1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms are permitted 6 * provided that this notice is preserved and that due credit is given 7 * to the University of California at Berkeley. The name of the University 8 * may not be used to endorse or promote products derived from this 9 * software without specific prior written permission. This software 10 * is provided ``as is'' without express or implied warranty. 11 * 12 * All recipients should regard themselves as participants in an ongoing 13 * research project and hence should feel obligated to report their 14 * experiences (good or bad) with these elementary function codes, using 15 * the sendbug(8) program, to the authors. 16 */ 17 18 #ifndef lint 19 static char sccsid[] = "@(#)asincos.c 5.2 (Berkeley) 04/29/88"; 20 #endif /* not lint */ 21 22 /* ASIN(X) 23 * RETURNS ARC SINE OF X 24 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits) 25 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85. 26 * 27 * Required system supported functions: 28 * copysign(x,y) 29 * sqrt(x) 30 * 31 * Required kernel function: 32 * atan2(y,x) 33 * 34 * Method : 35 * asin(x) = atan2(x,sqrt(1-x*x)); for better accuracy, 1-x*x is 36 * computed as follows 37 * 1-x*x if x < 0.5, 38 * 2*(1-|x|)-(1-|x|)*(1-|x|) if x >= 0.5. 39 * 40 * Special cases: 41 * if x is NaN, return x itself; 42 * if |x|>1, return NaN. 43 * 44 * Accuracy: 45 * 1) If atan2() uses machine PI, then 46 * 47 * asin(x) returns (PI/pi) * (the exact arc sine of x) nearly rounded; 48 * and PI is the exact pi rounded to machine precision (see atan2 for 49 * details): 50 * 51 * in decimal: 52 * pi = 3.141592653589793 23846264338327 ..... 53 * 53 bits PI = 3.141592653589793 115997963 ..... , 54 * 56 bits PI = 3.141592653589793 227020265 ..... , 55 * 56 * in hexadecimal: 57 * pi = 3.243F6A8885A308D313198A2E.... 58 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 59 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 60 * 61 * In a test run with more than 200,000 random arguments on a VAX, the 62 * maximum observed error in ulps (units in the last place) was 63 * 2.06 ulps. (comparing against (PI/pi)*(exact asin(x))); 64 * 65 * 2) If atan2() uses true pi, then 66 * 67 * asin(x) returns the exact asin(x) with error below about 2 ulps. 68 * 69 * In a test run with more than 1,024,000 random arguments on a VAX, the 70 * maximum observed error in ulps (units in the last place) was 71 * 1.99 ulps. 72 */ 73 74 double asin(x) 75 double x; 76 { 77 double s,t,copysign(),atan2(),sqrt(),one=1.0; 78 #if !defined(vax)&&!defined(tahoe) 79 if(x!=x) return(x); /* x is NaN */ 80 #endif /* !defined(vax)&&!defined(tahoe) */ 81 s=copysign(x,one); 82 if(s <= 0.5) 83 return(atan2(x,sqrt(one-x*x))); 84 else 85 { t=one-s; s=t+t; return(atan2(x,sqrt(s-t*t))); } 86 87 } 88 89 /* ACOS(X) 90 * RETURNS ARC COS OF X 91 * DOUBLE PRECISION (IEEE DOUBLE 53 bits, VAX D FORMAT 56 bits) 92 * CODED IN C BY K.C. NG, 4/16/85, REVISED ON 6/10/85. 93 * 94 * Required system supported functions: 95 * copysign(x,y) 96 * sqrt(x) 97 * 98 * Required kernel function: 99 * atan2(y,x) 100 * 101 * Method : 102 * ________ 103 * / 1 - x 104 * acos(x) = 2*atan2( / -------- , 1 ) . 105 * \/ 1 + x 106 * 107 * Special cases: 108 * if x is NaN, return x itself; 109 * if |x|>1, return NaN. 110 * 111 * Accuracy: 112 * 1) If atan2() uses machine PI, then 113 * 114 * acos(x) returns (PI/pi) * (the exact arc cosine of x) nearly rounded; 115 * and PI is the exact pi rounded to machine precision (see atan2 for 116 * details): 117 * 118 * in decimal: 119 * pi = 3.141592653589793 23846264338327 ..... 120 * 53 bits PI = 3.141592653589793 115997963 ..... , 121 * 56 bits PI = 3.141592653589793 227020265 ..... , 122 * 123 * in hexadecimal: 124 * pi = 3.243F6A8885A308D313198A2E.... 125 * 53 bits PI = 3.243F6A8885A30 = 2 * 1.921FB54442D18 error=.276ulps 126 * 56 bits PI = 3.243F6A8885A308 = 4 * .C90FDAA22168C2 error=.206ulps 127 * 128 * In a test run with more than 200,000 random arguments on a VAX, the 129 * maximum observed error in ulps (units in the last place) was 130 * 2.07 ulps. (comparing against (PI/pi)*(exact acos(x))); 131 * 132 * 2) If atan2() uses true pi, then 133 * 134 * acos(x) returns the exact acos(x) with error below about 2 ulps. 135 * 136 * In a test run with more than 1,024,000 random arguments on a VAX, the 137 * maximum observed error in ulps (units in the last place) was 138 * 2.15 ulps. 139 */ 140 141 double acos(x) 142 double x; 143 { 144 double t,copysign(),atan2(),sqrt(),one=1.0; 145 #if !defined(vax)&&!defined(tahoe) 146 if(x!=x) return(x); 147 #endif /* !defined(vax)&&!defined(tahoe) */ 148 if( x != -1.0) 149 t=atan2(sqrt((one-x)/(one+x)),one); 150 else 151 t=atan2(one,0.0); /* t = PI/2 */ 152 return(t+t); 153 } 154