sysdeps/ieee754/dbl-64/e_pow.c

   1 /*
   2  * IBM Accurate Mathematical Library
   3  * written by International Business Machines Corp.
   4  * Copyright (C) 2001-2014 Free Software Foundation, Inc.
   5  *
   6  * This program is free software; you can redistribute it and/or modify
   7  * it under the terms of the GNU Lesser General Public License as published by
   8  * the Free Software Foundation; either version 2.1 of the License, or
   9  * (at your option) any later version.
  10  *
  11  * This program is distributed in the hope that it will be useful,
  12  * but WITHOUT ANY WARRANTY; without even the implied warranty of
  13  * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
  14  * GNU Lesser General Public License for more details.
  15  *
  16  * You should have received a copy of the GNU Lesser General Public License
  17  * along with this program; if not, see <http://www.gnu.org/licenses/>.
  18  */
  19 /***************************************************************************/
  20 /*  MODULE_NAME: upow.c                                                    */
  21 /*                                                                         */
  22 /*  FUNCTIONS: upow                                                        */
  23 /*             power1                                                      */
  24 /*             my_log2                                                     */
  25 /*             log1                                                        */
  26 /*             checkint                                                    */
  27 /* FILES NEEDED: dla.h endian.h mpa.h mydefs.h                             */
  28 /*               halfulp.c mpexp.c mplog.c slowexp.c slowpow.c mpa.c       */
  29 /*                          uexp.c  upow.c                                 */
  30 /*               root.tbl uexp.tbl upow.tbl                                */
  31 /* An ultimate power routine. Given two IEEE double machine numbers y,x    */
  32 /* it computes the correctly rounded (to nearest) value of x^y.            */
  33 /* Assumption: Machine arithmetic operations are performed in              */
  34 /* round to nearest mode of IEEE 754 standard.                             */
  35 /*                                                                         */
  36 /***************************************************************************/
  37 #include "endian.h"
  38 #include "upow.h"
  39 #include <dla.h>
  40 #include "mydefs.h"
  41 #include "MathLib.h"
  42 #include "upow.tbl"
  43 #include <math_private.h>
  44 #include <fenv.h>
  45
  46 #ifndef SECTION
  47 # define SECTION
  48 #endif
  49
  50 static const double huge = 1.0e300, tiny = 1.0e-300;
  51
  52 double __exp1 (double x, double xx, double error);
  53 static double log1 (double x, double *delta, double *error);
  54 static double my_log2 (double x, double *delta, double *error);
  55 double __slowpow (double x, double y, double z);
  56 static double power1 (double x, double y);
  57 static int checkint (double x);
  58
  59 /* An ultimate power routine. Given two IEEE double machine numbers y, x it
  60    computes the correctly rounded (to nearest) value of X^y.  */
  61 double
  62 SECTION
