1 /*-
2 * Copyright (c) 2017, 2023 Steven G. Kargl
3 * All rights reserved.
4 *
5 * Redistribution and use in source and binary forms, with or without
6 * modification, are permitted provided that the following conditions
7 * are met:
8 * 1. Redistributions of source code must retain the above copyright
9 * notice unmodified, this list of conditions, and the following
10 * disclaimer.
11 * 2. Redistributions in binary form must reproduce the above copyright
12 * notice, this list of conditions and the following disclaimer in the
13 * documentation and/or other materials provided with the distribution.
14 *
15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
16 * IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
17 * OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
18 * IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
19 * INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
20 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
21 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
22 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
23 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
24 * THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
25 */
26
27 /**
28 * cospi(x) computes cos(pi*x) without multiplication by pi (almost). First,
29 * note that cospi(-x) = cospi(x), so the algorithm considers only |x|. The
30 * method used depends on the magnitude of x.
31 *
32 * 1. For small |x|, cospi(x) = 1 with FE_INEXACT raised where a sloppy
33 * threshold is used. The threshold is |x| < 0x1pN with N = -(P/2+M).
34 * P is the precision of the floating-point type and M = 2 to 4.
35 *
36 * 2. For |x| < 1, argument reduction is not required and sinpi(x) is
37 * computed by calling a kernel that leverages the kernels for sin(x)
38 * ans cos(x). See k_sinpi.c and k_cospi.c for details.
39 *
40 * 3. For 1 <= |x| < 0x1p(P-1), argument reduction is required where
41 * |x| = jj0 + r with jj0 an integer and the remainder r satisfies
42 * 0 <= r < 1. With the given domain, a simplified inline floor(x)
43 * is used. Also, note the following identity
44 *
45 * cospi(x) = cos(pi*(jj0+r))
46 * = cos(pi*jj0) * cos(pi*r) - sin(pi*jj0) * sin(pi*r)
47 * = cos(pi*jj0) * cos(pi*r)
48 * = +-cospi(r)
49 *
50 * If jj0 is even, then cos(pi*jj0) = 1. If jj0 is odd, then cos(pi*jj0) = -1.
51 * cospi(r) is then computed via an appropriate kernel.
52 *
53 * 4. For |x| >= 0x1p(P-1), |x| is integral and cospi(x) = 1.
54 *
55 * 5. Special cases:
56 *
57 * cospi(+-0) = 1.
58 * cospi(n.5) = 0 for n an integer.
59 * cospi(+-inf) = nan. Raises the "invalid" floating-point exception.
60 * cospi(nan) = nan. Raises the "invalid" floating-point exception.
61 */
62
63 #include <sys/cdefs.h>
64
65 #include "namespace.h"
66 __weak_alias(cospi, _cospi)
67
68 #include <float.h>
69 #include "math.h"
70 #include "math_private.h"
71
72 static const double
73 pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */
74 pi_lo =-2.7818135228334233e-08; /* 0xbe5dde97 0x3dcb3b3a */
75
76 #include "k_cospi.h"
77 #include "k_sinpi.h"
78
79 static volatile const double vzero = 0;
80
81 double
cospi(double x)82 cospi(double x)
83 {
84 double ax, c;
85 uint32_t hx, ix, jj0, lx;
86
87 EXTRACT_WORDS(hx, lx, x);
88 ix = hx & 0x7fffffff;
89 INSERT_WORDS(ax, ix, lx);
90
91 if (ix < 0x3ff00000) { /* |x| < 1 */
92 if (ix < 0x3fd00000) { /* |x| < 0.25 */
93 if (ix < 0x3e200000) { /* |x| < 0x1p-29 */
94 if ((int)ax == 0)
95 return (1);
96 }
97 return (__kernel_cospi(ax));
98 }
99
100 if (ix < 0x3fe00000) /* |x| < 0.5 */
101 c = __kernel_sinpi(0.5 - ax);
102 else if (ix < 0x3fe80000){ /* |x| < 0.75 */
103 if (ax == 0.5)
104 return (0);
105 c = -__kernel_sinpi(ax - 0.5);
106 } else
107 c = -__kernel_cospi(1 - ax);
108 return (c);
109 }
110
111 if (ix < 0x43300000) { /* 1 <= |x| < 0x1p52 */
112 FFLOOR(x, jj0, ix, lx); /* Integer part of ax. */
113 ax -= x;
114 EXTRACT_WORDS(ix, lx, ax);
115
116 if (ix < 0x3fe00000) { /* |x| < 0.5 */
117 if (ix < 0x3fd00000) /* |x| < 0.25 */
118 c = ix == 0 ? 1 : __kernel_cospi(ax);
119 else
120 c = __kernel_sinpi(0.5 - ax);
121 } else {
122 if (ix < 0x3fe80000) { /* |x| < 0.75 */
123 if (ax == 0.5)
124 return (0);
125 c = -__kernel_sinpi(ax - 0.5);
126 } else
127 c = -__kernel_cospi(1 - ax);
128 }
129
130 if (jj0 > 30)
131 x -= 0x1p30;
132 jj0 = (uint32_t)x;
133 return (jj0 & 1 ? -c : c);
134 }
135
136 /* x = +-inf or nan. */
137 if (ix >= 0x7ff00000)
138 return (vzero / vzero);
139
140 /*
141 * For 0x1p52 <= |x| < 0x1p53 need to determine if x is an even
142 * or odd integer to return +1 or -1.
143 * For |x| >= 0x1p53, it is always an even integer, so return 1.
144 */
145 return (ix < 0x43400000 ? ((lx & 1) ? -1 : 1) : 1);
146 }
147