usr/src/lib/libm/common_source/cosh.c

/*
 * Copyright (c) 1985 Regents of the University of California.
 * All rights reserved.
 *
 * %sccs.include.redist.c%
 */

#ifndef lint
static char sccsid[] = "@(#)cosh.c      5.6 (Berkeley) %G%";
#endif /* not lint */

/* COSH(X)
 * RETURN THE HYPERBOLIC COSINE OF X
 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
 * CODED IN C BY K.C. NG, 1/8/85;
 * REVISED BY K.C. NG on 2/8/85, 2/23/85, 3/7/85, 3/29/85, 4/16/85.
 *
 * Required system supported functions :
 *      copysign(x,y)
 *      scalb(x,N)
 *
 * Required kernel function:
 *      exp(x)
 *      exp__E(x,c)     ...return exp(x+c)-1-x for |x|<0.3465
 *
 * Method :
 *      1. Replace x by |x|.
 *      2.
 *                                                      [ exp(x) - 1 ]^2
 *          0        <= x <= 0.3465  :  cosh(x) := 1 + -------------------
 *                                                         2*exp(x)
 *
 *                                                 exp(x) +  1/exp(x)
 *          0.3465   <= x <= 22      :  cosh(x) := -------------------
 *                                                         2
 *          22       <= x <= lnovfl  :  cosh(x) := exp(x)/2
 *          lnovfl   <= x <= lnovfl+log(2)
 *                                   :  cosh(x) := exp(x)/2 (avoid overflow)
 *          log(2)+lnovfl <  x <  INF:  overflow to INF
 *
 *      Note: .3465 is a number near one half of ln2.
 *
 * Special cases:
 *      cosh(x) is x if x is +INF, -INF, or NaN.
 *      only cosh(0)=1 is exact for finite x.
 *
 * Accuracy:
 *      cosh(x) returns the exact hyperbolic cosine of x nearly rounded.
 *      In a test run with 768,000 random arguments on a VAX, the maximum
 *      observed error was 1.23 ulps (units in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following constants.
 * The decimal values may be used, provided that the compiler will convert
 * from decimal to binary accurately enough to produce the hexadecimal values
 * shown.
 */

#include "mathimpl.h"

vc(mln2hi, 8.8029691931113054792E1   ,0f33,43b0,2bdb,c7e2,   7, .B00F33C7E22BDB)
vc(mln2lo,-4.9650192275318476525E-16 ,1b60,a70f,582a,279e, -50,-.8F1B60279E582A)
vc(lnovfl, 8.8029691931113053016E1   ,0f33,43b0,2bda,c7e2,   7, .B00F33C7E22BDA)

ic(mln2hi, 7.0978271289338397310E2,    10, 1.62E42FEFA39EF)
ic(mln2lo, 2.3747039373786107478E-14, -45, 1.ABC9E3B39803F)
ic(lnovfl, 7.0978271289338397310E2,     9, 1.62E42FEFA39EF)

#ifdef vccast
#define   mln2hi    vccast(mln2hi)
#define   mln2lo    vccast(mln2lo)
#define   lnovfl    vccast(lnovfl)
#endif

#if defined(vax)||defined(tahoe)
static max = 126                      ;
#else   /* defined(vax)||defined(tahoe) */
static max = 1023                     ;
#endif  /* defined(vax)||defined(tahoe) */

double cosh(x)
double x;
{
        static const double half=1.0/2.0,
                one=1.0, small=1.0E-18; /* fl(1+small)==1 */
        double t;

#if !defined(vax)&&!defined(tahoe)
        if(x!=x) return(x);     /* x is NaN */
#endif  /* !defined(vax)&&!defined(tahoe) */
        if((x=copysign(x,one)) <= 22)
            if(x<0.3465)
                if(x<small) return(one+x);
                else {t=x+exp__E(x,0.0);x=t+t; return(one+t*t/(2.0+x)); }

            else /* for x lies in [0.3465,22] */
                { t=exp(x); return((t+one/t)*half); }

        if( lnovfl <= x && x <= (lnovfl+0.7))
        /* for x lies in [lnovfl, lnovfl+ln2], decrease x by ln(2^(max+1))
         * and return 2^max*exp(x) to avoid unnecessary overflow
         */
            return(scalb(exp((x-mln2hi)-mln2lo), max));

        else
            return(exp(x)*half);        /* for large x,  cosh(x)=exp(x)/2 */
}