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 *
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12 *
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15 * If applicable, add the following below this CDDL HEADER, with the
16 * fields enclosed by brackets "[]" replaced with your own identifying
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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 __cacos = cacos
31
32/* INDENT OFF */
33/*
34 * dcomplex cacos(dcomplex z);
35 *
36 * Alogrithm
37 * (based on T.E.Hull, Thomas F. Fairgrieve and Ping Tak Peter Tang's
38 * paper "Implementing the Complex Arcsine and Arccosine Functins Using
39 * Exception Handling", ACM TOMS, Vol 23, pp 299-335)
40 *
41 * The principal value of complex inverse cosine function cacos(z),
42 * where z = x+iy, can be defined by
43 *
44 * 	cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)),
45 *
46 * where the log function is the natural log, and
47 *             ____________           ____________
48 *       1    /     2    2      1    /     2    2
49 *  A = ---  / (x+1)  + y   +  ---  / (x-1)  + y
50 *       2 \/                   2 \/
51 *             ____________           ____________
52 *       1    /     2    2      1    /     2    2
53 *  B = ---  / (x+1)  + y   -  ---  / (x-1)  + y   .
54 *       2 \/                   2 \/
55 *
56 * The Branch cuts are on the real line from -inf to -1 and from 1 to inf.
57 * The real and imaginary parts are based on Abramowitz and Stegun
58 * [Handbook of Mathematic Functions, 1972].  The sign of the imaginary
59 * part is chosen to be the generally considered the principal value of
60 * this function.
61 *
62 * Notes:1. A is the average of the distances from z to the points (1,0)
63 *          and (-1,0) in the complex z-plane, and in particular A>=1.
64 *       2. B is in [-1,1], and A*B = x
65 *
66 * Basic relations
67 *    cacos(conj(z)) = conj(cacos(z))
68 *    cacos(-z)      = pi   - cacos(z)
69 *    cacos( z)      = pi/2 - casin(z)
70 *
71 * Special cases (conform to ISO/IEC 9899:1999(E)):
72 *    cacos(+-0  + i y  ) = pi/2 - i y for y is +-0, +-inf, NaN
73 *    cacos( x   + i inf) = pi/2 - i inf for all x
74 *    cacos( x   + i NaN) = NaN  + i NaN with invalid for non-zero finite x
75 *    cacos(-inf + i y  ) = pi   - i inf for finite +y
76 *    cacos( inf + i y  ) = 0    - i inf for finite +y
77 *    cacos(-inf + i inf) = 3pi/4- i inf
78 *    cacos( inf + i inf) = pi/4 - i inf
79 *    cacos(+-inf+ i NaN) = NaN  - i inf (sign of imaginary is unspecified)
80 *    cacos(NaN  + i y  ) = NaN  + i NaN with invalid for finite y
81 *    cacos(NaN  + i inf) = NaN  - i inf
82 *    cacos(NaN  + i NaN) = NaN  + i NaN
83 *
84 * Special Regions (better formula for accuracy and for avoiding spurious
85 * overflow or underflow) (all x and y are assumed nonnegative):
86 *  case 1: y = 0
87 *  case 2: tiny y relative to x-1: y <= ulp(0.5)*|x-1|
88 *  case 3: tiny y: y < 4 sqrt(u), where u = minimum normal number
89 *  case 4: huge y relative to x+1: y >= (1+x)/ulp(0.5)
90 *  case 5: huge x and y: x and y >= sqrt(M)/8, where M = maximum normal number
91 *  case 6: tiny x: x < 4 sqrt(u)
92 *  --------
93 *  case	1 & 2. y=0 or y/|x-1| is tiny. We have
94 *             ____________              _____________
95 *            /      2    2             /       y    2
96 *           / (x+-1)  + y   =  |x+-1| / 1 + (------)
97 *         \/                        \/       |x+-1|
98 *
99 *                                            1     y    2
100 *                           ~  |x+-1| ( 1 + --- (------)  )
101 *                                            2   |x+-1|
102 *
103 *                                          2
104 *                                         y
105 *                           = |x+-1| + --------.