  63 __ieee754_pow (double x, double y)
  64 {
  65   double z, a, aa, error, t, a1, a2, y1, y2;
  66   mynumber u, v;
  67   int k;
  68   int4 qx, qy;
  69   v.x = y;
  70   u.x = x;
  71   if (v.i[LOW_HALF] == 0)
  72     {                           /* of y */
  73       qx = u.i[HIGH_HALF] & 0x7fffffff;
  74       /* Is x a NaN?  */
  75       if (((qx == 0x7ff00000) && (u.i[LOW_HALF] != 0)) || (qx > 0x7ff00000))
  76         return x;
  77       if (y == 1.0)
  78         return x;
  79       if (y == 2.0)
  80         return x * x;
  81       if (y == -1.0)
  82         return 1.0 / x;
  83       if (y == 0)
  84         return 1.0;
  85     }
  86   /* else */
  87   if (((u.i[HIGH_HALF] > 0 && u.i[HIGH_HALF] < 0x7ff00000) ||   /* x>0 and not x->0 */
  88        (u.i[HIGH_HALF] == 0 && u.i[LOW_HALF] != 0)) &&
  89       /*   2^-1023< x<= 2^-1023 * 0x1.0000ffffffff */
  90       (v.i[HIGH_HALF] & 0x7fffffff) < 0x4ff00000)
  91     {                           /* if y<-1 or y>1   */
  92       double retval;
  93
  94       SET_RESTORE_ROUND (FE_TONEAREST);
  95
  96       /* Avoid internal underflow for tiny y.  The exact value of y does
  97          not matter if |y| <= 2**-64.  */
  98       if (ABS (y) < 0x1p-64)
  99         y = y < 0 ? -0x1p-64 : 0x1p-64;
 100       z = log1 (x, &aa, &error);        /* x^y  =e^(y log (X)) */
 101       t = y * CN;
 102       y1 = t - (t - y);
 103       y2 = y - y1;
 104       t = z * CN;
 105       a1 = t - (t - z);
 106       a2 = (z - a1) + aa;
 107       a = y1 * a1;
 108       aa = y2 * a1 + y * a2;
 109       a1 = a + aa;
 110       a2 = (a - a1) + aa;
 111       error = error * ABS (y);
 112       t = __exp1 (a1, a2, 1.9e16 * error);      /* return -10 or 0 if wasn't computed exactly */
 113       retval = (t > 0) ? t : power1 (x, y);
 114
 115       return retval;
 116     }
 117
 118   if (x == 0)
 119     {
 120       if (((v.i[HIGH_HALF] & 0x7fffffff) == 0x7ff00000 && v.i[LOW_HALF] != 0)
 121           || (v.i[HIGH_HALF] & 0x7fffffff) > 0x7ff00000)        /* NaN */
 122         return y;
 123       if (ABS (y) > 1.0e20)
 124         return (y > 0) ? 0 : 1.0 / 0.0;
 125       k = checkint (y);
 126       if (k == -1)
 127         return y < 0 ? 1.0 / x : x;
 128       else
 129         return y < 0 ? 1.0 / 0.0 : 0.0; /* return 0 */
 130     }
 131
 132   qx = u.i[HIGH_HALF] & 0x7fffffff;     /*   no sign   */
 133   qy = v.i[HIGH_HALF] & 0x7fffffff;     /*   no sign   */
 134
 135   if (qx >= 0x7ff00000 && (qx > 0x7ff00000 || u.i[LOW_HALF] != 0))      /* NaN */
 136     return x;
 137   if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0))      /* NaN */
 138     return x == 1.0 ? 1.0 : y;
 139
 140   /* if x<0 */
 141   if (u.i[HIGH_HALF] < 0)
 142     {
 143       k = checkint (y);
 144       if (k == 0)
 145         {
 146           if (qy == 0x7ff00000)
 147             {
 148               if (x == -1.0)
 149                 return 1.0;
 150               else if (x > -1.0)
 151                 return v.i[HIGH_HALF] < 0 ? INF.x : 0.0;
 152               else
 153                 return v.i[HIGH_HALF] < 0 ? 0.0 : INF.x;
 154             }
 155           else if (qx == 0x7ff00000)
 156             return y < 0 ? 0.0 : INF.x;
