1/*
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3 *
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15 * If applicable, add the following below this CDDL HEADER, with the
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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 __casin = casin
31
32/* INDENT OFF */
33/*
34 * dcomplex casin(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 sine function casin(z),
42 * where z = x+iy, can be defined by
43 *
44 * 	casin(z) = asin(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 * Special notes: if casin( x, y) = ( u, v), then
67 *		    casin(-x, y) = (-u, v),
68 *		    casin( x,-y) = ( u,-v),
69 *    in general, we have casin(conj(z))     =  conj(casin(z))
70 *                       casin(-z)          = -casin(z)
71 *			 casin(z)           =  pi/2 - cacos(z)
72 *
73 * EXCEPTION CASES (conform to ISO/IEC 9899:1999(E)):
74 *    casin( 0 + i 0   ) =  0    + i 0
75 *    casin( 0 + i NaN ) =  0    + i NaN
76 *    casin( x + i inf ) =  0    + i inf for finite x
77 *    casin( x + i NaN ) =  NaN  + i NaN with invalid for finite x != 0
78 *    casin(inf + iy   ) =  pi/2 + i inf finite y
79 *    casin(inf + i inf) =  pi/4 + i inf
80 *    casin(inf + i NaN) =  NaN  + i inf
81 *    casin(NaN + i y  ) =  NaN  + i NaN for finite y
82 *    casin(NaN + i inf) =  NaN  + i inf
83 *    casin(NaN + i NaN) =  NaN  + i NaN
84 *
85 * Special Regions (better formula for accuracy and for avoiding spurious
86 * overflow or underflow) (all x and y are assumed nonnegative):
87 *  case 1: y = 0
88 *  case 2: tiny y relative to x-1: y <= ulp(0.5)*|x-1|
89 *  case 3: tiny y: y < 4 sqrt(u), where u = minimum normal number
90 *  case 4: huge y relative to x+1: y >= (1+x)/ulp(0.5)
91 *  case 5: huge x and y: x and y >= sqrt(M)/8, where M = maximum normal number
92 *  case 6: tiny x: x < 4 sqrt(u)
93 *  --------
94 *  case	1 & 2. y=0 or y/|x-1| is tiny. We have
95 *             ____________              _____________
96 *            /      2    2             /       y    2
97 *           / (x+-1)  + y   =  |x+-1| / 1 + (------)
98 *         \/                        \/       |x+-1|
99 *
100 *                                            1      y   2
101 *                           ~  |x+-1| ( 1 + --- (------)  )
102 *                                            2   |x+-1|
103 *
104 *                                           2
105 *                                          y
106 *                           =  |x+-1| + --------.
107 *                                       2|x+-1|
108 *
109 *	Consequently, it is not difficult to see that
110 *                                 2
111 *                                y
112 *                    [ 1 + ------------ ,  if x < 1,
113 *                    [      2(1+x)(1-x)
114 *                    [
115 *                    [
116 *                    [ x,                 if x = 1 (y = 0),
117 *                    [
118 *		A ~=  [             2
119 *                    [        x * y
120 *                    [ x + ------------ ,  if x > 1
121 *                    [      2(1+x)(x-1)
122 *
123 *	and hence
124 *                      ______                                 2
125 *                     / 2                    y               y
126 *               A + \/ A  - 1  ~  1 + ---------------- + -----------, if x < 1,
127 *                                     sqrt((x+1)(1-x))   2(x+1)(1-x)
128 *
129 *
130 *			       ~  x + sqrt((x-1)*(x+1)),              if x >= 1.
131 *
132 *                                         2
133 *                                        y
134 *                          [ x(1 - ------------), if x < 1,
135 *                          [       2(1+x)(1-x)
136 *		B = x/A  ~  [
137 *                          [ 1,                  if x = 1,
138 *			    [
139 *                          [           2
140 *                          [          y
141 *                          [ 1 - ------------ ,   if x > 1,
142 *                          [      2(1+x)(1-x)
143 *	Thus
144 *                            [ asin(x) + i y/sqrt((x-1)*(x+1)), if x <  1
145 *		casin(x+i*y)=[
146 *                            [ pi/2    + i log(x+sqrt(x*x-1)),  if x >= 1
147 *
148 *  case 3. y < 4 sqrt(u), where u = minimum normal x.
149 *	After case 1 and 2, this will only occurs when x=1. When x=1, we have
150 *	   A = (sqrt(4+y*y)+y)/2 ~ 1 + y/2 + y^2/8 + ...
151 *	and
152 *	   B = 1/A = 1 - y/2 + y^2/8 + ...
153 * 	Since
154 *	   asin(x) = pi/2-2*asin(sqrt((1-x)/2))
155 *	   asin(x) = x + x^3/6 + x^5*3/40 + x^7*15/336 + ...
