[unix-history] / usr / src / lib / libm / ieee / cabs.c

/*
 * 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 { double x, y;} z;
{
	return hypot(z.x,z.y);
}

double
z_abs(z)
struct { double x,y;} *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
Commit	Line	Data
98cc7428	1	/*
f9fea09f	2	* Copyright (c) 1985 Regents of the University of California.
98cc7428 KB	3	* All rights reserved.
98cc7428 KB	4	*
af359dea C	5	* Redistribution and use in source and binary forms, with or without
	6	* modification, are permitted provided that the following conditions
	7	* are met:
	8	* 1. Redistributions of source code must retain the above copyright
	9	* notice, this list of conditions and the following disclaimer.
	10	* 2. Redistributions in binary form must reproduce the above copyright
	11	* notice, this list of conditions and the following disclaimer in the
	12	* documentation and/or other materials provided with the distribution.
	13	* 3. All advertising materials mentioning features or use of this software
	14	* must display the following acknowledgement:
	15	* This product includes software developed by the University of
	16	* California, Berkeley and its contributors.
	17	* 4. Neither the name of the University nor the names of its contributors
	18	* may be used to endorse or promote products derived from this software
	19	* without specific prior written permission.
98cc7428	20	*
af359dea C	21	* THIS SOFTWARE IS PROVIDED BY THE REGENTS AND CONTRIBUTORS ``AS IS'' AND
	22	* ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
	23	* IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
	24	* ARE DISCLAIMED. IN NO EVENT SHALL THE REGENTS OR CONTRIBUTORS BE LIABLE
	25	* FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
	26	* DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
	27	* OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
	28	* HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
	29	* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
	30	* OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
	31	* SUCH DAMAGE.
f9fea09f ZAL	32	*/
	33
	34	#ifndef lint
af359dea	35	static char sccsid[] = "@(#)cabs.c 5.6 (Berkeley) 10/9/90";
98cc7428	36	#endif /* not lint */
f9fea09f	37
f9fea09f ZAL	38	/* HYPOT(X,Y)
	39	* RETURN THE SQUARE ROOT OF X^2 + Y^2 WHERE Z=X+iY
	40	* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
	41	* CODED IN C BY K.C. NG, 11/28/84;
	42	* REVISED BY K.C. NG, 7/12/85.
	43	*
	44	* Required system supported functions :
	45	* copysign(x,y)
	46	* finite(x)
	47	* scalb(x,N)
	48	* sqrt(x)
	49	*
	50	* Method :
	51	* 1. replace x by \|x\| and y by \|y\|, and swap x and
	52	* y if y > x (hence x is never smaller than y).
	53	* 2. Hypot(x,y) is computed by:
	54	* Case I, x/y > 2
	55	*
	56	* y
	57	* hypot = x + -----------------------------
	58	* 2
	59	* sqrt ( 1 + [x/y] ) + x/y
	60	*
	61	* Case II, x/y <= 2
	62	* y
	63	* hypot = x + --------------------------------------------------
	64	* 2
	65	* [x/y] - 2
	66	* (sqrt(2)+1) + (x-y)/y + -----------------------------
	67	* 2
	68	* sqrt ( 1 + [x/y] ) + sqrt(2)
	69	*
	70	*
	71	*
	72	* Special cases:
	73	* hypot(x,y) is INF if x or y is +INF or -INF; else
	74	* hypot(x,y) is NAN if x or y is NAN.
	75	*
	76	* Accuracy:
	77	* hypot(x,y) returns the sqrt(x^2+y^2) with error less than 1 ulps (units
	78	* in the last place). See Kahan's "Interval Arithmetic Options in the
	79	* Proposed IEEE Floating Point Arithmetic Standard", Interval Mathematics
	80	* 1980, Edited by Karl L.E. Nickel, pp 99-128. (A faster but less accurate
	81	* code follows in comments.) In a test run with 500,000 random arguments
	82	* on a VAX, the maximum observed error was .959 ulps.
	83	*
	84	* Constants:
	85	* The hexadecimal values are the intended ones for the following constants.
	86	* The decimal values may be used, provided that the compiler will convert
	87	* from decimal to binary accurately enough to produce the hexadecimal values
	88	* shown.
