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/* From: @(#)k_cos.c 1.3 95/01/18 */
/*
* ====================================================
* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
* Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
*
* Developed at SunSoft, a Sun Microsystems, Inc. business.
* Permission to use, copy, modify, and distribute this
* software is freely granted, provided that this notice
* is preserved.
* ====================================================
*/
#include <sys/cdefs.h>
__FBSDID("$FreeBSD$");
/*
* ld80 version of k_cos.c. See ../src/k_cos.c for most comments.
*/
#include "math_private.h"
/*
* Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]:
* |cos(x) - c(x)| < 2**-75.1
*
* The coefficients of c(x) were generated by a pari-gp script using
* a Remez algorithm that searches for the best higher coefficients
* after rounding leading coefficients to a specified precision.
*
* Simpler methods like Chebyshev or basic Remez barely suffice for
* cos() in 64-bit precision, because we want the coefficient of x^2
* to be precisely -0.5 so that multiplying by it is exact, and plain
* rounding of the coefficients of a good polynomial approximation only
* gives this up to about 64-bit precision. Plain rounding also gives
* a mediocre approximation for the coefficient of x^4, but a rounding
* error of 0.5 ulps for this coefficient would only contribute ~0.01
* ulps to the final error, so this is unimportant. Rounding errors in
* higher coefficients are even less important.
*
* In fact, coefficients above the x^4 one only need to have 53-bit
* precision, and this is more efficient. We get this optimization
* almost for free from the complications needed to search for the best
* higher coefficients.
*/
static const double
one = 1.0;
#if defined(__amd64__) || defined(__i386__)
/* Long double constants are slow on these arches, and broken on i386. */
static const volatile double
C1hi = 0.041666666666666664, /* 0x15555555555555.0p-57 */
C1lo = 2.2598839032744733e-18; /* 0x14d80000000000.0p-111 */
#define C1 ((long double)C1hi + C1lo)
#else
static const long double
C1 = 0.0416666666666666666136L; /* 0xaaaaaaaaaaaaaa9b.0p-68 */
#endif
static const double
C2 = -0.0013888888888888874, /* -0x16c16c16c16c10.0p-62 */
C3 = 0.000024801587301571716, /* 0x1a01a01a018e22.0p-68 */
C4 = -0.00000027557319215507120, /* -0x127e4fb7602f22.0p-74 */
C5 = 0.0000000020876754400407278, /* 0x11eed8caaeccf1.0p-81 */
C6 = -1.1470297442401303e-11, /* -0x19393412bd1529.0p-89 */
C7 = 4.7383039476436467e-14; /* 0x1aac9d9af5c43e.0p-97 */
long double
__kernel_cosl(long double x, long double y)
{
long double hz,z,r,w;
z = x*x;
r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7))))));
hz = 0.5*z;
w = one-hz;
return w + (((one-w)-hz) + (z*r-x*y));
}
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