106 *                                      2|x+-1|
107 *
108 *	Consequently, it is not difficult to see that
109 *                                 2
110 *                                y
111 *                    [ 1 + ------------ ,     if x < 1,
112 *                    [      2(1+x)(1-x)
113 *                    [
114 *                    [
115 *                    [ x,                     if x = 1 (y = 0),
116 *                    [
117 *		A ~=  [             2
118 *                    [        x * y
119 *                    [ x + ------------ ~ x,  if x > 1
120 *                    [      2(x+1)(x-1)
121 *
122 *	and hence
123 *                      ______                                 2
124 *                     / 2                    y               y
125 *               A + \/ A  - 1  ~  1 + ---------------- + -----------, if x < 1,
126 *                                     sqrt((x+1)(1-x))   2(x+1)(1-x)
127 *
128 *
129 *			        ~  x + sqrt((x-1)*(x+1)),             if x >= 1.
130 *
131 *                                         2
132 *                                        y
133 *                          [ x(1 - -----------) ~ x,  if x < 1,
134 *                          [       2(1+x)(1-x)
135 *		B = x/A  ~  [
136 *                          [ 1,                       if x = 1,
137 *			    [
138 *                          [           2
139 *                          [          y
140 *                          [ 1 - ------------ ,       if x > 1,
141 *                          [      2(x+1)(x-1)
142 *	Thus
143 *                            [ acos(x) - i y/sqrt((x-1)*(x+1)),      if x < 1,
144 *                            [
145 *		cacos(x+i*y)~ [ 0 - i 0,                              if x = 1,
146 *                            [
147 *                            [ y/sqrt(x*x-1) - i log(x+sqrt(x*x-1)), if x > 1.
148 *
149 *      Note: y/sqrt(x*x-1) ~ y/x when x >= 2**26.
150 *  case 3. y < 4 sqrt(u), where u = minimum normal x.
151 *	After case 1 and 2, this will only occurs when x=1. When x=1, we have
152 *	   A = (sqrt(4+y*y)+y)/2 ~ 1 + y/2 + y^2/8 + ...
153 *	and
154 *	   B = 1/A = 1 - y/2 + y^2/8 + ...
155 * 	Since
156 *         cos(sqrt(y)) ~ 1 - y/2 + ...
157 *      we have, for the real part,
158 *         acos(B) ~ acos(1 - y/2) ~ sqrt(y)
159 *	For the imaginary part,
160 *	   log(A+sqrt(A*A-1)) ~ log(1+y/2+sqrt(2*y/2))
161 *	                      = log(1+y/2+sqrt(y))
162 *	                      = (y/2+sqrt(y)) - (y/2+sqrt(y))^2/2 + ...
163 *	                      ~ sqrt(y) - y*(sqrt(y)+y/2)/2
164 *	                      ~ sqrt(y)
165 *
166 *  case 4. y >= (x+1)/ulp(0.5). In this case, A ~ y and B ~ x/y. Thus
167 *	   real part = acos(B) ~ pi/2
168 * 	and
169 *	   imag part = log(y+sqrt(y*y-one))
170 *
171 *  case 5. Both x and y are large: x and y > sqrt(M)/8, where M = maximum x
172 *	In this case,
173 *	   A ~ sqrt(x*x+y*y)
174 *	   B ~ x/sqrt(x*x+y*y).
175 *	Thus
176 *	   real part = acos(B) = atan(y/x),
177 *	   imag part = log(A+sqrt(A*A-1)) ~ log(2A)
178 *	             = log(2) + 0.5*log(x*x+y*y)
179 *	             = log(2) + log(y) + 0.5*log(1+(x/y)^2)
180 *
181 *  case 6. x < 4 sqrt(u). In this case, we have
182 *	    A ~ sqrt(1+y*y), B = x/sqrt(1+y*y).
