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 (the "License").
6 * You may not use this file except in compliance with the License.
7 *
8 * You can obtain a copy of the license at usr/src/OPENSOLARIS.LICENSE
9 * or http://www.opensolaris.org/os/licensing.
10 * See the License for the specific language governing permissions
11 * and limitations under the License.
12 *
13 * When distributing Covered Code, include this CDDL HEADER in each
14 * file and include the License file at usr/src/OPENSOLARIS.LICENSE.
15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
17 * information: Portions Copyright [yyyy] [name of copyright owner]
18 *
19 * CDDL HEADER END
20 */
21
22/*
23 * Copyright 2011 Nexenta Systems, Inc.  All rights reserved.
24 */
25/*
26 * Copyright 2006 Sun Microsystems, Inc.  All rights reserved.
27 * Use is subject to license terms.
28 */
29
30#pragma weak __cpow = cpow
31
32/* INDENT OFF */
33/*
34 * dcomplex cpow(dcomplex z);
35 *
36 * z**w analytically equivalent to
37 *
38 * cpow(z,w) = cexp(w clog(z))
39 *
40 * Let z = x+iy, w = u+iv.
41 * Since
42 *                        _________
43 *                       / 2    2            -1   y
44 *     log(x+iy) = log(\/ x  + y    ) + i tan   (---)
45 *                                                x
46 *
47 *                  1       2    2         -1   y
48 *               = --- log(x  + y ) + i tan   (---)
49 *                  2                           x
50 *                       u       2    2         -1  y
51 * (u+iv)* log(x+iy) =  --- log(x  + y ) - v tan  (---)  +          (1)
52 *                       2                          x
53 *
54 *                            v       2    2         -1  y
55 *                     i * [ --- log(x  + y ) + u tan  (---) ]      (2)
56 *                            2                          x
57 *
58 *                   = r + i q
59 *
60 * Therefore,
61 *      w     r+iq    r
62 *     z  =  e     = e  (cos(q)+i*sin(q))
63 *                                   _______
64 *                                  / 2   2
65 *       r                        \/ x + y     -v*atan2(y,x)
66 * Here e  can be expressed as:  u          * e
67 *
68 * Special cases (in the order of appearance):
69 *      1.  (anything) ** 0  is 1
70 *      2.  (anything) ** 1  is itself
71 *      3.  When v = 0, y = 0:
72 *            If x is finite and negative, and u is finite, then
73 *               x ** u = exp(u*pi i) * pow(|x|, u);
74 *            otherwise,
75 *               x ** u = pow(x, u);
76 *      4.  When v = 0, x = 0 or |x| = |y| or x is inf or y is inf:
77 *               (x + y i) ** u = r * exp(q i)
78 *          where
79 *               r = hypot(x,y) ** u
80 *               q = u * atan2pi(y, x)
81 *
82 *      5.  otherwise, z**w is NAN if any x, y, u, v is a Nan or inf
83 *
84 *      Note: many results of special cases are obtained in terms of
85 *      polar coordinate. In the conversion from polar to rectangle:
86 *                  r exp(q i) = r * cos(q) + r * sin(q) i,
87 *      we regard r * 0 is 0 except when r is a NaN.
88 */
89/* INDENT ON */
90
91#include "libm.h"	/* atan2/exp/fabs/hypot/log/pow/scalbn */
92			/* atan2pi/exp2/sincos/sincospi/__k_clog_r/__k_atan2 */
93#include "complex_wrapper.h"
94
95extern void sincospi(double, double *, double *);
96
97static const double
98	huge = 1e300,
99	tiny = 1e-300,
100	invln2 = 1.44269504088896338700e+00,
101	ln2hi = 6.93147180369123816490e-01,   /* 0x3fe62e42, 0xfee00000 */
102	ln2lo = 1.90821492927058770002e-10,   /* 0x3dea39ef, 0x35793c76 */
103	one = 1.0,
104	zero = 0.0;
105
106static const int hiinf = 0x7ff00000;
107extern double atan2pi(double, double);
108
109/*
110 * Assuming |t[0]| > |t[1]| and |t[2]| > |t[3]|, sum4fp subroutine
111 * compute t[0] + t[1] + t[2] + t[3] into two double fp numbers.
