = 1, even if x is nan */
 217         if (y == 1.0)
 218                 return x;
 219         /* if y*log2(x) < log2(LDBL_TRUE_MIN)-1 then x^y uflows to 0
 220            if y*log2(x) > -log2(LDBL_TRUE_MIN)+1 > LDBL_MAX_EXP then x^y oflows
 221            if |x|!=1 then |log2(x)| > |log(x)| > LDBL_EPSILON/2 so
 222            x^y oflows/uflows if |y|*LDBL_EPSILON/2 > -log2(LDBL_TRUE_MIN)+1 */
 223         if (fabsl(y) > 2*(-LDBL_MIN_EXP+LDBL_MANT_DIG+1)/LDBL_EPSILON) {
 224                 /* y is not an odd int */
 225                 if (x == -1.0)
 226                         return 1.0;
 227                 if (y == INFINITY) {
 228                         if (x > 1.0 || x < -1.0)
 229                                 return INFINITY;
 230                         return 0.0;
 231                 }
 232                 if (y == -INFINITY) {
 233                         if (x > 1.0 || x < -1.0)
 234                                 return 0.0;
 235                         return INFINITY;
 236                 }
 237                 if ((x > 1.0 || x < -1.0) == (y > 0))
 238                         return huge * huge;
 239                 return twom10000 * twom10000;
 240         }
 241         if (x == INFINITY) {
 242                 if (y > 0.0)
 243                         return INFINITY;
 244                 return 0.0;
 245         }
 246
 247         w = floorl(y);
 248
 249         /* Set iyflg to 1 if y is an integer. */
 250         iyflg = 0;
 251         if (w == y)
 252                 iyflg = 1;
 253
 254         /* Test for odd integer y. */
 255         yoddint = 0;
 256         if (iyflg) {
 257                 ya = fabsl(y);
 258                 ya = floorl(0.5 * ya);
 259                 yb = 0.5 * fabsl(w);
 260                 if( ya != yb )
 261                         yoddint = 1;
 262         }
 263
 264         if (x == -INFINITY) {
 265                 if (y > 0.0) {
 266                         if (yoddint)
 267                                 return -INFINITY;
 268                         return INFINITY;
 269                 }
 270                 if (y < 0.0) {
 271                         if (yoddint)
 272                                 return -0.0;
 273                         return 0.0;
 274                 }
 275         }
 276         nflg = 0; /* (x<0)**(odd int) */
 277         if (x <= 0.0) {
 278                 if (x == 0.0) {
 279                         if (y < 0.0) {
 280                                 if (signbit(x) && yoddint)
 281                                         /* (-0.0)**(-odd int) = -inf, divbyzero */
 282                                         return -1.0/0.0;
 283                                 /* (+-0.0)**(negative) = inf, divbyzero */
 284                                 return 1.0/0.0;
 285                         }
 286                         if (signbit(x) && yoddint)
 287                                 return -0.0;
 288                         return 0.0;
 289                 }
 290                 if (iyflg == 0)
 291                         return (x - x) / (x - x); /* (x<0)**(non-int) is NaN */
 292                 /* (x<0)**(integer) */
 293                 if (yoddint)
 294                         nflg = 1; /* negate result */
 295                 x = -x;
 296         }
 297         /* (+integer)**(integer)  */
 298         if (iyflg && floorl(x) == x && fabsl(y) < 32768.0) {
 299                 w = powil(x, (int)y);
 300                 return nflg ? -w : w;
 301         }
 302
 303         /* separate significand from exponent */
 304         x = frexpl(x, &i);
 305         e = i;
 306
 307         /* find significand in antilog table A[] */
 308         i = 1;
 309         if (x <= A[17])
 310                 i = 17;
 311         if (x <= A[i+8])
 312                 i += 8;
 313         if (x <= A[i+4])
 314                 i += 4;
 315         if (x <= A[i+2])
 316                 i += 2;
 317         if (x >= A[1])
 318                 i = -1;
 319         i += 1;
 320
 321         /* Find (x - A[i])/A[i]
 322          * in order to compute log(x/A[i]):
 323          *
 324          * log(x) = log( a x/a ) = log(a) + log(x/a)
 325          *
 326          * log(x/a) = log(1+v),  v = x/a - 1 = (x-a)/a
 327          */
 328         x -= A[i];
 329         x -= B[i/2];
 330         x /= A[i];
 331
 332         /* rational approximation for log(1+v):
 333          *
 334          * log(1+v)  =  v  -  v**2/2  +  v**3 P(v) / Q(v)
 335          */
 336         z = x*x;
 337         w = x * (z * __polevll(x, P, 3) / __p1evll(x, Q, 3));
 338         w = w - 0.5*z;
 339
 340         /* Convert to base 2 logarithm:
 341          * multiply by log2(e) = 1 + LOG2EA
 342          */
 343         z = LOG2EA * w;
 344         z += w;
 345         z += LOG2EA * x;
 346         z += x;
 347
 348         /* Compute exponent term of the base 2 logarithm. */
 349         w = -i;
 350         w /= NXT;
 351         w += e;
 352         /* Now base 2 log of x is w + z. */
 353
 354         /* Multiply base 2 log by y, in extended precision. */
 355
 356         /* separate y into large part ya
 357          * and small part yb less than 1/NXT
 358          */
 359         ya = reducl(y);
 360         yb = y - ya;
 361
 362         /* (w+z)(ya+yb)
 363          * = w*ya + w*yb + z*y
 364          */
 365         F = z * y  +  w * yb;
 366         Fa = reducl(F);
 367         Fb = F - Fa;
 368
 369         G = Fa + w * ya;
 370         Ga = reducl(G);
 371         Gb = G - Ga;
 372
 373         H = Fb + Gb;
 374         Ha = reducl(H);
 375         w = (Ga + Ha) * NXT;
 376
 377         /* Test the power of 2 for overflow */
 378         if (w > MEXP)
 379                 return huge * huge;  /* overflow */
 380         if (w < MNEXP)
 381                 return twom10000 * twom10000;  /* underflow */
 382
 383         e = w;
 384         Hb = H - Ha;
 385
 386         if (Hb > 0.0) {
 387                 e += 1;
 388                 Hb -= 1.0/NXT;  /*0.0625L;*/
 389         }
 390
 391         /* Now the product y * log2(x)  =  Hb + e/NXT.