156 *	we have, for the real part asin(B),
157 *	   asin(1-y/2) ~ pi/2 - 2 asin(sqrt(y/4))
158 *	               ~ pi/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 = asin(B) ~ x/y (be careful, x/y may underflow)
168 * 	and
169 *	   imag part = log(y+sqrt(y*y-one))
170 *
171 *
172 *  case 5. Both x and y are large: x and y > sqrt(M)/8, where M = maximum x
173 *	In this case,
174 *	   A ~ sqrt(x*x+y*y)
175 *	   B ~ x/sqrt(x*x+y*y).
176 *	Thus
177 *	   real part = asin(B) = atan(x/y),
178 *	   imag part = log(A+sqrt(A*A-1)) ~ log(2A)
179 *	             = log(2) + 0.5*log(x*x+y*y)
180 *	             = log(2) + log(y) + 0.5*log(1+(x/y)^2)
181 *
182 *  case 6. x < 4 sqrt(u). In this case, we have
183 *	    A ~ sqrt(1+y*y), B = x/sqrt(1+y*y).
184 *	Since B is tiny, we have
185 *	    real part = asin(B) ~ B = x/sqrt(1+y*y)
186 *	    imag part = log(A+sqrt(A*A-1)) = log (A+sqrt(y*y))
187 *	              = log(y+sqrt(1+y*y))
188 *	              = 0.5*log(y^2+2ysqrt(1+y^2)+1+y^2)
189 *	              = 0.5*log(1+2y(y+sqrt(1+y^2)));
190 *	              = 0.5*log1p(2y(y+A));
191 *
192 * 	casin(z) = asin(B) + i sign(y) log (A + sqrt(A*A-1)),
193 */
194/* INDENT ON */
195
196#include "libm.h"		/* asin/atan/fabs/log/log1p/sqrt */
197#include "complex_wrapper.h"
198
199/* INDENT OFF */
200static const double
201	zero = 0.0,
202	one = 1.0,
203	E = 1.11022302462515654042e-16,			/* 2**-53 */
204	ln2 = 6.93147180559945286227e-01,
205	pi_2 = 1.570796326794896558e+00,
206	pi_2_l = 6.123233995736765886e-17,
207	pi_4 = 7.85398163397448278999e-01,
208	Foursqrtu = 5.96667258496016539463e-154,	/* 2**(-509) */
209	Acrossover = 1.5,
210	Bcrossover = 0.6417,
211	half = 0.5;
212/* INDENT ON */
213
214dcomplex
215casin(dcomplex z) {
216	double x, y, t, R, S, A, Am1, B, y2, xm1, xp1, Apx;
217	int ix, iy, hx, hy;
218	unsigned lx, ly;
219	dcomplex ans;
220
221	x = D_RE(z);
222	y = D_IM(z);
223	hx = HI_WORD(x);
224	lx = LO_WORD(x);
225	hy = HI_WORD(y);
226	ly = LO_WORD(y);
227	ix = hx & 0x7fffffff;
228	iy = hy & 0x7fffffff;
229	x = fabs(x);
230	y = fabs(y);
231
232	/* special cases */
233
234	/* x is inf or NaN */
235	if (ix >= 0x7ff00000) {	/* x is inf or NaN */
236		if (ISINF(ix, lx)) {	/* x is INF */
237			D_IM(ans) = x;
238			if (iy >= 0x7ff00000) {
239				if (ISINF(iy, ly))
240					/* casin(inf + i inf) = pi/4 + i inf */
241					D_RE(ans) = pi_4;
242				else	/* casin(inf + i NaN) = NaN  + i inf  */
243					D_RE(ans) = y + y;
244			} else	/* casin(inf + iy) = pi/2 + i inf */
245				D_RE(ans) = pi_2;
246		} else {		/* x is NaN */
247			if (iy >= 0x7ff00000) {
248				/* INDENT OFF */
249				/*
250				 * casin(NaN + i inf) = NaN + i inf
251				 * casin(NaN + i NaN) = NaN + i NaN
252				 */
253				/* INDENT ON */
254				D_IM(ans) = y + y;
255				D_RE(ans) = x + x;
256			} else {
257				/* casin(NaN + i y ) = NaN  + i NaN */
258				D_IM(ans) = D_RE(ans) = x + y;
259			}
260		}
261		if (hx < 0)
262			D_RE(ans) = -D_RE(ans);
263		if (hy < 0)
264			D_IM(ans) = -D_IM(ans);
265		return (ans);
266	}
267
268	/* casin(+0 + i 0  ) =  0   + i 0. */
269	if ((ix | lx | iy | ly) == 0)
270		return (z);
271
272	if (iy >= 0x7ff00000) {	/* y is inf or NaN */
