xref: /csrg-svn/lib/libm/ieee/cabs.c (revision 24719)
1 /*
2  * Copyright (c) 1985 Regents of the University of California.
3  *
4  * Use and reproduction of this software are granted  in  accordance  with
5  * the terms and conditions specified in  the  Berkeley  Software  License
6  * Agreement (in particular, this entails acknowledgement of the programs'
7  * source, and inclusion of this notice) with the additional understanding
8  * that  all  recipients  should regard themselves as participants  in  an
9  * ongoing  research  project and hence should  feel  obligated  to report
10  * their  experiences (good or bad) with these elementary function  codes,
11  * using "sendbug 4bsd-bugs@BERKELEY", to the authors.
12  */
13 
14 #ifndef lint
15 static char sccsid[] =
16 "@(#)cabs.c	1.2 (Berkeley) 8/21/85; 1.2 (ucb.elefunt) 09/12/85";
17 #endif not lint
18 
19 /* CABS(Z)
20  * RETURN THE ABSOLUTE VALUE OF THE COMPLEX NUMBER  Z = X + iY
21  * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
22  * CODED IN C BY K.C. NG, 11/28/84.
23  * REVISED BY K.C. NG, 7/12/85.
24  *
25  * Required kernel function :
26  *	hypot(x,y)
27  *
28  * Method :
29  *	cabs(z) = hypot(x,y) .
30  */
31 
32 double cabs(z)
33 struct { double x, y;} z;
34 {
35 	double hypot();
36 	return(hypot(z.x,z.y));
37 }
38 
39 
40 /* HYPOT(X,Y)
41  * RETURN THE SQUARE ROOT OF X^2 + Y^2  WHERE Z=X+iY
42  * DOUBLE PRECISION (VAX D format 56 bits, IEEE DOUBLE 53 BITS)
43  * CODED IN C BY K.C. NG, 11/28/84;
44  * REVISED BY K.C. NG, 7/12/85.
45  *
46  * Required system supported functions :
47  *	copysign(x,y)
48  *	finite(x)
49  *	scalb(x,N)
50  *	sqrt(x)
51  *
52  * Method :
53  *	1. replace x by |x| and y by |y|, and swap x and
54  *	   y if y > x (hence x is never smaller than y).
55  *	2. Hypot(x,y) is computed by:
56  *	   Case I, x/y > 2
57  *
58  *				       y
59  *		hypot = x + -----------------------------
60  *			 		    2
61  *			    sqrt ( 1 + [x/y]  )  +  x/y
62  *
63  *	   Case II, x/y <= 2
64  *				                   y
65  *		hypot = x + --------------------------------------------------
66  *				          		     2
67  *				     			[x/y]   -  2
68  *			   (sqrt(2)+1) + (x-y)/y + -----------------------------
69  *			 		    			  2
70  *			    			  sqrt ( 1 + [x/y]  )  + sqrt(2)
71  *
72  *
73  *
74  * Special cases:
75  *	hypot(x,y) is INF if x or y is +INF or -INF; else
76  *	hypot(x,y) is NAN if x or y is NAN.
77  *
78  * Accuracy:
79  * 	hypot(x,y) returns the sqrt(x^2+y^2) with error less than 1 ulps (units
80  *	in the last place). See Kahan's "Interval Arithmetic Options in the
81  *	Proposed IEEE Floating Point Arithmetic Standard", Interval Mathematics
82  *      1980, Edited by Karl L.E. Nickel, pp 99-128. (A faster but less accurate
83  *	code follows in	comments.) In a test run with 500,000 random arguments
84  *	on a VAX, the maximum observed error was .959 ulps.
85  *
86  * Constants:
87  * The hexadecimal values are the intended ones for the following constants.
88  * The decimal values may be used, provided that the compiler will convert
89  * from decimal to binary accurately enough to produce the hexadecimal values
90  * shown.
