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