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+/*
+ * Copyright (c) 1985, 1993
+ *	The Regents of the University of California.  All rights reserved.
+ *
+ * Redistribution and use in source and binary forms, with or without
+ * modification, are permitted provided that the following conditions
+ * are met:
+ * 1. Redistributions of source code must retain the above copyright
+ *    notice, this list of conditions and the following disclaimer.
+ * 2. Redistributions in binary form must reproduce the above copyright
+ *    notice, this list of conditions and the following disclaimer in the
+ *    documentation and/or other materials provided with the distribution.
+ * 3. All advertising materials mentioning features or use of this software
+ *    must display the following acknowledgement:
+ *	This product includes software developed by the University of
+ *	California, Berkeley and its contributors.
+ * 4. Neither the name of the University nor the names of its contributors
+ *    may be used to endorse or promote products derived from this software
+ *    without specific prior written permission.
+ *
+ * THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
+ * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
+ * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
+ * ARE DISCLAIMED.  IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
+ * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
+ * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
+ * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
+ * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
+ * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
+ * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
+ * SUCH DAMAGE.
+ */
+
+#ifndef lint
+static char sccsid[] = "@(#)cabs.c	8.1 (Berkeley) 6/4/93";
+#endif /* not lint */
+
+/* HYPOT(X,Y)
+ * RETURN THE SQUARE ROOT OF X^2 + Y^2  WHERE Z=X+iY
+ * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
+ * CODED IN C BY K.C. NG, 11/28/84; 
+ * REVISED BY K.C. NG, 7/12/85.
+ *
+ * Required system supported functions :
+ *	copysign(x,y)
+ *	finite(x)
+ *	scalb(x,N)
+ *	sqrt(x)
+ *
+ * Method :
+ *	1. replace x by |x| and y by |y|, and swap x and
+ *	   y if y > x (hence x is never smaller than y).
+ *	2. Hypot(x,y) is computed by:
+ *	   Case I, x/y > 2
+ *		
+ *				       y
+ *		hypot = x + -----------------------------
+ *			 		    2
+ *			    sqrt ( 1 + [x/y]  )  +  x/y
+ *
+ *	   Case II, x/y <= 2 
+ *				                   y
+ *		hypot = x + --------------------------------------------------
+ *				          		     2 
+ *				     			[x/y]   -  2
+ *			   (sqrt(2)+1) + (x-y)/y + -----------------------------
+ *			 		    			  2
+ *			    			  sqrt ( 1 + [x/y]  )  + sqrt(2)
+ *
+ *
+ *
+ * Special cases:
+ *	hypot(x,y) is INF if x or y is +INF or -INF; else
+ *	hypot(x,y) is NAN if x or y is NAN.
+ *
+ * Accuracy:
+ * 	hypot(x,y) returns the sqrt(x^2+y^2) with error less than 1 ulps (units
+ *	in the last place). See Kahan's "Interval Arithmetic Options in the
+ *	Proposed IEEE Floating Point Arithmetic Standard", Interval Mathematics
+ *      1980, Edited by Karl L.E. Nickel, pp 99-128. (A faster but less accurate
+ *	code follows in	comments.) In a test run with 500,000 random arguments
+ *	on a VAX, the maximum observed error was .959 ulps.
+ *
+ * Constants:
+ * The hexadecimal values are the intended ones for the following constants.
+ * The decimal values may be used, provided that the compiler will convert
+ * from decimal to binary accurately enough to produce the hexadecimal values
+ * shown.
