[unix-history] / usr / src / lib / libm / common_source / asincos.c

/*
 * Copyright (c) 1985 Regents of the University of California.
 * All rights reserved.
 *
 * Redistribution and use in source and binary forms are permitted
 * provided that the above copyright notice and this paragraph are
 * duplicated in all such forms and that any documentation,
 * advertising materials, and other materials related to such
 * distribution and use acknowledge that the software was developed
 * by the University of California, Berkeley.  The name of the
 * University may not be used to endorse or promote products derived
 * from this software without specific prior written permission.
 * THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
 * IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
 * WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
 *
 * 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
 * the sendbug(8) program, to the authors.
 */

#ifndef lint
static char sccsid[] = "@(#)asincos.c	5.3 (Berkeley) %G%";
#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;
#if !defined(vax)&&!defined(tahoe)
	if(x!=x) return(x);	/* x is NaN */
#endif	/* !defined(vax)&&!defined(tahoe) */
	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;
#if !defined(vax)&&!defined(tahoe)
	if(x!=x) return(x);
#endif	/* !defined(vax)&&!defined(tahoe) */
	if( x != -1.0)
	    t=atan2(sqrt((one-x)/(one+x)),one);
	else
	    t=atan2(one,0.0);	/* t = PI/2 */
	return(t+t);
}
Commit	Line	Data
4c182be8	1	/*
3cd1de16	2	* Copyright (c) 1985 Regents of the University of California.
4c182be8 KB	3	* All rights reserved.
	4	*
	5	* Redistribution and use in source and binary forms are permitted
a399f6c8 KB	6	* provided that the above copyright notice and this paragraph are
	7	* duplicated in all such forms and that any documentation,
	8	* advertising materials, and other materials related to such
	9	* distribution and use acknowledge that the software was developed
	10	* by the University of California, Berkeley. The name of the
	11	* University may not be used to endorse or promote products derived
	12	* from this software without specific prior written permission.
	13	* THIS SOFTWARE IS PROVIDED ``AS IS'' AND WITHOUT ANY EXPRESS OR
	14	* IMPLIED WARRANTIES, INCLUDING, WITHOUT LIMITATION, THE IMPLIED
	15	* WARRANTIES OF MERCHANTIBILITY AND FITNESS FOR A PARTICULAR PURPOSE.
4c182be8 KB	16	*
	17	* All recipients should regard themselves as participants in an ongoing
	18	* research project and hence should feel obligated to report their
	19	* experiences (good or bad) with these elementary function codes, using
	20	* the sendbug(8) program, to the authors.
3cd1de16 ZAL	21	*/
	22
	23	#ifndef lint
a399f6c8	24	static char sccsid[] = "@(#)asincos.c 5.3 (Berkeley) %G%";
4c182be8	25	#endif /* not lint */
3cd1de16 ZAL	26
	27	/* ASIN(X)
	28	* RETURNS ARC SINE OF X
	29	* DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
	30	* CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
	31	*
	32	* Required system supported functions:
	33	* copysign(x,y)
	34	* sqrt(x)
	35	*
	36	* Required kernel function:
	37	* atan2(y,x)
	38	*
	39	* Method :
	40	* asin(x) = atan2(x,sqrt(1-xx)); for better accuracy, 1-xx is
	41	* computed as follows
	42	* 1-x*x if x < 0.5,
	43	* 2(1-\|x\|)-(1-\|x\|)(1-\|x\|) if x >= 0.5.
	44	*
	45	* Special cases:
	46	* if x is NaN, return x itself;
	47	* if \|x\|>1, return NaN.
	48	*
	49	* Accuracy:
	50	* 1) If atan2() uses machine PI, then
	51	*
	52	* asin(x) returns (PI/pi) * (the exact arc sine of x) nearly rounded;
	53	* and PI is the exact pi rounded to machine precision (see atan2 for
	54	* details):
	55	*
	56	* in decimal:
	57	* pi = 3.141592653589793 23846264338327 .....