91  */
92 
93 #ifdef VAX	/* VAX D format */
94 /* static double */
95 /* r2p1hi =  2.4142135623730950345E0     , Hex  2^  2   *  .9A827999FCEF32 */
96 /* r2p1lo =  1.4349369327986523769E-17   , Hex  2^-55   *  .84597D89B3754B */
97 /* sqrt2  =  1.4142135623730950622E0     ; Hex  2^  1   *  .B504F333F9DE65 */
98 static long    r2p1hix[] = { 0x8279411a, 0xef3299fc};
99 static long    r2p1lox[] = { 0x597d2484, 0x754b89b3};
100 static long     sqrt2x[] = { 0x04f340b5, 0xde6533f9};
101 #define   r2p1hi    (*(double*)r2p1hix)
102 #define   r2p1lo    (*(double*)r2p1lox)
103 #define    sqrt2    (*(double*)sqrt2x)
104 #else		/* IEEE double format */
105 static double
106 r2p1hi =  2.4142135623730949234E0     , /*Hex  2^1     *  1.3504F333F9DE6 */
107 r2p1lo =  1.2537167179050217666E-16   , /*Hex  2^-53   *  1.21165F626CDD5 */
108 sqrt2  =  1.4142135623730951455E0     ; /*Hex  2^  0   *  1.6A09E667F3BCD */
109 #endif
110 
111 double hypot(x,y)
112 double x, y;
113 {
114 	static double zero=0, one=1,
115 		      small=1.0E-18;	/* fl(1+small)==1 */
116 	static ibig=30;	/* fl(1+2**(2*ibig))==1 */
117 	double copysign(),scalb(),logb(),sqrt(),t,r;
118 	int finite(), exp;
119 
120 	if(finite(x))
121 	    if(finite(y))
122 	    {
123 		x=copysign(x,one);
124 		y=copysign(y,one);
125 		if(y > x)
126 		    { t=x; x=y; y=t; }
127 		if(x == zero) return(zero);
128 		if(y == zero) return(x);
129 		exp= logb(x);
130 		if(exp-(int)logb(y) > ibig )
131 			/* raise inexact flag and return |x| */
132 		   { one+small; return(x); }
133 
134 	    /* start computing sqrt(x^2 + y^2) */
135 		r=x-y;
136 		if(r>y) { 	/* x/y > 2 */
137 		    r=x/y;
138 		    r=r+sqrt(one+r*r); }
139 		else {		/* 1 <= x/y <= 2 */
140 		    r/=y; t=r*(r+2.0);
141 		    r+=t/(sqrt2+sqrt(2.0+t));
142 		    r+=r2p1lo; r+=r2p1hi; }
143 
144 		r=y/r;
145 		return(x+r);
146 
147 	    }
148 
149 	    else if(y==y)   	   /* y is +-INF */
150 		     return(copysign(y,one));
151 	    else
152 		     return(y);	   /* y is NaN and x is finite */
153 
154 	else if(x==x) 		   /* x is +-INF */
155 	         return (copysign(x,one));
156 	else if(finite(y))
157 	         return(x);		   /* x is NaN, y is finite */
158 	else if(y!=y) return(y);  /* x and y is NaN */
159 	else return(copysign(y,one));   /* y is INF */
160 }
161 
162 /* A faster but less accurate version of cabs(x,y) */
163 #if 0
164 double hypot(x,y)
165 double x, y;
166 {
167 	static double zero=0, one=1;
168 		      small=1.0E-18;	/* fl(1+small)==1 */
169 	static ibig=30;	/* fl(1+2**(2*ibig))==1 */
170 	double copysign(),scalb(),logb(),sqrt(),temp;
171 	int finite(), exp;
172 
173 	if(finite(x))
174 	    if(finite(y))
175 	    {
176 		x=copysign(x,one);
177 		y=copysign(y,one);
178 		if(y > x)
179 		    { temp=x; x=y; y=temp; }
180 		if(x == zero) return(zero);
181 		if(y == zero) return(x);
182 		exp= logb(x);
183 		x=scalb(x,-exp);
184 		if(exp-(int)logb(y) > ibig )
185 			/* raise inexact flag and return |x| */
186 		   { one+small; return(scalb(x,exp)); }
187 		else y=scalb(y,-exp);
188 		return(scalb(sqrt(x*x+y*y),exp));
189 	    }
190 
191 	    else if(y==y)   	   /* y is +-INF */
192 		     return(copysign(y,one));
193 	    else
194 		     return(y);	   /* y is NaN and x is finite */
195 
196 	else if(x==x) 		   /* x is +-INF */
197 	         return (copysign(x,one));
198 	else if(finite(y))
199 	         return(x);		   /* x is NaN, y is finite */
200 	else if(y!=y) return(y);  	/* x and y is NaN */
201 	else return(copysign(y,one));   /* y is INF */
202 }
203 #endif
204