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/*
* Copyright (c) 1985 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 5.6 (Berkeley) 10/9/90";
#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) .
*/
double
cabs(z)
struct __cabs_complex z;
{
return hypot(z.__x,z.__y);
}
double
z_abs(z)
struct __cabs_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
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