1 /* $OpenBSD: s_casinl.c,v 1.3 2011/07/20 21:02:51 martynas Exp $ */
2
3 /*
4 * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
5 *
6 * Permission to use, copy, modify, and distribute this software for any
7 * purpose with or without fee is hereby granted, provided that the above
8 * copyright notice and this permission notice appear in all copies.
9 *
10 * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11 * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12 * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13 * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14 * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15 * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16 * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
17 */
18
19 /* casinl()
20 *
21 * Complex circular arc sine
22 *
23 *
24 *
25 * SYNOPSIS:
26 *
27 * long double complex casinl();
28 * long double complex z, w;
29 *
30 * w = casinl( z );
31 *
32 *
33 *
34 * DESCRIPTION:
35 *
36 * Inverse complex sine:
37 *
38 * 2
39 * w = -i clog( iz + csqrt( 1 - z ) ).
40 *
41 *
42 * ACCURACY:
43 *
44 * Relative error:
45 * arithmetic domain # trials peak rms
46 * DEC -10,+10 10100 2.1e-15 3.4e-16
47 * IEEE -10,+10 30000 2.2e-14 2.7e-15
48 * Larger relative error can be observed for z near zero.
49 * Also tested by csin(casin(z)) = z.
50 */
51
52 #include <complex.h>
53 #include <float.h>
54 #include <math.h>
55
56 #if LDBL_MANT_DIG == 64
57 static const long double MACHEPL= 5.42101086242752217003726400434970855712890625E-20L;
58 #elif LDBL_MANT_DIG == 113
59 static const long double MACHEPL = 9.629649721936179265279889712924636592690508e-35L;
60 #endif
61
62 static const long double PIO2L = 1.570796326794896619231321691639751442098585L;
63
64 long double complex
casinl(long double complex z)65 casinl(long double complex z)
66 {
67 long double complex w;
68 long double x, y, b;
69 static long double complex ca, ct, zz, z2;
70
71 x = creall(z);
72 y = cimagl(z);
73
74 if (y == 0.0L) {
75 if (fabsl(x) > 1.0L) {
76 w = PIO2L + 0.0L * I;
77 /*mtherr( "casinl", DOMAIN );*/
78 }
79 else {
80 w = asinl(x) + 0.0L * I;
81 }
82 return (w);
83 }
84
85 /* Power series expansion */
86 b = cabsl(z);
87 if (b < 0.125L) {
88 long double complex sum;
89 long double n, cn;
90
91 z2 = (x - y) * (x + y) + (2.0L * x * y) * I;
92 cn = 1.0L;
93 n = 1.0L;
94 ca = x + y * I;
95 sum = x + y * I;
96 do {
97 ct = z2 * ca;
98 ca = ct;
99
100 cn *= n;
101 n += 1.0L;
102 cn /= n;
103 n += 1.0L;
104 b = cn/n;
105
106 ct *= b;
107 sum += ct;
108 b = cabsl(ct);
109 }
110
111 while (b > MACHEPL);
112 w = sum;
113 return w;
114 }
115
116 ca = x + y * I;
117 ct = ca * I; /* iz */
118 /* sqrt(1 - z*z) */
119 /* cmul(&ca, &ca, &zz) */
120 /* x * x - y * y */
121 zz = (x - y) * (x + y) + (2.0L * x * y) * I;
122 zz = 1.0L - creall(zz) - cimagl(zz) * I;
123 z2 = csqrtl(zz);
124
125 zz = ct + z2;
126 zz = clogl(zz);
127 /* multiply by 1/i = -i */
128 w = zz * (-1.0L * I);
129 return (w);
130 }
131