* 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.
"@(#)cabs.c 1.2 (Berkeley) 8/21/85; 1.7 (ucb.elefunt) %G%";
* 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 :
struct { double x
, y
;} z
;
* 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 :
* 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:
* hypot = x + -----------------------------
* sqrt ( 1 + [x/y] ) + x/y
* hypot = x + --------------------------------------------------
* (sqrt(2)+1) + (x-y)/y + -----------------------------
* sqrt ( 1 + [x/y] ) + sqrt(2)
* 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.
* 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.
* 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
#if defined(vax)||defined(tahoe) /* VAX D format */
#define _0x(A,B) 0x/**/A/**/B
#define _0x(A,B) 0x/**/B/**/A
/* 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
[] = { _0x(8279,411a
), _0x(ef32
,99fc
)};
static long r2p1lox
[] = { _0x(597d
,2484), _0x(754b
,89b3
)};
static long sqrt2x
[] = { _0x(04f3
,40b5
), _0x(de65
,33f9
)};
#define r2p1hi (*(double*)r2p1hix)
#define r2p1lo (*(double*)r2p1lox)
#define sqrt2 (*(double*)sqrt2x)
#else /* defined(vax)||defined(tahoe) */
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 /* defined(vax)||defined(tahoe) */
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
;
if(x
== zero
) return(zero
);
if(exp
-(int)logb(y
) > ibig
)
/* raise inexact flag and return |x| */
{ one
+small
; return(x
); }
/* start computing sqrt(x^2 + y^2) */
else { /* 1 <= x/y <= 2 */
r
+=t
/(sqrt2
+sqrt(2.0+t
));
else if(y
==y
) /* y is +-INF */
return(y
); /* y is NaN and x is finite */
else if(x
==x
) /* x is +-INF */
return (copysign(x
,one
));
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 */
/* A faster but less accurate version of cabs(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
;
if(x
== zero
) return(zero
);
if(exp
-(int)logb(y
) > ibig
)
/* raise inexact flag and return |x| */
{ one
+small
; return(scalb(x
,exp
)); }
return(scalb(sqrt(x
*x
+y
*y
),exp
));
else if(y
==y
) /* y is +-INF */
return(y
); /* y is NaN and x is finite */
else if(x
==x
) /* x is +-INF */
return (copysign(x
,one
));
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 */