112 */
113static double
114sum4fp(double ta[], double *w) {
115	double t1, t2, t3, t4, w1, w2, t;
116	t1 = ta[0]; t2 = ta[1]; t3 = ta[2]; t4 = ta[3];
117	/*
118	 * Rearrange ti so that |t1| >= |t2| >= |t3| >= |t4|
119	 */
120	if (fabs(t4) > fabs(t1)) {
121		t = t1; t1 = t3; t3 = t;
122		t = t2; t2 = t4; t4 = t;
123	} else if (fabs(t3) > fabs(t1)) {
124		t = t1; t1 = t3;
125		if (fabs(t4) > fabs(t2)) {
126			t3 = t4; t4 = t2; t2 = t;
127		} else {
128			t3 = t2; t2 = t;
129		}
130	} else if (fabs(t3) > fabs(t2)) {
131		t = t2; t2 = t3;
132		if (fabs(t4) > fabs(t2)) {
133			t3 = t4; t4 = t;
134		} else
135			t3 = t;
136	}
137	/* summing r = t1 + t2 + t3 + t4 to w1 + w2 */
138	w1 = t3 + t4;
139	w2 = t4 - (w1 - t3);
140	t  = t2 + w1;
141	w2 += w1 - (t - t2);
142	w1 = t + w2;
143	w2 += t - w1;
144	t  = t1 + w1;
145	w2 += w1 - (t - t1);
146	w1 = t + w2;
147	*w = w2 - (w1 - t);
148	return (w1);
149}
150
151dcomplex
152cpow(dcomplex z, dcomplex w) {
153	dcomplex ans;
154	double x, y, u, v, t, c, s, r, x2, y2;
155	double b[4], t1, t2, t3, t4, w1, w2, u1, v1, x1, y1;
156	int ix, iy, hx, lx, hy, ly, hv, hu, iu, iv, lu, lv;
157	int i, j, k;
158
159	x = D_RE(z);
160	y = D_IM(z);
161	u = D_RE(w);
162	v = D_IM(w);
163	hx = ((int *) &x)[HIWORD];
164	lx = ((int *) &x)[LOWORD];
165	hy = ((int *) &y)[HIWORD];
166	ly = ((int *) &y)[LOWORD];
167	hu = ((int *) &u)[HIWORD];
168	lu = ((int *) &u)[LOWORD];
169	hv = ((int *) &v)[HIWORD];
170	lv = ((int *) &v)[LOWORD];
171	ix = hx & 0x7fffffff;
172	iy = hy & 0x7fffffff;
173	iu = hu & 0x7fffffff;
174	iv = hv & 0x7fffffff;
175
176	j = 0;
177	if ((iv | lv) == 0) {	/* z**(real) */
178		if (((hu - 0x3ff00000) | lu) == 0) {	/* z ** 1 = z */
179			D_RE(ans) = x;
180			D_IM(ans) = y;
181		} else if ((iu | lu) == 0) {	/* z ** 0 = 1 */
182			D_RE(ans) = one;
183			D_IM(ans) = zero;
184		} else if ((iy | ly) == 0) {	/* (real)**(real) */
185			D_IM(ans) = zero;
186			if (hx < 0 && ix < hiinf && iu < hiinf) {
187				/* -x ** u  is exp(i*pi*u)*pow(x,u) */
188				r = pow(-x, u);
189				sincospi(u, &s, &c);
190				D_RE(ans) = (c == zero)? c: c * r;
191				D_IM(ans) = (s == zero)? s: s * r;
192			} else
193				D_RE(ans) = pow(x, u);
194		} else if (((ix | lx) == 0) || ix >= hiinf || iy >= hiinf) {
195			if (isnan(x) || isnan(y) || isnan(u))
196				D_RE(ans) = D_IM(ans) = x + y + u;
197			else {
198				if ((ix | lx) == 0)
199					r = fabs(y);
200				else
201					r = fabs(x) + fabs(y);
202				t = atan2pi(y, x);
203				sincospi(t * u, &s, &c);
204				D_RE(ans) = (c == zero)? c: c * r;
205				D_IM(ans) = (s == zero)? s: s * r;
206			}
207		} else if (((ix - iy) | (lx - ly)) == 0) {   /* |x| = |y| */
208			if (hx >= 0) {
209				t = (hy >= 0)? 0.25 : -0.25;
210				sincospi(t * u, &s, &c);
211			} else if ((lu & 3) == 0) {
212				t = (hy >= 0)? 0.75 : -0.75;
213				sincospi(t * u, &s, &c);
214			} else {
215				r = (hy >= 0)? u : -u;
216				t = -0.25 * r;
217				w1 = r + t;
218				w2 = t - (w1 - r);
219				sincospi(w1, &t1, &t2);
220				sincospi(w2, &t3, &t4);
221				s = t1 * t4 + t3 * t2;
222				c = t2 * t4 - t1 * t3;
223			}
224			if (ix < 0x3fe00000)	/* |x| < 1/2 */
225				r = pow(fabs(x + x), u) * exp2(-0.5 * u);
226			else if (ix >= 0x3ff00000 || iu < 0x408ff800)
