/*- * 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
