sysdeps/ieee754/dbl-64/e_exp2.c

   1 /* Double-precision floating point 2^x.
   2    Copyright (C) 1997-2014 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 (__builtin_expect (isless (x, himark), 1))
  47     {
  48       /* Exceptional cases:  */
  49       if (__builtin_expect (!isgreaterequal (x, lomark), 0))
  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       {
  65         SET_RESTORE_ROUND_NOEX (FE_TONEAREST);
  66
  67         /* 1. Argument reduction.
  68            Choose integers ex, -256 <= t < 256, and some real
  69            -1/1024 <= x1 <= 1024 so that
  70            x = ex + t/512 + x1.
  71
  72            First, calculate rx = ex + t/512.  */
  73         rx = x + THREEp42;
  74         rx -= THREEp42;
  75         x -= rx;  /* Compute x=x1. */
  76         /* Compute tval = (ex*512 + t)+256.
  77            Now, t = (tval mod 512)-256 and ex=tval/512  [that's mod, NOT %;
  78            and /-round-to-nearest not the usual c integer /].  */
  79         tval = (int) (rx * 512.0 + 256.0);
  80
  81         /* 2. Adjust for accurate table entry.
  82            Find e so that
  83            x = ex + t/512 + e + x2
  84            where -1e6 < e < 1e6, and
  85            (double)(2^(t/512+e))
  86            is accurate to one part in 2^-64.  */
  87
  88         /* 'tval & 511' is the same as 'tval%512' except that it's always
  89            positive.
  90            Compute x = x2.  */
  91         x -= exp2_deltatable[tval & 511];
  92
  93         /* 3. Compute ex2 = 2^(t/512+e+ex).  */
  94         ex2_u.d = exp2_accuratetable[tval & 511];
  95         tval >>= 9;
  96         unsafe = abs (tval) >= -DBL_MIN_EXP - 1;
  97         ex2_u.ieee.exponent += tval >> unsafe;
  98         scale_u.d = 1.0;
  99         scale_u.ieee.exponent += tval - (tval >> unsafe);
 100
 101         /* 4. Approximate 2^x2 - 1, using a fourth-degree polynomial,
 102            with maximum error in [-2^-10-2^-30,2^-10+2^-30]
 103            less than 10^-19.  */
 104
 105         x22 = (((.0096181293647031180
 106                  * x + .055504110254308625)
 107                 * x + .240226506959100583)
 108                * x + .69314718055994495) * ex2_u.d;
 109         math_opt_barrier (x22);
 110       }
 111
 112       /* 5. Return (2^x2-1) * 2^(t/512+e+ex) + 2^(t/512+e+ex).  */
 113       result = x22 * x + ex2_u.d;
 114
 115       if (!unsafe)
 116         return result;
 117       else
 118         return result * scale_u.d;
 119     }
 120   else
 121     /* Return x, if x is a NaN or Inf; or overflow, otherwise.  */
 122     return TWO1023 * x;
 123 }
 124 strong_alias (__ieee754_exp2, __exp2_finite)