 392          *
 393          * Compute base 2 exponential of Hb,
 394          * where -0.0625 <= Hb <= 0.
 395          */
 396         z = Hb * __polevll(Hb, R, 6);  /*  z = 2**Hb - 1  */
 397
 398         /* Express e/NXT as an integer plus a negative number of (1/NXT)ths.
 399          * Find lookup table entry for the fractional power of 2.
 400          */
 401         if (e < 0)
 402                 i = 0;
 403         else
 404                 i = 1;
 405         i = e/NXT + i;
 406         e = NXT*i - e;
 407         w = A[e];
 408         z = w * z;  /*  2**-e * ( 1 + (2**Hb-1) )  */
 409         z = z + w;
 410         z = scalbnl(z, i);  /* multiply by integer power of 2 */
 411
 412         if (nflg)
 413                 z = -z;
 414         return z;
 415 }
 416
 417
 418 /* Find a multiple of 1/NXT that is within 1/NXT of x. */
 419 static long double reducl(long double x)
 420 {
 421         long double t;
 422
 423         t = x * NXT;
 424         t = floorl(t);
 425         t = t / NXT;
 426         return t;
 427 }
 428
 429 /*
 430  *      Positive real raised to integer power, long double precision
 431  *
 432  *
 433  * SYNOPSIS:
 434  *
 435  * long double x, y, powil();
 436  * int n;
 437  *
 438  * y = powil( x, n );
 439  *
 440  *
 441  * DESCRIPTION:
 442  *
 443  * Returns argument x>0 raised to the nth power.
 444  * The routine efficiently decomposes n as a sum of powers of
 445  * two. The desired power is a product of two-to-the-kth
 446  * powers of x.  Thus to compute the 32767 power of x requires
 447  * 28 multiplications instead of 32767 multiplications.
 448  *
 449  *
 450  * ACCURACY:
 451  *
 452  *                      Relative error:
 453  * arithmetic   x domain   n domain  # trials      peak         rms
 454  *    IEEE     .001,1000  -1022,1023  50000       4.3e-17     7.8e-18
 455  *    IEEE        1,2     -1022,1023  20000       3.9e-17     7.6e-18
 456  *    IEEE     .99,1.01     0,8700    10000       3.6e-16     7.2e-17
 457  *
 458  * Returns MAXNUM on overflow, zero on underflow.
 459  */
 460
 461 static long double powil(long double x, int nn)
 462 {
 463         long double ww, y;
 464         long double s;
 465         int n, e, sign, lx;
 466
 467         if (nn == 0)
 468                 return 1.0;
 469
 470         if (nn < 0) {
 471                 sign = -1;
 472                 n = -nn;
 473         } else {
 474                 sign = 1;
 475                 n = nn;
 476         }
 477
 478         /* Overflow detection */
 479
 480         /* Calculate approximate logarithm of answer */
 481         s = x;
 482         s = frexpl( s, &lx);
 483         e = (lx - 1)*n;
 484         if ((e == 0) || (e > 64) || (e < -64)) {
 485                 s = (s - 7.0710678118654752e-1L) / (s +  7.0710678118654752e-1L);
 486                 s = (2.9142135623730950L * s - 0.5 + lx) * nn * LOGE2L;
 487         } else {
 488                 s = LOGE2L * e;
 489         }
 490
 491         if (s > MAXLOGL)
 492                 return huge * huge;  /* overflow */
 493
 494         if (s < MINLOGL)
 495                 return twom10000 * twom10000;  /* underflow */
 496         /* Handle tiny denormal answer, but with less accuracy
 497          * since roundoff error in 1.0/x will be amplified.
 498          * The precise demarcation should be the gradual underflow threshold.
 499          */
 500         if (s < -MAXLOGL+2.0) {
 501                 x = 1.0/x;
 502                 sign = -sign;
 503         }
 504
 505         /* First bit of the power */
 506         if (n & 1)
 507                 y = x;
 508         else
 509                 y = 1.0;
 510
 511         ww = x;
 512         n >>= 1;
 513         while (n) {
 514                 ww = ww * ww;   /* arg to the 2-to-the-kth power */
 515                 if (n & 1)     /* if that bit is set, then include in product */
 516                         y *= ww;
 517                 n >>= 1;
 518         }
 519
 520         if (sign < 0)
 521                 y = 1.0/y;
 522         return y;
 523 }
 524 #elif LDBL_MANT_DIG == 113 && LDBL_MAX_EXP == 16384
 525 // TODO: broken implementation to make things compile
 526 long double powl(long double x, long double y)
 527 {
 528         return pow(x, y);
 529 }
 530 #endif