1 /* $OpenBSD: b_exp__D.c,v 1.5 2009/10/27 23:59:29 deraadt Exp $ */
2 /*
3 * Copyright (c) 1985, 1993
4 * The Regents of the University of California. All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 * notice, this list of conditions and the following disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 * notice, this list of conditions and the following disclaimer in the
13 * documentation and/or other materials provided with the distribution.
14 * 3. Neither the name of the University nor the names of its contributors
15 * may be used to endorse or promote products derived from this software
16 * without specific prior written permission.
17 *
18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
21 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
28 * SUCH DAMAGE.
29 */
30
31 /* EXP(X)
32 * RETURN THE EXPONENTIAL OF X
33 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
34 * CODED IN C BY K.C. NG, 1/19/85;
35 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
36 *
37 * Required system supported functions:
38 * scalb(x,n)
39 * copysign(x,y)
40 * finite(x)
41 *
42 * Method:
43 * 1. Argument Reduction: given the input x, find r and integer k such
44 * that
45 * x = k*ln2 + r, |r| <= 0.5*ln2 .
46 * r will be represented as r := z+c for better accuracy.
47 *
48 * 2. Compute exp(r) by
49 *
50 * exp(r) = 1 + r + r*R1/(2-R1),
51 * where
52 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
53 *
54 * 3. exp(x) = 2^k * exp(r) .
55 *
56 * Special cases:
57 * exp(INF) is INF, exp(NaN) is NaN;
58 * exp(-INF)= 0;
59 * for finite argument, only exp(0)=1 is exact.
60 *
61 * Accuracy:
62 * exp(x) returns the exponential of x nearly rounded. In a test run
63 * with 1,156,000 random arguments on a VAX, the maximum observed
64 * error was 0.869 ulps (units in the last place).
65 */
66
67 #include "math.h"
68 #include "math_private.h"
69
70 static const double p1 = 0x1.555555555553ep-3;
71 static const double p2 = -0x1.6c16c16bebd93p-9;
72 static const double p3 = 0x1.1566aaf25de2cp-14;
73 static const double p4 = -0x1.bbd41c5d26bf1p-20;
74 static const double p5 = 0x1.6376972bea4d0p-25;
75 static const double ln2hi = 0x1.62e42fee00000p-1;
76 static const double ln2lo = 0x1.a39ef35793c76p-33;
77 static const double lnhuge = 0x1.6602b15b7ecf2p9;
78 static const double lntiny = -0x1.77af8ebeae354p9;
79 static const double invln2 = 0x1.71547652b82fep0;
80
81 /* returns exp(r = x + c) for |c| < |x| with no overlap. */
82
83 double
__exp__D(double x,double c)84 __exp__D(double x, double c)
85 {
86 double z, hi, lo;
87 int k;
88
89 if (isnan(x)) /* x is NaN */
90 return(x);
91 if ( x <= lnhuge ) {
92 if ( x >= lntiny ) {
93
94 /* argument reduction : x --> x - k*ln2 */
95 z = invln2*x;
96 k = z + copysign(.5, x);
97
98 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
99
100 hi=(x-k*ln2hi); /* Exact. */
101 x= hi - (lo = k*ln2lo-c);
102 /* return 2^k*[1+x+x*c/(2+c)] */
103 z=x*x;
104 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
105 c = (x*c)/(2.0-c);
106
107 return scalb(1.+(hi-(lo - c)), k);
108 }
109 /* end of x > lntiny */
110
111 else
112 /* exp(-big#) underflows to zero */
113 if(finite(x)) return(scalb(1.0,-5000));
114
115 /* exp(-INF) is zero */
116 else return(0.0);
117 }
118 /* end of x < lnhuge */
119
120 else
121 /* exp(INF) is INF, exp(+big#) overflows to INF */
122 return( finite(x) ? scalb(1.0,5000) : x);
123 }
124