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 * 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
70static const double
71pi_hi = 3.1415926814079285e+00,	/* 0x400921fb 0x58000000 */
72pi_lo =-2.7818135228334233e-08;	/* 0xbe5dde97 0x3dcb3b3a */
73
74#include "k_cospi.h"
75#include "k_sinpi.h"
76
77volatile static const double vzero = 0;
78
79double
80sinpi(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		FFLOOR(x, j0, ix, lx);	/* Integer part of ax. */
122		ax -= x;
123		EXTRACT_WORDS(ix, lx, ax);
124
125		if (ix == 0)
126			s = 0;
127		else {
128			if (ix < 0x3fe00000) {		/* |x| < 0.5 */
129				if (ix < 0x3fd00000)	/* |x| < 0.25 */
130					s = __kernel_sinpi(ax);
131				else
132					s = __kernel_cospi(0.5 - ax);
133			} else {
134				if (ix < 0x3fe80000)	/* |x| < 0.75 */
135					s = __kernel_cospi(ax - 0.5);
136				else
137					s = __kernel_sinpi(1 - ax);
138			}
139
140			if (j0 > 30)
141				x -= 0x1p30;
142			j0 = (uint32_t)x;
143			if (j0 & 1) s = -s;
144		}
145
146		return ((hx & 0x80000000) ? -s : s);
147	}
148
149	/* x = +-inf or nan. */
150	if (ix >= 0x7ff00000)
151		return (vzero / vzero);
152
153	/*
154	 * |x| >= 0x1p52 is always an integer, so return +-0.
155	 */
156	return (copysign(0, x));
157}
158
159#if LDBL_MANT_DIG == 53
160__weak_reference(sinpi, sinpil);
161#endif
162