Date: Tue, 2 Nov 2021 09:07:02 GMT From: Konstantin Belousov <kib@FreeBSD.org> To: src-committers@FreeBSD.org, dev-commits-src-all@FreeBSD.org, dev-commits-src-main@FreeBSD.org Subject: git: 6d04e1422e70 - main - cosl(): fix polynomial approximation coefficients for ld128 version Message-ID: <202111020907.1A297227004617@gitrepo.freebsd.org>
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The branch main has been updated by kib: URL: https://cgit.FreeBSD.org/src/commit/?id=6d04e1422e70ca0a77552782c01c291f90716773 commit 6d04e1422e70ca0a77552782c01c291f90716773 Author: Steve Kargl <kargl@FreeBSD.org> AuthorDate: 2021-11-02 08:54:10 +0000 Commit: Konstantin Belousov <kib@FreeBSD.org> CommitDate: 2021-11-02 08:54:10 +0000 cosl(): fix polynomial approximation coefficients for ld128 version 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 --- lib/msun/ld128/k_cosl.c | 32 +++++++++++++++----------------- 1 file changed, 15 insertions(+), 17 deletions(-) diff --git a/lib/msun/ld128/k_cosl.c b/lib/msun/ld128/k_cosl.c index fe57773a1348..422357bf8a6c 100644 --- a/lib/msun/ld128/k_cosl.c +++ b/lib/msun/ld128/k_cosl.c @@ -21,8 +21,8 @@ __FBSDID("$FreeBSD$"); #include "math_private.h" /* - * Domain [-0.7854, 0.7854], range ~[-1.80e-37, 1.79e-37]: - * |cos(x) - c(x))| < 2**-122.0 + * Domain [-0.7854, 0.7854], range ~[-1.17e-39, 1.19e-39]: + * |cos(x) - c(x))| < 2**-129.3 * * 113-bit precision requires more care than 64-bit precision, since * simple methods give a minimax polynomial with coefficient for x^2 @@ -31,21 +31,19 @@ __FBSDID("$FreeBSD$"); */ static const double one = 1.0; - static const long double -C1 = 0.04166666666666666666666666666666658424671L, -C2 = -0.001388888888888888888888888888863490893732L, -C3 = 0.00002480158730158730158730158600795304914210L, -C4 = -0.2755731922398589065255474947078934284324e-6L, -C5 = 0.2087675698786809897659225313136400793948e-8L, -C6 = -0.1147074559772972315817149986812031204775e-10L, -C7 = 0.4779477332386808976875457937252120293400e-13L; - -static const double -C8 = -0.1561920696721507929516718307820958119868e-15, -C9 = 0.4110317413744594971475941557607804508039e-18, -C10 = -0.8896592467191938803288521958313920156409e-21, -C11 = 0.1601061435794535138244346256065192782581e-23; +C1 = 4.16666666666666666666666666666666667e-02L, +C2 = -1.38888888888888888888888888888888834e-03L, +C3 = 2.48015873015873015873015873015446795e-05L, +C4 = -2.75573192239858906525573190949988493e-07L, +C5 = 2.08767569878680989792098886701451072e-09L, +C6 = -1.14707455977297247136657111139971865e-11L, +C7 = 4.77947733238738518870113294139830239e-14L, +C8 = -1.56192069685858079920640872925306403e-16L, +C9 = 4.11031762320473354032038893429515732e-19L, +C10= -8.89679121027589608738005163931958096e-22L, +C11= 1.61171797801314301767074036661901531e-24L, +C12= -2.46748624357670948912574279501044295e-27L; long double __kernel_cosl(long double x, long double y) @@ -54,7 +52,7 @@ __kernel_cosl(long double x, long double y) z = x*x; r = z*(C1+z*(C2+z*(C3+z*(C4+z*(C5+z*(C6+z*(C7+ - z*(C8+z*(C9+z*(C10+z*C11)))))))))); + z*(C8+z*(C9+z*(C10+z*(C11+z*C12))))))))))); hz = 0.5*z; w = one-hz; return w + (((one-w)-hz) + (z*r-x*y));
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