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1.1 root 1: /*
2: * Copyright (c) 1985 Regents of the University of California.
3: * All rights reserved.
4: *
5: * Redistribution and use in source and binary forms are permitted
6: * provided that the above copyright notice and this paragraph are
7: * duplicated in all such forms and that any documentation,
8: * advertising materials, and other materials related to such
9: * distribution and use acknowledge that the software was developed
10: * by the University of California, Berkeley. The name of the
11: * University may not be used to endorse or promote products derived
12: * from this software without specific prior written permission.
13: * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
14: * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
15: * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
16: *
17: * All recipients should regard themselves as participants in an ongoing
18: * research project and hence should feel obligated to report their
19: * experiences (good or bad) with these elementary function codes, using
20: * the sendbug(8) program, to the authors.
21: */
22:
23: #ifndef lint
24: static char sccsid[] = "@(#)exp.c 5.3 (Berkeley) 6/30/88";
25: #endif /* not lint */
26:
27: /* EXP(X)
28: * RETURN THE EXPONENTIAL OF X
29: * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
30: * CODED IN C BY K.C. NG, 1/19/85;
31: * REVISED BY K.C. NG on 2/6/85, 2/15/85, 3/7/85, 3/24/85, 4/16/85, 6/14/86.
32: *
33: * Required system supported functions:
34: * scalb(x,n)
35: * copysign(x,y)
36: * finite(x)
37: *
38: * Method:
39: * 1. Argument Reduction: given the input x, find r and integer k such
40: * that
41: * x = k*ln2 + r, |r| <= 0.5*ln2 .
42: * r will be represented as r := z+c for better accuracy.
43: *
44: * 2. Compute exp(r) by
45: *
46: * exp(r) = 1 + r + r*R1/(2-R1),
47: * where
48: * R1 = x - x^2*(p1+x^2*(p2+x^2*(p3+x^2*(p4+p5*x^2)))).
49: *
50: * 3. exp(x) = 2^k * exp(r) .
51: *
52: * Special cases:
53: * exp(INF) is INF, exp(NaN) is NaN;
54: * exp(-INF)= 0;
55: * for finite argument, only exp(0)=1 is exact.
56: *
57: * Accuracy:
58: * exp(x) returns the exponential of x nearly rounded. In a test run
59: * with 1,156,000 random arguments on a VAX, the maximum observed
60: * error was 0.869 ulps (units in the last place).
61: *
62: * Constants:
63: * The hexadecimal values are the intended ones for the following constants.
64: * The decimal values may be used, provided that the compiler will convert
65: * from decimal to binary accurately enough to produce the hexadecimal values
66: * shown.
67: */
68:
69: #if defined(vax)||defined(tahoe) /* VAX D format */
70: #ifdef vax
71: #define _0x(A,B) 0x/**/A/**/B
72: #else /* vax */
73: #define _0x(A,B) 0x/**/B/**/A
74: #endif /* vax */
75: /* static double */
76: /* ln2hi = 6.9314718055829871446E-1 , Hex 2^ 0 * .B17217F7D00000 */
77: /* ln2lo = 1.6465949582897081279E-12 , Hex 2^-39 * .E7BCD5E4F1D9CC */
78: /* lnhuge = 9.4961163736712506989E1 , Hex 2^ 7 * .BDEC1DA73E9010 */
79: /* lntiny = -9.5654310917272452386E1 , Hex 2^ 7 * -.BF4F01D72E33AF */
80: /* invln2 = 1.4426950408889634148E0 ; Hex 2^ 1 * .B8AA3B295C17F1 */
