1 /* $NetBSD: n_asincos.c,v 1.7 2003/08/07 16:44:50 agc 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 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 /* 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 */ 157 158 double 159 acos(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