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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: (1) source distributions retain this entire copyright
7: * notice and comment, and (2) distributions including binaries display
8: * the following acknowledgement: ``This product includes software
9: * developed by the University of California, Berkeley and its contributors''
10: * in the documentation or other materials provided with the distribution
11: * and in all advertising materials mentioning features or use of this
12: * software. Neither the name of the University nor the names of its
13: * contributors may be used to endorse or promote products derived
14: * from this software without specific prior written permission.
15: * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
16: * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
17: * WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
18: *
19: * All recipients should regard themselves as participants in an ongoing
20: * research project and hence should feel obligated to report their
21: * experiences (good or bad) with these elementary function codes, using
22: * the sendbug(8) program, to the authors.
23: */
24:
25: #ifndef lint
26: static char sccsid[] = "@(#)exp__E.c 5.5 (Berkeley) 6/1/90";
27: #endif /* not lint */
28:
29: /* exp__E(x,c)
30: * ASSUMPTION: c << x SO THAT fl(x+c)=x.
31: * (c is the correction term for x)
32: * exp__E RETURNS
33: *
34: * / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736
35: * exp__E(x,c) = |
36: * \ 0 , |x| < 1E-19.
37: *
38: * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
39: * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS
40: * CODED IN C BY K.C. NG, 1/31/85;
41: * REVISED BY K.C. NG on 3/16/85, 4/16/85.
42: *
43: * Required system supported function:
44: * copysign(x,y)
45: *
46: * Method:
47: * 1. Rational approximation. Let r=x+c.
48: * Based on
49: * 2 * sinh(r/2)
50: * exp(r) - 1 = ---------------------- ,
51: * cosh(r/2) - sinh(r/2)
52: * exp__E(r) is computed using
53: * x*x (x/2)*W - ( Q - ( 2*P + x*P ) )
54: * --- + (c + x*[---------------------------------- + c ])
55: * 2 1 - W
56: * where P := p1*x^2 + p2*x^4,
57: * Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6)
58: * W := x/2-(Q-x*P),
59: *
60: * (See the listing below for the values of p1,p2,q1,q2,q3. The poly-
61: * nomials P and Q may be regarded as the approximations to sinh
62: * and cosh :
63: * sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . )
64: *
65: * The coefficients were obtained by a special Remez algorithm.
66: *
67: * Approximation error:
68: *
69: * | exp(x) - 1 | 2**(-57), (IEEE double)
70: * | ------------ - (exp__E(x,0)+x)/x | <=
71: * | x | 2**(-69). (VAX D)
72: *
73: * Constants:
74: * The hexadecimal values are the intended ones for the following constants.
75: * The decimal values may be used, provided that the compiler will convert
76: * from decimal to binary accurately enough to produce the hexadecimal values
77: * shown.
78: */
79:
80: #include "mathimpl.h"
81:
82: vc(p1, 1.5150724356786683059E-2 ,3abe,3d78,066a,67e1, -6, .F83ABE67E1066A)
83: vc(p2, 6.3112487873718332688E-5 ,5b42,3984,0173,48cd, -13, .845B4248CD0173)
84: vc(q1, 1.1363478204690669916E-1 ,b95a,3ee8,ec45,44a2, -3, .E8B95A44A2EC45)
85: vc(q2, 1.2624568129896839182E-3 ,7905,3ba5,f5e7,72e4, -9, .A5790572E4F5E7)
86: vc(q3, 1.5021856115869022674E-6 ,9eb4,36c9,c395,604a, -19, .C99EB4604AC395)
87:
88: ic(p1, 1.3887401997267371720E-2, -7, 1.C70FF8B3CC2CF)
89: ic(p2, 3.3044019718331897649E-5, -15, 1.15317DF4526C4)
90: ic(q1, 1.1110813732786649355E-1, -4, 1.C719538248597)
91: ic(q2, 9.9176615021572857300E-4, -10, 1.03FC4CB8C98E8)
92:
93: #ifdef vccast
94: #define p1 vccast(p1)
95: #define p2 vccast(p2)
96: #define q1 vccast(q1)
97: #define q2 vccast(q2)
98: #define q3 vccast(q3)
99: #endif
100:
101: double exp__E(x,c)
102: double x,c;
103: {
104: const static double zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19;
105: double z,p,q,xp,xh,w;
106: if(copysign(x,one)>small) {
107: z = x*x ;
108: p = z*( p1 +z* p2 );
109: #if defined(vax)||defined(tahoe)
110: q = z*( q1 +z*( q2 +z* q3 ));
111: #else /* defined(vax)||defined(tahoe) */
112: q = z*( q1 +z* q2 );
113: #endif /* defined(vax)||defined(tahoe) */
114: xp= x*p ;
115: xh= x*half ;
116: w = xh-(q-xp) ;
117: p = p+p;
118: c += x*((xh*w-(q-(p+xp)))/(one-w)+c);
119: return(z*half+c);
120: }
121: /* end of |x| > small */
122:
123: else {
124: if(x!=zero) one+small; /* raise the inexact flag */
125: return(copysign(zero,x));
126: }
127: }
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