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| = j0 + r with j0 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*(j0+r))
46 * = cos(pi*j0) * cos(pi*r) - sin(pi*j0) * sin(pi*r)
47 * = cos(pi*j0) * cos(pi*r)
48 * = +-cospi(r)
49 *
50 * If j0 is even, then cos(pi*j0) = 1. If j0 is odd, then cos(pi*j0) = -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 <float.h>
64 #include "math.h"
65 #include "math_private.h"
66
67 static const double
68 pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */
69 pi_lo =-2.7818135228334233e-08; /* 0xbe5dde97 0x3dcb3b3a */
70
71 #include "k_cospi.h"
72 #include "k_sinpi.h"
73
74 volatile static const double vzero = 0;
75
76 double
cospi(double x)77 cospi(double x)
78 {
79 double ax, c;
80 uint32_t hx, ix, j0, lx;
81
82 EXTRACT_WORDS(hx, lx, x);
83 ix = hx & 0x7fffffff;
84 INSERT_WORDS(ax, ix, lx);
85
86 if (ix < 0x3ff00000) { /* |x| < 1 */
87 if (ix < 0x3fd00000) { /* |x| < 0.25 */
88 if (ix < 0x3e200000) { /* |x| < 0x1p-29 */
89 if ((int)ax == 0)
90 return (1);
91 }
92 return (__kernel_cospi(ax));
93 }
94
95 if (ix < 0x3fe00000) /* |x| < 0.5 */
96 c = __kernel_sinpi(0.5 - ax);
97 else if (ix < 0x3fe80000){ /* |x| < 0.75 */
98 if (ax == 0.5)
99 return (0);
100 c = -__kernel_sinpi(ax - 0.5);
101 } else
102 c = -__kernel_cospi(1 - ax);
103 return (c);
104 }
105
106 if (ix < 0x43300000) { /* 1 <= |x| < 0x1p52 */
107 FFLOOR(x, j0, ix, lx); /* Integer part of ax. */
108 ax -= x;
109 EXTRACT_WORDS(ix, lx, ax);
110
111 if (ix < 0x3fe00000) { /* |x| < 0.5 */
112 if (ix < 0x3fd00000) /* |x| < 0.25 */
113 c = ix == 0 ? 1 : __kernel_cospi(ax);
114 else
115 c = __kernel_sinpi(0.5 - ax);
116 } else {
117 if (ix < 0x3fe80000) { /* |x| < 0.75 */
118 if (ax == 0.5)
119 return (0);
120 c = -__kernel_sinpi(ax - 0.5);
121 } else
122 c = -__kernel_cospi(1 - ax);
123 }
124
125 if (j0 > 30)
126 x -= 0x1p30;
127 j0 = (uint32_t)x;
128 return (j0 & 1 ? -c : c);
129 }
130
131 /* x = +-inf or nan. */
132 if (ix >= 0x7ff00000)
133 return (vzero / vzero);
134
135 /*
136 * For 0x1p52 <= |x| < 0x1p53 need to determine if x is an even
137 * or odd integer to return +1 or -1.
138 * For |x| >= 0x1p53, it is always an even integer, so return 1.
139 */
140 return (ix < 0x43400000 ? ((lx & 1) ? -1 : 1) : 1);
141 }
142
143 #if LDBL_MANT_DIG == 53
144 __weak_reference(cospi, cospil);
145 #endif
146