 157           return (x - x) / (x - x);     /* y not integer and x<0 */
 158         }
 159       else if (qx == 0x7ff00000)
 160         {
 161           if (k < 0)
 162             return y < 0 ? nZERO.x : nINF.x;
 163           else
 164             return y < 0 ? 0.0 : INF.x;
 165         }
 166       /* if y even or odd */
 167       return (k == 1) ? __ieee754_pow (-x, y) : -__ieee754_pow (-x, y);
 168     }
 169   /* x>0 */
 170
 171   if (qx == 0x7ff00000)         /* x= 2^-0x3ff */
 172     return y > 0 ? x : 0;
 173
 174   if (qy > 0x45f00000 && qy < 0x7ff00000)
 175     {
 176       if (x == 1.0)
 177         return 1.0;
 178       if (y > 0)
 179         return (x > 1.0) ? huge * huge : tiny * tiny;
 180       if (y < 0)
 181         return (x < 1.0) ? huge * huge : tiny * tiny;
 182     }
 183
 184   if (x == 1.0)
 185     return 1.0;
 186   if (y > 0)
 187     return (x > 1.0) ? INF.x : 0;
 188   if (y < 0)
 189     return (x < 1.0) ? INF.x : 0;
 190   return 0;                     /* unreachable, to make the compiler happy */
 191 }
 192
 193 #ifndef __ieee754_pow
 194 strong_alias (__ieee754_pow, __pow_finite)
 195 #endif
 196
 197 /* Compute x^y using more accurate but more slow log routine.  */
 198 static double
 199 SECTION
 200 power1 (double x, double y)
 201 {
 202   double z, a, aa, error, t, a1, a2, y1, y2;
 203   z = my_log2 (x, &aa, &error);
 204   t = y * CN;
 205   y1 = t - (t - y);
 206   y2 = y - y1;
 207   t = z * CN;
 208   a1 = t - (t - z);
 209   a2 = z - a1;
 210   a = y * z;
 211   aa = ((y1 * a1 - a) + y1 * a2 + y2 * a1) + y2 * a2 + aa * y;
 212   a1 = a + aa;
 213   a2 = (a - a1) + aa;
 214   error = error * ABS (y);
 215   t = __exp1 (a1, a2, 1.9e16 * error);
 216   return (t >= 0) ? t : __slowpow (x, y, z);
 217 }
 218
 219 /* Compute log(x) (x is left argument). The result is the returned double + the
 220    parameter DELTA.  The result is bounded by ERROR.  */
 221 static double
 222 SECTION
 223 log1 (double x, double *delta, double *error)
 224 {
 225   int i, j, m;
 226   double uu, vv, eps, nx, e, e1, e2, t, t1, t2, res, add = 0;
 227   mynumber u, v;
 228 #ifdef BIG_ENDI
 229   mynumber /**/ two52 = {{0x43300000, 0x00000000}};     /* 2**52  */
 230 #else
 231 # ifdef LITTLE_ENDI
 232   mynumber /**/ two52 = {{0x00000000, 0x43300000}};     /* 2**52  */
 233 # endif
 234 #endif
 235
 236   u.x = x;
 237   m = u.i[HIGH_HALF];
 238   *error = 0;
 239   *delta = 0;
 240   if (m < 0x00100000)           /*  1<x<2^-1007 */
 241     {
 242       x = x * t52.x;
 243       add = -52.0;
 244       u.x = x;
 245       m = u.i[HIGH_HALF];
 246     }
 247
 248   if ((m & 0x000fffff) < 0x0006a09e)
 249     {
 250       u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3ff00000;
 251       two52.i[LOW_HALF] = (m >> 20);
 252     }
 253   else
 254     {
 255       u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3fe00000;
 256       two52.i[LOW_HALF] = (m >> 20) + 1;
 257     }
 258
 259   v.x = u.x + bigu.x;
 260   uu = v.x - bigu.x;
 261   i = (v.i[LOW_HALF] & 0x000003ff) << 2;
 262   if (two52.i[LOW_HALF] == 1023)        /* nx = 0              */
 263     {
 264       if (i > 1192 && i < 1208) /* |x-1| < 1.5*2**-10  */
 265         {
 266           t = x - 1.0;