183 *	Since B is tiny, we have
184 *	    real part = acos(B) ~ pi/2
185 *	    imag part = log(A+sqrt(A*A-1)) = log (A+sqrt(y*y))
186 *	              = log(y+sqrt(1+y*y))
187 *	              = 0.5*log(y^2+2ysqrt(1+y^2)+1+y^2)
188 *	              = 0.5*log(1+2y(y+sqrt(1+y^2)));
189 *	              = 0.5*log1p(2y(y+A));
190 *
191 * 	cacos(z) = acos(B) - i sign(y) log (A + sqrt(A*A-1)),
192 */
193/* INDENT ON */
194
195#include "libm.h"
196#include "complex_wrapper.h"
197
198/* INDENT OFF */
199static const double
200	zero = 0.0,
201	one = 1.0,
202	E = 1.11022302462515654042e-16,			/* 2**-53 */
203	ln2 = 6.93147180559945286227e-01,
204	pi = 3.1415926535897931159979634685,
205	pi_l = 1.224646799147353177e-16,
206	pi_2 = 1.570796326794896558e+00,
207	pi_2_l = 6.123233995736765886e-17,
208	pi_4 = 0.78539816339744827899949,
209	pi_4_l = 3.061616997868382943e-17,
210	pi3_4 = 2.356194490192344836998,
211	pi3_4_l = 9.184850993605148829195e-17,
212	Foursqrtu = 5.96667258496016539463e-154,	/* 2**(-509) */
213	Acrossover = 1.5,
214	Bcrossover = 0.6417,
215	half = 0.5;
216/* INDENT ON */
217
218dcomplex
219cacos(dcomplex z) {
220	double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx;
221	int ix, iy, hx, hy;
222	unsigned lx, ly;
223	dcomplex ans;
224
225	x = D_RE(z);
226	y = D_IM(z);
227	hx = HI_WORD(x);
228	lx = LO_WORD(x);
229	hy = HI_WORD(y);
230	ly = LO_WORD(y);
231	ix = hx & 0x7fffffff;
232	iy = hy & 0x7fffffff;
233
234	/* x is 0 */
235	if ((ix | lx) == 0) {
236		if (((iy | ly) == 0) || (iy >= 0x7ff00000)) {
237			D_RE(ans) = pi_2;
238			D_IM(ans) = -y;
239			return (ans);
240		}
241	}
242
243	/* |y| is inf or NaN */
244	if (iy >= 0x7ff00000) {
245		if (ISINF(iy, ly)) {	/* cacos(x + i inf) = pi/2  - i inf */
246			D_IM(ans) = -y;
247			if (ix < 0x7ff00000) {
248				D_RE(ans) = pi_2 + pi_2_l;
249			} else if (ISINF(ix, lx)) {
250				if (hx >= 0)
251					D_RE(ans) = pi_4 + pi_4_l;
252				else
253					D_RE(ans) = pi3_4 + pi3_4_l;
254			} else {
255				D_RE(ans) = x;
256			}
257		} else {		/* cacos(x + i NaN) = NaN  + i NaN */
258			D_RE(ans) = y + x;
259			if (ISINF(ix, lx))
260				D_IM(ans) = -fabs(x);
261			else
262				D_IM(ans) = y;
263		}
264		return (ans);
265	}
266
267	x = fabs(x);
268	y = fabs(y);
269
270	/* x is inf or NaN */
271	if (ix >= 0x7ff00000) {	/* x is inf or NaN */
272		if (ISINF(ix, lx)) {	/* x is INF */
273			D_IM(ans) = -x;
274			if (iy >= 0x7ff00000) {
275				if (ISINF(iy, ly)) {
276					/* INDENT OFF */
277					/* cacos(inf + i inf) = pi/4 - i inf */
278					/* cacos(-inf+ i inf) =3pi/4 - i inf */
279					/* INDENT ON */
280					if (hx >= 0)
281						D_RE(ans) = pi_4 + pi_4_l;
282					else
283						D_RE(ans) = pi3_4 + pi3_4_l;
284				} else
285					/* INDENT OFF */
286					/* cacos(inf + i NaN) = NaN  - i inf  */
287					/* INDENT ON */
288					D_RE(ans) = y + y;
289			} else
290				/* INDENT OFF */
291				/* cacos(inf + iy ) = 0  - i inf */
292				/* cacos(-inf+ iy  ) = pi - i inf */
293				/* INDENT ON */
294			if (hx >= 0)
295				D_RE(ans) = zero;
296			else
297				D_RE(ans) = pi + pi_l;
298		} else {		/* x is NaN */
299			/* INDENT OFF */
300			/*
