1 /*-
2 * SPDX-License-Identifier: BSD-3-Clause
3 *
4 * Copyright (c) 1985, 1993
5 * The Regents of the University of California. All rights reserved.
6 *
7 * Redistribution and use in source and binary forms, with or without
8 * modification, are permitted provided that the following conditions
9 * are met:
10 * 1. Redistributions of source code must retain the above copyright
11 * notice, this list of conditions and the following disclaimer.
12 * 2. Redistributions in binary form must reproduce the above copyright
13 * notice, this list of conditions and the following disclaimer in the
14 * documentation and/or other materials provided with the distribution.
15 * 3. Neither the name of the University nor the names of its contributors
16 * may be used to endorse or promote products derived from this software
17 * without specific prior written permission.
18 *
19 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
20 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
21 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
22 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
23 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
24 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
25 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
26 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
27 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
28 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
29 * SUCH DAMAGE.
30 */
31
32 /* EXP(X)
33 * RETURN THE EXPONENTIAL OF X
34 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
35 * CODED IN C BY K.C. NG, 1/19/85;
36 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
37 *
38 * Required system supported functions:
39 * ldexp(x,n)
40 * copysign(x,y)
41 * isfinite(x)
42 *
43 * Method:
44 * 1. Argument Reduction: given the input x, find r and integer k such
45 * that
46 * x = k*ln2 + r, |r| <= 0.5*ln2.
47 * r will be represented as r := z+c for better accuracy.
48 *
49 * 2. Compute exp(r) by
50 *
51 * exp(r) = 1 + r + r*R1/(2-R1),
52 * where
53 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
54 *
55 * 3. exp(x) = 2^k * exp(r) .
56 *
57 * Special cases:
58 * exp(INF) is INF, exp(NaN) is NaN;
59 * exp(-INF)= 0;
60 * for finite argument, only exp(0)=1 is exact.
61 *
62 * Accuracy:
63 * exp(x) returns the exponential of x nearly rounded. In a test run
64 * with 1,156,000 random arguments on a VAX, the maximum observed
65 * error was 0.869 ulps (units in the last place).
66 */
67 static const double
68 p1 = 1.6666666666666660e-01, /* 0x3fc55555, 0x55555553 */
69 p2 = -2.7777777777564776e-03, /* 0xbf66c16c, 0x16c0ac3c */
70 p3 = 6.6137564717940088e-05, /* 0x3f11566a, 0xb5c2ba0d */
71 p4 = -1.6534060280704225e-06, /* 0xbebbbd53, 0x273e8fb7 */
72 p5 = 4.1437773411069054e-08; /* 0x3e663f2a, 0x09c94b6c */
73
74 static const double
75 ln2hi = 0x1.62e42fee00000p-1, /* High 32 bits round-down. */
76 ln2lo = 0x1.a39ef35793c76p-33; /* Next 53 bits round-to-nearst. */
77
78 static const double
79 lnhuge = 0x1.6602b15b7ecf2p9, /* (DBL_MAX_EXP + 9) * log(2.) */
80 lntiny = -0x1.77af8ebeae354p9, /* (DBL_MIN_EXP - 53 - 10) * log(2.) */
81 invln2 = 0x1.71547652b82fep0; /* 1 / log(2.) */
82
83 /* returns exp(r = x + c) for |c| < |x| with no overlap. */
84
85 static double
__exp__D(double x,double c)86 __exp__D(double x, double c)
87 {
88 double hi, lo, z;
89 int k;
90
91 if (x != x) /* x is NaN. */
92 return(x);
93
94 if (x <= lnhuge) {
95 if (x >= lntiny) {
96 /* argument reduction: x --> x - k*ln2 */
97 z = invln2 * x;
98 k = z + copysign(0.5, x);
99
100 /*
101 * Express (x + c) - k * ln2 as hi - lo.
102 * Let x = hi - lo rounded.
103 */
104 hi = x - k * ln2hi; /* Exact. */
105 lo = k * ln2lo - c;
106 x = hi - lo;
107
108 /* Return 2^k*[1+x+x*c/(2+c)] */
109 z = x * x;
110 c = x - z * (p1 + z * (p2 + z * (p3 + z * (p4 +
111 z * p5))));
112 c = (x * c) / (2 - c);
113
114 return (ldexp(1 + (hi - (lo - c)), k));
115 } else {
116 /* exp(-INF) is 0. exp(-big) underflows to 0. */
117 return (isfinite(x) ? ldexp(1., -5000) : 0);
118 }
119 } else
120 /* exp(INF) is INF, exp(+big#) overflows to INF */
121 return (isfinite(x) ? ldexp(1., 5000) : x);
122 }
123