 267           t1 = (t + 5.0e6) - 5.0e6;
 268           t2 = t - t1;
 269           e1 = t - 0.5 * t1 * t1;
 270           e2 = (t * t * t * (r3 + t * (r4 + t * (r5 + t * (r6 + t
 271                                                            * (r7 + t * r8)))))
 272                 - 0.5 * t2 * (t + t1));
 273           res = e1 + e2;
 274           *error = 1.0e-21 * ABS (t);
 275           *delta = (e1 - res) + e2;
 276           return res;
 277         }                       /* |x-1| < 1.5*2**-10  */
 278       else
 279         {
 280           v.x = u.x * (ui.x[i] + ui.x[i + 1]) + bigv.x;
 281           vv = v.x - bigv.x;
 282           j = v.i[LOW_HALF] & 0x0007ffff;
 283           j = j + j + j;
 284           eps = u.x - uu * vv;
 285           e1 = eps * ui.x[i];
 286           e2 = eps * (ui.x[i + 1] + vj.x[j] * (ui.x[i] + ui.x[i + 1]));
 287           e = e1 + e2;
 288           e2 = ((e1 - e) + e2);
 289           t = ui.x[i + 2] + vj.x[j + 1];
 290           t1 = t + e;
 291           t2 = ((((t - t1) + e) + (ui.x[i + 3] + vj.x[j + 2])) + e2 + e * e
 292                 * (p2 + e * (p3 + e * p4)));
 293           res = t1 + t2;
 294           *error = 1.0e-24;
 295           *delta = (t1 - res) + t2;
 296           return res;
 297         }
 298     }                           /* nx = 0 */
 299   else                          /* nx != 0   */
 300     {
 301       eps = u.x - uu;
 302       nx = (two52.x - two52e.x) + add;
 303       e1 = eps * ui.x[i];
 304       e2 = eps * ui.x[i + 1];
 305       e = e1 + e2;
 306       e2 = (e1 - e) + e2;
 307       t = nx * ln2a.x + ui.x[i + 2];
 308       t1 = t + e;
 309       t2 = ((((t - t1) + e) + nx * ln2b.x + ui.x[i + 3] + e2) + e * e
 310             * (q2 + e * (q3 + e * (q4 + e * (q5 + e * q6)))));
 311       res = t1 + t2;
 312       *error = 1.0e-21;
 313       *delta = (t1 - res) + t2;
 314       return res;
 315     }                           /* nx != 0   */
 316 }
 317
 318 /* Slower but more accurate routine of log.  The returned result is double +
 319    DELTA.  The result is bounded by ERROR.  */
 320 static double
 321 SECTION
 322 my_log2 (double x, double *delta, double *error)
 323 {
 324   int i, j, m;
 325   double uu, vv, eps, nx, e, e1, e2, t, t1, t2, res, add = 0;
 326   double ou1, ou2, lu1, lu2, ov, lv1, lv2, a, a1, a2;
 327   double y, yy, z, zz, j1, j2, j7, j8;
 328 #ifndef DLA_FMS
 329   double j3, j4, j5, j6;
 330 #endif
 331   mynumber u, v;
 332 #ifdef BIG_ENDI
 333   mynumber /**/ two52 = {{0x43300000, 0x00000000}};     /* 2**52  */
 334 #else
 335 # ifdef LITTLE_ENDI
 336   mynumber /**/ two52 = {{0x00000000, 0x43300000}};     /* 2**52  */
 337 # endif
 338 #endif
 339
 340   u.x = x;
 341   m = u.i[HIGH_HALF];
 342   *error = 0;
 343   *delta = 0;
 344   add = 0;
 345   if (m < 0x00100000)
 346     {                           /* x < 2^-1022 */
 347       x = x * t52.x;
 348       add = -52.0;
 349       u.x = x;
 350       m = u.i[HIGH_HALF];
 351     }
 352
 353   if ((m & 0x000fffff) < 0x0006a09e)
 354     {
 355       u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3ff00000;
 356       two52.i[LOW_HALF] = (m >> 20);
 357     }
 358   else
 359     {
 360       u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3fe00000;