	89	*/
9eda3584	90	#include "mathimpl.h"
f9fea09f	91
9eda3584 KB	92	vc(r2p1hi, 2.4142135623730950345E0 ,8279,411a,ef32,99fc, 2, .9A827999FCEF32)
	93	vc(r2p1lo, 1.4349369327986523769E-17 ,597d,2484,754b,89b3, -55, .84597D89B3754B)
	94	vc(sqrt2, 1.4142135623730950622E0 ,04f3,40b5,de65,33f9, 1, .B504F333F9DE65)
	95
	96	ic(r2p1hi, 2.4142135623730949234E0 , 1, 1.3504F333F9DE6)
	97	ic(r2p1lo, 1.2537167179050217666E-16 , -53, 1.21165F626CDD5)
	98	ic(sqrt2, 1.4142135623730951455E0 , 0, 1.6A09E667F3BCD)
	99
	100	#ifdef vccast
	101	#define r2p1hi vccast(r2p1hi)
	102	#define r2p1lo vccast(r2p1lo)
	103	#define sqrt2 vccast(sqrt2)
	104	#endif
f9fea09f	105
74920388 ZAL	106	double
74920388 ZAL	107	hypot(x,y)
f9fea09f ZAL	108	double x, y;
f9fea09f ZAL	109	{
9eda3584	110	static const double zero=0, one=1,
f9fea09f	111	small=1.0E-18; /* fl(1+small)==1 */
9eda3584 KB	112	static const ibig=30; /* fl(1+2*(2ibig))==1 */
	113	double t,r;
	114	int exp;
f9fea09f ZAL	115
	116	if(finite(x))
	117	if(finite(y))
	118	{
	119	x=copysign(x,one);
	120	y=copysign(y,one);
	121	if(y > x)
	122	{ t=x; x=y; y=t; }
	123	if(x == zero) return(zero);
	124	if(y == zero) return(x);
	125	exp= logb(x);
	126	if(exp-(int)logb(y) > ibig )
	127	/* raise inexact flag and return \|x\| */
	128	{ one+small; return(x); }
	129
	130	/* start computing sqrt(x^2 + y^2) */
	131	r=x-y;
	132	if(r>y) { /* x/y > 2 */
	133	r=x/y;
	134	r=r+sqrt(one+r*r); }
	135	else { /* 1 <= x/y <= 2 */
	136	r/=y; t=r*(r+2.0);
	137	r+=t/(sqrt2+sqrt(2.0+t));
	138	r+=r2p1lo; r+=r2p1hi; }
	139
	140	r=y/r;
	141	return(x+r);
	142
	143	}
	144
	145	else if(y==y) /* y is +-INF */
	146	return(copysign(y,one));
	147	else
	148	return(y); /* y is NaN and x is finite */
	149
	150	else if(x==x) /* x is +-INF */
	151	return (copysign(x,one));
	152	else if(finite(y))
	153	return(x); /* x is NaN, y is finite */
859dc438	154	#if !defined(vax)&&!defined(tahoe)
f9fea09f	155	else if(y!=y) return(y); /* x and y is NaN */
859dc438	156	#endif /* !defined(vax)&&!defined(tahoe) */
f9fea09f ZAL	157	else return(copysign(y,one)); /* y is INF */
	158	}
	159
74920388 ZAL	160	/* CABS(Z)
	161	* RETURN THE ABSOLUTE VALUE OF THE COMPLEX NUMBER Z = X + iY
	162	* DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
	163	* CODED IN C BY K.C. NG, 11/28/84.
	164	* REVISED BY K.C. NG, 7/12/85.
	165	*
	166	* Required kernel function :
	167	* hypot(x,y)
	168	*
	169	* Method :
	170	* cabs(z) = hypot(x,y) .
	171	*/
	172
	173	double
	174	cabs(z)
	175	struct { double x, y;} z;
	176	{
	177	return hypot(z.x,z.y);
	178	}
	179
	180	double
	181	z_abs(z)
	182	struct { double x,y;} *z;
	183	{
	184	return hypot(z->x,z->y);
	185	}
	186
f9fea09f ZAL	187	/* A faster but less accurate version of cabs(x,y) */
	188	#if 0
	189	double hypot(x,y)
	190	double x, y;
	191	{
9eda3584	192	static const double zero=0, one=1;
f9fea09f	193	small=1.0E-18; /* fl(1+small)==1 */
9eda3584 KB	194	static const ibig=30; /* fl(1+2*(2ibig))==1 */
	195	double temp;
	196	int exp;
f9fea09f ZAL	197
	198	if(finite(x))
	199	if(finite(y))
	200	{
	201	x=copysign(x,one);
	202	y=copysign(y,one);
	203	if(y > x)
	204	{ temp=x; x=y; y=temp; }
	205	if(x == zero) return(zero);
	206	if(y == zero) return(x);
	207	exp= logb(x);
	208	x=scalb(x,-exp);
	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(xx+yy),exp));
	214	}
	215
	216	else if(y==y) /* y is +-INF */
	217	return(copysign(y,one));
	218	else
	219	return(y); /* y is NaN and x is finite */
	220
	221	else if(x==x) /* x is +-INF */
	222	return (copysign(x,one));
	223	else if(finite(y))
	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 */
	227	}
	228	#endif