sysdeps/ieee754/dbl-64/e_pow.c

   1 /*
   2  * IBM Accurate Mathematical Library
   3  * written by International Business Machines Corp.
   4  * Copyright (C) 2001-2018 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 /*             log1                                                        */
  24 /*             checkint                                                    */
  25 /* FILES NEEDED: dla.h endian.h mpa.h mydefs.h                             */
  26 /*               root.tbl uexp.tbl upow.tbl                                */
  27 /* An ultimate power routine. Given two IEEE double machine numbers y,x    */
  28 /* it computes the correctly rounded (to nearest) value of x^y.            */
  29 /* Assumption: Machine arithmetic operations are performed in              */
  30 /* round to nearest mode of IEEE 754 standard.                             */
  31 /*                                                                         */
  32 /***************************************************************************/
  33 #include <math.h>
  34 #include "endian.h"
  35 #include "upow.h"
  36 #include <dla.h>
  37 #include "mydefs.h"
  38 #include "MathLib.h"
  39 #include "upow.tbl"
  40 #include <math_private.h>
  41 #include <fenv_private.h>
  42 #include <math-underflow.h>
  43 #include <fenv.h>
  44
  45 #ifndef SECTION
  46 # define SECTION
  47 #endif
  48
  49 static const double huge = 1.0e300, tiny = 1.0e-300;
  50
  51 double __exp1 (double x, double xx);
  52 static double log1 (double x, double *delta);
  53 static int checkint (double x);
  54
  55 /* An ultimate power routine. Given two IEEE double machine numbers y, x it
  56    computes the correctly rounded (to nearest) value of X^y.  */
  57 double
  58 SECTION
  59 __ieee754_pow (double x, double y)
  60 {
  61   double z, a, aa, t, a1, a2, y1, y2;
  62   mynumber u, v;
  63   int k;
  64   int4 qx, qy;
  65   v.x = y;
  66   u.x = x;
  67   if (v.i[LOW_HALF] == 0)
  68     {                           /* of y */
  69       qx = u.i[HIGH_HALF] & 0x7fffffff;
  70       /* Is x a NaN?  */
  71       if ((((qx == 0x7ff00000) && (u.i[LOW_HALF] != 0)) || (qx > 0x7ff00000))
  72           && (y != 0 || issignaling (x)))
  73         return x + x;
  74       if (y == 1.0)
  75         return x;
  76       if (y == 2.0)
  77         return x * x;
  78       if (y == -1.0)
  79         return 1.0 / x;
  80       if (y == 0)
  81         return 1.0;
  82     }
  83   /* else */
  84   if (((u.i[HIGH_HALF] > 0 && u.i[HIGH_HALF] < 0x7ff00000) ||   /* x>0 and not x->0 */
  85        (u.i[HIGH_HALF] == 0 && u.i[LOW_HALF] != 0)) &&
  86       /*   2^-1023< x<= 2^-1023 * 0x1.0000ffffffff */
  87       (v.i[HIGH_HALF] & 0x7fffffff) < 0x4ff00000)
  88     {                           /* if y<-1 or y>1   */
  89       double retval;
  90
  91       {
  92         SET_RESTORE_ROUND (FE_TONEAREST);
  93
  94         /* Avoid internal underflow for tiny y.  The exact value of y does
  95            not matter if |y| <= 2**-64.  */
  96         if (fabs (y) < 0x1p-64)
  97           y = y < 0 ? -0x1p-64 : 0x1p-64;
  98         z = log1 (x, &aa);      /* x^y  =e^(y log (X)) */
  99         t = y * CN;
 100         y1 = t - (t - y);
 101         y2 = y - y1;
 102         t = z * CN;
 103         a1 = t - (t - z);
 104         a2 = (z - a1) + aa;
 105         a = y1 * a1;
 106         aa = y2 * a1 + y * a2;
 107         a1 = a + aa;
 108         a2 = (a - a1) + aa;
 109
 110         /* Maximum relative error RElog of log1 is 1.0e-21 (69.7 bits).
 111            Maximum relative error REexp of __exp1 is 1.0e-18 (59.8 bits).
 112            We actually compute exp ((1 + RElog) * log (x) * y) * (1 + REexp).
 113            Since RElog/REexp are tiny and log (x) * y is at most log (DBL_MAX),
 114            this is equivalent to pow (x, y) * (1 + 710 * RElog + REexp).
