1/* Return arc hyperbole sine for long double value, with the imaginary
2   part of the result possibly adjusted for use in computing other
3   functions.
4   Copyright (C) 1997-2015 Free Software Foundation, Inc.
5   This file is part of the GNU C Library.
6
7   The GNU C Library is free software; you can redistribute it and/or
8   modify it under the terms of the GNU Lesser General Public
9   License as published by the Free Software Foundation; either
10   version 2.1 of the License, or (at your option) any later version.
11
12   The GNU C Library is distributed in the hope that it will be useful,
13   but WITHOUT ANY WARRANTY; without even the implied warranty of
14   MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
15   Lesser General Public License for more details.
16
17   You should have received a copy of the GNU Lesser General Public
18   License along with the GNU C Library; if not, see
19   <http://www.gnu.org/licenses/>.  */
20
21#include <complex.h>
22#include <math.h>
23#include <math_private.h>
24#include <float.h>
25
26/* To avoid spurious overflows, use this definition to treat IBM long
27   double as approximating an IEEE-style format.  */
28#if LDBL_MANT_DIG == 106
29# undef LDBL_EPSILON
30# define LDBL_EPSILON 0x1p-106L
31#endif
32
33/* Return the complex inverse hyperbolic sine of finite nonzero Z,
34   with the imaginary part of the result subtracted from pi/2 if ADJ
35   is nonzero.  */
36
37__complex__ long double
38__kernel_casinhl (__complex__ long double x, int adj)
39{
40  __complex__ long double res;
41  long double rx, ix;
42  __complex__ long double y;
43
44  /* Avoid cancellation by reducing to the first quadrant.  */
45  rx = fabsl (__real__ x);
46  ix = fabsl (__imag__ x);
47
48  if (rx >= 1.0L / LDBL_EPSILON || ix >= 1.0L / LDBL_EPSILON)
49    {
50      /* For large x in the first quadrant, x + csqrt (1 + x * x)
51	 is sufficiently close to 2 * x to make no significant
52	 difference to the result; avoid possible overflow from
53	 the squaring and addition.  */
54      __real__ y = rx;
55      __imag__ y = ix;
56
57      if (adj)
58	{
59	  long double t = __real__ y;
60	  __real__ y = copysignl (__imag__ y, __imag__ x);
61	  __imag__ y = t;
62	}
63
64      res = clogl (y);
65      __real__ res += M_LN2l;
66    }
67  else if (rx >= 0.5L && ix < LDBL_EPSILON / 8.0L)
68    {
69      long double s = hypotl (1.0L, rx);
70
71      __real__ res = logl (rx + s);
72      if (adj)
73	__imag__ res = atan2l (s, __imag__ x);
74      else
75	__imag__ res = atan2l (ix, s);
76    }
77  else if (rx < LDBL_EPSILON / 8.0L && ix >= 1.5L)
78    {
79      long double s = sqrtl ((ix + 1.0L) * (ix - 1.0L));
80
81      __real__ res = logl (ix + s);
82      if (adj)
83	__imag__ res = atan2l (rx, copysignl (s, __imag__ x));
84      else
85	__imag__ res = atan2l (s, rx);
86    }
87  else if (ix > 1.0L && ix < 1.5L && rx < 0.5L)
88    {
89      if (rx < LDBL_EPSILON * LDBL_EPSILON)
90	{
91	  long double ix2m1 = (ix + 1.0L) * (ix - 1.0L);
92	  long double s = sqrtl (ix2m1);
93
94	  __real__ res = log1pl (2.0L * (ix2m1 + ix * s)) / 2.0L;
95	  if (adj)
96	    __imag__ res = atan2l (rx, copysignl (s, __imag__ x));
97	  else
98	    __imag__ res = atan2l (s, rx);
99	}
100      else
101	{
102	  long double ix2m1 = (ix + 1.0L) * (ix - 1.0L);
103	  long double rx2 = rx * rx;
104	  long double f = rx2 * (2.0L + rx2 + 2.0L * ix * ix);
