1/*	$NetBSD: n_asincos.c,v 1.6 2002/06/15 00:10:17 matt Exp $	*/
2/*
3 * Copyright (c) 1985, 1993
4 *	The Regents of the University of California.  All rights reserved.
5 *
6 * Redistribution and use in source and binary forms, with or without
7 * modification, are permitted provided that the following conditions
8 * are met:
9 * 1. Redistributions of source code must retain the above copyright
10 *    notice, this list of conditions and the following disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 *    notice, this list of conditions and the following disclaimer in the
13 *    documentation and/or other materials provided with the distribution.
14 * 3. Neither the name of the University nor the names of its contributors
15 *    may be used to endorse or promote products derived from this software
16 *    without specific prior written permission.
17 *
18 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
19 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
20 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
21 * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
22 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
23 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
24 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
25 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
26 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
27 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
28 * SUCH DAMAGE.
29 */
30
31#ifndef lint
32#if 0
33static char sccsid[] = "@(#)asincos.c	8.1 (Berkeley) 6/4/93";
34#endif
35#endif /* not lint */
36
37/* ASIN(X)
38 * RETURNS ARC SINE OF X
39 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
40 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
41 *
42 * Required system supported functions:
43 *	copysign(x,y)
44 *	sqrt(x)
45 *
46 * Required kernel function:
47 *	atan2(y,x)
48 *
49 * Method :
50 *	asin(x) = atan2(x,sqrt(1-x*x)); for better accuracy, 1-x*x is
51 *		  computed as follows
52 *			1-x*x                     if x <  0.5,
53 *			2*(1-|x|)-(1-|x|)*(1-|x|) if x >= 0.5.
54 *
55 * Special cases:
56 *	if x is NaN, return x itself;
57 *	if |x|>1, return NaN.
58 *
59 * Accuracy:
60 * 1)  If atan2() uses machine PI, then
61 *
62 *	asin(x) returns (PI/pi) * (the exact arc sine of x) nearly rounded;
63 *	and PI is the exact pi rounded to machine precision (see atan2 for
64 *      details):
65 *
66 *	in decimal:
67 *		pi = 3.141592653589793 23846264338327 .....
68 *    53 bits   PI = 3.141592653589793 115997963 ..... ,
69 *    56 bits   PI = 3.141592653589793 227020265 ..... ,
70 *
71 *	in hexadecimal:
72 *		pi = 3.243F6A8885A308D313198A2E....
73 *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18	error=.276ulps
74 *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
75 *
76 *	In a test run with more than 200,000 random arguments on a VAX, the
77 *	maximum observed error in ulps (units in the last place) was
78 *	2.06 ulps.      (comparing against (PI/pi)*(exact asin(x)));
79 *
80 * 2)  If atan2() uses true pi, then
81 *
82 *	asin(x) returns the exact asin(x) with error below about 2 ulps.
83 *
84 *	In a test run with more than 1,024,000 random arguments on a VAX, the
85 *	maximum observed error in ulps (units in the last place) was
86 *      1.99 ulps.
87 */
88
89#include "mathimpl.h"
90
91double
92asin(double x)
93{
94	double s,t,one=1.0;
95#if !defined(__vax__)&&!defined(tahoe)
96	if(x!=x) return(x);	/* x is NaN */
97#endif	/* !defined(__vax__)&&!defined(tahoe) */
98	s=copysign(x,one);
99	if(s <= 0.5)
100	    return(atan2(x,sqrt(one-x*x)));
101	else
102	    { t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); }
103
104}
105
106/* ACOS(X)
107 * RETURNS ARC COS OF X
108 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
109 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
110 *
111 * Required system supported functions:
112 *	copysign(x,y)
113 *	sqrt(x)
114 *
115 * Required kernel function:
116 *	atan2(y,x)
117 *
118 * Method :
119 *			      ________
120 *                           / 1 - x
121 *	acos(x) = 2*atan2(  / -------- , 1 ) .
122 *                        \/   1 + x
123 *
124 * Special cases:
125 *	if x is NaN, return x itself;
126 *	if |x|>1, return NaN.
127 *
128 * Accuracy:
129 * 1)  If atan2() uses machine PI, then
130 *
131 *	acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded;
132 *	and PI is the exact pi rounded to machine precision (see atan2 for
133 *      details):
134 *
135 *	in decimal:
136 *		pi = 3.141592653589793 23846264338327 .....
137 *    53 bits   PI = 3.141592653589793 115997963 ..... ,
138 *    56 bits   PI = 3.141592653589793 227020265 ..... ,
139 *
140 *	in hexadecimal:
141 *		pi = 3.243F6A8885A308D313198A2E....
142 *    53 bits   PI = 3.243F6A8885A30  =  2 * 1.921FB54442D18	error=.276ulps
143 *    56 bits   PI = 3.243F6A8885A308 =  4 * .C90FDAA22168C2    error=.206ulps
144 *
145 *	In a test run with more than 200,000 random arguments on a VAX, the
146 *	maximum observed error in ulps (units in the last place) was
147 *	2.07 ulps.      (comparing against (PI/pi)*(exact acos(x)));
148 *
149 * 2)  If atan2() uses true pi, then
150 *
151 *	acos(x) returns the exact acos(x) with error below about 2 ulps.
152 *
153 *	In a test run with more than 1,024,000 random arguments on a VAX, the
154 *	maximum observed error in ulps (units in the last place) was
155 *	2.15 ulps.
156 */
157
158double
159acos(double x)
160{
161	double t,one=1.0;
162#if !defined(__vax__)&&!defined(tahoe)
163	if(x!=x) return(x);
164#endif	/* !defined(__vax__)&&!defined(tahoe) */
165	if( x != -1.0)
166	    t=atan2(sqrt((one-x)/(one+x)),one);
167	else
168	    t=atan2(one,0.0);	/* t = PI/2 */
169	return(t+t);
170}
171