1 /* $NetBSD: n_asincos.c,v 1.9 2016/09/21 14:11:40 christos 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 #include <sys/cdefs.h> 31 __RCSID("$NetBSD: n_asincos.c,v 1.9 2016/09/21 14:11:40 christos Exp $"); 32 33 #ifndef lint 34 #if 0 35 static char sccsid[] = "@(#)asincos.c 8.1 (Berkeley) 6/4/93"; 36 #endif 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 "namespace.h" 92 #include "mathimpl.h" 93 94 #ifdef __weak_alias 95 __weak_alias(asinf, _asinf) 96 #endif 97 #ifdef __weak_alias 98 __weak_alias(asin, _asin) 99 #endif 100 101 double 102 asin(double x) 103 { 104 double s,t,one=1.0; 105 #if !defined(__vax__)&&!defined(tahoe) 106 if(x!=x) return(x); /* x is NaN */ 107 #endif /* !defined(__vax__)&&!defined(tahoe) */ 108 s=copysign(x,one); 109 if(s <= 0.5) 110 return(atan2(x,sqrt(one-x*x))); 111 else 112 { t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); } 113 114 } 115 116 float 117 asinf(float x) 118 { 119 return (float)asin(x); 120 } 121 122 /* ACOS(X) 123 * RETURNS ARC COS OF X 124 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits) 125 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85. 126 * 127 * Required system supported functions: 128 * copysign(x,y) 129 * sqrt(x) 130 * 131 * Required kernel function: 132 * atan2(y,x) 133 * 134 * Method : 135 * ________ 136 * / 1 - x 137 * acos(x) = 2*atan2( / -------- , 1 ) . 138 * \/ 1 + x 139 * 140 * Special cases: 141 * if x is NaN, return x itself; 142 * if |x|>1, return NaN. 143 * 144 * Accuracy: 145 * 1) If atan2() uses machine PI, then 146 * 147 * acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded; 148 * and PI is the exact pi rounded to machine precision (see atan2 for 149 * details): 150 * 151 * in decimal: 152 * pi = 3.141592653589793 23846264338327 ..... 153 * 53 bits PI = 3.141592653589793 115997963 ..... , 154 * 56 bits PI = 3.141592653589793 227020265 ..... , 155 * 156 * in hexadecimal: 157 * pi = 3.243F6A8885A308D313198A2E.... 158 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 159 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 160 * 161 * In a test run with more than 200,000 random arguments on a VAX, the 162 * maximum observed error in ulps (units in the last place) was 163 * 2.07 ulps. (comparing against (PI/pi)*(exact acos(x))); 164 * 165 * 2) If atan2() uses true pi, then 166 * 167 * acos(x) returns the exact acos(x) with error below about 2 ulps. 168 * 169 * In a test run with more than 1,024,000 random arguments on a VAX, the 170 * maximum observed error in ulps (units in the last place) was 171 * 2.15 ulps. 172 */ 173 174 double 175 acos(double x) 176 { 177 double t,one=1.0; 178 #if !defined(__vax__)&&!defined(tahoe) 179 if(x!=x) return(x); 180 #endif /* !defined(__vax__)&&!defined(tahoe) */ 181 if( x != -1.0) 182 t=atan2(sqrt((one-x)/(one+x)),one); 183 else 184 t=atan2(one,0.0); /* t = PI/2 */ 185 return(t+t); 186 } 187 188 float 189 acosf(float x) 190 { 191 return (float)acos(x); 192 } 193