2 * Copyright (c) 1985, 1993
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33 * @(#)cabs.c 8.1 (Berkeley) 6/4/93
37 * RETURN THE SQUARE ROOT OF X^2 + Y^2 WHERE Z=X+iY
38 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
39 * CODED IN C BY K.C. NG, 11/28/84;
40 * REVISED BY K.C. NG, 7/12/85.
42 * Required system supported functions :
49 * 1. replace x by |x| and y by |y|, and swap x and
50 * y if y > x (hence x is never smaller than y).
51 * 2. Hypot(x,y) is computed by:
55 * hypot = x + -----------------------------
57 * sqrt ( 1 + [x/y] ) + x/y
61 * hypot = x + --------------------------------------------------
64 * (sqrt(2)+1) + (x-y)/y + -----------------------------
66 * sqrt ( 1 + [x/y] ) + sqrt(2)
71 * hypot(x,y) is INF if x or y is +INF or -INF; else
72 * hypot(x,y) is NAN if x or y is NAN.
75 * hypot(x,y) returns the sqrt(x^2+y^2) with error less than 1 ulps (units
76 * in the last place). See Kahan's "Interval Arithmetic Options in the
77 * Proposed IEEE Floating Point Arithmetic Standard", Interval Mathematics
78 * 1980, Edited by Karl L.E. Nickel, pp 99-128. (A faster but less accurate
79 * code follows in comments.) In a test run with 500,000 random arguments
80 * on a VAX, the maximum observed error was .959 ulps.
83 * The hexadecimal values are the intended ones for the following constants.
84 * The decimal values may be used, provided that the compiler will convert
85 * from decimal to binary accurately enough to produce the hexadecimal values
90 vc(r2p1hi, 2.4142135623730950345E0 ,8279,411a,ef32,99fc, 2, .9A827999FCEF32)
91 vc(r2p1lo, 1.4349369327986523769E-17 ,597d,2484,754b,89b3, -55, .84597D89B3754B)
92 vc(sqrt2, 1.4142135623730950622E0 ,04f3,40b5,de65,33f9, 1, .B504F333F9DE65)
94 ic(r2p1hi, 2.4142135623730949234E0 , 1, 1.3504F333F9DE6)
95 ic(r2p1lo, 1.2537167179050217666E-16 , -53, 1.21165F626CDD5)
96 ic(sqrt2, 1.4142135623730951455E0 , 0, 1.6A09E667F3BCD)
99 #define r2p1hi vccast(r2p1hi)
100 #define r2p1lo vccast(r2p1lo)
101 #define sqrt2 vccast(sqrt2)
108 static const double zero=0, one=1,
109 small=1.0E-18; /* fl(1+small)==1 */
110 static const ibig=30; /* fl(1+2**(2*ibig))==1 */
121 if(x == zero) return(zero);
122 if(y == zero) return(x);
124 if(exp-(int)logb(y) > ibig )
125 /* raise inexact flag and return |x| */
126 { one+small; return(x); }
128 /* start computing sqrt(x^2 + y^2) */
130 if(r>y) { /* x/y > 2 */
133 else { /* 1 <= x/y <= 2 */
135 r+=t/(sqrt2+sqrt(2.0+t));
136 r+=r2p1lo; r+=r2p1hi; }
143 else if(y==y) /* y is +-INF */
144 return(copysign(y,one));
146 return(y); /* y is NaN and x is finite */
148 else if(x==x) /* x is +-INF */
149 return (copysign(x,one));
151 return(x); /* x is NaN, y is finite */
152 #if !defined(vax)&&!defined(tahoe)
153 else if(y!=y) return(y); /* x and y is NaN */
154 #endif /* !defined(vax)&&!defined(tahoe) */
155 else return(copysign(y,one)); /* y is INF */
159 * RETURN THE ABSOLUTE VALUE OF THE COMPLEX NUMBER Z = X + iY
160 * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
161 * CODED IN C BY K.C. NG, 11/28/84.
162 * REVISED BY K.C. NG, 7/12/85.
164 * Required kernel function :
168 * cabs(z) = hypot(x,y) .
171 struct complex { double x, y; };
177 return hypot(z.x,z.y);
184 return hypot(z->x,z->y);
187 /* A faster but less accurate version of cabs(x,y) */
192 static const double zero=0, one=1;
193 small=1.0E-18; /* fl(1+small)==1 */
194 static const ibig=30; /* fl(1+2**(2*ibig))==1 */
204 { temp=x; x=y; y=temp; }
205 if(x == zero) return(zero);
206 if(y == zero) return(x);
209 if(exp-(int)logb(y) > ibig )
210 /* raise inexact flag and return |x| */
211 { one+small; return(scalb(x,exp)); }
212 else y=scalb(y,-exp);
213 return(scalb(sqrt(x*x+y*y),exp));
216 else if(y==y) /* y is +-INF */
217 return(copysign(y,one));
219 return(y); /* y is NaN and x is finite */
221 else if(x==x) /* x is +-INF */
222 return (copysign(x,one));
224 return(x); /* x is NaN, y is finite */
225 else if(y!=y) return(y); /* x and y is NaN */
226 else return(copysign(y,one)); /* y is INF */