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/*-
* Copyright (c) 2017, 2023 Steven G. Kargl
* All rights reserved.
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
* 1. Redistributions of source code must retain the above copyright
* notice unmodified, this list of conditions, and the following
* disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in the
* documentation and/or other materials provided with the distribution.
*
* THIS SOFTWARE IS PROVIDED BY THE AUTHOR ``AS IS'' AND ANY EXPRESS OR
* IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES
* OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.
* IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR ANY DIRECT, INDIRECT,
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT
* NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
* DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
* THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
* (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF
* THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
*/
/**
* sinpi(x) computes sin(pi*x) without multiplication by pi (almost). First,
* note that sinpi(-x) = -sinpi(x), so the algorithm considers only |x| and
* includes reflection symmetry by considering the sign of x on output. The
* method used depends on the magnitude of x.
*
* 1. For small |x|, sinpi(x) = pi * x where a sloppy threshold is used. The
* threshold is |x| < 0x1pN with N = -(P/2+M). P is the precision of the
* floating-point type and M = 2 to 4. To achieve high accuracy, pi is
* decomposed into high and low parts with the high part containing a
* number of trailing zero bits. x is also split into high and low parts.
*
* 2. For |x| < 1, argument reduction is not required and sinpi(x) is
* computed by calling a kernel that leverages the kernels for sin(x)
* ans cos(x). See k_sinpi.c and k_cospi.c for details.
*
* 3. For 1 <= |x| < 0x1p(P-1), argument reduction is required where
* |x| = j0 + r with j0 an integer and the remainder r satisfies
* 0 <= r < 1. With the given domain, a simplified inline floor(x)
* is used. Also, note the following identity
*
* sinpi(x) = sin(pi*(j0+r))
* = sin(pi*j0) * cos(pi*r) + cos(pi*j0) * sin(pi*r)
* = cos(pi*j0) * sin(pi*r)
* = +-sinpi(r)
*
* If j0 is even, then cos(pi*j0) = 1. If j0 is odd, then cos(pi*j0) = -1.
* sinpi(r) is then computed via an appropriate kernel.
*
* 4. For |x| >= 0x1p(P-1), |x| is integral and sinpi(x) = copysign(0,x).
*
* 5. Special cases:
*
* sinpi(+-0) = +-0
* sinpi(+-n) = +-0, for positive integers n.
* sinpi(+-inf) = nan. Raises the "invalid" floating-point exception.
* sinpi(nan) = nan. Raises the "invalid" floating-point exception.
*/
#include <float.h>
#include "math.h"
#include "math_private.h"
static const double
pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */
pi_lo =-2.7818135228334233e-08; /* 0xbe5dde97 0x3dcb3b3a */
#include "k_cospi.h"
#include "k_sinpi.h"
volatile static const double vzero = 0;
double
sinpi(double x)
{
double ax, hi, lo, s;
uint32_t hx, ix, j0, lx;
EXTRACT_WORDS(hx, lx, x);
ix = hx & 0x7fffffff;
INSERT_WORDS(ax, ix, lx);
if (ix < 0x3ff00000) { /* |x| < 1 */
if (ix < 0x3fd00000) { /* |x| < 0.25 */
if (ix < 0x3e200000) { /* |x| < 0x1p-29 */
if (x == 0)
return (x);
/*
* To avoid issues with subnormal values,
* scale the computation and rescale on
* return.
*/
INSERT_WORDS(hi, hx, 0);
hi *= 0x1p53;
lo = x * 0x1p53 - hi;
s = (pi_lo + pi_hi) * lo + pi_lo * hi +
pi_hi * hi;
return (s * 0x1p-53);
}
s = __kernel_sinpi(ax);
return ((hx & 0x80000000) ? -s : s);
}
if (ix < 0x3fe00000) /* |x| < 0.5 */
s = __kernel_cospi(0.5 - ax);
else if (ix < 0x3fe80000) /* |x| < 0.75 */
s = __kernel_cospi(ax - 0.5);
else
s = __kernel_sinpi(1 - ax);
return ((hx & 0x80000000) ? -s : s);
}
if (ix < 0x43300000) { /* 1 <= |x| < 0x1p52 */
FFLOOR(x, j0, ix, lx); /* Integer part of ax. */
ax -= x;
EXTRACT_WORDS(ix, lx, ax);
if (ix == 0)
s = 0;
else {
if (ix < 0x3fe00000) { /* |x| < 0.5 */
if (ix < 0x3fd00000) /* |x| < 0.25 */
s = __kernel_sinpi(ax);
else
s = __kernel_cospi(0.5 - ax);
} else {
if (ix < 0x3fe80000) /* |x| < 0.75 */
s = __kernel_cospi(ax - 0.5);
else
s = __kernel_sinpi(1 - ax);
}
if (j0 > 30)
x -= 0x1p30;
j0 = (uint32_t)x;
if (j0 & 1) s = -s;
}
return ((hx & 0x80000000) ? -s : s);
}
/* x = +-inf or nan. */
if (ix >= 0x7ff00000)
return (vzero / vzero);
/*
* |x| >= 0x1p52 is always an integer, so return +-0.
*/
return (copysign(0, x));
}
#if LDBL_MANT_DIG == 53
__weak_reference(sinpi, sinpil);
#endif
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