sysdeps/ieee754/dbl-64/e_exp2.c

   1 /* Double-precision floating point 2^x.
   2    Copyright (C) 1997-2015 Free Software Foundation, Inc.
   3    This file is part of the GNU C Library.
   4    Contributed by Geoffrey Keating <geoffk@ozemail.com.au>
   5
   6    The GNU C Library is free software; you can redistribute it and/or
   7    modify it under the terms of the GNU Lesser General Public
   8    License as published by the Free Software Foundation; either
   9    version 2.1 of the License, or (at your option) any later version.
  10
  11    The GNU C Library 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 GNU
  14    Lesser General Public License for more details.
  15
  16    You should have received a copy of the GNU Lesser General Public
  17    License along with the GNU C Library; if not, see
  18    <http://www.gnu.org/licenses/>.  */
  19
  20 /* The basic design here is from
  21    Shmuel Gal and Boris Bachelis, "An Accurate Elementary Mathematical
  22    Library for the IEEE Floating Point Standard", ACM Trans. Math. Soft.,
  23    17 (1), March 1991, pp. 26-45.
  24    It has been slightly modified to compute 2^x instead of e^x.
  25    */
  26 #include <stdlib.h>
  27 #include <float.h>
  28 #include <ieee754.h>
  29 #include <math.h>
  30 #include <fenv.h>
  31 #include <inttypes.h>
  32 #include <math_private.h>
  33
  34 #include "t_exp2.h"
  35
  36 static const double TWO1023 = 8.988465674311579539e+307;
  37 static const double TWOM1000 = 9.3326361850321887899e-302;
  38
  39 double
  40 __ieee754_exp2 (double x)
  41 {
  42   static const double himark = (double) DBL_MAX_EXP;
  43   static const double lomark = (double) (DBL_MIN_EXP - DBL_MANT_DIG - 1);
  44
  45   /* Check for usual case.  */
  46   if (__glibc_likely (isless (x, himark)))
  47     {
  48       /* Exceptional cases:  */
  49       if (__glibc_unlikely (!isgreaterequal (x, lomark)))
  50         {
  51           if (isinf (x))
  52             /* e^-inf == 0, with no error.  */
  53             return 0;
  54           else
  55             /* Underflow */
  56             return TWOM1000 * TWOM1000;
  57         }
  58
  59       static const double THREEp42 = 13194139533312.0;
  60       int tval, unsafe;
  61       double rx, x22, result;
  62       union ieee754_double ex2_u, scale_u;
  63
  64       if (fabs (x) < DBL_EPSILON / 4.0)
  65         return 1.0 + x;
  66
  67       {
  68         SET_RESTORE_ROUND_NOEX (FE_TONEAREST);
  69
  70         /* 1. Argument reduction.
  71            Choose integers ex, -256 <= t < 256, and some real
  72            -1/1024 <= x1 <= 1024 so that
  73            x = ex + t/512 + x1.
  74
  75            First, calculate rx = ex + t/512.  */
  76         rx = x + THREEp42;
  77         rx -= THREEp42;
  78         x -= rx;  /* Compute x=x1. */
  79         /* Compute tval = (ex*512 + t)+256.
  80            Now, t = (tval mod 512)-256 and ex=tval/512  [that's mod, NOT %;
  81            and /-round-to-nearest not the usual c integer /].  */
  82         tval = (int) (rx * 512.0 + 256.0);
  83
  84         /* 2. Adjust for accurate table entry.
  85            Find e so that
  86            x = ex + t/512 + e + x2
  87            where -1e6 < e < 1e6, and
  88            (double)(2^(t/512+e))
  89            is accurate to one part in 2^-64.  */
  90
  91         /* 'tval & 511' is the same as 'tval%512' except that it's always
  92            positive.
  93            Compute x = x2.  */
  94         x -= exp2_deltatable[tval & 511];
  95
  96         /* 3. Compute ex2 = 2^(t/512+e+ex).  */
  97         ex2_u.d = exp2_accuratetable[tval & 511];
  98         tval >>= 9;
  99         unsafe = abs (tval) >= -DBL_MIN_EXP - 1;
 100         ex2_u.ieee.exponent += tval >> unsafe;
 101         scale_u.d = 1.0;
 102         scale_u.ieee.exponent += tval - (tval >> unsafe);
 103
 104         /* 4. Approximate 2^x2 - 1, using a fourth-degree polynomial,
 105            with maximum error in [-2^-10-2^-30,2^-10+2^-30]
 106            less than 10^-19.  */
 107
 108         x22 = (((.0096181293647031180
 109                  * x + .055504110254308625)
 110                 * x + .240226506959100583)
 111                * x + .69314718055994495) * ex2_u.d;
 112         math_opt_barrier (x22);
 113       }
 114
 115       /* 5. Return (2^x2-1) * 2^(t/512+e+ex) + 2^(t/512+e+ex).  */
 116       result = x22 * x + ex2_u.d;
 117
 118       if (!unsafe)
 119         return result;
 120       else
 121         return result * scale_u.d;
 122     }
 123   else
 124     /* Return x, if x is a NaN or Inf; or overflow, otherwise.  */
 125     return TWO1023 * x;
 126 }
 127 strong_alias (__ieee754_exp2, __exp2_finite)