compiler/stdc/math/ld80/e_expl.c

   1 /*      $OpenBSD: e_expl.c,v 1.3 2013/11/12 20:35:19 martynas Exp $     */
   2
   3 /*
   4  * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
   5  *
   6  * Permission to use, copy, modify, and distribute this software for any
   7  * purpose with or without fee is hereby granted, provided that the above
   8  * copyright notice and this permission notice appear in all copies.
   9  *
  10  * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
  11  * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
  12  * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
  13  * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
  14  * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
  15  * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
  16  * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
  17  */
  18
  19 /*                                                      expl.c
  20  *
  21  *      Exponential function, long double precision
  22  *
  23  *
  24  *
  25  * SYNOPSIS:
  26  *
  27  * long double x, y, expl();
  28  *
  29  * y = expl( x );
  30  *
  31  *
  32  *
  33  * DESCRIPTION:
  34  *
  35  * Returns e (2.71828...) raised to the x power.
  36  *
  37  * Range reduction is accomplished by separating the argument
  38  * into an integer k and fraction f such that
  39  *
  40  *     x    k  f
  41  *    e  = 2  e.
  42  *
  43  * A Pade' form of degree 2/3 is used to approximate exp(f) - 1
  44  * in the basic range [-0.5 ln 2, 0.5 ln 2].
  45  *
  46  *
  47  * ACCURACY:
  48  *
  49  *                      Relative error:
  50  * arithmetic   domain     # trials      peak         rms
  51  *    IEEE      +-10000     50000       1.12e-19    2.81e-20
  52  *
  53  *
  54  * Error amplification in the exponential function can be
  55  * a serious matter.  The error propagation involves
  56  * exp( X(1+delta) ) = exp(X) ( 1 + X*delta + ... ),
  57  * which shows that a 1 lsb error in representing X produces
  58  * a relative error of X times 1 lsb in the function.
  59  * While the routine gives an accurate result for arguments
  60  * that are exactly represented by a long double precision
  61  * computer number, the result contains amplified roundoff
  62  * error for large arguments not exactly represented.
  63  *
  64  *
  65  * ERROR MESSAGES:
  66  *
  67  *   message         condition      value returned
  68  * exp underflow    x < MINLOG         0.0
  69  * exp overflow     x > MAXLOG         MAXNUM
  70  *
  71  */
  72
  73 /*      Exponential function    */
  74
  75 #include "math.h"
  76
  77 #include "math_private.h"
  78
  79 static const long double P[3] = {
  80  1.2617719307481059087798E-4L,
  81  3.0299440770744196129956E-2L,
  82  9.9999999999999999991025E-1L,
  83 };
  84 static const long double Q[4] = {
  85  3.0019850513866445504159E-6L,
  86  2.5244834034968410419224E-3L,
  87  2.2726554820815502876593E-1L,
  88  2.0000000000000000000897E0L,
  89 };
  90 static const long double C1 = 6.9314575195312500000000E-1L;
  91 static const long double C2 = 1.4286068203094172321215E-6L;
  92 static const long double MAXLOGL = 1.1356523406294143949492E4L;
  93 static const long double MINLOGL = -1.13994985314888605586758E4L;
  94 static const long double LOG2EL = 1.4426950408889634073599E0L;
  95
  96 long double
  97 expl(long double x)
  98 {
  99 long double px, xx;
 100 int n;
 101
 102 if( isnan(x) )
 103         return(x);
 104 if( x > MAXLOGL)
 105         return( INFINITY );
 106
 107 if( x < MINLOGL )
 108         return(0.0L);
 109
 110 /* Express e**x = e**g 2**n
 111  *   = e**g e**( n loge(2) )
 112  *   = e**( g + n loge(2) )
 113  */
 114 px = floorl( LOG2EL * x + 0.5L ); /* floor() truncates toward -infinity. */
 115 n = px;
 116 x -= px * C1;
 117 x -= px * C2;
 118
 119
 120 /* rational approximation for exponential
 121  * of the fractional part:
 122  * e**x =  1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
 123  */
 124 xx = x * x;
 125 px = x * __polevll( xx, (void *)P, 2 );
 126 x =  px/( __polevll( xx, (void *)Q, 3 ) - px );
 127 x = 1.0L + ldexpl( x, 1 );
 128
 129 x = ldexpl( x, n );
 130 return(x);
 131 }