273		if (ISINF(iy, ly)) {	/* casin(x + i inf) =  0   + i inf */
274			D_IM(ans) = y;
275			D_RE(ans) = zero;
276		} else {		/* casin(x + i NaN) = NaN  + i NaN */
277			D_IM(ans) = x + y;
278			if ((ix | lx) == 0)
279				D_RE(ans) = x;
280			else
281				D_RE(ans) = y;
282		}
283		if (hx < 0)
284			D_RE(ans) = -D_RE(ans);
285		if (hy < 0)
286			D_IM(ans) = -D_IM(ans);
287		return (ans);
288	}
289
290	if ((iy | ly) == 0) {	/* region 1: y=0 */
291		if (ix < 0x3ff00000) {	/* |x| < 1 */
292			D_RE(ans) = asin(x);
293			D_IM(ans) = zero;
294		} else {
295			D_RE(ans) = pi_2;
296			if (ix >= 0x43500000)	/* |x| >= 2**54 */
297				D_IM(ans) = ln2 + log(x);
298			else if (ix >= 0x3ff80000)	/* x > Acrossover */
299				D_IM(ans) = log(x + sqrt((x - one) * (x +
300					one)));
301			else {
302				xm1 = x - one;
303				D_IM(ans) = log1p(xm1 + sqrt(xm1 * (x + one)));
304			}
305		}
306	} else if (y <= E * fabs(x - one)) {	/* region 2: y < tiny*|x-1| */
307		if (ix < 0x3ff00000) {	/* x < 1 */
308			D_RE(ans) = asin(x);
309			D_IM(ans) = y / sqrt((one + x) * (one - x));
310		} else {
311			D_RE(ans) = pi_2;
312			if (ix >= 0x43500000) {	/* |x| >= 2**54 */
313				D_IM(ans) = ln2 + log(x);
314			} else if (ix >= 0x3ff80000)	/* x > Acrossover */
315				D_IM(ans) = log(x + sqrt((x - one) * (x +
316					one)));
317			else
318				D_IM(ans) = log1p((x - one) + sqrt((x - one) *
319					(x + one)));
320		}
321	} else if (y < Foursqrtu) {	/* region 3 */
322		t = sqrt(y);
323		D_RE(ans) = pi_2 - (t - pi_2_l);
324		D_IM(ans) = t;
325	} else if (E * y - one >= x) {	/* region 4 */
326		D_RE(ans) = x / y;	/* need to fix underflow cases */
327		D_IM(ans) = ln2 + log(y);
328	} else if (ix >= 0x5fc00000 || iy >= 0x5fc00000) {	/* x,y>2**509 */
329		/* region 5: x+1 or y is very large (>= sqrt(max)/8) */
330		t = x / y;
331		D_RE(ans) = atan(t);
332		D_IM(ans) = ln2 + log(y) + half * log1p(t * t);
333	} else if (x < Foursqrtu) {
334		/* region 6: x is very small, < 4sqrt(min) */
335		A = sqrt(one + y * y);
336		D_RE(ans) = x / A;	/* may underflow */
337		if (iy >= 0x3ff80000)	/* if y > Acrossover */
338			D_IM(ans) = log(y + A);
339		else
340			D_IM(ans) = half * log1p((y + y) * (y + A));
341	} else {	/* safe region */
342		y2 = y * y;
343		xp1 = x + one;
344		xm1 = x - one;
345		R = sqrt(xp1 * xp1 + y2);
346		S = sqrt(xm1 * xm1 + y2);
347		A = half * (R + S);
348		B = x / A;
349
350		if (B <= Bcrossover)
351			D_RE(ans) = asin(B);
352		else {		/* use atan and an accurate approx to a-x */
353			Apx = A + x;
354			if (x <= one)
355				D_RE(ans) = atan(x / sqrt(half * Apx * (y2 /
356					(R + xp1) + (S - xm1))));
357			else
358				D_RE(ans) = atan(x / (y * sqrt(half * (Apx /
359					(R + xp1) + Apx / (S + xm1)))));
360		}
361		if (A <= Acrossover) {
362			/* use log1p and an accurate approx to A-1 */
363			if (x < one)
364				Am1 = half * (y2 / (R + xp1) + y2 / (S - xm1));
365			else
366				Am1 = half * (y2 / (R + xp1) + (S + xm1));
367			D_IM(ans) = log1p(Am1 + sqrt(Am1 * (A + one)));
368		} else {
369			D_IM(ans) = log(A + sqrt(A * A - one));
370		}
371	}
372
373	if (hx < 0)
374		D_RE(ans) = -D_RE(ans);
375	if (hy < 0)
376		D_IM(ans) = -D_IM(ans);
377
378	return (ans);
379}
380