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      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| = jj0 + r with jj0 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*(jj0+r))
     46  *             = cos(pi*jj0) * cos(pi*r) - sin(pi*jj0) * sin(pi*r)
     47  *             = cos(pi*jj0) * cos(pi*r)
     48  *             = +-cospi(r)
     49  *
     50  *    If jj0 is even, then cos(pi*jj0) = 1. If jj0 is odd, then cos(pi*jj0) = -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 <sys/cdefs.h>
     64 
     65 #include "namespace.h"
     66 __weak_alias(cospi, _cospi)
     67 
     68 #include <float.h>
     69 #include "math.h"
     70 #include "math_private.h"
     71 
     72 static const double
     73 pi_hi = 3.1415926814079285e+00,	/* 0x400921fb 0x58000000 */
     74 pi_lo =-2.7818135228334233e-08;	/* 0xbe5dde97 0x3dcb3b3a */
     75 
     76 #include "k_cospi.h"
     77 #include "k_sinpi.h"
     78 
     79 static volatile const double vzero = 0;
     80 
     81 double
     82 cospi(double x)
     83 {
     84 	double ax, c;
     85 	uint32_t hx, ix, jj0, lx;
     86 
     87 	EXTRACT_WORDS(hx, lx, x);
     88 	ix = hx & 0x7fffffff;
     89 	INSERT_WORDS(ax, ix, lx);
     90 
     91 	if (ix < 0x3ff00000) {			/* |x| < 1 */
     92 		if (ix < 0x3fd00000) {		/* |x| < 0.25 */
     93 			if (ix < 0x3e200000) {	/* |x| < 0x1p-29 */
     94 				if ((int)ax == 0)
     95 					return (1);
     96 			}
     97 			return (__kernel_cospi(ax));
     98 		}
     99 
    100 		if (ix < 0x3fe00000)		/* |x| < 0.5 */
    101 			c = __kernel_sinpi(0.5 - ax);
    102 		else if (ix < 0x3fe80000){	/* |x| < 0.75 */
    103 			if (ax == 0.5)
    104 				return (0);
    105 			c = -__kernel_sinpi(ax - 0.5);
    106 		} else
    107 			c = -__kernel_cospi(1 - ax);
    108 		return (c);
    109 	}
    110 
    111 	if (ix < 0x43300000) {		/* 1 <= |x| < 0x1p52 */
    112 		FFLOOR(x, jj0, ix, lx);	/* Integer part of ax. */
    113 		ax -= x;
    114 		EXTRACT_WORDS(ix, lx, ax);
    115 
    116 		if (ix < 0x3fe00000) {		/* |x| < 0.5 */
    117 			if (ix < 0x3fd00000)	/* |x| < 0.25 */
    118 				c = ix == 0 ? 1 : __kernel_cospi(ax);
    119 			else
    120 				c = __kernel_sinpi(0.5 - ax);
    121 		} else {
    122 			if (ix < 0x3fe80000) {	/* |x| < 0.75 */
    123 				if (ax == 0.5)
    124 					return (0);
    125 				c = -__kernel_sinpi(ax - 0.5);
    126 			} else
    127 				c = -__kernel_cospi(1 - ax);
    128 		}
    129 
    130 		if (jj0 > 30)
    131 			x -= 0x1p30;
    132 		jj0 = (uint32_t)x;
    133 		return (jj0 & 1 ? -c : c);
    134 	}
    135 
    136 	/* x = +-inf or nan. */
    137 	if (ix >= 0x7ff00000)
    138 		return (vzero / vzero);
    139 
    140 	/*
    141 	 * For 0x1p52 <= |x| < 0x1p53 need to determine if x is an even
    142 	 * or odd integer to return +1 or -1.
    143 	 * For |x| >= 0x1p53, it is always an even integer, so return 1.
    144 	 */
    145 	return (ix < 0x43400000 ? ((lx & 1) ? -1 : 1) : 1);
    146 }
    147