asinh.c

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/*
 * Copyright (c) 1985, 1993
 *	The Regents of the University of California.  All rights reserved.
 *
 * Redistribution and use in source and binary forms, with or without
 * modification, are permitted provided that the following conditions
 * are met:
 * 1. Redistributions of source code must retain the above copyright
 *    notice, this list of conditions and the following disclaimer.
 * 2. Redistributions in binary form must reproduce the above copyright
 *    notice, this list of conditions and the following disclaimer in the
 *    documentation and/or other materials provided with the distribution.
 * 3. All advertising materials mentioning features or use of this software
 *    must display the following acknowledgement:
 *	This product includes software developed by the University of
 *	California, Berkeley and its contributors.
 * 4. Neither the name of the University nor the names of its contributors
 *    may be used to endorse or promote products derived from this software
 *    without specific prior written permission.
 *
 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
 * SUCH DAMAGE.
 */

#ifndef lint
static char sccsid[] = "@(#)asinh.c	8.1 (Berkeley) 6/4/93";
#endif /* not lint */

/* ASINH(X)
 * RETURN THE INVERSE HYPERBOLIC SINE OF X
 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
 * CODED IN C BY K.C. NG, 2/16/85;
 * REVISED BY K.C. NG on 3/7/85, 3/24/85, 4/16/85.
 *
 * Required system supported functions :
 *	copysign(x,y)
 *	sqrt(x)
 *
 * Required kernel function:
 *	log1p(x) 		...return log(1+x)
 *
 * Method :
 *	Based on 
 *		asinh(x) = sign(x) * log [ |x| + sqrt(x*x+1) ]
 *	we have
 *	asinh(x) := x  if  1+x*x=1,
 *		 := sign(x)*(log1p(x)+ln2))	 if sqrt(1+x*x)=x, else
 *		 := sign(x)*log1p(|x| + |x|/(1/|x| + sqrt(1+(1/|x|)^2)) )  
 *
 * Accuracy:
 *	asinh(x) returns the exact inverse hyperbolic sine of x nearly rounded.
 *	In a test run with 52,000 random arguments on a VAX, the maximum 
 *	observed error was 1.58 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(ln2hi, 6.9314718055829871446E-1  ,7217,4031,0000,f7d0,   0, .B17217F7D00000)
vc(ln2lo, 1.6465949582897081279E-12 ,bcd5,2ce7,d9cc,e4f1, -39, .E7BCD5E4F1D9CC)

ic(ln2hi, 6.9314718036912381649E-1,   -1, 1.62E42FEE00000)
ic(ln2lo, 1.9082149292705877000E-10, -33, 1.A39EF35793C76)

#ifdef vccast
#define    ln2hi    vccast(ln2hi)
#define    ln2lo    vccast(ln2lo)
#endif

double asinh(x)
double x;
{	
	double t,s;
	const static double	small=1.0E-10,	/* fl(1+small*small) == 1 */
				big  =1.0E20,	/* fl(1+big) == big */
				one  =1.0   ;	

#if !defined(vax)&&!defined(tahoe)
	if(x!=x) return(x);	/* x is NaN */
#endif	/* !defined(vax)&&!defined(tahoe) */
	if((t=copysign(x,one))>small) 
	    if(t<big) {
	     	s=one/t; return(copysign(log1p(t+t/(s+sqrt(one+s*s))),x)); }
	    else	/* if |x| > big */
		{s=log1p(t)+ln2lo; return(copysign(s+ln2hi,x));}
	else	/* if |x| < small */
	    return(x);
}