1 /*
2  * CDDL HEADER START
3  *
4  * The contents of this file are subject to the terms of the
5  * Common Development and Distribution License, Version 1.0 only
6  * (the "License").  You may not use this file except in compliance
7  * with the License.
8  *
9  * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
10  * or http://www.opensolaris.org/os/licensing.
11  * See the License for the specific language governing permissions
12  * and limitations under the License.
13  *
14  * When distributing Covered Code, include this CDDL HEADER in each
15  * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
16  * If applicable, add the following below this CDDL HEADER, with the
17  * fields enclosed by brackets "[]" replaced with your own identifying
18  * information: Portions Copyright [yyyy] [name of copyright owner]
19  *
20  * CDDL HEADER END
21  */
22 /*
23  * Copyright 2003 Sun Microsystems, Inc.  All rights reserved.
24  * Use is subject to license terms.
25  */
26 
27 #pragma ident	"%Z%%M%	%I%	%E% SMI"
28 
29 /*
30  * _D_cplx_div(z, w) returns z / w with infinities handled according
31  * to C99.
32  *
33  * If z and w are both finite and w is nonzero, _D_cplx_div(z, w)
34  * delivers the complex quotient q according to the usual formula:
35  * let a = Re(z), b = Im(z), c = Re(w), and d = Im(w); then q = x +
36  * I * y where x = (a * c + b * d) / r and y = (b * c - a * d) / r
37  * with r = c * c + d * d.  This implementation scales to avoid
38  * premature underflow or overflow.
39  *
40  * If z is neither NaN nor zero and w is zero, or if z is infinite
41  * and w is finite and nonzero, _D_cplx_div delivers an infinite
42  * result.  If z is finite and w is infinite, _D_cplx_div delivers
43  * a zero result.
44  *
45  * If z and w are both zero or both infinite, or if either z or w is
46  * a complex NaN, _D_cplx_div delivers NaN + I * NaN.  C99 doesn't
47  * specify these cases.
48  *
49  * This implementation can raise spurious underflow, overflow, in-
50  * valid operation, inexact, and division-by-zero exceptions.  C99
51  * allows this.
52  *
53  * Warning: Do not attempt to "optimize" this code by removing multi-
54  * plications by zero.
55  */
56 
57 #if !defined(sparc) && !defined(__sparc)
58 #error This code is for SPARC only
59 #endif
60 
61 static union {
62 	int	i[2];
63 	double	d;
64 } inf = {
65 	0x7ff00000, 0
66 };
67 
68 /*
69  * Return +1 if x is +Inf, -1 if x is -Inf, and 0 otherwise
70  */
71 static int
testinf(double x)72 testinf(double x)
73 {
74 	union {
75 		int	i[2];
76 		double	d;
77 	} xx;
78 
79 	xx.d = x;
80 	return (((((xx.i[0] << 1) - 0xffe00000) | xx.i[1]) == 0)?
81 		(1 | (xx.i[0] >> 31)) : 0);
82 }
83 
84 double _Complex
_D_cplx_div(double _Complex z,double _Complex w)85 _D_cplx_div(double _Complex z, double _Complex w)
86 {
87 	double _Complex	v;
88 	union {
89 		int	i[2];
90 		double	d;
91 	} aa, bb, cc, dd, ss;
92 	double		a, b, c, d, r;
93 	int		ha, hb, hc, hd, hz, hw, hs, i, j;
94 
95 	/*
96 	 * The following is equivalent to
97 	 *
98 	 *  a = creal(z); b = cimag(z);
99 	 *  c = creal(w); d = cimag(w);
100 	 */
101 	a = ((double *)&z)[0];
102 	b = ((double *)&z)[1];
103 	c = ((double *)&w)[0];
104 	d = ((double *)&w)[1];
105 
106 	/* extract high-order words to estimate |z| and |w| */
107 	aa.d = a;
108 	bb.d = b;
109 	ha = aa.i[0] & ~0x80000000;
110 	hb = bb.i[0] & ~0x80000000;
111 	hz = (ha > hb)? ha : hb;
112 
113 	cc.d = c;
114 	dd.d = d;
115 	hc = cc.i[0] & ~0x80000000;
116 	hd = dd.i[0] & ~0x80000000;
117 	hw = (hc > hd)? hc : hd;
118 
119 	/* check for special cases */
120 	if (hw >= 0x7ff00000) { /* w is inf or nan */
121 		r = 0.0;
122 		i = testinf(c);
123 		j = testinf(d);
124 		if (i | j) { /* w is infinite */
125 			/*
126 			 * "factor out" infinity, being careful to preserve
127 			 * signs of finite values
128 			 */
129 			c = i? i : ((cc.i[0] < 0)? -0.0 : 0.0);
130 			d = j? j : ((dd.i[0] < 0)? -0.0 : 0.0);
131 			if (hz >= 0x7fe00000) {
132 				/* scale to avoid overflow below */
133 				c *= 0.5;
134 				d *= 0.5;
135 			}
136 		}
137 		((double *)&v)[0] = (a * c + b * d) * r;
138 		((double *)&v)[1] = (b * c - a * d) * r;
139 		return (v);
140 	}
141 
142 	if (hw < 0x00100000) {
143 		/*
144 		 * This nonsense is needed to work around some SPARC
145 		 * implementations of nonstandard mode; if both parts
146 		 * of w are subnormal, multiply them by one to force
147 		 * them to be flushed to zero when nonstandard mode
148 		 * is enabled.  Sheesh.