 361       two52.i[LOW_HALF] = (m >> 20) + 1;
 362     }
 363
 364   v.x = u.x + bigu.x;
 365   uu = v.x - bigu.x;
 366   i = (v.i[LOW_HALF] & 0x000003ff) << 2;
 367   /*------------------------------------- |x-1| < 2**-11-------------------------------  */
 368   if ((two52.i[LOW_HALF] == 1023) && (i == 1200))
 369     {
 370       t = x - 1.0;
 371       EMULV (t, s3, y, yy, j1, j2, j3, j4, j5);
 372       ADD2 (-0.5, 0, y, yy, z, zz, j1, j2);
 373       MUL2 (t, 0, z, zz, y, yy, j1, j2, j3, j4, j5, j6, j7, j8);
 374       MUL2 (t, 0, y, yy, z, zz, j1, j2, j3, j4, j5, j6, j7, j8);
 375
 376       e1 = t + z;
 377       e2 = ((((t - e1) + z) + zz) + t * t * t
 378             * (ss3 + t * (s4 + t * (s5 + t * (s6 + t * (s7 + t * s8))))));
 379       res = e1 + e2;
 380       *error = 1.0e-25 * ABS (t);
 381       *delta = (e1 - res) + e2;
 382       return res;
 383     }
 384   /*----------------------------- |x-1| > 2**-11  --------------------------  */
 385   else
 386     {                           /*Computing log(x) according to log table                        */
 387       nx = (two52.x - two52e.x) + add;
 388       ou1 = ui.x[i];
 389       ou2 = ui.x[i + 1];
 390       lu1 = ui.x[i + 2];
 391       lu2 = ui.x[i + 3];
 392       v.x = u.x * (ou1 + ou2) + bigv.x;
 393       vv = v.x - bigv.x;
 394       j = v.i[LOW_HALF] & 0x0007ffff;
 395       j = j + j + j;
 396       eps = u.x - uu * vv;
 397       ov = vj.x[j];
 398       lv1 = vj.x[j + 1];
 399       lv2 = vj.x[j + 2];
 400       a = (ou1 + ou2) * (1.0 + ov);
 401       a1 = (a + 1.0e10) - 1.0e10;
 402       a2 = a * (1.0 - a1 * uu * vv);
 403       e1 = eps * a1;
 404       e2 = eps * a2;
 405       e = e1 + e2;
 406       e2 = (e1 - e) + e2;
 407       t = nx * ln2a.x + lu1 + lv1;
 408       t1 = t + e;
 409       t2 = ((((t - t1) + e) + (lu2 + lv2 + nx * ln2b.x + e2)) + e * e
 410             * (p2 + e * (p3 + e * p4)));
 411       res = t1 + t2;
 412       *error = 1.0e-27;
 413       *delta = (t1 - res) + t2;
 414       return res;
 415     }
 416 }
 417
 418 /* This function receives a double x and checks if it is an integer.  If not,
 419    it returns 0, else it returns 1 if even or -1 if odd.  */
 420 static int
 421 SECTION
 422 checkint (double x)
 423 {
 424   union
 425   {
 426     int4 i[2];
 427     double x;
 428   } u;
 429   int k, m, n;
 430   u.x = x;
 431   m = u.i[HIGH_HALF] & 0x7fffffff;      /* no sign */
 432   if (m >= 0x7ff00000)
 433     return 0;                   /*  x is +/-inf or NaN  */
 434   if (m >= 0x43400000)
 435     return 1;                   /*  |x| >= 2**53   */
 436   if (m < 0x40000000)
 437     return 0;                   /* |x| < 2,  can not be 0 or 1  */
 438   n = u.i[LOW_HALF];
 439   k = (m >> 20) - 1023;         /*  1 <= k <= 52   */
 440   if (k == 52)
 441     return (n & 1) ? -1 : 1;    /* odd or even */
 442   if (k > 20)
 443     {
 444       if (n << (k - 20))
 445         return 0;               /* if not integer */
 446       return (n << (k - 21)) ? -1 : 1;
 447     }
 448   if (n)
 449     return 0;                   /*if  not integer */
 450   if (k == 20)
 451     return (m & 1) ? -1 : 1;
 452   if (m << (k + 12))
 453     return 0;
 454   return (m << (k + 11)) ? -1 : 1;
 455 }