134124Sbostic /* 224591Szliu * Copyright (c) 1985 Regents of the University of California. 334124Sbostic * All rights reserved. 434124Sbostic * 542657Sbostic * %sccs.include.redist.c% 624591Szliu */ 724591Szliu 824591Szliu #ifndef lint 9*57452Sbostic static char sccsid[] = "@(#)cosh.c 5.7 (Berkeley) 01/10/93"; 1034124Sbostic #endif /* not lint */ 1124591Szliu 1224591Szliu /* COSH(X) 1324591Szliu * RETURN THE HYPERBOLIC COSINE OF X 1424591Szliu * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 1524591Szliu * CODED IN C BY K.C. NG, 1/8/85; 1624591Szliu * REVISED BY K.C. NG on 2/8/85, 2/23/85, 3/7/85, 3/29/85, 4/16/85. 1724591Szliu * 1824591Szliu * Required system supported functions : 1924591Szliu * copysign(x,y) 2024591Szliu * scalb(x,N) 2124591Szliu * 2224591Szliu * Required kernel function: 2324591Szliu * exp(x) 2424591Szliu * exp__E(x,c) ...return exp(x+c)-1-x for |x|<0.3465 2524591Szliu * 2624591Szliu * Method : 2724591Szliu * 1. Replace x by |x|. 2824591Szliu * 2. 2924591Szliu * [ exp(x) - 1 ]^2 3024591Szliu * 0 <= x <= 0.3465 : cosh(x) := 1 + ------------------- 3124591Szliu * 2*exp(x) 3224591Szliu * 3324591Szliu * exp(x) + 1/exp(x) 3424591Szliu * 0.3465 <= x <= 22 : cosh(x) := ------------------- 3524591Szliu * 2 3624591Szliu * 22 <= x <= lnovfl : cosh(x) := exp(x)/2 3724591Szliu * lnovfl <= x <= lnovfl+log(2) 3824591Szliu * : cosh(x) := exp(x)/2 (avoid overflow) 3924591Szliu * log(2)+lnovfl < x < INF: overflow to INF 4024591Szliu * 4124591Szliu * Note: .3465 is a number near one half of ln2. 4224591Szliu * 4324591Szliu * Special cases: 4424591Szliu * cosh(x) is x if x is +INF, -INF, or NaN. 4524591Szliu * only cosh(0)=1 is exact for finite x. 4624591Szliu * 4724591Szliu * Accuracy: 4824591Szliu * cosh(x) returns the exact hyperbolic cosine of x nearly rounded. 4924591Szliu * In a test run with 768,000 random arguments on a VAX, the maximum 5024591Szliu * observed error was 1.23 ulps (units in the last place). 5124591Szliu * 5224591Szliu * Constants: 5324591Szliu * The hexadecimal values are the intended ones for the following constants. 5424591Szliu * The decimal values may be used, provided that the compiler will convert 5524591Szliu * from decimal to binary accurately enough to produce the hexadecimal values 5624591Szliu * shown. 5724591Szliu */ 5824591Szliu 5935679Sbostic #include "mathimpl.h" 6024591Szliu 6135679Sbostic vc(mln2hi, 8.8029691931113054792E1 ,0f33,43b0,2bdb,c7e2, 7, .B00F33C7E22BDB) 6235679Sbostic vc(mln2lo,-4.9650192275318476525E-16 ,1b60,a70f,582a,279e, -50,-.8F1B60279E582A) 6335679Sbostic vc(lnovfl, 8.8029691931113053016E1 ,0f33,43b0,2bda,c7e2, 7, .B00F33C7E22BDA) 6435679Sbostic 6535679Sbostic ic(mln2hi, 7.0978271289338397310E2, 10, 1.62E42FEFA39EF) 6635679Sbostic ic(mln2lo, 2.3747039373786107478E-14, -45, 1.ABC9E3B39803F) 6735679Sbostic ic(lnovfl, 7.0978271289338397310E2, 9, 1.62E42FEFA39EF) 6835679Sbostic 6935679Sbostic #ifdef vccast 7035679Sbostic #define mln2hi vccast(mln2hi) 7135679Sbostic #define mln2lo vccast(mln2lo) 7235679Sbostic #define lnovfl vccast(lnovfl) 7335679Sbostic #endif 7435679Sbostic 7531853Szliu #if defined(vax)||defined(tahoe) 7624591Szliu static max = 126 ; 7731853Szliu #else /* defined(vax)||defined(tahoe) */ 7824591Szliu static max = 1023 ; 7931853Szliu #endif /* defined(vax)||defined(tahoe) */ 8024591Szliu 8124591Szliu double cosh(x) 8224591Szliu double x; 8324591Szliu { 8435679Sbostic static const double half=1.0/2.0, 8535679Sbostic one=1.0, small=1.0E-18; /* fl(1+small)==1 */ 8635679Sbostic double t; 8724591Szliu 8831853Szliu #if !defined(vax)&&!defined(tahoe) 8924591Szliu if(x!=x) return(x); /* x is NaN */ 9031853Szliu #endif /* !defined(vax)&&!defined(tahoe) */ 9124591Szliu if((x=copysign(x,one)) <= 22) 9224591Szliu if(x<0.3465) 9324591Szliu if(x<small) return(one+x); 94*57452Sbostic else {t=x+__exp__E(x,0.0);x=t+t; return(one+t*t/(2.0+x)); } 9524591Szliu 9624591Szliu else /* for x lies in [0.3465,22] */ 9724591Szliu { t=exp(x); return((t+one/t)*half); } 9824591Szliu 9924591Szliu if( lnovfl <= x && x <= (lnovfl+0.7)) 10024591Szliu /* for x lies in [lnovfl, lnovfl+ln2], decrease x by ln(2^(max+1)) 10124591Szliu * and return 2^max*exp(x) to avoid unnecessary overflow 10224591Szliu */ 10324591Szliu return(scalb(exp((x-mln2hi)-mln2lo), max)); 10424591Szliu 10524591Szliu else 10624591Szliu return(exp(x)*half); /* for large x, cosh(x)=exp(x)/2 */ 10724591Szliu } 108