1*05a0b428SJohn Marino /* $OpenBSD: s_csqrtl.c,v 1.2 2011/07/20 19:28:33 martynas Exp $ */
2*05a0b428SJohn Marino
3*05a0b428SJohn Marino /*
4*05a0b428SJohn Marino * Copyright (c) 2008 Stephen L. Moshier <steve@moshier.net>
5*05a0b428SJohn Marino *
6*05a0b428SJohn Marino * Permission to use, copy, modify, and distribute this software for any
7*05a0b428SJohn Marino * purpose with or without fee is hereby granted, provided that the above
8*05a0b428SJohn Marino * copyright notice and this permission notice appear in all copies.
9*05a0b428SJohn Marino *
10*05a0b428SJohn Marino * THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
11*05a0b428SJohn Marino * WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
12*05a0b428SJohn Marino * MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
13*05a0b428SJohn Marino * ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
14*05a0b428SJohn Marino * WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
15*05a0b428SJohn Marino * ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
16*05a0b428SJohn Marino * OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
17*05a0b428SJohn Marino */
18*05a0b428SJohn Marino
19*05a0b428SJohn Marino /* csqrtl()
20*05a0b428SJohn Marino *
21*05a0b428SJohn Marino * Complex square root
22*05a0b428SJohn Marino *
23*05a0b428SJohn Marino *
24*05a0b428SJohn Marino *
25*05a0b428SJohn Marino * SYNOPSIS:
26*05a0b428SJohn Marino *
27*05a0b428SJohn Marino * long double complex csqrtl();
28*05a0b428SJohn Marino * long double complex z, w;
29*05a0b428SJohn Marino *
30*05a0b428SJohn Marino * w = csqrtl( z );
31*05a0b428SJohn Marino *
32*05a0b428SJohn Marino *
33*05a0b428SJohn Marino *
34*05a0b428SJohn Marino * DESCRIPTION:
35*05a0b428SJohn Marino *
36*05a0b428SJohn Marino *
37*05a0b428SJohn Marino * If z = x + iy, r = |z|, then
38*05a0b428SJohn Marino *
39*05a0b428SJohn Marino * 1/2
40*05a0b428SJohn Marino * Re w = [ (r + x)/2 ] ,
41*05a0b428SJohn Marino *
42*05a0b428SJohn Marino * 1/2
43*05a0b428SJohn Marino * Im w = [ (r - x)/2 ] .
44*05a0b428SJohn Marino *
45*05a0b428SJohn Marino * Cancellation error in r-x or r+x is avoided by using the
46*05a0b428SJohn Marino * identity 2 Re w Im w = y.
47*05a0b428SJohn Marino *
48*05a0b428SJohn Marino * Note that -w is also a square root of z. The root chosen
49*05a0b428SJohn Marino * is always in the right half plane and Im w has the same sign as y.
50*05a0b428SJohn Marino *
51*05a0b428SJohn Marino *
52*05a0b428SJohn Marino *
53*05a0b428SJohn Marino * ACCURACY:
54*05a0b428SJohn Marino *
55*05a0b428SJohn Marino * Relative error:
56*05a0b428SJohn Marino * arithmetic domain # trials peak rms
57*05a0b428SJohn Marino * IEEE -10,+10 500000 1.1e-19 3.0e-20
58*05a0b428SJohn Marino *
59*05a0b428SJohn Marino */
60*05a0b428SJohn Marino
61*05a0b428SJohn Marino #include <complex.h>
62*05a0b428SJohn Marino #include <math.h>
63*05a0b428SJohn Marino
64*05a0b428SJohn Marino long double complex
csqrtl(long double complex z)65*05a0b428SJohn Marino csqrtl(long double complex z)
66*05a0b428SJohn Marino {
67*05a0b428SJohn Marino long double complex w;
68*05a0b428SJohn Marino long double x, y, r, t, scale;
69*05a0b428SJohn Marino
70*05a0b428SJohn Marino x = creall(z);
71*05a0b428SJohn Marino y = cimagl(z);
72*05a0b428SJohn Marino
73*05a0b428SJohn Marino if (y == 0.0L) {
74*05a0b428SJohn Marino if (x < 0.0L) {
75*05a0b428SJohn Marino w = 0.0L + sqrtl(-x) * I;
76*05a0b428SJohn Marino return (w);
77*05a0b428SJohn Marino }
78*05a0b428SJohn Marino else {
79*05a0b428SJohn Marino w = sqrtl(x) + 0.0L * I;
80*05a0b428SJohn Marino return (w);
81*05a0b428SJohn Marino }
82*05a0b428SJohn Marino }
83*05a0b428SJohn Marino
84*05a0b428SJohn Marino if (x == 0.0L) {
85*05a0b428SJohn Marino r = fabsl(y);
86*05a0b428SJohn Marino r = sqrtl(0.5L * r);
87*05a0b428SJohn Marino if (y > 0.0L)
88*05a0b428SJohn Marino w = r + r * I;
89*05a0b428SJohn Marino else
90*05a0b428SJohn Marino w = r - r * I;
91*05a0b428SJohn Marino return (w);
92*05a0b428SJohn Marino }
93*05a0b428SJohn Marino
94*05a0b428SJohn Marino /* Rescale to avoid internal overflow or underflow. */
95*05a0b428SJohn Marino if ((fabsl(x) > 4.0L) || (fabsl(y) > 4.0L)) {
96*05a0b428SJohn Marino x *= 0.25L;
97*05a0b428SJohn Marino y *= 0.25L;
98*05a0b428SJohn Marino scale = 2.0L;
99*05a0b428SJohn Marino }
100*05a0b428SJohn Marino else {
101*05a0b428SJohn Marino #if 1
102*05a0b428SJohn Marino x *= 7.3786976294838206464e19; /* 2^66 */
103*05a0b428SJohn Marino y *= 7.3786976294838206464e19;
104*05a0b428SJohn Marino scale = 1.16415321826934814453125e-10; /* 2^-33 */
105*05a0b428SJohn Marino #else
106*05a0b428SJohn Marino x *= 4.0L;
107*05a0b428SJohn Marino y *= 4.0L;
108*05a0b428SJohn Marino scale = 0.5L;
109*05a0b428SJohn Marino #endif
110*05a0b428SJohn Marino }
111*05a0b428SJohn Marino w = x + y * I;
112*05a0b428SJohn Marino r = cabsl(w);
113*05a0b428SJohn Marino if (x > 0) {
114*05a0b428SJohn Marino t = sqrtl(0.5L * r + 0.5L * x);
115*05a0b428SJohn Marino r = scale * fabsl((0.5L * y) / t);
116*05a0b428SJohn Marino t *= scale;
117*05a0b428SJohn Marino }
118*05a0b428SJohn Marino else {
119*05a0b428SJohn Marino r = sqrtl(0.5L * r - 0.5L * x);
120*05a0b428SJohn Marino t = scale * fabsl((0.5L * y) / r);
121*05a0b428SJohn Marino r *= scale;
122*05a0b428SJohn Marino }
123*05a0b428SJohn Marino if (y < 0)
124*05a0b428SJohn Marino w = t - r * I;
125*05a0b428SJohn Marino else
126*05a0b428SJohn Marino w = t + r * I;
127*05a0b428SJohn Marino return (w);
128*05a0b428SJohn Marino }
129