1 /* $NetBSD: n_asincos.c,v 1.1 1995/10/10 23:36:34 ragge 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 static char sccsid[] = "@(#)asincos.c 8.1 (Berkeley) 6/4/93"; 37 #endif /* not lint */ 38 39 /* ASIN(X) 40 * RETURNS ARC SINE OF X 41 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits) 42 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85. 43 * 44 * Required system supported functions: 45 * copysign(x,y) 46 * sqrt(x) 47 * 48 * Required kernel function: 49 * atan2(y,x) 50 * 51 * Method : 52 * asin(x) = atan2(x,sqrt(1-x*x)); for better accuracy, 1-x*x is 53 * computed as follows 54 * 1-x*x if x < 0.5, 55 * 2*(1-|x|)-(1-|x|)*(1-|x|) if x >= 0.5. 56 * 57 * Special cases: 58 * if x is NaN, return x itself; 59 * if |x|>1, return NaN. 60 * 61 * Accuracy: 62 * 1) If atan2() uses machine PI, then 63 * 64 * asin(x) returns (PI/pi) * (the exact arc sine of x) nearly rounded; 65 * and PI is the exact pi rounded to machine precision (see atan2 for 66 * details): 67 * 68 * in decimal: 69 * pi = 3.141592653589793 23846264338327 ..... 70 * 53 bits PI = 3.141592653589793 115997963 ..... , 71 * 56 bits PI = 3.141592653589793 227020265 ..... , 72 * 73 * in hexadecimal: 74 * pi = 3.243F6A8885A308D313198A2E.... 75 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 76 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 77 * 78 * In a test run with more than 200,000 random arguments on a VAX, the 79 * maximum observed error in ulps (units in the last place) was 80 * 2.06 ulps. (comparing against (PI/pi)*(exact asin(x))); 81 * 82 * 2) If atan2() uses true pi, then 83 * 84 * asin(x) returns the exact asin(x) with error below about 2 ulps. 85 * 86 * In a test run with more than 1,024,000 random arguments on a VAX, the 87 * maximum observed error in ulps (units in the last place) was 88 * 1.99 ulps. 89 */ 90 91 #include "mathimpl.h" 92 93 double 94 asin(x) 95 double x; 96 { 97 double s,t,copysign(),atan2(),sqrt(),one=1.0; 98 #if !defined(vax)&&!defined(tahoe) 99 if(x!=x) return(x); /* x is NaN */ 100 #endif /* !defined(vax)&&!defined(tahoe) */ 101 s=copysign(x,one); 102 if(s <= 0.5) 103 return(atan2(x,sqrt(one-x*x))); 104 else 105 { t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); } 106 107 } 108 109 /* ACOS(X) 110 * RETURNS ARC COS OF X 111 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits) 112 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85. 113 * 114 * Required system supported functions: 115 * copysign(x,y) 116 * sqrt(x) 117 * 118 * Required kernel function: 119 * atan2(y,x) 120 * 121 * Method : 122 * ________ 123 * / 1 - x 124 * acos(x) = 2*atan2( / -------- , 1 ) . 125 * \/ 1 + x 126 * 127 * Special cases: 128 * if x is NaN, return x itself; 129 * if |x|>1, return NaN. 130 * 131 * Accuracy: 132 * 1) If atan2() uses machine PI, then 133 * 134 * acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded; 135 * and PI is the exact pi rounded to machine precision (see atan2 for 136 * details): 137 * 138 * in decimal: 139 * pi = 3.141592653589793 23846264338327 ..... 140 * 53 bits PI = 3.141592653589793 115997963 ..... , 141 * 56 bits PI = 3.141592653589793 227020265 ..... , 142 * 143 * in hexadecimal: 144 * pi = 3.243F6A8885A308D313198A2E.... 145 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 146 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 147 * 148 * In a test run with more than 200,000 random arguments on a VAX, the 149 * maximum observed error in ulps (units in the last place) was 150 * 2.07 ulps. (comparing against (PI/pi)*(exact acos(x))); 151 * 152 * 2) If atan2() uses true pi, then 153 * 154 * acos(x) returns the exact acos(x) with error below about 2 ulps. 155 * 156 * In a test run with more than 1,024,000 random arguments on a VAX, the 157 * maximum observed error in ulps (units in the last place) was 158 * 2.15 ulps. 159 */ 160 161 double 162 acos(x) 163 double x; 164 { 165 double t,copysign(),atan2(),sqrt(),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