1/*	$OpenBSD: e_expl.c,v 1.4 2016/09/12 19:47:02 guenther 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
79static long double P[3] = {
80 1.2617719307481059087798E-4L,
81 3.0299440770744196129956E-2L,
82 9.9999999999999999991025E-1L,
83};
84static long double Q[4] = {
85 3.0019850513866445504159E-6L,
86 2.5244834034968410419224E-3L,
87 2.2726554820815502876593E-1L,
88 2.0000000000000000000897E0L,
89};
90static const long double C1 = 6.9314575195312500000000E-1L;
91static const long double C2 = 1.4286068203094172321215E-6L;
92static const long double MAXLOGL = 1.1356523406294143949492E4L;
93static const long double MINLOGL = -1.13994985314888605586758E4L;
94static const long double LOG2EL = 1.4426950408889634073599E0L;
95
96long double
97expl(long double x)
98{
99long double px, xx;
100int n;
101
102if( isnan(x) )
103	return(x);
104if( x > MAXLOGL)
105	return( INFINITY );
106
107if( 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 */
114px = floorl( LOG2EL * x + 0.5L ); /* floor() truncates toward -infinity. */
115n = px;
116x -= px * C1;
117x -= 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 */
124xx = x * x;
125px = x * __polevll( xx, P, 2 );
126x =  px/( __polevll( xx, Q, 3 ) - px );
127x = 1.0L + ldexpl( x, 1 );
128
129x = ldexpl( x, n );
130return(x);
131}
132DEF_STD(expl);
133