1 /* $NetBSD: n_cabs.c,v 1.3 1999/07/02 15:37:36 simonb 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[] = "@(#)cabs.c 8.1 (Berkeley) 6/4/93"; 37 #endif /* not lint */ 38 39 /* HYPOT(X,Y) 40 * RETURN THE SQUARE ROOT OF X^2 + Y^2 WHERE Z=X+iY 41 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 42 * CODED IN C BY K.C. NG, 11/28/84; 43 * REVISED BY K.C. NG, 7/12/85. 44 * 45 * Required system supported functions : 46 * copysign(x,y) 47 * finite(x) 48 * scalb(x,N) 49 * sqrt(x) 50 * 51 * Method : 52 * 1. replace x by |x| and y by |y|, and swap x and 53 * y if y > x (hence x is never smaller than y). 54 * 2. Hypot(x,y) is computed by: 55 * Case I, x/y > 2 56 * 57 * y 58 * hypot = x + ----------------------------- 59 * 2 60 * sqrt ( 1 + [x/y] ) + x/y 61 * 62 * Case II, x/y <= 2 63 * y 64 * hypot = x + -------------------------------------------------- 65 * 2 66 * [x/y] - 2 67 * (sqrt(2)+1) + (x-y)/y + ----------------------------- 68 * 2 69 * sqrt ( 1 + [x/y] ) + sqrt(2) 70 * 71 * 72 * 73 * Special cases: 74 * hypot(x,y) is INF if x or y is +INF or -INF; else 75 * hypot(x,y) is NAN if x or y is NAN. 76 * 77 * Accuracy: 78 * hypot(x,y) returns the sqrt(x^2+y^2) with error less than 1 ulps (units 79 * in the last place). See Kahan's "Interval Arithmetic Options in the 80 * Proposed IEEE Floating Point Arithmetic Standard", Interval Mathematics 81 * 1980, Edited by Karl L.E. Nickel, pp 99-128. (A faster but less accurate 82 * code follows in comments.) In a test run with 500,000 random arguments 83 * on a VAX, the maximum observed error was .959 ulps. 84 * 85 * Constants: 86 * The hexadecimal values are the intended ones for the following constants. 87 * The decimal values may be used, provided that the compiler will convert 88 * from decimal to binary accurately enough to produce the hexadecimal values 89 * shown. 90 */ 91 #include "mathimpl.h" 92 93 vc(r2p1hi, 2.4142135623730950345E0 ,8279,411a,ef32,99fc, 2, .9A827999FCEF32) 94 vc(r2p1lo, 1.4349369327986523769E-17 ,597d,2484,754b,89b3, -55, .84597D89B3754B) 95 vc(sqrt2, 1.4142135623730950622E0 ,04f3,40b5,de65,33f9, 1, .B504F333F9DE65) 96 97 ic(r2p1hi, 2.4142135623730949234E0 , 1, 1.3504F333F9DE6) 98 ic(r2p1lo, 1.2537167179050217666E-16 , -53, 1.21165F626CDD5) 99 ic(sqrt2, 1.4142135623730951455E0 , 0, 1.6A09E667F3BCD) 100 101 #ifdef vccast 102 #define r2p1hi vccast(r2p1hi) 103 #define r2p1lo vccast(r2p1lo) 104 #define sqrt2 vccast(sqrt2) 105 #endif 106 107 double 108 hypot(x,y) 109 double x, y; 110 { 111 static const double zero=0, one=1, 112 small=1.0E-18; /* fl(1+small)==1 */ 113 static const ibig=30; /* fl(1+2**(2*ibig))==1 */ 114 double t,r; 115 int exp; 116 117 if(finite(x)) 118 if(finite(y)) 119 { 120 x=copysign(x,one); 121 y=copysign(y,one); 122 if(y > x) 123 { t=x; x=y; y=t; } 124 if(x == zero) return(zero); 125 if(y == zero) return(x); 126 exp= logb(x); 127 if(exp-(int)logb(y) > ibig ) 128 /* raise inexact flag and return |x| */ 129 { one+small; return(x); } 130 131 /* start computing sqrt(x^2 + y^2) */ 132 r=x-y; 133 if(r>y) { /* x/y > 2 */ 134 r=x/y; 135 r=r+sqrt(one+r*r); } 136 else { /* 1 <= x/y <= 2 */ 137 r/=y; t=r*(r+2.0); 138 r+=t/(sqrt2+sqrt(2.0+t)); 139 r+=r2p1lo; r+=r2p1hi; } 140 141 r=y/r; 142 return(x+r); 143 144 } 145 146 else if(y==y) /* y is +-INF */ 147 return(copysign(y,one)); 148 else 149 return(y); /* y is NaN and x is finite */ 150 151 else if(x==x) /* x is +-INF */ 152 return (copysign(x,one)); 153 else if(finite(y)) 154 return(x); /* x is NaN, y is finite */ 155 #if !defined(__vax__)&&!defined(tahoe) 156 else if(y!=y) return(y); /* x and y is NaN */ 157 #endif /* !defined(__vax__)&&!defined(tahoe) */ 158 else return(copysign(y,one)); /* y is INF */ 159 } 160 161 /* CABS(Z) 162 * RETURN THE ABSOLUTE VALUE OF THE COMPLEX NUMBER Z = X + iY 163 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS) 164 * CODED IN C BY K.C. NG, 11/28/84. 165 * REVISED BY K.C. NG, 7/12/85. 166 * 167 * Required kernel function : 168 * hypot(x,y) 169 * 170 * Method : 171 * cabs(z) = hypot(x,y) . 172 */ 173 174 struct complex { double x, y; }; 175 176 double 177 cabs(z) 178 struct complex z; 179 { 180 return hypot(z.x,z.y); 181 } 182 183 double 184 z_abs(z) 185 struct complex *z; 186 { 187 return hypot(z->x,z->y); 188 } 189 190 /* A faster but less accurate version of cabs(x,y) */ 191 #if 0 192 double hypot(x,y) 193 double x, y; 194 { 195 static const double zero=0, one=1; 196 small=1.0E-18; /* fl(1+small)==1 */ 197 static const ibig=30; /* fl(1+2**(2*ibig))==1 */ 198 double temp; 199 int exp; 200 201 if(finite(x)) 202 if(finite(y)) 203 { 204 x=copysign(x,one); 205 y=copysign(y,one); 206 if(y > x) 207 { temp=x; x=y; y=temp; } 208 if(x == zero) return(zero); 209 if(y == zero) return(x); 210 exp= logb(x); 211 x=scalb(x,-exp); 212 if(exp-(int)logb(y) > ibig ) 213 /* raise inexact flag and return |x| */ 214 { one+small; return(scalb(x,exp)); } 215 else y=scalb(y,-exp); 216 return(scalb(sqrt(x*x+y*y),exp)); 217 } 218 219 else if(y==y) /* y is +-INF */ 220 return(copysign(y,one)); 221 else 222 return(y); /* y is NaN and x is finite */ 223 224 else if(x==x) /* x is +-INF */ 225 return (copysign(x,one)); 226 else if(finite(y)) 227 return(x); /* x is NaN, y is finite */ 228 else if(y!=y) return(y); /* x and y is NaN */ 229 else return(copysign(y,one)); /* y is INF */ 230 } 231 #endif 232