4.3BSD/usr/src/usr.lib/libm/IEEE/cabs.c
/*
* Copyright (c) 1985 Regents of the University of California.
*
* Use and reproduction of this software are granted in accordance with
* the terms and conditions specified in the Berkeley Software License
* Agreement (in particular, this entails acknowledgement of the programs'
* source, and inclusion of this notice) with the additional understanding
* that all recipients should regard themselves as participants in an
* ongoing research project and hence should feel obligated to report
* their experiences (good or bad) with these elementary function codes,
* using "sendbug 4bsd-bugs@BERKELEY", to the authors.
*/
#ifndef lint
static char sccsid[] = "@(#)cabs.c 1.2 (Berkeley) 8/21/85";
#endif not lint
/* 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 { double x, y;} z;
{
double hypot();
return(hypot(z.x,z.y));
}
/* 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.
*/
#ifdef VAX /* VAX D format */
/* static double */
/* r2p1hi = 2.4142135623730950345E0 , Hex 2^ 2 * .9A827999FCEF32 */
/* r2p1lo = 1.4349369327986523769E-17 , Hex 2^-55 * .84597D89B3754B */
/* sqrt2 = 1.4142135623730950622E0 ; Hex 2^ 1 * .B504F333F9DE65 */
static long r2p1hix[] = { 0x8279411a, 0xef3299fc};
static long r2p1lox[] = { 0x597d2484, 0x754b89b3};
static long sqrt2x[] = { 0x04f340b5, 0xde6533f9};
#define r2p1hi (*(double*)r2p1hix)
#define r2p1lo (*(double*)r2p1lox)
#define sqrt2 (*(double*)sqrt2x)
#else /* IEEE double format */
static double
r2p1hi = 2.4142135623730949234E0 , /*Hex 2^1 * 1.3504F333F9DE6 */
r2p1lo = 1.2537167179050217666E-16 , /*Hex 2^-53 * 1.21165F626CDD5 */
sqrt2 = 1.4142135623730951455E0 ; /*Hex 2^ 0 * 1.6A09E667F3BCD */
#endif
double hypot(x,y)
double x, y;
{
static double zero=0, one=1,
small=1.0E-18; /* fl(1+small)==1 */
static ibig=30; /* fl(1+2**(2*ibig))==1 */
double copysign(),scalb(),logb(),sqrt(),t,r;
int finite(), 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 */
else if(y!=y) return(y); /* x and y is NaN */
else return(copysign(y,one)); /* y is INF */
}
/* A faster but less accurate version of cabs(x,y) */
#if 0
double hypot(x,y)
double x, y;
{
static double zero=0, one=1;
small=1.0E-18; /* fl(1+small)==1 */
static ibig=30; /* fl(1+2**(2*ibig))==1 */
double copysign(),scalb(),logb(),sqrt(),temp;
int finite(), 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