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.c 8.1 (Berkeley) 6/4/93 */
33 #include <sys/cdefs.h>
34 __FBSDID("$FreeBSD$");
35
36
37 /* EXP(X)
38 * RETURN THE EXPONENTIAL OF X
39 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
40 * CODED IN C BY K.C. NG, 1/19/85;
41 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
42 *
43 * Required system supported functions:
44 * scalb(x,n)
45 * copysign(x,y)
46 * finite(x)
47 *
48 * Method:
49 * 1. Argument Reduction: given the input x, find r and integer k such
50 * that
51 * x = k*ln2 + r, |r| <= 0.5*ln2 .
52 * r will be represented as r := z+c for better accuracy.
53 *
54 * 2. Compute exp(r) by
55 *
56 * exp(r) = 1 + r + r*R1/(2-R1),
57 * where
58 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
59 *
60 * 3. exp(x) = 2^k * exp(r) .
61 *
62 * Special cases:
63 * exp(INF) is INF, exp(NaN) is NaN;
64 * exp(-INF)= 0;
65 * for finite argument, only exp(0)=1 is exact.
66 *
67 * Accuracy:
68 * exp(x) returns the exponential of x nearly rounded. In a test run
69 * with 1,156,000 random arguments on a VAX, the maximum observed
70 * error was 0.869 ulps (units in the last place).
71 */
72
73 #include "mathimpl.h"
74
75 static const double p1 = 0x1.555555555553ep-3;
76 static const double p2 = -0x1.6c16c16bebd93p-9;
77 static const double p3 = 0x1.1566aaf25de2cp-14;
78 static const double p4 = -0x1.bbd41c5d26bf1p-20;
79 static const double p5 = 0x1.6376972bea4d0p-25;
80 static const double ln2hi = 0x1.62e42fee00000p-1;
81 static const double ln2lo = 0x1.a39ef35793c76p-33;
82 static const double lnhuge = 0x1.6602b15b7ecf2p9;
83 static const double lntiny = -0x1.77af8ebeae354p9;
84 static const double invln2 = 0x1.71547652b82fep0;
85
86 #if 0
87 double exp(x)
88 double x;
89 {
90 double z,hi,lo,c;
91 int k;
92
93 #if !defined(vax)&&!defined(tahoe)
94 if(x!=x) return(x); /* x is NaN */
95 #endif /* !defined(vax)&&!defined(tahoe) */
96 if( x <= lnhuge ) {
97 if( x >= lntiny ) {
98
99 /* argument reduction : x --> x - k*ln2 */
100
101 k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
102
103 /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
104
105 hi=x-k*ln2hi;
106 x=hi-(lo=k*ln2lo);
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+z*p5))));
111 return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
112
113 }
114 /* end of x > lntiny */
115
116 else
117 /* exp(-big#) underflows to zero */
118 if(finite(x)) return(scalb(1.0,-5000));
119
120 /* exp(-INF) is zero */
121 else return(0.0);
122 }
123 /* end of x < lnhuge */
124
125 else
126 /* exp(INF) is INF, exp(+big#) overflows to INF */
127 return( finite(x) ? scalb(1.0,5000) : x);
128 }
129 #endif
130
131 /* returns exp(r = x + c) for |c| < |x| with no overlap. */
132
__exp__D(x,c)133 double __exp__D(x, c)
134 double x, c;
135 {
136 double z,hi,lo;
137 int k;
138
139 if (x != x) /* x is NaN */
140 return(x);
141 if ( x <= lnhuge ) {
142 if ( x >= lntiny ) {
143
144 /* argument reduction : x --> x - k*ln2 */
145 z = invln2*x;
146 k = z + copysign(.5, x);
147
148 /* express (x+c)-k*ln2 as hi-lo and let x=hi-lo rounded */
149
150 hi=(x-k*ln2hi); /* Exact. */
151 x= hi - (lo = k*ln2lo-c);
152 /* return 2^k*[1+x+x*c/(2+c)] */
153 z=x*x;
154 c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
155 c = (x*c)/(2.0-c);
156
157 return scalb(1.+(hi-(lo - c)), k);
158 }
159 /* end of x > lntiny */
160
161 else
162 /* exp(-big#) underflows to zero */
163 if(finite(x)) return(scalb(1.0,-5000));
164
165 /* exp(-INF) is zero */
166 else return(0.0);
167 }
168 /* end of x < lnhuge */
169
170 else
171 /* exp(INF) is INF, exp(+big#) overflows to INF */
172 return( finite(x) ? scalb(1.0,5000) : x);
173 }
174