1 /*
2 * Copyright (c) 1985, 1993
3 * The Regents of the University of California. All rights reserved.
4 *
5 * %sccs.include.redist.c%
6 */
7
8 #ifndef lint
9 static char sccsid[] = "@(#)exp.c 8.1 (Berkeley) 06/04/93";
10 #endif /* not lint */
11
12 /* EXP(X)
13 * RETURN THE EXPONENTIAL OF X
14 * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
15 * CODED IN C BY K.C. NG, 1/19/85;
16 * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
17 *
18 * Required system supported functions:
19 * scalb(x,n)
20 * copysign(x,y)
21 * finite(x)
22 *
23 * Method:
24 * 1. Argument Reduction: given the input x, find r and integer k such
25 * that
26 * x = k*ln2 + r, |r| <= 0.5*ln2 .
27 * r will be represented as r := z+c for better accuracy.
28 *
29 * 2. Compute exp(r) by
30 *
31 * exp(r) = 1 + r + r*R1/(2-R1),
32 * where
33 * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
34 *
35 * 3. exp(x) = 2^k * exp(r) .
36 *
37 * Special cases:
38 * exp(INF) is INF, exp(NaN) is NaN;
39 * exp(-INF)= 0;
40 * for finite argument, only exp(0)=1 is exact.
41 *
42 * Accuracy:
43 * exp(x) returns the exponential of x nearly rounded. In a test run
44 * with 1,156,000 random arguments on a VAX, the maximum observed
45 * error was 0.869 ulps (units in the last place).
46 *
47 * Constants:
48 * The hexadecimal values are the intended ones for the following constants.
49 * The decimal values may be used, provided that the compiler will convert
50 * from decimal to binary accurately enough to produce the hexadecimal values
51 * shown.
52 */
53
54 #include "mathimpl.h"
55
56 vc(ln2hi, 6.9314718055829871446E-1 ,7217,4031,0000,f7d0, 0, .B17217F7D00000)
57 vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)
58 vc(lnhuge, 9.4961163736712506989E1 ,ec1d,43bd,9010,a73e, 7, .BDEC1DA73E9010)
59 vc(lntiny,-9.5654310917272452386E1 ,4f01,c3bf,33af,d72e, 7,-.BF4F01D72E33AF)
60 vc(invln2, 1.4426950408889634148E0 ,aa3b,40b8,17f1,295c, 1, .B8AA3B295C17F1)
61 vc(p1, 1.6666666666666602251E-1 ,aaaa,3f2a,a9f1,aaaa, -2, .AAAAAAAAAAA9F1)
62 vc(p2, -2.7777777777015591216E-3 ,0b60,bc36,ec94,b5f5, -8,-.B60B60B5F5EC94)
63 vc(p3, 6.6137563214379341918E-5 ,b355,398a,f15f,792e, -13, .8AB355792EF15F)
64 vc(p4, -1.6533902205465250480E-6 ,ea0e,b6dd,5f84,2e93, -19,-.DDEA0E2E935F84)
65 vc(p5, 4.1381367970572387085E-8 ,bb4b,3431,2683,95f5, -24, .B1BB4B95F52683)
66
67 #ifdef vccast
68 #define ln2hi vccast(ln2hi)
69 #define ln2lo vccast(ln2lo)
70 #define lnhuge vccast(lnhuge)
71 #define lntiny vccast(lntiny)
72 #define invln2 vccast(invln2)
73 #define p1 vccast(p1)
74 #define p2 vccast(p2)
75 #define p3 vccast(p3)
76 #define p4 vccast(p4)
77 #define p5 vccast(p5)
78 #endif
79
80 ic(p1, 1.6666666666666601904E-1, -3, 1.555555555553E)
81 ic(p2, -2.7777777777015593384E-3, -9, -1.6C16C16BEBD93)
82 ic(p3, 6.6137563214379343612E-5, -14, 1.1566AAF25DE2C)
83 ic(p4, -1.6533902205465251539E-6, -20, -1.BBD41C5D26BF1)
84 ic(p5, 4.1381367970572384604E-8, -25, 1.6376972BEA4D0)
85 ic(ln2hi, 6.9314718036912381649E-1, -1, 1.62E42FEE00000)
86 ic(ln2lo, 1.9082149292705877000E-10,-33, 1.A39EF35793C76)
87 ic(lnhuge, 7.1602103751842355450E2, 9, 1.6602B15B7ECF2)
88 ic(lntiny,-7.5137154372698068983E2, 9, -1.77AF8EBEAE354)
89 ic(invln2, 1.4426950408889633870E0, 0, 1.71547652B82FE)
90
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
134 /* returns exp(r = x + c) for |c| < |x| with no overlap. */
135
__exp__D(x,c)136 double __exp__D(x, c)
137 double x, c;
138 {
139 double z,hi,lo, t;
140 int k;
141
142 #if !defined(vax)&&!defined(tahoe)
143 if (x!=x) return(x); /* x is NaN */
144 #endif /* !defined(vax)&&!defined(tahoe) */
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