1 /*
2  * Copyright (c) 1985 Regents of the University of California.
3  * All rights reserved.
4  *
5  * %sccs.include.redist.c%
6  */
7 
8 #ifndef lint
9 static char sccsid[] = "@(#)exp__E.c	5.6 (Berkeley) 10/09/90";
10 #endif /* not lint */
11 
12 /* exp__E(x,c)
13  * ASSUMPTION: c << x  SO THAT  fl(x+c)=x.
14  * (c is the correction term for x)
15  * exp__E RETURNS
16  *
17  *			 /  exp(x+c) - 1 - x ,  1E-19 < |x| < .3465736
18  *       exp__E(x,c) = 	|
19  *			 \  0 ,  |x| < 1E-19.
20  *
21  * DOUBLE PRECISION (IEEE 53 bits, VAX D FORMAT 56 BITS)
22  * KERNEL FUNCTION OF EXP, EXPM1, POW FUNCTIONS
23  * CODED IN C BY K.C. NG, 1/31/85;
24  * REVISED BY K.C. NG on 3/16/85, 4/16/85.
25  *
26  * Required system supported function:
27  *	copysign(x,y)
28  *
29  * Method:
30  *	1. Rational approximation. Let r=x+c.
31  *	   Based on
32  *                                   2 * sinh(r/2)
33  *                exp(r) - 1 =   ----------------------   ,
34  *                               cosh(r/2) - sinh(r/2)
35  *	   exp__E(r) is computed using
36  *                   x*x            (x/2)*W - ( Q - ( 2*P  + x*P ) )
37  *                   --- + (c + x*[---------------------------------- + c ])
38  *                    2                          1 - W
39  * 	   where  P := p1*x^2 + p2*x^4,
40  *	          Q := q1*x^2 + q2*x^4 (for 56 bits precision, add q3*x^6)
41  *	          W := x/2-(Q-x*P),
42  *
43  *	   (See the listing below for the values of p1,p2,q1,q2,q3. The poly-
44  *	    nomials P and Q may be regarded as the approximations to sinh
45  *	    and cosh :
46  *		sinh(r/2) =  r/2 + r * P  ,  cosh(r/2) =  1 + Q . )
47  *
48  *         The coefficients were obtained by a special Remez algorithm.
49  *
50  * Approximation error:
51  *
52  *   |	exp(x) - 1			   |        2**(-57),  (IEEE double)
53  *   | ------------  -  (exp__E(x,0)+x)/x  |  <=
54  *   |	     x			           |	    2**(-69).  (VAX D)
55  *
56  * Constants:
57  * The hexadecimal values are the intended ones for the following constants.
58  * The decimal values may be used, provided that the compiler will convert
59  * from decimal to binary accurately enough to produce the hexadecimal values
60  * shown.
61  */
62 
63 #include "mathimpl.h"
64 
65 vc(p1, 1.5150724356786683059E-2 ,3abe,3d78,066a,67e1,  -6, .F83ABE67E1066A)
66 vc(p2, 6.3112487873718332688E-5 ,5b42,3984,0173,48cd, -13, .845B4248CD0173)
67 vc(q1, 1.1363478204690669916E-1 ,b95a,3ee8,ec45,44a2,  -3, .E8B95A44A2EC45)
68 vc(q2, 1.2624568129896839182E-3 ,7905,3ba5,f5e7,72e4,  -9, .A5790572E4F5E7)
69 vc(q3, 1.5021856115869022674E-6 ,9eb4,36c9,c395,604a, -19, .C99EB4604AC395)
70 
71 ic(p1, 1.3887401997267371720E-2,  -7, 1.C70FF8B3CC2CF)
72 ic(p2, 3.3044019718331897649E-5, -15, 1.15317DF4526C4)
73 ic(q1, 1.1110813732786649355E-1,  -4, 1.C719538248597)
74 ic(q2, 9.9176615021572857300E-4, -10, 1.03FC4CB8C98E8)
75 
76 #ifdef vccast
77 #define       p1    vccast(p1)
78 #define       p2    vccast(p2)
79 #define       q1    vccast(q1)
80 #define       q2    vccast(q2)
81 #define       q3    vccast(q3)
82 #endif
83 
84 double exp__E(x,c)
85 double x,c;
86 {
87 	const static double zero=0.0, one=1.0, half=1.0/2.0, small=1.0E-19;
88 	double z,p,q,xp,xh,w;
89 	if(copysign(x,one)>small) {
90            z = x*x  ;
91 	   p = z*( p1 +z* p2 );
92 #if defined(vax)||defined(tahoe)
93            q = z*( q1 +z*( q2 +z* q3 ));
94 #else	/* defined(vax)||defined(tahoe) */
95            q = z*( q1 +z*  q2 );
96 #endif	/* defined(vax)||defined(tahoe) */
97            xp= x*p     ;
98 	   xh= x*half  ;
99            w = xh-(q-xp)  ;
100 	   p = p+p;
101 	   c += x*((xh*w-(q-(p+xp)))/(one-w)+c);
102 	   return(z*half+c);
103 	}
104 	/* end of |x| > small */
105 
106 	else {
107 	    if(x!=zero) one+small;	/* raise the inexact flag */
108 	    return(copysign(zero,x));
109 	}
110 }
111