105	  long double d = sqrtl (ix2m1 * ix2m1 + f);
106	  long double dp = d + ix2m1;
107	  long double dm = f / dp;
108	  long double r1 = sqrtl ((dm + rx2) / 2.0L);
109	  long double r2 = rx * ix / r1;
110
111	  __real__ res
112	    = log1pl (rx2 + dp + 2.0L * (rx * r1 + ix * r2)) / 2.0L;
113	  if (adj)
114	    __imag__ res = atan2l (rx + r1, copysignl (ix + r2,
115								   __imag__ x));
116	  else
117	    __imag__ res = atan2l (ix + r2, rx + r1);
118	}
119    }
120  else if (ix == 1.0L && rx < 0.5L)
121    {
122      if (rx < LDBL_EPSILON / 8.0L)
123	{
124	  __real__ res = log1pl (2.0L * (rx + sqrtl (rx))) / 2.0L;
125	  if (adj)
126	    __imag__ res = atan2l (sqrtl (rx),
127					     copysignl (1.0L, __imag__ x));
128	  else
129	    __imag__ res = atan2l (1.0L, sqrtl (rx));
130	}
131      else
132	{
133	  long double d = rx * sqrtl (4.0L + rx * rx);
134	  long double s1 = sqrtl ((d + rx * rx) / 2.0L);
135	  long double s2 = sqrtl ((d - rx * rx) / 2.0L);
136
137	  __real__ res = log1pl (rx * rx + d + 2.0L * (rx * s1 + s2)) / 2.0L;
138	  if (adj)
139	    __imag__ res = atan2l (rx + s1,
140					     copysignl (1.0L + s2,
141							  __imag__ x));
142	  else
143	    __imag__ res = atan2l (1.0L + s2, rx + s1);
144	}
145    }
146  else if (ix < 1.0L && rx < 0.5L)
147    {
148      if (ix >= LDBL_EPSILON)
149	{
150	  if (rx < LDBL_EPSILON * LDBL_EPSILON)
151	    {
152	      long double onemix2 = (1.0L + ix) * (1.0L - ix);
153	      long double s = sqrtl (onemix2);
154
155	      __real__ res = log1pl (2.0L * rx / s) / 2.0L;
156	      if (adj)
157		__imag__ res = atan2l (s, __imag__ x);
158	      else
159		__imag__ res = atan2l (ix, s);
160	    }
161	  else
162	    {
163	      long double onemix2 = (1.0L + ix) * (1.0L - ix);
164	      long double rx2 = rx * rx;
165	      long double f = rx2 * (2.0L + rx2 + 2.0L * ix * ix);
166	      long double d = sqrtl (onemix2 * onemix2 + f);
167	      long double dp = d + onemix2;
168	      long double dm = f / dp;
169	      long double r1 = sqrtl ((dp + rx2) / 2.0L);
170	      long double r2 = rx * ix / r1;
171
172	      __real__ res
173		= log1pl (rx2 + dm + 2.0L * (rx * r1 + ix * r2)) / 2.0L;
174	      if (adj)
175		__imag__ res = atan2l (rx + r1,
176						 copysignl (ix + r2,
177							      __imag__ x));
178	      else
179		__imag__ res = atan2l (ix + r2, rx + r1);
180	    }
181	}
182      else
183	{
184	  long double s = hypotl (1.0L, rx);
185
186	  __real__ res = log1pl (2.0L * rx * (rx + s)) / 2.0L;
187	  if (adj)
188	    __imag__ res = atan2l (s, __imag__ x);
189	  else
190	    __imag__ res = atan2l (ix, s);
191	}
192      if (__real__ res < LDBL_MIN)
193	{
194	  volatile long double force_underflow = __real__ res * __real__ res;
195	  (void) force_underflow;
196	}
197    }
198  else
199    {
200      __real__ y = (rx - ix) * (rx + ix) + 1.0L;
201      __imag__ y = 2.0L * rx * ix;
202
203      y = csqrtl (y);
204
205      __real__ y += rx;
206      __imag__ y += ix;
207
208      if (adj)
209	{
210	  long double t = __real__ y;
211	  __real__ y = copysignl (__imag__ y, __imag__ x);
212	  __imag__ y = t;
213	}
214
215      res = clogl (y);
216    }
217
218  /* Give results the correct sign for the original argument.  */
219  __real__ res = copysignl (__real__ res, __real__ x);
220  __imag__ res = copysignl (__imag__ res, (adj ? 1.0L : __imag__ x));
221
222  return res;
223}
224