	58	* 53 bits PI = 3.141592653589793 115997963 ..... ,
	59	* 56 bits PI = 3.141592653589793 227020265 ..... ,
	60	*
	61	* in hexadecimal:
	62	* pi = 3.243F6A8885A308D313198A2E....
	63	* 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps
	64	* 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps
	65	*
	66	* In a test run with more than 200,000 random arguments on a VAX, the
	67	* maximum observed error in ulps (units in the last place) was
	68	* 2.06 ulps. (comparing against (PI/pi)*(exact asin(x)));
	69	*
	70	* 2) If atan2() uses true pi, then
	71	*
	72	* asin(x) returns the exact asin(x) with error below about 2 ulps.
	73	*
	74	* In a test run with more than 1,024,000 random arguments on a VAX, the
	75	* maximum observed error in ulps (units in the last place) was
	76	* 1.99 ulps.
	77	*/
	78
	79	double asin(x)
	80	double x;
	81	{
	82	double s,t,copysign(),atan2(),sqrt(),one=1.0;
859dc438	83	#if !defined(vax)&&!defined(tahoe)
3cd1de16	84	if(x!=x) return(x); /* x is NaN */
859dc438	85	#endif /* !defined(vax)&&!defined(tahoe) */
3cd1de16 ZAL	86	s=copysign(x,one);
	87	if(s <= 0.5)
	88	return(atan2(x,sqrt(one-x*x)));
	89	else
	90	{ t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); }
	91
	92	}
	93
	94	/* ACOS(X)
	95	* RETURNS ARC COS OF X
	96	* DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
	97	* CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
	98	*
	99	* Required system supported functions:
	100	* copysign(x,y)
	101	* sqrt(x)
	102	*
	103	* Required kernel function:
	104	* atan2(y,x)
	105	*
	106	* Method :
	107	* ________
	108	* / 1 - x
	109	* acos(x) = 2*atan2( / -------- , 1 ) .
	110	* \/ 1 + x
	111	*
	112	* Special cases:
	113	* if x is NaN, return x itself;
	114	* if \|x\|>1, return NaN.
	115	*
	116	* Accuracy:
	117	* 1) If atan2() uses machine PI, then
	118	*
	119	* acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded;
	120	* and PI is the exact pi rounded to machine precision (see atan2 for
	121	* details):
	122	*
	123	* in decimal:
	124	* pi = 3.141592653589793 23846264338327 .....
	125	* 53 bits PI = 3.141592653589793 115997963 ..... ,
	126	* 56 bits PI = 3.141592653589793 227020265 ..... ,
	127	*
	128	* in hexadecimal:
	129	* pi = 3.243F6A8885A308D313198A2E....
	130	* 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps
	131	* 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps
	132	*
	133	* In a test run with more than 200,000 random arguments on a VAX, the
	134	* maximum observed error in ulps (units in the last place) was
	135	* 2.07 ulps. (comparing against (PI/pi)*(exact acos(x)));
	136	*
	137	* 2) If atan2() uses true pi, then
	138	*
	139	* acos(x) returns the exact acos(x) with error below about 2 ulps.
	140	*
	141	* In a test run with more than 1,024,000 random arguments on a VAX, the
	142	* maximum observed error in ulps (units in the last place) was
	143	* 2.15 ulps.
	144	*/
	145
	146	double acos(x)
	147	double x;
	148	{
	149	double t,copysign(),atan2(),sqrt(),one=1.0;
859dc438	150	#if !defined(vax)&&!defined(tahoe)
3cd1de16	151	if(x!=x) return(x);
859dc438	152	#endif /* !defined(vax)&&!defined(tahoe) */
3cd1de16 ZAL	153	if( x != -1.0)
	154	t=atan2(sqrt((one-x)/(one+x)),one);
	155	else
	156	t=atan2(one,0.0); /* t = PI/2 */
	157	return(t+t);
	158	}