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) 1997-2016 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 = __ieee754_hypotl (1.0L, rx);
70
71 __real__ res = __ieee754_logl (rx + s);
72 if (adj)
73 __imag__ res = __ieee754_atan2l (s, __imag__ x);
74 else
75 __imag__ res = __ieee754_atan2l (ix, s);
76 }
77 else if (rx < LDBL_EPSILON / 8.0L && ix >= 1.5L)
78 {
79 long double s = __ieee754_sqrtl ((ix + 1.0L) * (ix - 1.0L));
80
81 __real__ res = __ieee754_logl (ix + s);
82 if (adj)
83 __imag__ res = __ieee754_atan2l (rx, __copysignl (s, __imag__ x));
84 else
85 __imag__ res = __ieee754_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 = __ieee754_sqrtl (ix2m1);
93
94 __real__ res = __log1pl (2.0L * (ix2m1 + ix * s)) / 2.0L;
95 if (adj)
96 __imag__ res = __ieee754_atan2l (rx, __copysignl (s, __imag__ x));
97 else
98 __imag__ res = __ieee754_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 = __ieee754_sqrtl (ix2m1 * ix2m1 + f);
106 long double dp = d + ix2m1;
107 long double dm = f / dp;
108 long double r1 = __ieee754_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 = __ieee754_atan2l (rx + r1, __copysignl (ix + r2,
115 __imag__ x));
116 else
117 __imag__ res = __ieee754_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 + __ieee754_sqrtl (rx))) / 2.0L;
125 if (adj)
126 __imag__ res = __ieee754_atan2l (__ieee754_sqrtl (rx),
127 __copysignl (1.0L, __imag__ x));
128 else
129 __imag__ res = __ieee754_atan2l (1.0L, __ieee754_sqrtl (rx));
130 }
131 else
132 {
133 long double d = rx * __ieee754_sqrtl (4.0L + rx * rx);
134 long double s1 = __ieee754_sqrtl ((d + rx * rx) / 2.0L);
135 long double s2 = __ieee754_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 = __ieee754_atan2l (rx + s1,
140 __copysignl (1.0L + s2,
141 __imag__ x));
142 else
143 __imag__ res = __ieee754_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 = __ieee754_sqrtl (onemix2);
154
155 __real__ res = __log1pl (2.0L * rx / s) / 2.0L;
156 if (adj)
157 __imag__ res = __ieee754_atan2l (s, __imag__ x);
158 else
159 __imag__ res = __ieee754_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 = __ieee754_sqrtl (onemix2 * onemix2 + f);
167 long double dp = d + onemix2;
168 long double dm = f / dp;
169 long double r1 = __ieee754_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 = __ieee754_atan2l (rx + r1,
176 __copysignl (ix + r2,
177 __imag__ x));
178 else
179 __imag__ res = __ieee754_atan2l (ix + r2, rx + r1);
180 }
181 }
182 else
183 {
184 long double s = __ieee754_hypotl (1.0L, rx);
185
186 __real__ res = __log1pl (2.0L * rx * (rx + s)) / 2.0L;
187 if (adj)
188 __imag__ res = __ieee754_atan2l (s, __imag__ x);
189 else
190 __imag__ res = __ieee754_atan2l (ix, s);
191 }
192 math_check_force_underflow_nonneg (__real__ res);
193 }
194 else
195 {
196 __real__ y = (rx - ix) * (rx + ix) + 1.0L;
197 __imag__ y = 2.0L * rx * ix;
198
199 y = __csqrtl (y);
200
201 __real__ y += rx;
202 __imag__ y += ix;
203
204 if (adj)
205 {
206 long double t = __real__ y;
207 __real__ y = __copysignl (__imag__ y, __imag__ x);
208 __imag__ y = t;
209 }
210
211 res = __clogl (y);
212 }
213
214 /* Give results the correct sign for the original argument. */
215 __real__ res = __copysignl (__real__ res, __real__ x);
216 __imag__ res = __copysignl (__imag__ res, (adj ? 1.0L : __imag__ x));
217
218 return res;
219}
220