1 /*
2 * Copyright (c) 1985, 1993
3 * The Regents of the University of California. All rights reserved.
4 *
5 * %sccs.include.redist.c%
6 */
7
8 #ifndef lint
9 static char sccsid[] = "@(#)cosh.c 8.1 (Berkeley) 06/04/93";
10 #endif /* not lint */
11
12 /* COSH(X)
13 * RETURN THE HYPERBOLIC COSINE OF X
14 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
15 * CODED IN C BY K.C. NG, 1/8/85;
16 * REVISED BY K.C. NG on 2/8/85, 2/23/85, 3/7/85, 3/29/85, 4/16/85.
17 *
18 * Required system supported functions :
19 * copysign(x,y)
20 * scalb(x,N)
21 *
22 * Required kernel function:
23 * exp(x)
24 * exp__E(x,c) ...return exp(x+c)-1-x for |x|<0.3465
25 *
26 * Method :
27 * 1. Replace x by |x|.
28 * 2.
29 * [ exp(x) - 1 ]^2
30 * 0 <= x <= 0.3465 : cosh(x) := 1 + -------------------
31 * 2*exp(x)
32 *
33 * exp(x) + 1/exp(x)
34 * 0.3465 <= x <= 22 : cosh(x) := -------------------
35 * 2
36 * 22 <= x <= lnovfl : cosh(x) := exp(x)/2
37 * lnovfl <= x <= lnovfl+log(2)
38 * : cosh(x) := exp(x)/2 (avoid overflow)
39 * log(2)+lnovfl < x < INF: overflow to INF
40 *
41 * Note: .3465 is a number near one half of ln2.
42 *
43 * Special cases:
44 * cosh(x) is x if x is +INF, -INF, or NaN.
45 * only cosh(0)=1 is exact for finite x.
46 *
47 * Accuracy:
48 * cosh(x) returns the exact hyperbolic cosine of x nearly rounded.
49 * In a test run with 768,000 random arguments on a VAX, the maximum
50 * observed error was 1.23 ulps (units in the last place).
51 *
52 * Constants:
53 * The hexadecimal values are the intended ones for the following constants.
54 * The decimal values may be used, provided that the compiler will convert
55 * from decimal to binary accurately enough to produce the hexadecimal values
56 * shown.
57 */
58
59 #include "mathimpl.h"
60
61 vc(mln2hi, 8.8029691931113054792E1 ,0f33,43b0,2bdb,c7e2, 7, .B00F33C7E22BDB)
62 vc(mln2lo,-4.9650192275318476525E-16 ,1b60,a70f,582a,279e, -50,-.8F1B60279E582A)
63 vc(lnovfl, 8.8029691931113053016E1 ,0f33,43b0,2bda,c7e2, 7, .B00F33C7E22BDA)
64
65 ic(mln2hi, 7.0978271289338397310E2, 10, 1.62E42FEFA39EF)
66 ic(mln2lo, 2.3747039373786107478E-14, -45, 1.ABC9E3B39803F)
67 ic(lnovfl, 7.0978271289338397310E2, 9, 1.62E42FEFA39EF)
68
69 #ifdef vccast
70 #define mln2hi vccast(mln2hi)
71 #define mln2lo vccast(mln2lo)
72 #define lnovfl vccast(lnovfl)
73 #endif
74
75 #if defined(vax)||defined(tahoe)
76 static max = 126 ;
77 #else /* defined(vax)||defined(tahoe) */
78 static max = 1023 ;
79 #endif /* defined(vax)||defined(tahoe) */
80
cosh(x)81 double cosh(x)
82 double x;
83 {
84 static const double half=1.0/2.0,
85 one=1.0, small=1.0E-18; /* fl(1+small)==1 */
86 double t;
87
88 #if !defined(vax)&&!defined(tahoe)
89 if(x!=x) return(x); /* x is NaN */
90 #endif /* !defined(vax)&&!defined(tahoe) */
91 if((x=copysign(x,one)) <= 22)
92 if(x<0.3465)
93 if(x<small) return(one+x);
94 else {t=x+__exp__E(x,0.0);x=t+t; return(one+t*t/(2.0+x)); }
95
96 else /* for x lies in [0.3465,22] */
97 { t=exp(x); return((t+one/t)*half); }
98
99 if( lnovfl <= x && x <= (lnovfl+0.7))
100 /* for x lies in [lnovfl, lnovfl+ln2], decrease x by ln(2^(max+1))
101 * and return 2^max*exp(x) to avoid unnecessary overflow
102 */
103 return(scalb(exp((x-mln2hi)-mln2lo), max));
104
105 else
106 return(exp(x)*half); /* for large x, cosh(x)=exp(x)/2 */
107 }
108