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. All advertising materials mentioning features or use of this software 15 * must display the following acknowledgement: 16 * This product includes software developed by the University of 17 * California, Berkeley and its contributors. 18 * 4. Neither the name of the University nor the names of its contributors 19 * may be used to endorse or promote products derived from this software 20 * without specific prior written permission. 21 * 22 * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND 23 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 24 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 25 * ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE 26 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 27 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 28 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 29 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 30 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 31 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 32 * SUCH DAMAGE. 33 */ 34 35 #ifndef lint 36 #if 0 37 static char sccsid[] = "@(#)asincos.c 8.1 (Berkeley) 6/4/93"; 38 #endif 39 #endif /* not lint */ 40 41 /* ASIN(X) 42 * RETURNS ARC SINE OF X 43 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits) 44 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85. 45 * 46 * Required system supported functions: 47 * copysign(x,y) 48 * sqrt(x) 49 * 50 * Required kernel function: 51 * atan2(y,x) 52 * 53 * Method : 54 * asin(x) = atan2(x,sqrt(1-x*x)); for better accuracy, 1-x*x is 55 * computed as follows 56 * 1-x*x if x < 0.5, 57 * 2*(1-|x|)-(1-|x|)*(1-|x|) if x >= 0.5. 58 * 59 * Special cases: 60 * if x is NaN, return x itself; 61 * if |x|>1, return NaN. 62 * 63 * Accuracy: 64 * 1) If atan2() uses machine PI, then 65 * 66 * asin(x) returns (PI/pi) * (the exact arc sine of x) nearly rounded; 67 * and PI is the exact pi rounded to machine precision (see atan2 for 68 * details): 69 * 70 * in decimal: 71 * pi = 3.141592653589793 23846264338327 ..... 72 * 53 bits PI = 3.141592653589793 115997963 ..... , 73 * 56 bits PI = 3.141592653589793 227020265 ..... , 74 * 75 * in hexadecimal: 76 * pi = 3.243F6A8885A308D313198A2E.... 77 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 78 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 79 * 80 * In a test run with more than 200,000 random arguments on a VAX, the 81 * maximum observed error in ulps (units in the last place) was 82 * 2.06 ulps. (comparing against (PI/pi)*(exact asin(x))); 83 * 84 * 2) If atan2() uses true pi, then 85 * 86 * asin(x) returns the exact asin(x) with error below about 2 ulps. 87 * 88 * In a test run with more than 1,024,000 random arguments on a VAX, the 89 * maximum observed error in ulps (units in the last place) was 90 * 1.99 ulps. 91 */ 92 93 #include "mathimpl.h" 94 95 double 96 asin(double x) 97 { 98 double s,t,one=1.0; 99 #if !defined(__vax__)&&!defined(tahoe) 100 if(x!=x) return(x); /* x is NaN */ 101 #endif /* !defined(__vax__)&&!defined(tahoe) */ 102 s=copysign(x,one); 103 if(s <= 0.5) 104 return(atan2(x,sqrt(one-x*x))); 105 else 106 { t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); } 107 108 } 109 110 /* ACOS(X) 111 * RETURNS ARC COS OF X 112 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits) 113 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85. 114 * 115 * Required system supported functions: 116 * copysign(x,y) 117 * sqrt(x) 118 * 119 * Required kernel function: 120 * atan2(y,x) 121 * 122 * Method : 123 * ________ 124 * / 1 - x 125 * acos(x) = 2*atan2( / -------- , 1 ) . 126 * \/ 1 + x 127 * 128 * Special cases: 129 * if x is NaN, return x itself; 130 * if |x|>1, return NaN. 131 * 132 * Accuracy: 133 * 1) If atan2() uses machine PI, then 134 * 135 * acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded; 136 * and PI is the exact pi rounded to machine precision (see atan2 for 137 * details): 138 * 139 * in decimal: 140 * pi = 3.141592653589793 23846264338327 ..... 141 * 53 bits PI = 3.141592653589793 115997963 ..... , 142 * 56 bits PI = 3.141592653589793 227020265 ..... , 143 * 144 * in hexadecimal: 145 * pi = 3.243F6A8885A308D313198A2E.... 146 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 147 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 148 * 149 * In a test run with more than 200,000 random arguments on a VAX, the 150 * maximum observed error in ulps (units in the last place) was 151 * 2.07 ulps. (comparing against (PI/pi)*(exact acos(x))); 152 * 153 * 2) If atan2() uses true pi, then 154 * 155 * acos(x) returns the exact acos(x) with error below about 2 ulps. 156 * 157 * In a test run with more than 1,024,000 random arguments on a VAX, the 158 * maximum observed error in ulps (units in the last place) was 159 * 2.15 ulps. 160 */ 161 162 double 163 acos(double x) 164 { 165 double t,one=1.0; 166 #if !defined(__vax__)&&!defined(tahoe) 167 if(x!=x) return(x); 168 #endif /* !defined(__vax__)&&!defined(tahoe) */ 169 if( x != -1.0) 170 t=atan2(sqrt((one-x)/(one+x)),one); 171 else 172 t=atan2(one,0.0); /* t = PI/2 */ 173 return(t+t); 174 } 175