1 /* $NetBSD: n_asincos.c,v 1.8 2013/11/24 14:41:53 martin 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
33 static 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
91 double
asin(double x)92 asin(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 float
asinf(float x)107 asinf(float x)
108 {
109 return (float)asin(x);
110 }
111
112 /* ACOS(X)
113 * RETURNS ARC COS OF X
114 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits)
115 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85.
116 *
117 * Required system supported functions:
118 * copysign(x,y)
119 * sqrt(x)
120 *
121 * Required kernel function:
122 * atan2(y,x)
123 *
124 * Method :
125 * ________
126 * / 1 - x
127 * acos(x) = 2*atan2( / -------- , 1 ) .
128 * \/ 1 + x
129 *
130 * Special cases:
131 * if x is NaN, return x itself;
132 * if |x|>1, return NaN.
133 *
134 * Accuracy:
135 * 1) If atan2() uses machine PI, then
136 *
137 * acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded;
138 * and PI is the exact pi rounded to machine precision (see atan2 for
139 * details):
140 *
141 * in decimal:
142 * pi = 3.141592653589793 23846264338327 .....
143 * 53 bits PI = 3.141592653589793 115997963 ..... ,
144 * 56 bits PI = 3.141592653589793 227020265 ..... ,
145 *
146 * in hexadecimal:
147 * pi = 3.243F6A8885A308D313198A2E....
148 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps
149 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps
150 *
151 * In a test run with more than 200,000 random arguments on a VAX, the
152 * maximum observed error in ulps (units in the last place) was
153 * 2.07 ulps. (comparing against (PI/pi)*(exact acos(x)));
154 *
155 * 2) If atan2() uses true pi, then
156 *
157 * acos(x) returns the exact acos(x) with error below about 2 ulps.
158 *
159 * In a test run with more than 1,024,000 random arguments on a VAX, the
160 * maximum observed error in ulps (units in the last place) was
161 * 2.15 ulps.
162 */
163
164 double
acos(double x)165 acos(double x)
166 {
167 double t,one=1.0;
168 #if !defined(__vax__)&&!defined(tahoe)
169 if(x!=x) return(x);
170 #endif /* !defined(__vax__)&&!defined(tahoe) */
171 if( x != -1.0)
172 t=atan2(sqrt((one-x)/(one+x)),one);
173 else
174 t=atan2(one,0.0); /* t = PI/2 */
175 return(t+t);
176 }
177
178 float
acosf(float x)179 acosf(float x)
180 {
181 return (float)acos(x);
182 }
183