1 /* 2 * Copyright (c) 1985 Regents of the University of California. 3 * 4 * Use and reproduction of this software are granted in accordance with 5 * the terms and conditions specified in the Berkeley Software License 6 * Agreement (in particular, this entails acknowledgement of the programs' 7 * source, and inclusion of this notice) with the additional understanding 8 * that all recipients should regard themselves as participants in an 9 * ongoing research project and hence should feel obligated to report 10 * their experiences (good or bad) with these elementary function codes, 11 * using "sendbug 4bsd-bugs@BERKELEY", to the authors. 12 */ 13 14 #ifndef lint 15 static char sccsid[] = 16 "@(#)cabs.c 1.2 (Berkeley) 8/21/85; 5.1 (ucb.elefunt) 11/30/87"; 17 #endif /* not lint */ 18 19 /* HYPOT(X,Y) 20 * RETURN THE SQUARE ROOT OF X^2 + Y^2 WHERE Z=X+iY 21 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 22 * CODED IN C BY K.C. NG, 11/28/84; 23 * REVISED BY K.C. NG, 7/12/85. 24 * 25 * Required system supported functions : 26 * copysign(x,y) 27 * finite(x) 28 * scalb(x,N) 29 * sqrt(x) 30 * 31 * Method : 32 * 1. replace x by |x| and y by |y|, and swap x and 33 * y if y > x (hence x is never smaller than y). 34 * 2. Hypot(x,y) is computed by: 35 * Case I, x/y > 2 36 * 37 * y 38 * hypot = x + ----------------------------- 39 * 2 40 * sqrt ( 1 + [x/y] ) + x/y 41 * 42 * Case II, x/y <= 2 43 * y 44 * hypot = x + -------------------------------------------------- 45 * 2 46 * [x/y] - 2 47 * (sqrt(2)+1) + (x-y)/y + ----------------------------- 48 * 2 49 * sqrt ( 1 + [x/y] ) + sqrt(2) 50 * 51 * 52 * 53 * Special cases: 54 * hypot(x,y) is INF if x or y is +INF or -INF; else 55 * hypot(x,y) is NAN if x or y is NAN. 56 * 57 * Accuracy: 58 * hypot(x,y) returns the sqrt(x^2+y^2) with error less than 1 ulps (units 59 * in the last place). See Kahan's "Interval Arithmetic Options in the 60 * Proposed IEEE Floating Point Arithmetic Standard", Interval Mathematics 61 * 1980, Edited by Karl L.E. Nickel, pp 99-128. (A faster but less accurate 62 * code follows in comments.) In a test run with 500,000 random arguments 63 * on a VAX, the maximum observed error was .959 ulps. 64 * 65 * Constants: 66 * The hexadecimal values are the intended ones for the following constants. 67 * The decimal values may be used, provided that the compiler will convert 68 * from decimal to binary accurately enough to produce the hexadecimal values 69 * shown. 70 */ 71 72 #if defined(vax)||defined(tahoe) /* VAX D format */ 73 #ifdef vax 74 #define _0x(A,B) 0x/**/A/**/B 75 #else /* vax */ 76 #define _0x(A,B) 0x/**/B/**/A 77 #endif /* vax */ 78 /* static double */ 79 /* r2p1hi = 2.4142135623730950345E0 , Hex 2^ 2 * .9A827999FCEF32 */ 80 /* r2p1lo = 1.4349369327986523769E-17 , Hex 2^-55 * .84597D89B3754B */ 81 /* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */ 82 static long r2p1hix[] = { _0x(8279,411a), _0x(ef32,99fc)}; 83 static long r2p1lox[] = { _0x(597d,2484), _0x(754b,89b3)}; 84 static long sqrt2x[] = { _0x(04f3,40b5), _0x(de65,33f9)}; 85 #define r2p1hi (*(double*)r2p1hix) 86 #define r2p1lo (*(double*)r2p1lox) 87 #define sqrt2 (*(double*)sqrt2x) 88 #else /* defined(vax)||defined(tahoe) */ 89 static double 90 r2p1hi = 2.4142135623730949234E0 , /*Hex 2^1 * 1.3504F333F9DE6 */ 91 r2p1lo = 1.2537167179050217666E-16 , /*Hex 2^-53 * 1.21165F626CDD5 */ 92 sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */ 93 #endif /* defined(vax)||defined(tahoe) */ 94 95 double 96 hypot(x,y) 97 double x, y; 98 { 99 static double zero=0, one=1, 100 small=1.0E-18; /* fl(1+small)==1 */ 101 static ibig=30; /* fl(1+2**(2*ibig))==1 */ 102 double copysign(),scalb(),logb(),sqrt(),t,r; 103 int finite(), exp; 104 105 if(finite(x)) 106 if(finite(y)) 107 { 108 x=copysign(x,one); 109 y=copysign(y,one); 110 if(y > x) 111 { t=x; x=y; y=t; } 112 if(x == zero) return(zero); 113 if(y == zero) return(x); 114 exp= logb(x); 115 if(exp-(int)logb(y) > ibig ) 116 /* raise inexact flag and return |x| */ 117 { one+small; return(x); } 118 119 /* start computing sqrt(x^2 + y^2) */ 120 r=x-y; 121 if(r>y) { /* x/y > 2 */ 122 r=x/y; 123 r=r+sqrt(one+r*r); } 124 else { /* 1 <= x/y <= 2 */ 125 r/=y; t=r*(r+2.0); 126 r+=t/(sqrt2+sqrt(2.0+t)); 127 r+=r2p1lo; r+=r2p1hi; } 128 129 r=y/r; 130 return(x+r); 131 132 } 133 134 else if(y==y) /* y is +-INF */ 135 return(copysign(y,one)); 136 else 137 return(y); /* y is NaN and x is finite */ 138 139 else if(x==x) /* x is +-INF */ 140 return (copysign(x,one)); 141 else if(finite(y)) 142 return(x); /* x is NaN, y is finite */ 143 #if !defined(vax)&&!defined(tahoe) 144 else if(y!=y) return(y); /* x and y is NaN */ 145 #endif /* !defined(vax)&&!defined(tahoe) */ 146 else return(copysign(y,one)); /* y is INF */ 147 } 148 149 /* CABS(Z) 150 * RETURN THE ABSOLUTE VALUE OF THE COMPLEX NUMBER Z = X + iY 151 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 152 * CODED IN C BY K.C. NG, 11/28/84. 153 * REVISED BY K.C. NG, 7/12/85. 154 * 155 * Required kernel function : 156 * hypot(x,y) 157 * 158 * Method : 159 * cabs(z) = hypot(x,y) . 160 */ 161 162 double 163 cabs(z) 164 struct { double x, y;} z; 165 { 166 return hypot(z.x,z.y); 167 } 168 169 double 170 z_abs(z) 171 struct { double x,y;} *z; 172 { 173 return hypot(z->x,z->y); 174 } 175 176 /* A faster but less accurate version of cabs(x,y) */ 177 #if 0 178 double hypot(x,y) 179 double x, y; 180 { 181 static double zero=0, one=1; 182 small=1.0E-18; /* fl(1+small)==1 */ 183 static ibig=30; /* fl(1+2**(2*ibig))==1 */ 184 double copysign(),scalb(),logb(),sqrt(),temp; 185 int finite(), exp; 186 187 if(finite(x)) 188 if(finite(y)) 189 { 190 x=copysign(x,one); 191 y=copysign(y,one); 192 if(y > x) 193 { temp=x; x=y; y=temp; } 194 if(x == zero) return(zero); 195 if(y == zero) return(x); 196 exp= logb(x); 197 x=scalb(x,-exp); 198 if(exp-(int)logb(y) > ibig ) 199 /* raise inexact flag and return |x| */ 200 { one+small; return(scalb(x,exp)); } 201 else y=scalb(y,-exp); 202 return(scalb(sqrt(x*x+y*y),exp)); 203 } 204 205 else if(y==y) /* y is +-INF */ 206 return(copysign(y,one)); 207 else 208 return(y); /* y is NaN and x is finite */ 209 210 else if(x==x) /* x is +-INF */ 211 return (copysign(x,one)); 212 else if(finite(y)) 213 return(x); /* x is NaN, y is finite */ 214 else if(y!=y) return(y); /* x and y is NaN */ 215 else return(copysign(y,one)); /* y is INF */ 216 } 217 #endif 218