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
76
77 static long double P[3] = {
78 1.2617719307481059087798E-4L,
79 3.0299440770744196129956E-2L,
80 9.9999999999999999991025E-1L,
81 };
82 static long double Q[4] = {
83 3.0019850513866445504159E-6L,
84 2.5244834034968410419224E-3L,
85 2.2726554820815502876593E-1L,
86 2.0000000000000000000897E0L,
87 };
88 static const long double C1 = 6.9314575195312500000000E-1L;
89 static const long double C2 = 1.4286068203094172321215E-6L;
90 static const long double MAXLOGL = 1.1356523406294143949492E4L;
91 static const long double MINLOGL = -1.13994985314888605586758E4L;
92 static const long double LOG2EL = 1.4426950408889634073599E0L;
93
94 long double
expl(long double x)95 expl(long double x)
96 {
97 long double px, xx;
98 int n;
99
100 if( isnan(x) )
101 return(x + x);
102 if( x > MAXLOGL) {
103 if (isinf(x))
104 return x;
105 return __math_oflowl(0);
106 }
107
108 if( x < MINLOGL ) {
109 if (isinf(x))
110 return 0.0L;
111 return __math_uflowl(0);
112 }
113
114 /* Express e**x = e**g 2**n
115 * = e**g e**( n loge(2) )
116 * = e**( g + n loge(2) )
117 */
118 px = floorl( LOG2EL * x + 0.5L ); /* floor() truncates toward -infinity. */
119 n = px;
120 x -= px * C1;
121 x -= px * C2;
122
123
124 /* rational approximation for exponential
125 * of the fractional part:
126 * e**x = 1 + 2x P(x**2)/( Q(x**2) - P(x**2) )
127 */
128 xx = x * x;
129 px = x * __polevll( xx, P, 2 );
130 x = px/( __polevll( xx, Q, 3 ) - px );
131 x = 1.0L + ldexpl( x, 1 );
132
133 x = ldexpl( x, n );
134 return(x);
135 }
136
