1 /*-
2 * Copyright (c) 2017 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 * sinpi(x) computes sin(pi*x) without multiplication by pi (almost). First,
29 * note that sinpi(-x) = -sinpi(x), so the algorithm considers only |x| and
30 * includes reflection symmetry by considering the sign of x on output. The
31 * method used depends on the magnitude of x.
32 *
33 * 1. For small |x|, sinpi(x) = pi * x where a sloppy threshold is used. The
34 * threshold is |x| < 0x1pN with N = -(P/2+M). P is the precision of the
35 * floating-point type and M = 2 to 4. To achieve high accuracy, pi is
36 * decomposed into high and low parts with the high part containing a
37 * number of trailing zero bits. x is also split into high and low parts.
38 *
39 * 2. For |x| < 1, argument reduction is not required and sinpi(x) is
40 * computed by calling a kernel that leverages the kernels for sin(x)
41 * ans cos(x). See k_sinpi.c and k_cospi.c for details.
42 *
43 * 3. For 1 <= |x| < 0x1p(P-1), argument reduction is required where
44 * |x| = j0 + r with j0 an integer and the remainder r satisfies
45 * 0 <= r < 1. With the given domain, a simplified inline floor(x)
46 * is used. Also, note the following identity
47 *
48 * sinpi(x) = sin(pi*(j0+r))
49 * = sin(pi*j0) * cos(pi*r) + cos(pi*j0) * sin(pi*r)
50 * = cos(pi*j0) * sin(pi*r)
51 * = +-sinpi(r)
52 *
53 * If j0 is even, then cos(pi*j0) = 1. If j0 is odd, then cos(pi*j0) = -1.
54 * sinpi(r) is then computed via an appropriate kernel.
55 *
56 * 4. For |x| >= 0x1p(P-1), |x| is integral and sinpi(x) = copysign(0,x).
57 *
58 * 5. Special cases:
59 *
60 * sinpi(+-0) = +-0
61 * sinpi(+-n) = +-0, for positive integers n.
62 * sinpi(+-inf) = nan. Raises the "invalid" floating-point exception.
63 * sinpi(nan) = nan. Raises the "invalid" floating-point exception.
64 */
65
66 #include <float.h>
67 #include "math.h"
68 #include "math_private.h"
69
70 static const double
71 pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */
72 pi_lo =-2.7818135228334233e-08; /* 0xbe5dde97 0x3dcb3b3a */
73
74 #include "k_cospi.h"
75 #include "k_sinpi.h"
76
77 volatile static const double vzero = 0;
78
79 double
sinpi(double x)80 sinpi(double x)
81 {
82 double ax, hi, lo, s;
83 uint32_t hx, ix, j0, lx;
84
85 EXTRACT_WORDS(hx, lx, x);
86 ix = hx & 0x7fffffff;
87 INSERT_WORDS(ax, ix, lx);
88
89 if (ix < 0x3ff00000) { /* |x| < 1 */
90 if (ix < 0x3fd00000) { /* |x| < 0.25 */
91 if (ix < 0x3e200000) { /* |x| < 0x1p-29 */
92 if (x == 0)
93 return (x);
94 /*
95 * To avoid issues with subnormal values,
96 * scale the computation and rescale on
97 * return.
98 */
99 INSERT_WORDS(hi, hx, 0);
100 hi *= 0x1p53;
101 lo = x * 0x1p53 - hi;
102 s = (pi_lo + pi_hi) * lo + pi_lo * hi +
103 pi_hi * hi;
104 return (s * 0x1p-53);
105 }
106
107 s = __kernel_sinpi(ax);
108 return ((hx & 0x80000000) ? -s : s);
109 }
110
111 if (ix < 0x3fe00000) /* |x| < 0.5 */
112 s = __kernel_cospi(0.5 - ax);
113 else if (ix < 0x3fe80000) /* |x| < 0.75 */
114 s = __kernel_cospi(ax - 0.5);
115 else
116 s = __kernel_sinpi(1 - ax);
117 return ((hx & 0x80000000) ? -s : s);
118 }
119
120 if (ix < 0x43300000) { /* 1 <= |x| < 0x1p52 */
121 /* Determine integer part of ax. */
122 j0 = ((ix >> 20) & 0x7ff) - 0x3ff;
123 if (j0 < 20) {
124 ix &= ~(0x000fffff >> j0);
125 lx = 0;
126 } else {
127 lx &= ~((uint32_t)0xffffffff >> (j0 - 20));
128 }
129 INSERT_WORDS(x, ix, lx);
130
131 ax -= x;
132 EXTRACT_WORDS(ix, lx, ax);
133
134 if (ix == 0)
135 s = 0;
136 else {
137 if (ix < 0x3fe00000) { /* |x| < 0.5 */
138 if (ix < 0x3fd00000) /* |x| < 0.25 */
139 s = __kernel_sinpi(ax);
140 else
141 s = __kernel_cospi(0.5 - ax);
142 } else {
143 if (ix < 0x3fe80000) /* |x| < 0.75 */
144 s = __kernel_cospi(ax - 0.5);
145 else
146 s = __kernel_sinpi(1 - ax);
147 }
148
149 if (j0 > 30)
150 x -= 0x1p30;
151 j0 = (uint32_t)x;
152 if (j0 & 1) s = -s;
153 }
154
155 return ((hx & 0x80000000) ? -s : s);
156 }
157
158 if (ix >= 0x7f800000)
159 return (vzero / vzero);
160
161 /*
162 * |x| >= 0x1p52 is always an integer, so return +-0.
163 */
164 return (copysign(0, x));
165 }
166
167 #if LDBL_MANT_DIG == 53
168 __weak_reference(sinpi, sinpil);
169 #endif
170