 115            So the relative error is 710 * 1.0e-21 + 1.0e-18 = 1.7e-18
 116            (59 bits).  The worst-case ULP error is 0.515.  */
 117
 118         retval = __exp1 (a1, a2);
 119       }
 120
 121       if (isinf (retval))
 122         retval = huge * huge;
 123       else if (retval == 0)
 124         retval = tiny * tiny;
 125       else
 126         math_check_force_underflow_nonneg (retval);
 127       return retval;
 128     }
 129
 130   if (x == 0)
 131     {
 132       if (((v.i[HIGH_HALF] & 0x7fffffff) == 0x7ff00000 && v.i[LOW_HALF] != 0)
 133           || (v.i[HIGH_HALF] & 0x7fffffff) > 0x7ff00000)        /* NaN */
 134         return y + y;
 135       if (fabs (y) > 1.0e20)
 136         return (y > 0) ? 0 : 1.0 / 0.0;
 137       k = checkint (y);
 138       if (k == -1)
 139         return y < 0 ? 1.0 / x : x;
 140       else
 141         return y < 0 ? 1.0 / 0.0 : 0.0; /* return 0 */
 142     }
 143
 144   qx = u.i[HIGH_HALF] & 0x7fffffff;     /*   no sign   */
 145   qy = v.i[HIGH_HALF] & 0x7fffffff;     /*   no sign   */
 146
 147   if (qx >= 0x7ff00000 && (qx > 0x7ff00000 || u.i[LOW_HALF] != 0))      /* NaN */
 148     return x + y;
 149   if (qy >= 0x7ff00000 && (qy > 0x7ff00000 || v.i[LOW_HALF] != 0))      /* NaN */
 150     return x == 1.0 && !issignaling (y) ? 1.0 : y + y;
 151
 152   /* if x<0 */
 153   if (u.i[HIGH_HALF] < 0)
 154     {
 155       k = checkint (y);
 156       if (k == 0)
 157         {
 158           if (qy == 0x7ff00000)
 159             {
 160               if (x == -1.0)
 161                 return 1.0;
 162               else if (x > -1.0)
 163                 return v.i[HIGH_HALF] < 0 ? INF.x : 0.0;
 164               else
 165                 return v.i[HIGH_HALF] < 0 ? 0.0 : INF.x;
 166             }
 167           else if (qx == 0x7ff00000)
 168             return y < 0 ? 0.0 : INF.x;
 169           return (x - x) / (x - x);     /* y not integer and x<0 */
 170         }
 171       else if (qx == 0x7ff00000)
 172         {
 173           if (k < 0)
 174             return y < 0 ? nZERO.x : nINF.x;
 175           else
 176             return y < 0 ? 0.0 : INF.x;
 177         }
 178       /* if y even or odd */
 179       if (k == 1)
 180         return __ieee754_pow (-x, y);
 181       else
 182         {
 183           double retval;
 184           {
 185             SET_RESTORE_ROUND (FE_TONEAREST);
 186             retval = -__ieee754_pow (-x, y);
 187           }
 188           if (isinf (retval))
 189             retval = -huge * huge;
 190           else if (retval == 0)
 191             retval = -tiny * tiny;
 192           return retval;
 193         }
 194     }
 195   /* x>0 */
 196
 197   if (qx == 0x7ff00000)         /* x= 2^-0x3ff */
 198     return y > 0 ? x : 0;
 199
 200   if (qy > 0x45f00000 && qy < 0x7ff00000)
 201     {
 202       if (x == 1.0)
 203         return 1.0;
 204       if (y > 0)
 205         return (x > 1.0) ? huge * huge : tiny * tiny;
 206       if (y < 0)
 207         return (x < 1.0) ? huge * huge : tiny * tiny;
 208     }
 209
 210   if (x == 1.0)
 211     return 1.0;
 212   if (y > 0)
 213     return (x > 1.0) ? INF.x : 0;
 214   if (y < 0)
 215     return (x < 1.0) ? INF.x : 0;
 216   return 0;                     /* unreachable, to make the compiler happy */
 217 }
 218
 219 #ifndef __ieee754_pow
 220 strong_alias (__ieee754_pow, __pow_finite)
 221 #endif
 222
 223 /* Compute log(x) (x is left argument). The result is the returned double + the
 224    parameter DELTA.  */
 225 static double
 226 SECTION
 227 log1 (double x, double *delta)
 228 {
 229   unsigned int i, j;
 230   int m;
 231   double uu, vv, eps, nx, e, e1, e2, t, t1, t2, res, add = 0;
 232   mynumber u, v;
 233 #ifdef BIG_ENDI
 234   mynumber /**/ two52 = {{0x43300000, 0x00000000}};     /* 2**52  */
 235 #else
 236 # ifdef LITTLE_ENDI
 237   mynumber /**/ two52 = {{0x00000000, 0x43300000}};     /* 2**52  */
 238 # endif
 239 #endif
 240
 241   u.x = x;
 242   m = u.i[HIGH_HALF];
 243   if (m < 0x00100000)           /* Handle denormal x.  */