227				/* |x| >= 1 or |u| < 1023 */
228				r = pow(fabs(x), u) * exp2(0.5 * u);
229			else   /* special treatment */
230				j = 2;
231			if (j == 0) {
232				D_RE(ans) = (c == zero)? c: c * r;
233				D_IM(ans) = (s == zero)? s: s * r;
234			}
235		} else
236			j = 1;
237		if (j == 0)
238			return (ans);
239	}
240	if (iu >= hiinf || iv >= hiinf || ix >= hiinf || iy >= hiinf) {
241		/*
242		 * non-zero imag part(s) with inf component(s) yields NaN
243		 */
244		t = fabs(x) + fabs(y) + fabs(u) + fabs(v);
245		D_RE(ans) = D_IM(ans) = t - t;
246	} else {
247		k = 0;	/* no scaling */
248		if (iu > 0x7f000000 || iv > 0x7f000000) {
249			u *= .0009765625; /* scale 2**-10 to avoid overflow */
250			v *= .0009765625;
251			k = 1;	/* scale by 2**-10 */
252		}
253		/*
254		 * Use similated higher precision arithmetic to compute:
255		 * r = u * log(hypot(x, y)) - v * atan2(y, x)
256		 * q = u * atan2(y, x) + v * log(hypot(x, y))
257		 */
258		t1 = __k_clog_r(x, y, &t2);
259		t3 = __k_atan2(y, x, &t4);
260		x1 = t1;
261		y1 = t3;
262		u1 = u;
263		v1 = v;
264		((int *) &u1)[LOWORD] &= 0xf8000000;
265		((int *) &v1)[LOWORD] &= 0xf8000000;
266		((int *) &x1)[LOWORD] &= 0xf8000000;
267		((int *) &y1)[LOWORD] &= 0xf8000000;
268		x2 = t2 - (x1 - t1);	/* log(hypot(x,y)) = x1 + x2 */
269		y2 = t4 - (y1 - t3);	/* atan2(y,x) = y1 + y2 */
270		/* compute q = u * atan2(y, x) + v * log(hypot(x, y)) */
271		if (j != 2) {
272			b[0] = u1 * y1;
273			b[1] = (u - u1) * y1 + u * y2;
274			if (j == 1) {	/* v = 0 */
275				w1 = b[0] + b[1];
276				w2 = b[1] - (w1 - b[0]);
277			} else {
278				b[2] = v1 * x1;
279				b[3] = (v - v1) * x1 + v * x2;
280				w1 = sum4fp(b, &w2);
281			}
282			sincos(w1, &t1, &t2);
283			sincos(w2, &t3, &t4);
284			s = t1 * t4 + t3 * t2;
285			c = t2 * t4 - t1 * t3;
286			if (k == 1)
287			/*
288			 * square (cos(q) + i sin(q)) k times to get
289			 * (cos(2^k * q + i sin(2^k * q)
290			 */
291				for (i = 0; i < 10; i++) {
292					t1 = s * c;
293					c = (c + s) * (c - s);
294					s = t1 + t1;
295				}
296		}
297		/* compute r = u * (t1, t2) - v * (t3, t4) */
298		b[0] = u1 * x1;
299		b[1] = (u - u1) * x1 + u * x2;
300		if (j == 1) {	/* v = 0 */
301			w1 = b[0] + b[1];
302			w2 = b[1] - (w1 - b[0]);
303		} else {
304			b[2] = -v1 * y1;
305			b[3] = (v1 - v) * y1 - v * y2;
306			w1 = sum4fp(b, &w2);
307		}
308		/* check over/underflow for exp(w1 + w2) */
309		if (k && fabs(w1) < 1000.0) {
310			w1 *= 1024; w2 *= 1024; k = 0;
311		}
312		hx = ((int *) &w1)[HIWORD];
313		lx = ((int *) &w1)[LOWORD];
314		ix = hx & 0x7fffffff;
315		/* compute exp(w1 + w2) */
316		if (ix < 0x3c900000) /* exp(tiny < 2**-54) = 1 */
317			r = one;
318		else if (ix >= 0x40880000) /* overflow/underflow */
319			r = (hx < 0)? tiny * tiny : huge * huge;
320		else {	/* compute exp(w1 + w2) */
321			k = (int) (invln2 * w1 + ((hx >= 0)? 0.5 : -0.5));
322			t1 = (double) k;
323			t2 = w1 - t1 * ln2hi;
324			t3 = w2 - t1 * ln2lo;
325			r = exp(t2 + t3);
326		}
327		if (c != zero) c *= r;
328		if (s != zero) s *= r;
329		if (k != 0) {
330			c = scalbn(c, k);
331			s = scalbn(s, k);
332		}
333		D_RE(ans) = c;
334		D_IM(ans) = s;
335	}
336	return (ans);
337}
338