1 /*
2 * Copyright (c) 1985, 1993
3 * The Regents of the University of California. All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 * notice, this list of conditions and the following disclaimer.
10 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
13 * 3. All advertising materials mentioning features or use of this software
14 * must display the following acknowledgement:
15 * This product includes software developed by the University of
16 * California, Berkeley and its contributors.
17 * 4. Neither the name of the University nor the names of its contributors
18 * may be used to endorse or promote products derived from this software
19 * without specific prior written permission.
20 *
21 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
22 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
23 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
24 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
25 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
26 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
27 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
28 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
29 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
30 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
31 * SUCH DAMAGE.
32 */
33
34 #ifndef lint
35 static char sccsid[] = "@(#)exp.c 8.1 (Berkeley) 6/4/93";
36 #endif /* not lint */
37 #include <sys/cdefs.h>
38 /* __FBSDID("$FreeBSD: src/lib/msun/bsdsrc/b_exp.c,v 1.7 2004/12/16 20:40:37 das Exp $"); */
39
40
41 /* EXP(X)
42 * RETURN THE EXPONENTIAL OF X
43 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
44 * CODED IN C BY K.C. NG, 1/19/85;
45 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
46 *
47 * Required system supported functions:
48 * scalb(x,n)
49 * copysign(x,y)
50 * finite(x)
51 *
52 * Method:
53 * 1. Argument Reduction: given the input x, find r and integer k such
54 * that
55 * x = k*ln2 + r, |r| <= 0.5*ln2 .
56 * r will be represented as r := z+c for better accuracy.
57 *
58 * 2. Compute exp(r) by
59 *
60 * exp(r) = 1 + r + r*R1/(2-R1),
61 * where
62 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
63 *
64 * 3. exp(x) = 2^k * exp(r) .
65 *
66 * Special cases:
67 * exp(INF) is INF, exp(NaN) is NaN;
68 * exp(-INF)= 0;
69 * for finite argument, only exp(0)=1 is exact.
70 *
71 * Accuracy:
72 * exp(x) returns the exponential of x nearly rounded. In a test run
73 * with 1,156,000 random arguments on a VAX, the maximum observed
74 * error was 0.869 ulps (units in the last place).
75 */
76
77 #include "mathimpl.h"
78
79 const static double p1 = 0x1.555555555553ep-3;
80 const static double p2 = -0x1.6c16c16bebd93p-9;
81 const static double p3 = 0x1.1566aaf25de2cp-14;
82 const static double p4 = -0x1.bbd41c5d26bf1p-20;
83 const static double p5 = 0x1.6376972bea4d0p-25;
84 const static double ln2hi = 0x1.62e42fee00000p-1;
85 const static double ln2lo = 0x1.a39ef35793c76p-33;
86 const static double lnhuge = 0x1.6602b15b7ecf2p9;
87 const static double lntiny = -0x1.77af8ebeae354p9;
88 const static double invln2 = 0x1.71547652b82fep0;
89
90 #if 0
91 double exp(x)
92 double x;
93 {
94 double z,hi,lo,c;
95 int k;
96
97 #if !defined(vax)&&!defined(tahoe)
98 if(x!=x) return(x); /* x is NaN */
99 #endif /* !defined(vax)&&!defined(tahoe) */
100 if( x <= lnhuge ) {
101 if( x >= lntiny ) {
102
103 /* argument reduction : x --> x - k*ln2 */
104
105 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
106
107 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
108
109 hi=x-k*ln2hi;
110 x=hi-(lo=k*ln2lo);
111
112 /* return 2^k*[1+x+x*c/(2+c)] */
113 z=x*x;
114 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
115 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
116
117 }
118 /* end of x > lntiny */
119
120 else
121 /* exp(-big#) underflows to zero */
122 if(finite(x)) return(scalb(1.0,-5000));
123
124 /* exp(-INF) is zero */
125 else return(0.0);
126 }
127 /* end of x < lnhuge */
128
129 else
130 /* exp(INF) is INF, exp(+big#) overflows to INF */
131 return( finite(x) ? scalb(1.0,5000) : x);
132 }
133 #endif
134
135 /* returns exp(r = x + c) for |c| < |x| with no overlap. */
136
__exp__D(x,c)137 double __exp__D(x, c)
138 double x, c;
139 {
140 double z,hi,lo;
141 int k;
142
143 if (x != x) /* x is NaN */
144 return(x);
145 if ( x <= lnhuge ) {
146 if ( x >= lntiny ) {
147
148 /* argument reduction : x --> x - k*ln2 */
149 z = invln2*x;
150 k = z + copysign(.5, x);
151
152 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
153
154 hi=(x-k*ln2hi); /* Exact. */
155 x= hi - (lo = k*ln2lo-c);
156 /* return 2^k*[1+x+x*c/(2+c)] */
157 z=x*x;
158 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
159 c = (x*c)/(2.0-c);
160
161 return scalb(1.+(hi-(lo - c)), k);
162 }
163 /* end of x > lntiny */
164
165 else
166 /* exp(-big#) underflows to zero */
167 if(finite(x)) return(scalb(1.0,-5000));
168
169 /* exp(-INF) is zero */
170 else return(0.0);
171 }
172 /* end of x < lnhuge */
173
174 else
175 /* exp(INF) is INF, exp(+big#) overflows to INF */
176 return( finite(x) ? scalb(1.0,5000) : x);
177 }
178