6d04e1422e
As mention previously, the minmax polynomial approximation in the kernel for cosl() seem to have a bad set of coefficients. In testing, cosl() in the interval [0.785, pi/4] for 1 million values and pi/4 written to 37 decimal digits. The old version on an aarch64 system gave % tlibm/tlibm_lmath -l -x 0.78 -X 7.85398163397448309615660845819875721e-1L cos Interval tested for cosl: [0.78,0.785398] count: 1000000 xm = 7.80213913234863919029058821396125599e-01L libm = 7.10763080972549562455058499280609083e-01L mpfr = 7.10763080972549562455058499280608983e-01L ULP = 1.04431 The max ULP exceeds 1, which is not good. So, I rinsed off a 10 year code and recomputed coefficients. The new minmax polynomial now yields % tlibm/tlibm_lmath -l -x 0.78 -X 7.85398163397448309615660845819875721e-1L cos Interval tested for cosl: [0.78,0.785398] count: 1000000 xm = 7.82916198746768272588844890973704219e-01L libm = 7.08859615479571058183956453286628396e-01L mpfr = 7.08859615479571058183956453286628469e-01L ULP = 0.75407 which is very good. PR: 218514 MFC after: 1 week
60 lines
1.9 KiB
C
60 lines
1.9 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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* ld128 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 ~[-1.17e-39, 1.19e-39]:
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* |cos(x) - c(x))| < 2**-129.3
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*
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* 113-bit precision requires more care than 64-bit precision, since
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* simple methods give a minimax polynomial with coefficient for x^2
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* that is 1 ulp below 0.5, but we want it to be precisely 0.5. See
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* ../ld80/k_cosl.c for more details.
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*/
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static const double
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one = 1.0;
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static const long double
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C1 = 4.16666666666666666666666666666666667e-02L,
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C2 = -1.38888888888888888888888888888888834e-03L,
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C3 = 2.48015873015873015873015873015446795e-05L,
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C4 = -2.75573192239858906525573190949988493e-07L,
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C5 = 2.08767569878680989792098886701451072e-09L,
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C6 = -1.14707455977297247136657111139971865e-11L,
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C7 = 4.77947733238738518870113294139830239e-14L,
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C8 = -1.56192069685858079920640872925306403e-16L,
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C9 = 4.11031762320473354032038893429515732e-19L,
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C10= -8.89679121027589608738005163931958096e-22L,
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C11= 1.61171797801314301767074036661901531e-24L,
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C12= -2.46748624357670948912574279501044295e-27L;
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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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z*(C8+z*(C9+z*(C10+z*(C11+z*C12)))))))))));
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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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