sysdeps/ieee754/dbl-64/e_exp2.c

   1 /* Double-precision floating point 2^x.
   2    Copyright (C) 1997-2018 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-barriers.h>
  33 #include <math_private.h>
  34 #include <fenv_private.h>
  35 #include <math-underflow.h>
  36
  37 #include "t_exp2.h"
  38
  39 static const double TWO1023 = 8.988465674311579539e+307;
  40 static const double TWOM1000 = 9.3326361850321887899e-302;
  41
  42 double
  43 __ieee754_exp2 (double x)
  44 {
  45   static const double himark = (double) DBL_MAX_EXP;
  46   static const double lomark = (double) (DBL_MIN_EXP - DBL_MANT_DIG - 1);
  47
  48   /* Check for usual case.  */
  49   if (__glibc_likely (isless (x, himark)))
  50     {
  51       /* Exceptional cases:  */
  52       if (__glibc_unlikely (!isgreaterequal (x, lomark)))
  53         {
  54           if (isinf (x))
  55             /* e^-inf == 0, with no error.  */
  56             return 0;
  57           else
  58             /* Underflow */
  59             return TWOM1000 * TWOM1000;
  60         }
  61
  62       static const double THREEp42 = 13194139533312.0;
  63       int tval, unsafe;
  64       double rx, x22, result;
  65       union ieee754_double ex2_u, scale_u;
  66
  67       if (fabs (x) < DBL_EPSILON / 4.0)
  68         return 1.0 + x;
  69
  70       {
  71         SET_RESTORE_ROUND_NOEX (FE_TONEAREST);
  72
  73         /* 1. Argument reduction.
  74            Choose integers ex, -256 <= t < 256, and some real
  75            -1/1024 <= x1 <= 1024 so that
  76            x = ex + t/512 + x1.
  77
  78            First, calculate rx = ex + t/512.  */
  79         rx = x + THREEp42;
  80         rx -= THREEp42;
  81         x -= rx;  /* Compute x=x1. */
  82         /* Compute tval = (ex*512 + t)+256.
  83            Now, t = (tval mod 512)-256 and ex=tval/512  [that's mod, NOT %;
  84            and /-round-to-nearest not the usual c integer /].  */
  85         tval = (int) (rx * 512.0 + 256.0);
  86
  87         /* 2. Adjust for accurate table entry.
  88            Find e so that
  89            x = ex + t/512 + e + x2
  90            where -1e6 < e < 1e6, and
  91            (double)(2^(t/512+e))
  92            is accurate to one part in 2^-64.  */
  93
  94         /* 'tval & 511' is the same as 'tval%512' except that it's always
  95            positive.
  96            Compute x = x2.  */
  97         x -= exp2_deltatable[tval & 511];
  98
  99         /* 3. Compute ex2 = 2^(t/512+e+ex).  */
 100         ex2_u.d = exp2_accuratetable[tval & 511];
 101         tval >>= 9;
 102         /* x2 is an integer multiple of 2^-54; avoid intermediate
 103            underflow from the calculation of x22 * x.  */
 104         unsafe = abs (tval) >= -DBL_MIN_EXP - 56;
 105         ex2_u.ieee.exponent += tval >> unsafe;
 106         scale_u.d = 1.0;
 107         scale_u.ieee.exponent += tval - (tval >> unsafe);
 108
 109         /* 4. Approximate 2^x2 - 1, using a fourth-degree polynomial,
 110            with maximum error in [-2^-10-2^-30,2^-10+2^-30]
 111            less than 10^-19.  */
 112
 113         x22 = (((.0096181293647031180
 114                  * x + .055504110254308625)
 115                 * x + .240226506959100583)
 116                * x + .69314718055994495) * ex2_u.d;
 117         math_opt_barrier (x22);
 118       }
 119
 120       /* 5. Return (2^x2-1) * 2^(t/512+e+ex) + 2^(t/512+e+ex).  */
 121       result = x22 * x + ex2_u.d;
 122
 123       if (!unsafe)
 124         return result;
 125       else
 126         {
 127           result *= scale_u.d;
 128           math_check_force_underflow_nonneg (result);
 129           return result;
 130         }
 131     }
 132   else
 133     /* Return x, if x is a NaN or Inf; or overflow, otherwise.  */
 134     return TWO1023 * x;
 135 }
 136 strong_alias (__ieee754_exp2, __exp2_finite)