xref: /csrg-svn/old/libm/libm/asincos.c (revision 24354)

/*
 * Copyright (c) 1985 Regents of the University of California.
 *
 * Use and reproduction of this software are granted  in  accordance  with
 * the terms and conditions specified in  the  Berkeley  Software  License
 * Agreement (in particular, this entails acknowledgement of the programs'
 * source, and inclusion of this notice) with the additional understanding
 * that  all  recipients  should regard themselves as participants  in  an
 * ongoing  research  project and hence should  feel  obligated  to report
 * their  experiences (good or bad) with these elementary function  codes,
 * using "sendbug 4bsd-bugs@BERKELEY", to the authors.
 */

#ifndef lint
static char sccsid[] = "@(#)asincos.c	1.1 (Berkeley) 08/21/85";
#endif not lint

/* ASIN(X)
 * RETURNS ARC SINE OF X
 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
 *
 * Required system supported functions:
 *	copysign(x,y)
 *	sqrt(x)
 *
 * Required kernel function:
 *	atan2(y,x)
 *
 * Method :
 *	asin(x) = atan2(x,sqrt(1-x*x)); for better accuracy, 1-x*x is
 *		  computed as follows
 *			1-x*x                     if x <  0.5,
 *			2*(1-|x|)-(1-|x|)*(1-|x|) if x >= 0.5.
 *
 * Special cases:
 *	if x is NaN, return x itself;
 *	if |x|>1, return NaN.
 *
 * Accuracy:
 * 1)  If atan2() uses machine PI, then
 *
 *	asin(x) returns (PI/pi) * (the exact arc sine of x) nearly rounded;
 *	and PI is the exact pi rounded to machine precision (see atan2 for
 *      details):
 *
 *	in decimal:
 *		pi = 3.141592653589793 23846264338327 .....
 *    53 bits   PI = 3.141592653589793 115997963 ..... ,
 *    56 bits   PI = 3.141592653589793 227020265 ..... ,
 *
 *	in hexadecimal:
 *		pi = 3.243F6A8885A308D313198A2E....
 *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18	error=.276ulps
 *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
 *
 *	In a test run with more than 200,000 random arguments on a VAX, the
 *	maximum observed error in ulps (units in the last place) was
 *	2.06 ulps.      (comparing against (PI/pi)*(exact asin(x)));
 *
 * 2)  If atan2() uses true pi, then
 *
 *	asin(x) returns the exact asin(x) with error below about 2 ulps.
 *
 *	In a test run with more than 1,024,000 random arguments on a VAX, the
 *	maximum observed error in ulps (units in the last place) was
 *      1.99 ulps.
 */

double asin(x)
double x;
{
	double s,t,copysign(),atan2(),sqrt(),one=1.0;
#ifndef VAX
	if(x!=x) return(x);	/* x is NaN */
#endif
	s=copysign(x,one);
	if(s <= 0.5)
	    return(atan2(x,sqrt(one-x*x)));
	else
	    { t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); }

}

/* ACOS(X)
 * RETURNS ARC COS OF X
 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
 *
 * Required system supported functions:
 *	copysign(x,y)
 *	sqrt(x)
 *
 * Required kernel function:
 *	atan2(y,x)
 *
 * Method :
 *			      ________
 *                           / 1 - x
 *	acos(x) = 2*atan2(  / -------- , 1 ) .
 *                        \/   1 + x
 *
 * Special cases:
 *	if x is NaN, return x itself;
 *	if |x|>1, return NaN.
 *
 * Accuracy:
 * 1)  If atan2() uses machine PI, then
 *
 *	acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded;
 *	and PI is the exact pi rounded to machine precision (see atan2 for
 *      details):
 *
 *	in decimal:
 *		pi = 3.141592653589793 23846264338327 .....
 *    53 bits   PI = 3.141592653589793 115997963 ..... ,
 *    56 bits   PI = 3.141592653589793 227020265 ..... ,
 *
 *	in hexadecimal:
 *		pi = 3.243F6A8885A308D313198A2E....
 *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18	error=.276ulps
 *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
 *
 *	In a test run with more than 200,000 random arguments on a VAX, the
 *	maximum observed error in ulps (units in the last place) was
 *	2.07 ulps.      (comparing against (PI/pi)*(exact acos(x)));
 *
 * 2)  If atan2() uses true pi, then
 *
 *	acos(x) returns the exact acos(x) with error below about 2 ulps.
 *
 *	In a test run with more than 1,024,000 random arguments on a VAX, the
 *	maximum observed error in ulps (units in the last place) was
 *	2.15 ulps.
 */

double acos(x)
double x;
{
	double t,copysign(),atan2(),sqrt(),one=1.0;
#ifndef VAX
	if(x!=x) return(x);
#endif
	if( x != -1.0)
	    t=atan2(sqrt((one-x)/(one+x)),one);
	else
	    t=atan2(one,0.0);	/* t = PI/2 */
	return(t+t);
}