150476Speter/*-
22337Scsgr * SPDX-License-Identifier: BSD-3-Clause
32337Scsgr *
4241777Sed * Copyright (c) 1985, 1993
574814Sru *	The Regents of the University of California.  All rights reserved.
62337Scsgr *
7275077Sbapt * Redistribution and use in source and binary forms, with or without
82337Scsgr * modification, are permitted provided that the following conditions
919416Sjkh * are met:
1019404Storstenb * 1. Redistributions of source code must retain the above copyright
1119416Sjkh *    notice, this list of conditions and the following disclaimer.
1219404Storstenb * 2. Redistributions in binary form must reproduce the above copyright
132337Scsgr *    notice, this list of conditions and the following disclaimer in the
142337Scsgr *    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 */
67static 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
74static const double
75    ln2hi = 0x1.62e42fee00000p-1,   /* High 32 bits round-down. */
76    ln2lo = 0x1.a39ef35793c76p-33;  /* Next 53 bits round-to-nearst. */
77
78static 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
85static double
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