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1.1 root 1: /*
2: * Copyright (c) 1985 Regents of the University of California.
3: *
4: * Use and reproduction of this software are granted in accordance with
5: * the terms and conditions specified in the Berkeley Software License
6: * Agreement (in particular, this entails acknowledgement of the programs'
7: * source, and inclusion of this notice) with the additional understanding
8: * that all recipients should regard themselves as participants in an
9: * ongoing research project and hence should feel obligated to report
10: * their experiences (good or bad) with these elementary function codes,
11: * using "sendbug 4bsd-bugs@BERKELEY", to the authors.
12: */
13:
14: #ifndef lint
15: static char sccsid[] = "@(#)exp__E.c 1.2 (Berkeley) 8/21/85";
16: #endif not lint
17:
18: /* exp__E(x,c)
19: * ASSUMPTION: c << x SO THAT fl(x+c)=x.
20: * (c is the correction term for x)
21: * exp__E RETURNS
22: *
23: * / exp(x+c) - 1 - x , 1E-19 < |x| < .3465736
24: * exp__E(x,c) = |
25: * \ 0 , |x| < 1E-19.
26: *
27: * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
28: * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS
29: * CODED IN C BY K.C. NG, 1/31/85;
30: * REVISED BY K.C. NG on 3/16/85, 4/16/85.
31: *
32: * Required system supported function:
33: * copysign(x,y)
34: *
35: * Method:
36: * 1. Rational approximation. Let r=x+c.
37: * Based on
38: * 2 * sinh(r/2)
39: * exp(r) - 1 = ---------------------- ,
40: * cosh(r/2) - sinh(r/2)
41: * exp__E(r) is computed using
42: * x*x (x/2)*W - ( Q - ( 2*P + x*P ) )
43: * --- + (c + x*[---------------------------------- + c ])
44: * 2 1 - W
45: * where P := p1*x^2 + p2*x^4,
46: * Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6)
47: * W := x/2-(Q-x*P),
48: *
49: * (See the listing below for the values of p1,p2,q1,q2,q3. The poly-
50: * nomials P and Q may be regarded as the approximations to sinh
51: * and cosh :
52: * sinh(r/2) = r/2 + r * P , cosh(r/2) = 1 + Q . )
53: *
54: * The coefficients were obtained by a special Remez algorithm.
55: *
56: * Approximation error:
57: *
58: * | exp(x) - 1 | 2**(-57), (IEEE double)
59: * | ------------ - (exp__E(x,0)+x)/x | <=
60: * | x | 2**(-69). (VAX D)
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: #ifdef VAX /* VAX D format */
70: /* static double */
71: /* p1 = 1.5150724356786683059E-2 , Hex 2^ -6 * .F83ABE67E1066A */
72: /* p2 = 6.3112487873718332688E-5 , Hex 2^-13 * .845B4248CD0173 */
73: /* q1 = 1.1363478204690669916E-1 , Hex 2^ -3 * .E8B95A44A2EC45 */
74: /* q2 = 1.2624568129896839182E-3 , Hex 2^ -9 * .A5790572E4F5E7 */
75: /* q3 = 1.5021856115869022674E-6 ; Hex 2^-19 * .C99EB4604AC395 */
76: static long p1x[] = { 0x3abe3d78, 0x066a67e1};
77: static long p2x[] = { 0x5b423984, 0x017348cd};
78: static long q1x[] = { 0xb95a3ee8, 0xec4544a2};
79: static long q2x[] = { 0x79053ba5, 0xf5e772e4};
80: static long q3x[] = { 0x9eb436c9, 0xc395604a};
81: #define p1 (*(double*)p1x)
82: #define p2 (*(double*)p2x)
83: #define q1 (*(double*)q1x)
84: #define q2 (*(double*)q2x)
85: #define q3 (*(double*)q3x)
86: #else /* IEEE double */
87: static double
88: p1 = 1.3887401997267371720E-2 , /*Hex 2^ -7 * 1.C70FF8B3CC2CF */
89: p2 = 3.3044019718331897649E-5 , /*Hex 2^-15 * 1.15317DF4526C4 */
90: q1 = 1.1110813732786649355E-1 , /*Hex 2^ -4 * 1.C719538248597 */
91: q2 = 9.9176615021572857300E-4 ; /*Hex 2^-10 * 1.03FC4CB8C98E8 */
92: #endif
93:
94: double exp__E(x,c)
95: double x,c;
96: {
97: double static zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19;
98: double copysign(),z,p,q,xp,xh,w;
99: if(copysign(x,one)>small) {
100: z = x*x ;
101: p = z*( p1 +z* p2 );
102: #ifdef VAX
103: q = z*( q1 +z*( q2 +z* q3 ));
104: #else /* IEEE double */
105: q = z*( q1 +z* q2 );
106: #endif
107: xp= x*p ;
108: xh= x*half ;
109: w = xh-(q-xp) ;
110: p = p+p;
111: c += x*((xh*w-(q-(p+xp)))/(one-w)+c);
112: return(z*half+c);
113: }
114: /* end of |x| > small */
115:
116: else {
117: if(x!=zero) one+small; /* raise the inexact flag */
118: return(copysign(zero,x));
119: }
120: }
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