2 * Copyright (c) 2017, 2023 Steven G. Kargl
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6 * modification, are permitted provided that the following conditions
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9 * notice unmodified, this list of conditions, and the following
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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.
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.
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.
40 * 3. For 1 <= |x| < 0x1p(P-1), argument reduction is required where
41 * |x| = j0 + r with j0 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
45 * cospi(x) = cos(pi*(j0+r))
46 * = cos(pi*j0) * cos(pi*r) - sin(pi*j0) * sin(pi*r)
47 * = cos(pi*j0) * cos(pi*r)
50 * If j0 is even, then cos(pi*j0) = 1. If j0 is odd, then cos(pi*j0) = -1.
51 * cospi(r) is then computed via an appropriate kernel.
53 * 4. For |x| >= 0x1p(P-1), |x| is integral and cospi(x) = 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.
65 #include "math_private.h"
68 pi_hi = 3.1415926814079285e+00, /* 0x400921fb 0x58000000 */
69 pi_lo =-2.7818135228334233e-08; /* 0xbe5dde97 0x3dcb3b3a */
74 volatile static const double vzero = 0;
80 uint32_t hx, ix, j0, lx;
82 EXTRACT_WORDS(hx, lx, x);
84 INSERT_WORDS(ax, ix, lx);
86 if (ix < 0x3ff00000) { /* |x| < 1 */
87 if (ix < 0x3fd00000) { /* |x| < 0.25 */
88 if (ix < 0x3e200000) { /* |x| < 0x1p-29 */
92 return (__kernel_cospi(ax));
95 if (ix < 0x3fe00000) /* |x| < 0.5 */
96 c = __kernel_sinpi(0.5 - ax);
97 else if (ix < 0x3fe80000){ /* |x| < 0.75 */
100 c = -__kernel_sinpi(ax - 0.5);
102 c = -__kernel_cospi(1 - ax);
106 if (ix < 0x43300000) { /* 1 <= |x| < 0x1p52 */
107 FFLOOR(x, j0, ix, lx); /* Integer part of ax. */
109 EXTRACT_WORDS(ix, lx, ax);
111 if (ix < 0x3fe00000) { /* |x| < 0.5 */
112 if (ix < 0x3fd00000) /* |x| < 0.25 */
113 c = ix == 0 ? 1 : __kernel_cospi(ax);
115 c = __kernel_sinpi(0.5 - ax);
117 if (ix < 0x3fe80000) { /* |x| < 0.75 */
120 c = -__kernel_sinpi(ax - 0.5);
122 c = -__kernel_cospi(1 - ax);
128 return (j0 & 1 ? -c : c);
131 /* x = +-inf or nan. */
132 if (ix >= 0x7ff00000)
133 return (vzero / vzero);
136 * For 0x1p52 <= |x| < 0x1p53 need to determine if x is an even
137 * or odd integer to return +1 or -1.
138 * For |x| >= 0x1p53, it is always an even integer, so return 1.
140 return (ix < 0x43400000 ? ((lx & 1) ? -1 : 1) : 1);
143 #if LDBL_MANT_DIG == 53
144 __weak_reference(cospi, cospil);