301			 * cacos(NaN + i inf) = NaN  - i inf
302			 * cacos(NaN + i y  ) = NaN  + i NaN
303			 * cacos(NaN + i NaN) = NaN  + i NaN
304			 */
305			/* INDENT ON */
306			D_RE(ans) = x + y;
307			if (iy >= 0x7ff00000) {
308				D_IM(ans) = -y;
309			} else {
310				D_IM(ans) = x;
311			}
312		}
313		if (hy < 0)
314			D_IM(ans) = -D_IM(ans);
315		return (ans);
316	}
317
318	if ((iy | ly) == 0) {	/* region 1: y=0 */
319		if (ix < 0x3ff00000) {	/* |x| < 1 */
320			D_RE(ans) = acos(x);
321			D_IM(ans) = zero;
322		} else {
323			D_RE(ans) = zero;
324			if (ix >= 0x43500000)	/* |x| >= 2**54 */
325				D_IM(ans) = ln2 + log(x);
326			else if (ix >= 0x3ff80000)	/* x > Acrossover */
327				D_IM(ans) = log(x + sqrt((x - one) * (x +
328					one)));
329			else {
330				xm1 = x - one;
331				D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one)));
332			}
333		}
334	} else if (y <= E * fabs(x - one)) {	/* region 2: y < tiny*|x-1| */
335		if (ix < 0x3ff00000) {	/* x < 1 */
336			D_RE(ans) = acos(x);
337			D_IM(ans) = y / sqrt((one + x) * (one - x));
338		} else if (ix >= 0x43500000) {	/* |x| >= 2**54 */
339			D_RE(ans) = y / x;
340			D_IM(ans) = ln2 + log(x);
341		} else {
342			t = sqrt((x - one) * (x + one));
343			D_RE(ans) = y / t;
344			if (ix >= 0x3ff80000)	/* x > Acrossover */
345				D_IM(ans) = log(x + t);
346			else
347				D_IM(ans) = log1p((x - one) + t);
348		}
349	} else if (y < Foursqrtu) {	/* region 3 */
350		t = sqrt(y);
351		D_RE(ans) = t;
352		D_IM(ans) = t;
353	} else if (E * y - one >= x) {	/* region 4 */
354		D_RE(ans) = pi_2;
355		D_IM(ans) = ln2 + log(y);
356	} else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) {	/* x,y>2**509 */
357		/* region 5: x+1 or y is very large (>= sqrt(max)/8) */
358		t = x / y;
359		D_RE(ans) = atan(y / x);
360		D_IM(ans) = ln2 + log(y) + half * log1p(t * t);
361	} else if (x < Foursqrtu) {
362		/* region 6: x is very small, < 4sqrt(min) */
363		D_RE(ans) = pi_2;
364		A = sqrt(one + y * y);
365		if (iy >= 0x3ff80000)	/* if y > Acrossover */
366			D_IM(ans) = log(y + A);
367		else
368			D_IM(ans) = half * log1p((y + y) * (y + A));
369	} else {	/* safe region */
370		y2 = y * y;
371		xp1 = x + one;
372		xm1 = x - one;
373		R = sqrt(xp1 * xp1 + y2);
374		S = sqrt(xm1 * xm1 + y2);
375		A = half * (R + S);
376		B = x / A;
377		if (B <= Bcrossover)
378			D_RE(ans) = acos(B);
379		else {		/* use atan and an accurate approx to a-x */
380			Apx = A + x;
381			if (x <= one)
382				D_RE(ans) = atan(sqrt(half * Apx * (y2 / (R +
383					xp1) + (S - xm1))) / x);
384			else
385				D_RE(ans) = atan((y * sqrt(half * (Apx / (R +
386					xp1) + Apx / (S + xm1)))) / x);
387		}
388		if (A <= Acrossover) {
389			/* use log1p and an accurate approx to A-1 */
390			if (x < one)
391				Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1));
392			else
393				Am1 = half * (y2 / (R + xp1) + (S + xm1));
394			D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one)));
395		} else {
396			D_IM(ans) = log(A + sqrt(A * A - one));
397		}
398	}
399	if (hx < 0)
400		D_RE(ans) = pi - D_RE(ans);
401	if (hy >= 0)
402		D_IM(ans) = -D_IM(ans);
403	return (ans);
404}
405