81: /* p1 = 1.6666666666666602251E-1 , Hex 2^-2 * .AAAAAAAAAAA9F1 */
82: /* p2 = -2.7777777777015591216E-3 , Hex 2^-8 * -.B60B60B5F5EC94 */
83: /* p3 = 6.6137563214379341918E-5 , Hex 2^-13 * .8AB355792EF15F */
84: /* p4 = -1.6533902205465250480E-6 , Hex 2^-19 * -.DDEA0E2E935F84 */
85: /* p5 = 4.1381367970572387085E-8 , Hex 2^-24 * .B1BB4B95F52683 */
86: static long ln2hix[] = { _0x(7217,4031), _0x(0000,f7d0)};
87: static long ln2lox[] = { _0x(bcd5,2ce7), _0x(d9cc,e4f1)};
88: static long lnhugex[] = { _0x(ec1d,43bd), _0x(9010,a73e)};
89: static long lntinyx[] = { _0x(4f01,c3bf), _0x(33af,d72e)};
90: static long invln2x[] = { _0x(aa3b,40b8), _0x(17f1,295c)};
91: static long p1x[] = { _0x(aaaa,3f2a), _0x(a9f1,aaaa)};
92: static long p2x[] = { _0x(0b60,bc36), _0x(ec94,b5f5)};
93: static long p3x[] = { _0x(b355,398a), _0x(f15f,792e)};
94: static long p4x[] = { _0x(ea0e,b6dd), _0x(5f84,2e93)};
95: static long p5x[] = { _0x(bb4b,3431), _0x(2683,95f5)};
96: #define ln2hi (*(double*)ln2hix)
97: #define ln2lo (*(double*)ln2lox)
98: #define lnhuge (*(double*)lnhugex)
99: #define lntiny (*(double*)lntinyx)
100: #define invln2 (*(double*)invln2x)
101: #define p1 (*(double*)p1x)
102: #define p2 (*(double*)p2x)
103: #define p3 (*(double*)p3x)
104: #define p4 (*(double*)p4x)
105: #define p5 (*(double*)p5x)
106:
107: #else /* defined(vax)||defined(tahoe) */
108: static double
109: p1 = 1.6666666666666601904E-1 , /*Hex 2^-3 * 1.555555555553E */
110: p2 = -2.7777777777015593384E-3 , /*Hex 2^-9 * -1.6C16C16BEBD93 */
111: p3 = 6.6137563214379343612E-5 , /*Hex 2^-14 * 1.1566AAF25DE2C */
112: p4 = -1.6533902205465251539E-6 , /*Hex 2^-20 * -1.BBD41C5D26BF1 */
113: p5 = 4.1381367970572384604E-8 , /*Hex 2^-25 * 1.6376972BEA4D0 */
114: ln2hi = 6.9314718036912381649E-1 , /*Hex 2^ -1 * 1.62E42FEE00000 */
115: ln2lo = 1.9082149292705877000E-10 , /*Hex 2^-33 * 1.A39EF35793C76 */
116: lnhuge = 7.1602103751842355450E2 , /*Hex 2^ 9 * 1.6602B15B7ECF2 */
117: lntiny = -7.5137154372698068983E2 , /*Hex 2^ 9 * -1.77AF8EBEAE354 */
118: invln2 = 1.4426950408889633870E0 ; /*Hex 2^ 0 * 1.71547652B82FE */
119: #endif /* defined(vax)||defined(tahoe) */
120:
121: double exp(x)
122: double x;
123: {
124: double scalb(), copysign(), z,hi,lo,c;
125: int k,finite();
126:
127: #if !defined(vax)&&!defined(tahoe)
128: if(x!=x) return(x); /* x is NaN */
129: #endif /* !defined(vax)&&!defined(tahoe) */
130: if( x <= lnhuge ) {
131: if( x >= lntiny ) {
132:
133: /* argument reduction : x --> x - k*ln2 */
134:
135: k=invln2*x+copysign(0.5,x); /* k=NINT(x/ln2) */
136:
137: /* express x-k*ln2 as hi-lo and let x=hi-lo rounded */
138:
139: hi=x-k*ln2hi;
140: x=hi-(lo=k*ln2lo);
141:
142: /* return 2^k*[1+x+x*c/(2+c)] */
143: z=x*x;
144: c= x - z*(p1+z*(p2+z*(p3+z*(p4+z*p5))));
145: return scalb(1.0+(hi-(lo-(x*c)/(2.0-c))),k);
146:
147: }
148: /* end of x > lntiny */
149:
150: else
151: /* exp(-big#) underflows to zero */
152: if(finite(x)) return(scalb(1.0,-5000));
153:
154: /* exp(-INF) is zero */
155: else return(0.0);
156: }
157: /* end of x < lnhuge */
158:
159: else
160: /* exp(INF) is INF, exp(+big#) overflows to INF */
161: return( finite(x) ? scalb(1.0,5000) : x);
162: }
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