+ */
+#include "mathimpl.h"
+
+vc(r2p1hi, 2.4142135623730950345E0   ,8279,411a,ef32,99fc,   2, .9A827999FCEF32)
+vc(r2p1lo, 1.4349369327986523769E-17 ,597d,2484,754b,89b3, -55, .84597D89B3754B)
+vc(sqrt2,  1.4142135623730950622E0   ,04f3,40b5,de65,33f9,   1, .B504F333F9DE65)
+
+ic(r2p1hi, 2.4142135623730949234E0   ,   1, 1.3504F333F9DE6)
+ic(r2p1lo, 1.2537167179050217666E-16 , -53, 1.21165F626CDD5)
+ic(sqrt2,  1.4142135623730951455E0   ,   0, 1.6A09E667F3BCD)
+
+#ifdef vccast
+#define	r2p1hi	vccast(r2p1hi)
+#define	r2p1lo	vccast(r2p1lo)
+#define	sqrt2	vccast(sqrt2)
+#endif
+
+double
+hypot(x,y)
+double x, y;
+{
+	static const double zero=0, one=1, 
+		      small=1.0E-18;	/* fl(1+small)==1 */
+	static const ibig=30;	/* fl(1+2**(2*ibig))==1 */
+	double t,r;
+	int exp;
+
+	if(finite(x))
+	    if(finite(y))
+	    {	
+		x=copysign(x,one);
+		y=copysign(y,one);
+		if(y > x) 
+		    { t=x; x=y; y=t; }
+		if(x == zero) return(zero);
+		if(y == zero) return(x);
+		exp= logb(x);
+		if(exp-(int)logb(y) > ibig ) 	
+			/* raise inexact flag and return |x| */
+		   { one+small; return(x); }
+
+	    /* start computing sqrt(x^2 + y^2) */
+		r=x-y;
+		if(r>y) { 	/* x/y > 2 */
+		    r=x/y;
+		    r=r+sqrt(one+r*r); }
+		else {		/* 1 <= x/y <= 2 */
+		    r/=y; t=r*(r+2.0);
+		    r+=t/(sqrt2+sqrt(2.0+t));
+		    r+=r2p1lo; r+=r2p1hi; }
+
+		r=y/r;
+		return(x+r);
+
+	    }
+
+	    else if(y==y)   	   /* y is +-INF */
+		     return(copysign(y,one));
+	    else 
+		     return(y);	   /* y is NaN and x is finite */
+
+	else if(x==x) 		   /* x is +-INF */
+	         return (copysign(x,one));
+	else if(finite(y))
+	         return(x);		   /* x is NaN, y is finite */
+#if !defined(vax)&&!defined(tahoe)
+	else if(y!=y) return(y);  /* x and y is NaN */
+#endif	/* !defined(vax)&&!defined(tahoe) */
+	else return(copysign(y,one));   /* y is INF */
+}
+
+/* CABS(Z)
+ * RETURN THE ABSOLUTE VALUE OF THE COMPLEX NUMBER  Z = X + iY
+ * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
+ * CODED IN C BY K.C. NG, 11/28/84.
+ * REVISED BY K.C. NG, 7/12/85.
+ *
+ * Required kernel function :
+ *	hypot(x,y)
+ *
+ * Method :
+ *	cabs(z) = hypot(x,y) .
+ */
+
+struct complex { double x, y; };
+
+double
+cabs(z)
+struct complex z;
+{
+	return hypot(z.x,z.y);
+}
+
+double
+z_abs(z)
+struct complex *z;
+{
+	return hypot(z->x,z->y);
+}
+
+/* A faster but less accurate version of cabs(x,y) */
+#if 0
+double hypot(x,y)
+double x, y;
+{
+	static const double zero=0, one=1;
+		      small=1.0E-18;	/* fl(1+small)==1 */
+	static const ibig=30;	/* fl(1+2**(2*ibig))==1 */
+	double temp;
+	int exp;
+
+	if(finite(x))
+	    if(finite(y))
+	    {	
+		x=copysign(x,one);
+		y=copysign(y,one);
+		if(y > x) 
+		    { temp=x; x=y; y=temp; }
+		if(x == zero) return(zero);
+		if(y == zero) return(x);
+		exp= logb(x);
+		x=scalb(x,-exp);
+		if(exp-(int)logb(y) > ibig ) 
+			/* raise inexact flag and return |x| */
+		   { one+small; return(scalb(x,exp)); }
+		else y=scalb(y,-exp);
+		return(scalb(sqrt(x*x+y*y),exp));
+	    }
+
+	    else if(y==y)   	   /* y is +-INF */
+		     return(copysign(y,one));
+	    else 
+		     return(y);	   /* y is NaN and x is finite */
+
+	else if(x==x) 		   /* x is +-INF */
+	         return (copysign(x,one));
+	else if(finite(y))
+	         return(x);		   /* x is NaN, y is finite */
+	else if(y!=y) return(y);  	/* x and y is NaN */
+	else return(copysign(y,one));   /* y is INF */
+}
+#endif