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) 08/21/85";
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
exp__E(x,c)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 }
121