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 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