149 		 */
150 		cc.d = c = c * 1.0;
151 		dd.d = d = d * 1.0;
152 		hc = cc.i[0] & ~0x80000000;
153 		hd = dd.i[0] & ~0x80000000;
154 		hw = (hc > hd)? hc : hd;
155 	}
156 
157 	if (hw == 0 && (cc.i[1] | dd.i[1]) == 0) {
158 		/* w is zero; multiply z by 1/Re(w) - I * Im(w) */
159 		c = 1.0 / c;
160 		i = testinf(a);
161 		j = testinf(b);
162 		if (i | j) { /* z is infinite */
163 			a = i;
164 			b = j;
165 		}
166 		((double *)&v)[0] = a * c + b * d;
167 		((double *)&v)[1] = b * c - a * d;
168 		return (v);
169 	}
170 
171 	if (hz >= 0x7ff00000) { /* z is inf or nan */
172 		r = 1.0;
173 		i = testinf(a);
174 		j = testinf(b);
175 		if (i | j) { /* z is infinite */
176 			a = i;
177 			b = j;
178 			r = inf.d;
179 		}
180 		((double *)&v)[0] = (a * c + b * d) * r;
181 		((double *)&v)[1] = (b * c - a * d) * r;
182 		return (v);
183 	}
184 
185 	/*
186 	 * Scale c and d to compute 1/|w|^2 and the real and imaginary
187 	 * parts of the quotient.
188 	 *
189 	 * Note that for any s, if we let c' = sc, d' = sd, c'' = sc',
190 	 * and d'' = sd', then
191 	 *
192 	 *  (ac'' + bd'') / (c'^2 + d'^2) = (ac + bd) / (c^2 + d^2)
193 	 *
194 	 * and similarly for the imaginary part of the quotient.  We want
195 	 * to choose s such that (i) r := 1/(c'^2 + d'^2) can be computed
196 	 * without overflow or harmful underflow, and (ii) (ac'' + bd'')
197 	 * and (bc'' - ad'') can be computed without spurious overflow or
198 	 * harmful underflow.  To avoid unnecessary rounding, we restrict
199 	 * s to a power of two.
200 	 *
201 	 * To satisfy (i), we need to choose s such that max(|c'|,|d'|)
202 	 * is not too far from one.  To satisfy (ii), we need to choose
203 	 * s such that max(|c''|,|d''|) is also not too far from one.
204 	 * There is some leeway in our choice, but to keep the logic
205 	 * from getting overly complicated, we simply attempt to roughly
206 	 * balance these constraints by choosing s so as to make r about
207 	 * the same size as max(|c''|,|d''|).  This corresponds to choos-
208 	 * ing s to be a power of two near |w|^(-3/4).
209 	 *
210 	 * Regarding overflow, observe that if max(|c''|,|d''|) <= 1/2,
211 	 * then the computation of (ac'' + bd'') and (bc'' - ad'') can-
212 	 * not overflow; otherwise, the computation of either of these
213 	 * values can only incur overflow if the true result would be
214 	 * within a factor of two of the overflow threshold.  In other
215 	 * words, if we bias the choice of s such that at least one of
216 	 *
217 	 *  max(|c''|,|d''|) <= 1/2   or   r >= 2
218 	 *
219 	 * always holds, then no undeserved overflow can occur.
220 	 *
221 	 * To cope with underflow, note that if r < 2^-53, then any
222 	 * intermediate results that underflow are insignificant; either
223 	 * they will be added to normal results, rendering the under-
224 	 * flow no worse than ordinary roundoff, or they will contribute
225 	 * to a final result that is smaller than the smallest subnormal
226 	 * number.  Therefore, we need only modify the preceding logic
227 	 * when z is very small and w is not too far from one.  In that
228 	 * case, we can reduce the effect of any intermediate underflow
229 	 * to no worse than ordinary roundoff error by choosing s so as
230 	 * to make max(|c''|,|d''|) large enough that at least one of
231 	 * (ac'' + bd'') or (bc'' - ad'') is normal.
232 	 */
233 	hs = (((hw >> 2) - hw) + 0x6fd7ffff) & 0xfff00000;
234 	if (hz < 0x07200000) { /* |z| < 2^-909 */
235 		if (((hw - 0x32800000) | (0x47100000 - hw)) >= 0)
236 			hs = (((0x47100000 - hw) >> 1) & 0xfff00000)
237 				+ 0x3ff00000;
238 	}
239 	ss.i[0] = hs;
240 	ss.i[1] = 0;
241 
242 	c *= ss.d;
243 	d *= ss.d;
244 	r = 1.0 / (c * c + d * d);
245 
246 	c *= ss.d;
247 	d *= ss.d;
248 	((double *)&v)[0] = (a * c + b * d) * r;
249 	((double *)&v)[1] = (b * c - a * d) * r;
250 	return (v);
251 }
252