 244     {
 245       x = x * t52.x;
 246       add = -52.0;
 247       u.x = x;
 248       m = u.i[HIGH_HALF];
 249     }
 250
 251   if ((m & 0x000fffff) < 0x0006a09e)
 252     {
 253       u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3ff00000;
 254       two52.i[LOW_HALF] = (m >> 20);
 255     }
 256   else
 257     {
 258       u.i[HIGH_HALF] = (m & 0x000fffff) | 0x3fe00000;
 259       two52.i[LOW_HALF] = (m >> 20) + 1;
 260     }
 261
 262   v.x = u.x + bigu.x;
 263   uu = v.x - bigu.x;
 264   i = (v.i[LOW_HALF] & 0x000003ff) << 2;
 265   if (two52.i[LOW_HALF] == 1023)        /* Exponent of x is 0.  */
 266     {
 267       if (i > 1192 && i < 1208) /* |x-1| < 1.5*2**-10  */
 268         {
 269           t = x - 1.0;
 270           t1 = (t + 5.0e6) - 5.0e6;
 271           t2 = t - t1;
 272           e1 = t - 0.5 * t1 * t1;
 273           e2 = (t * t * t * (r3 + t * (r4 + t * (r5 + t * (r6 + t
 274                                                            * (r7 + t * r8)))))
 275                 - 0.5 * t2 * (t + t1));
 276           res = e1 + e2;
 277           *delta = (e1 - res) + e2;
 278           /* Max relative error is 1.464844e-24, so accurate to 79.1 bits.  */
 279           return res;
 280         }                       /* |x-1| < 1.5*2**-10  */
 281       else
 282         {
 283           v.x = u.x * (ui.x[i] + ui.x[i + 1]) + bigv.x;
 284           vv = v.x - bigv.x;
 285           j = v.i[LOW_HALF] & 0x0007ffff;
 286           j = j + j + j;
 287           eps = u.x - uu * vv;
 288           e1 = eps * ui.x[i];
 289           e2 = eps * (ui.x[i + 1] + vj.x[j] * (ui.x[i] + ui.x[i + 1]));
 290           e = e1 + e2;
 291           e2 = ((e1 - e) + e2);
 292           t = ui.x[i + 2] + vj.x[j + 1];
 293           t1 = t + e;
 294           t2 = ((((t - t1) + e) + (ui.x[i + 3] + vj.x[j + 2])) + e2 + e * e
 295                 * (p2 + e * (p3 + e * p4)));
 296           res = t1 + t2;
 297           *delta = (t1 - res) + t2;
 298           /* Max relative error is 1.0e-24, so accurate to 79.7 bits.  */
 299           return res;
 300         }
 301     }
 302   else                          /* Exponent of x != 0.  */
 303     {
 304       eps = u.x - uu;
 305       nx = (two52.x - two52e.x) + add;
 306       e1 = eps * ui.x[i];
 307       e2 = eps * ui.x[i + 1];
 308       e = e1 + e2;
 309       e2 = (e1 - e) + e2;
 310       t = nx * ln2a.x + ui.x[i + 2];
 311       t1 = t + e;
 312       t2 = ((((t - t1) + e) + nx * ln2b.x + ui.x[i + 3] + e2) + e * e
 313             * (q2 + e * (q3 + e * (q4 + e * (q5 + e * q6)))));
 314       res = t1 + t2;
 315       *delta = (t1 - res) + t2;
 316       /* Max relative error is 1.0e-21, so accurate to 69.7 bits.  */
 317       return res;
 318     }
 319 }
 320
 321
 322 /* This function receives a double x and checks if it is an integer.  If not,
 323    it returns 0, else it returns 1 if even or -1 if odd.  */
 324 static int
 325 SECTION
 326 checkint (double x)
 327 {
 328   union
 329   {
 330     int4 i[2];
 331     double x;
 332   } u;
 333   int k;
 334   unsigned int m, n;
 335   u.x = x;
 336   m = u.i[HIGH_HALF] & 0x7fffffff;      /* no sign */
 337   if (m >= 0x7ff00000)
 338     return 0;                   /*  x is +/-inf or NaN  */
 339   if (m >= 0x43400000)
 340     return 1;                   /*  |x| >= 2**53   */
 341   if (m < 0x40000000)
 342     return 0;                   /* |x| < 2,  can not be 0 or 1  */
 343   n = u.i[LOW_HALF];
 344   k = (m >> 20) - 1023;         /*  1 <= k <= 52   */
 345   if (k == 52)
 346     return (n & 1) ? -1 : 1;    /* odd or even */
 347   if (k > 20)
 348     {
 349       if (n << (k - 20) != 0)
 350         return 0;               /* if not integer */
 351       return (n << (k - 21) != 0) ? -1 : 1;
 352     }
 353   if (n)
 354     return 0;                   /*if  not integer */
 355   if (k == 20)
 356     return (m & 1) ? -1 : 1;
 357   if (m << (k + 12) != 0)
 358     return 0;
 359   return (m << (k + 11) != 0) ? -1 : 1;
 360 }