1*34124Sbostic /* 224591Szliu * Copyright (c) 1985 Regents of the University of California. 3*34124Sbostic * All rights reserved. 4*34124Sbostic * 5*34124Sbostic * Redistribution and use in source and binary forms are permitted 6*34124Sbostic * provided that this notice is preserved and that due credit is given 7*34124Sbostic * to the University of California at Berkeley. The name of the University 8*34124Sbostic * may not be used to endorse or promote products derived from this 9*34124Sbostic * software without specific prior written permission. This software 10*34124Sbostic * is provided ``as is'' without express or implied warranty. 11*34124Sbostic * 12*34124Sbostic * All recipients should regard themselves as participants in an ongoing 13*34124Sbostic * research project and hence should feel obligated to report their 14*34124Sbostic * experiences (good or bad) with these elementary function codes, using 15*34124Sbostic * the sendbug(8) program, to the authors. 1624591Szliu */ 1724591Szliu 1824591Szliu #ifndef lint 19*34124Sbostic static char sccsid[] = "@(#)cosh.c 5.2 (Berkeley) 04/29/88"; 20*34124Sbostic #endif /* not lint */ 2124591Szliu 2224591Szliu /* COSH(X) 2324591Szliu * RETURN THE HYPERBOLIC COSINE OF X 2424591Szliu * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 2524591Szliu * CODED IN C BY K.C. NG, 1/8/85; 2624591Szliu * REVISED BY K.C. NG on 2/8/85, 2/23/85, 3/7/85, 3/29/85, 4/16/85. 2724591Szliu * 2824591Szliu * Required system supported functions : 2924591Szliu * copysign(x,y) 3024591Szliu * scalb(x,N) 3124591Szliu * 3224591Szliu * Required kernel function: 3324591Szliu * exp(x) 3424591Szliu * exp__E(x,c) ...return exp(x+c)-1-x for |x|<0.3465 3524591Szliu * 3624591Szliu * Method : 3724591Szliu * 1. Replace x by |x|. 3824591Szliu * 2. 3924591Szliu * [ exp(x) - 1 ]^2 4024591Szliu * 0 <= x <= 0.3465 : cosh(x) := 1 + ------------------- 4124591Szliu * 2*exp(x) 4224591Szliu * 4324591Szliu * exp(x) + 1/exp(x) 4424591Szliu * 0.3465 <= x <= 22 : cosh(x) := ------------------- 4524591Szliu * 2 4624591Szliu * 22 <= x <= lnovfl : cosh(x) := exp(x)/2 4724591Szliu * lnovfl <= x <= lnovfl+log(2) 4824591Szliu * : cosh(x) := exp(x)/2 (avoid overflow) 4924591Szliu * log(2)+lnovfl < x < INF: overflow to INF 5024591Szliu * 5124591Szliu * Note: .3465 is a number near one half of ln2. 5224591Szliu * 5324591Szliu * Special cases: 5424591Szliu * cosh(x) is x if x is +INF, -INF, or NaN. 5524591Szliu * only cosh(0)=1 is exact for finite x. 5624591Szliu * 5724591Szliu * Accuracy: 5824591Szliu * cosh(x) returns the exact hyperbolic cosine of x nearly rounded. 5924591Szliu * In a test run with 768,000 random arguments on a VAX, the maximum 6024591Szliu * observed error was 1.23 ulps (units in the last place). 6124591Szliu * 6224591Szliu * Constants: 6324591Szliu * The hexadecimal values are the intended ones for the following constants. 6424591Szliu * The decimal values may be used, provided that the compiler will convert 6524591Szliu * from decimal to binary accurately enough to produce the hexadecimal values 6624591Szliu * shown. 6724591Szliu */ 6824591Szliu 6931853Szliu #if defined(vax)||defined(tahoe) 7031853Szliu #ifdef vax 7131812Szliu #define _0x(A,B) 0x/**/A/**/B 7231853Szliu #else /* vax */ 7331812Szliu #define _0x(A,B) 0x/**/B/**/A 7431853Szliu #endif /* vax */ 7526893Selefunt /* static double */ 7624591Szliu /* mln2hi = 8.8029691931113054792E1 , Hex 2^ 7 * .B00F33C7E22BDB */ 7724591Szliu /* mln2lo = -4.9650192275318476525E-16 , Hex 2^-50 * -.8F1B60279E582A */ 7824591Szliu /* lnovfl = 8.8029691931113053016E1 ; Hex 2^ 7 * .B00F33C7E22BDA */ 7931812Szliu static long mln2hix[] = { _0x(0f33,43b0), _0x(2bdb,c7e2)}; 8031812Szliu static long mln2lox[] = { _0x(1b60,a70f), _0x(582a,279e)}; 8131812Szliu static long lnovflx[] = { _0x(0f33,43b0), _0x(2bda,c7e2)}; 8224591Szliu #define mln2hi (*(double*)mln2hix) 8324591Szliu #define mln2lo (*(double*)mln2lox) 8424591Szliu #define lnovfl (*(double*)lnovflx) 8531853Szliu #else /* defined(vax)||defined(tahoe) */ 8626893Selefunt static double 8724591Szliu mln2hi = 7.0978271289338397310E2 , /*Hex 2^ 10 * 1.62E42FEFA39EF */ 8824591Szliu mln2lo = 2.3747039373786107478E-14 , /*Hex 2^-45 * 1.ABC9E3B39803F */ 8924591Szliu lnovfl = 7.0978271289338397310E2 ; /*Hex 2^ 9 * 1.62E42FEFA39EF */ 9031853Szliu #endif /* defined(vax)||defined(tahoe) */ 9124591Szliu 9231853Szliu #if defined(vax)||defined(tahoe) 9324591Szliu static max = 126 ; 9431853Szliu #else /* defined(vax)||defined(tahoe) */ 9524591Szliu static max = 1023 ; 9631853Szliu #endif /* defined(vax)||defined(tahoe) */ 9724591Szliu 9824591Szliu double cosh(x) 9924591Szliu double x; 10024591Szliu { 10124591Szliu static double half=1.0/2.0,one=1.0, small=1.0E-18; /* fl(1+small)==1 */ 10224591Szliu double scalb(),copysign(),exp(),exp__E(),t; 10324591Szliu 10431853Szliu #if !defined(vax)&&!defined(tahoe) 10524591Szliu if(x!=x) return(x); /* x is NaN */ 10631853Szliu #endif /* !defined(vax)&&!defined(tahoe) */ 10724591Szliu if((x=copysign(x,one)) <= 22) 10824591Szliu if(x<0.3465) 10924591Szliu if(x<small) return(one+x); 11024591Szliu else {t=x+exp__E(x,0.0);x=t+t; return(one+t*t/(2.0+x)); } 11124591Szliu 11224591Szliu else /* for x lies in [0.3465,22] */ 11324591Szliu { t=exp(x); return((t+one/t)*half); } 11424591Szliu 11524591Szliu if( lnovfl <= x && x <= (lnovfl+0.7)) 11624591Szliu /* for x lies in [lnovfl, lnovfl+ln2], decrease x by ln(2^(max+1)) 11724591Szliu * and return 2^max*exp(x) to avoid unnecessary overflow 11824591Szliu */ 11924591Szliu return(scalb(exp((x-mln2hi)-mln2lo), max)); 12024591Szliu 12124591Szliu else 12224591Szliu return(exp(x)*half); /* for large x, cosh(x)=exp(x)/2 */ 12324591Szliu } 124