1 /* Return arc hyperbolic sine for a complex float type, with the
2 imaginary part of the result possibly adjusted for use in
3 computing other functions.
4 Copyright (C) 1997-2018 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 "quadmath-imp.h"
22
23 /* Return the complex inverse hyperbolic sine of finite nonzero Z,
24 with the imaginary part of the result subtracted from pi/2 if ADJ
25 is nonzero. */
26
27 __complex128
28 __quadmath_kernel_casinhq (__complex128 x, int adj)
29 {
30 __complex128 res;
31 __float128 rx, ix;
32 __complex128 y;
33
34 /* Avoid cancellation by reducing to the first quadrant. */
35 rx = fabsq (__real__ x);
36 ix = fabsq (__imag__ x);
37
38 if (rx >= 1 / FLT128_EPSILON || ix >= 1 / FLT128_EPSILON)
39 {
40 /* For large x in the first quadrant, x + csqrt (1 + x * x)
41 is sufficiently close to 2 * x to make no significant
42 difference to the result; avoid possible overflow from
43 the squaring and addition. */
44 __real__ y = rx;
45 __imag__ y = ix;
46
47 if (adj)
48 {
49 __float128 t = __real__ y;
50 __real__ y = copysignq (__imag__ y, __imag__ x);
51 __imag__ y = t;
52 }
53
54 res = clogq (y);
55 __real__ res += (__float128) M_LN2q;
56 }
57 else if (rx >= 0.5Q && ix < FLT128_EPSILON / 8)
58 {
59 __float128 s = hypotq (1, rx);
60
61 __real__ res = logq (rx + s);
62 if (adj)
63 __imag__ res = atan2q (s, __imag__ x);
64 else
65 __imag__ res = atan2q (ix, s);
66 }
67 else if (rx < FLT128_EPSILON / 8 && ix >= 1.5Q)
68 {
69 __float128 s = sqrtq ((ix + 1) * (ix - 1));
70
71 __real__ res = logq (ix + s);
72 if (adj)
73 __imag__ res = atan2q (rx, copysignq (s, __imag__ x));
74 else
75 __imag__ res = atan2q (s, rx);
76 }
77 else if (ix > 1 && ix < 1.5Q && rx < 0.5Q)
78 {
79 if (rx < FLT128_EPSILON * FLT128_EPSILON)
80 {
81 __float128 ix2m1 = (ix + 1) * (ix - 1);
82 __float128 s = sqrtq (ix2m1);
83
84 __real__ res = log1pq (2 * (ix2m1 + ix * s)) / 2;
85 if (adj)
86 __imag__ res = atan2q (rx, copysignq (s, __imag__ x));
87 else
88 __imag__ res = atan2q (s, rx);
89 }
90 else
91 {
92 __float128 ix2m1 = (ix + 1) * (ix - 1);
93 __float128 rx2 = rx * rx;
94 __float128 f = rx2 * (2 + rx2 + 2 * ix * ix);
95 __float128 d = sqrtq (ix2m1 * ix2m1 + f);
96 __float128 dp = d + ix2m1;
97 __float128 dm = f / dp;
98 __float128 r1 = sqrtq ((dm + rx2) / 2);
99 __float128 r2 = rx * ix / r1;
100
101 __real__ res = log1pq (rx2 + dp + 2 * (rx * r1 + ix * r2)) / 2;
102 if (adj)
103 __imag__ res = atan2q (rx + r1, copysignq (ix + r2, __imag__ x));
104 else
105 __imag__ res = atan2q (ix + r2, rx + r1);
106 }
107 }
108 else if (ix == 1 && rx < 0.5Q)
109 {
110 if (rx < FLT128_EPSILON / 8)
111 {
112 __real__ res = log1pq (2 * (rx + sqrtq (rx))) / 2;
113 if (adj)
114 __imag__ res = atan2q (sqrtq (rx), copysignq (1, __imag__ x));
115 else
116 __imag__ res = atan2q (1, sqrtq (rx));
117 }
118 else
119 {
120 __float128 d = rx * sqrtq (4 + rx * rx);
121 __float128 s1 = sqrtq ((d + rx * rx) / 2);
122 __float128 s2 = sqrtq ((d - rx * rx) / 2);
123
124 __real__ res = log1pq (rx * rx + d + 2 * (rx * s1 + s2)) / 2;
125 if (adj)
126 __imag__ res = atan2q (rx + s1, copysignq (1 + s2, __imag__ x));
127 else
128 __imag__ res = atan2q (1 + s2, rx + s1);
129 }
130 }
131 else if (ix < 1 && rx < 0.5Q)
132 {
133 if (ix >= FLT128_EPSILON)
134 {
135 if (rx < FLT128_EPSILON * FLT128_EPSILON)
136 {
137 __float128 onemix2 = (1 + ix) * (1 - ix);
138 __float128 s = sqrtq (onemix2);
139
140 __real__ res = log1pq (2 * rx / s) / 2;
141 if (adj)
142 __imag__ res = atan2q (s, __imag__ x);
143 else
144 __imag__ res = atan2q (ix, s);
145 }
146 else
147 {
148 __float128 onemix2 = (1 + ix) * (1 - ix);
149 __float128 rx2 = rx * rx;
150 __float128 f = rx2 * (2 + rx2 + 2 * ix * ix);
151 __float128 d = sqrtq (onemix2 * onemix2 + f);
152 __float128 dp = d + onemix2;
153 __float128 dm = f / dp;
154 __float128 r1 = sqrtq ((dp + rx2) / 2);
155 __float128 r2 = rx * ix / r1;
156
157 __real__ res = log1pq (rx2 + dm + 2 * (rx * r1 + ix * r2)) / 2;
158 if (adj)
159 __imag__ res = atan2q (rx + r1, copysignq (ix + r2,
160 __imag__ x));
161 else
162 __imag__ res = atan2q (ix + r2, rx + r1);
163 }
164 }
165 else
166 {
167 __float128 s = hypotq (1, rx);
168
169 __real__ res = log1pq (2 * rx * (rx + s)) / 2;
170 if (adj)
171 __imag__ res = atan2q (s, __imag__ x);
172 else
173 __imag__ res = atan2q (ix, s);
174 }
175 math_check_force_underflow_nonneg (__real__ res);
176 }
177 else
178 {
179 __real__ y = (rx - ix) * (rx + ix) + 1;
180 __imag__ y = 2 * rx * ix;
181
182 y = csqrtq (y);
183
184 __real__ y += rx;
185 __imag__ y += ix;
186
187 if (adj)
188 {
189 __float128 t = __real__ y;
190 __real__ y = copysignq (__imag__ y, __imag__ x);
191 __imag__ y = t;
192 }
193
194 res = clogq (y);
195 }
196
197 /* Give results the correct sign for the original argument. */
198 __real__ res = copysignq (__real__ res, __real__ x);
199 __imag__ res = copysignq (__imag__ res, (adj ? 1 : __imag__ x));
200
201 return res;
202 }