de336b0c5e
Bruce for putting lots of effort into these; getting them right isn't easy, and they went through many iterations. Submitted by: Steve Kargl <sgk@apl.washington.edu> with revisions from bde
79 lines
2.8 KiB
C
79 lines
2.8 KiB
C
/* From: @(#)k_cos.c 1.3 95/01/18 */
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/*
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* ====================================================
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* Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
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* Copyright (c) 2008 Steven G. Kargl, David Schultz, Bruce D. Evans.
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*
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* Developed at SunSoft, a Sun Microsystems, Inc. business.
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* Permission to use, copy, modify, and distribute this
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* software is freely granted, provided that this notice
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* is preserved.
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* ====================================================
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*/
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#include <sys/cdefs.h>
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__FBSDID("$FreeBSD$");
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/*
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* ld80 version of k_cos.c. See ../src/k_cos.c for most comments.
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*/
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#include "math_private.h"
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/*
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* Domain [-0.7854, 0.7854], range ~[-2.43e-23, 2.425e-23]:
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* |cos(x) - c(x)| < 2**-75.1
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*
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* The coefficients of c(x) were generated by a pari-gp script using
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* a Remez algorithm that searches for the best higher coefficients
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* after rounding leading coefficients to a specified precision.
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*
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* Simpler methods like Chebyshev or basic Remez barely suffice for
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* cos() in 64-bit precision, because we want the coefficient of x^2
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* to be precisely -0.5 so that multiplying by it is exact, and plain
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* rounding of the coefficients of a good polynomial approximation only
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* gives this up to about 64-bit precision. Plain rounding also gives
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* a mediocre approximation for the coefficient of x^4, but a rounding
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* error of 0.5 ulps for this coefficient would only contribute ~0.01
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* ulps to the final error, so this is unimportant. Rounding errors in
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* higher coefficients are even less important.
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*
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* In fact, coefficients above the x^4 one only need to have 53-bit
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* precision, and this is more efficient. We get this optimization
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* almost for free from the complications needed to search for the best
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* higher coefficients.
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*/
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static const double
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one = 1.0;
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#if defined(__amd64__) || defined(__i386__)
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/* Long double constants are slow on these arches, and broken on i386. */
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static const volatile double
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C1hi = 0.041666666666666664, /* 0x15555555555555.0p-57 */
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C1lo = 2.2598839032744733e-18; /* 0x14d80000000000.0p-111 */
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#define C1 ((long double)C1hi + C1lo)
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#else
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static const long double
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C1 = 0.0416666666666666666136L; /* 0xaaaaaaaaaaaaaa9b.0p-68 */
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#endif
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static const double
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C2 = -0.0013888888888888874, /* -0x16c16c16c16c10.0p-62 */
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C3 = 0.000024801587301571716, /* 0x1a01a01a018e22.0p-68 */
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C4 = -0.00000027557319215507120, /* -0x127e4fb7602f22.0p-74 */
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C5 = 0.0000000020876754400407278, /* 0x11eed8caaeccf1.0p-81 */
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C6 = -1.1470297442401303e-11, /* -0x19393412bd1529.0p-89 */
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C7 = 4.7383039476436467e-14; /* 0x1aac9d9af5c43e.0p-97 */
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long double
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__kernel_cosl(long double x, long double y)
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{
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long double hz,z,r,w;
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z = x*x;
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r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*C7))))));
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hz = 0.5*z;
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w = one-hz;
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return w + (((one-w)-hz) + (z*r-x*y));
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}
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