1/* Return arc hyperbole sine for double value, with the imaginary part
2 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/* Return the complex inverse hyperbolic sine of finite nonzero Z,
27 with the imaginary part of the result subtracted from pi/2 if ADJ
28 is nonzero. */
29
30__complex__ double
31__kernel_casinh (__complex__ double x, int adj)
32{
33 __complex__ double res;
34 double rx, ix;
35 __complex__ double y;
36
37 /* Avoid cancellation by reducing to the first quadrant. */
38 rx = fabs (__real__ x);
39 ix = fabs (__imag__ x);
40
41 if (rx >= 1.0 / DBL_EPSILON || ix >= 1.0 / DBL_EPSILON)
42 {
43 /* For large x in the first quadrant, x + csqrt (1 + x * x)
44 is sufficiently close to 2 * x to make no significant
45 difference to the result; avoid possible overflow from
46 the squaring and addition. */
47 __real__ y = rx;
48 __imag__ y = ix;
49
50 if (adj)
51 {
52 double t = __real__ y;
53 __real__ y = __copysign (__imag__ y, __imag__ x);
54 __imag__ y = t;
55 }
56
57 res = __clog (y);
58 __real__ res += M_LN2;
59 }
60 else if (rx >= 0.5 && ix < DBL_EPSILON / 8.0)
61 {
62 double s = __ieee754_hypot (1.0, rx);
63
64 __real__ res = __ieee754_log (rx + s);
65 if (adj)
66 __imag__ res = __ieee754_atan2 (s, __imag__ x);
67 else
68 __imag__ res = __ieee754_atan2 (ix, s);
69 }
70 else if (rx < DBL_EPSILON / 8.0 && ix >= 1.5)
71 {
72 double s = __ieee754_sqrt ((ix + 1.0) * (ix - 1.0));
73
74 __real__ res = __ieee754_log (ix + s);
75 if (adj)
76 __imag__ res = __ieee754_atan2 (rx, __copysign (s, __imag__ x));
77 else
78 __imag__ res = __ieee754_atan2 (s, rx);
79 }
80 else if (ix > 1.0 && ix < 1.5 && rx < 0.5)
81 {
82 if (rx < DBL_EPSILON * DBL_EPSILON)
83 {
84 double ix2m1 = (ix + 1.0) * (ix - 1.0);
85 double s = __ieee754_sqrt (ix2m1);
86
87 __real__ res = __log1p (2.0 * (ix2m1 + ix * s)) / 2.0;
88 if (adj)
89 __imag__ res = __ieee754_atan2 (rx, __copysign (s, __imag__ x));
90 else
91 __imag__ res = __ieee754_atan2 (s, rx);
92 }
93 else
94 {
95 double ix2m1 = (ix + 1.0) * (ix - 1.0);
96 double rx2 = rx * rx;
97 double f = rx2 * (2.0 + rx2 + 2.0 * ix * ix);
98 double d = __ieee754_sqrt (ix2m1 * ix2m1 + f);
99 double dp = d + ix2m1;
100 double dm = f / dp;
101 double r1 = __ieee754_sqrt ((dm + rx2) / 2.0);
102 double r2 = rx * ix / r1;
103
104 __real__ res = __log1p (rx2 + dp + 2.0 * (rx * r1 + ix * r2)) / 2.0;
105 if (adj)
106 __imag__ res = __ieee754_atan2 (rx + r1, __copysign (ix + r2,
107 __imag__ x));
108 else
109 __imag__ res = __ieee754_atan2 (ix + r2, rx + r1);
110 }
111 }
112 else if (ix == 1.0 && rx < 0.5)
113 {
114 if (rx < DBL_EPSILON / 8.0)
115 {
116 __real__ res = __log1p (2.0 * (rx + __ieee754_sqrt (rx))) / 2.0;
117 if (adj)
118 __imag__ res = __ieee754_atan2 (__ieee754_sqrt (rx),
119 __copysign (1.0, __imag__ x));
120 else
121 __imag__ res = __ieee754_atan2 (1.0, __ieee754_sqrt (rx));
122 }
123 else
124 {
125 double d = rx * __ieee754_sqrt (4.0 + rx * rx);
126 double s1 = __ieee754_sqrt ((d + rx * rx) / 2.0);
127 double s2 = __ieee754_sqrt ((d - rx * rx) / 2.0);
128
129 __real__ res = __log1p (rx * rx + d + 2.0 * (rx * s1 + s2)) / 2.0;
130 if (adj)
131 __imag__ res = __ieee754_atan2 (rx + s1, __copysign (1.0 + s2,
132 __imag__ x));
133 else
134 __imag__ res = __ieee754_atan2 (1.0 + s2, rx + s1);
135 }
136 }
137 else if (ix < 1.0 && rx < 0.5)
138 {
139 if (ix >= DBL_EPSILON)
140 {
141 if (rx < DBL_EPSILON * DBL_EPSILON)
142 {
143 double onemix2 = (1.0 + ix) * (1.0 - ix);
144 double s = __ieee754_sqrt (onemix2);
145
146 __real__ res = __log1p (2.0 * rx / s) / 2.0;
147 if (adj)
148 __imag__ res = __ieee754_atan2 (s, __imag__ x);
149 else
150 __imag__ res = __ieee754_atan2 (ix, s);
151 }
152 else
153 {
154 double onemix2 = (1.0 + ix) * (1.0 - ix);
155 double rx2 = rx * rx;
156 double f = rx2 * (2.0 + rx2 + 2.0 * ix * ix);
157 double d = __ieee754_sqrt (onemix2 * onemix2 + f);
158 double dp = d + onemix2;
159 double dm = f / dp;
160 double r1 = __ieee754_sqrt ((dp + rx2) / 2.0);
161 double r2 = rx * ix / r1;
162
163 __real__ res
164 = __log1p (rx2 + dm + 2.0 * (rx * r1 + ix * r2)) / 2.0;
165 if (adj)
166 __imag__ res = __ieee754_atan2 (rx + r1,
167 __copysign (ix + r2,
168 __imag__ x));
169 else
170 __imag__ res = __ieee754_atan2 (ix + r2, rx + r1);
171 }
172 }
173 else
174 {
175 double s = __ieee754_hypot (1.0, rx);
176
177 __real__ res = __log1p (2.0 * rx * (rx + s)) / 2.0;
178 if (adj)
179 __imag__ res = __ieee754_atan2 (s, __imag__ x);
180 else
181 __imag__ res = __ieee754_atan2 (ix, s);
182 }
183 math_check_force_underflow_nonneg (__real__ res);
184 }
185 else
186 {
187 __real__ y = (rx - ix) * (rx + ix) + 1.0;
188 __imag__ y = 2.0 * rx * ix;
189
190 y = __csqrt (y);
191
192 __real__ y += rx;
193 __imag__ y += ix;
194
195 if (adj)
196 {
197 double t = __real__ y;
198 __real__ y = __copysign (__imag__ y, __imag__ x);
199 __imag__ y = t;
200 }
201
202 res = __clog (y);
203 }
204
205 /* Give results the correct sign for the original argument. */
206 __real__ res = __copysign (__real__ res, __real__ x);
207 __imag__ res = __copysign (__imag__ res, (adj ? 1.0 : __imag__ x));
208
209 return res;
210}
211