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129 lines
4.7 KiB
C
129 lines
4.7 KiB
C
/* Double-precision vector (SVE) cbrt function
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Copyright (C) 2024 Free Software Foundation, Inc.
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This file is part of the GNU C Library.
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The GNU C Library is free software; you can redistribute it and/or
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modify it under the terms of the GNU Lesser General Public
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License as published by the Free Software Foundation; either
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version 2.1 of the License, or (at your option) any later version.
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The GNU C Library is distributed in the hope that it will be useful,
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but WITHOUT ANY WARRANTY; without even the implied warranty of
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MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
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Lesser General Public License for more details.
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You should have received a copy of the GNU Lesser General Public
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License along with the GNU C Library; if not, see
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<https://www.gnu.org/licenses/>. */
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#include "sv_math.h"
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#include "poly_sve_f64.h"
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const static struct data
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{
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float64_t poly[4];
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float64_t table[5];
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float64_t one_third, two_thirds, shift;
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int64_t exp_bias;
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uint64_t tiny_bound, thresh;
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} data = {
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/* Generated with FPMinimax in [0.5, 1]. */
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.poly = { 0x1.c14e8ee44767p-2, 0x1.dd2d3f99e4c0ep-1, -0x1.08e83026b7e74p-1,
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0x1.2c74eaa3ba428p-3, },
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/* table[i] = 2^((i - 2) / 3). */
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.table = { 0x1.428a2f98d728bp-1, 0x1.965fea53d6e3dp-1, 0x1p0,
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0x1.428a2f98d728bp0, 0x1.965fea53d6e3dp0, },
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.one_third = 0x1.5555555555555p-2,
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.two_thirds = 0x1.5555555555555p-1,
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.shift = 0x1.8p52,
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.exp_bias = 1022,
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.tiny_bound = 0x0010000000000000, /* Smallest normal. */
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.thresh = 0x7fe0000000000000, /* asuint64 (infinity) - tiny_bound. */
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};
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#define MantissaMask 0x000fffffffffffff
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#define HalfExp 0x3fe0000000000000
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static svfloat64_t NOINLINE
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special_case (svfloat64_t x, svfloat64_t y, svbool_t special)
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{
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return sv_call_f64 (cbrt, x, y, special);
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}
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static inline svfloat64_t
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shifted_lookup (const svbool_t pg, const float64_t *table, svint64_t i)
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{
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return svld1_gather_index (pg, table, svadd_x (pg, i, 2));
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}
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/* Approximation for double-precision vector cbrt(x), using low-order
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polynomial and two Newton iterations. Greatest observed error is 1.79 ULP.
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Errors repeat according to the exponent, for instance an error observed for
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double value m * 2^e will be observed for any input m * 2^(e + 3*i), where i
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is an integer.
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_ZGVsMxv_cbrt (0x0.3fffb8d4413f3p-1022) got 0x1.965f53b0e5d97p-342
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want 0x1.965f53b0e5d95p-342. */
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svfloat64_t SV_NAME_D1 (cbrt) (svfloat64_t x, const svbool_t pg)
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{
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const struct data *d = ptr_barrier (&data);
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svfloat64_t ax = svabs_x (pg, x);
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svuint64_t iax = svreinterpret_u64 (ax);
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svuint64_t sign = sveor_x (pg, svreinterpret_u64 (x), iax);
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/* Subnormal, +/-0 and special values. */
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svbool_t special = svcmpge (pg, svsub_x (pg, iax, d->tiny_bound), d->thresh);
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/* Decompose |x| into m * 2^e, where m is in [0.5, 1.0]. This is a vector
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version of frexp, which gets subnormal values wrong - these have to be
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special-cased as a result. */
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svfloat64_t m = svreinterpret_f64 (svorr_x (
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pg, svand_x (pg, svreinterpret_u64 (x), MantissaMask), HalfExp));
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svint64_t e
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= svsub_x (pg, svreinterpret_s64 (svlsr_x (pg, iax, 52)), d->exp_bias);
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/* Calculate rough approximation for cbrt(m) in [0.5, 1.0], starting point
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for Newton iterations. */
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svfloat64_t p
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= sv_pairwise_poly_3_f64_x (pg, m, svmul_x (pg, m, m), d->poly);
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/* Two iterations of Newton's method for iteratively approximating cbrt. */
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svfloat64_t m_by_3 = svmul_x (pg, m, d->one_third);
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svfloat64_t a = svmla_x (pg, svdiv_x (pg, m_by_3, svmul_x (pg, p, p)), p,
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d->two_thirds);
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a = svmla_x (pg, svdiv_x (pg, m_by_3, svmul_x (pg, a, a)), a, d->two_thirds);
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/* Assemble the result by the following:
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cbrt(x) = cbrt(m) * 2 ^ (e / 3).
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We can get 2 ^ round(e / 3) using ldexp and integer divide, but since e is
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not necessarily a multiple of 3 we lose some information.
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Let q = 2 ^ round(e / 3), then t = 2 ^ (e / 3) / q.
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Then we know t = 2 ^ (i / 3), where i is the remainder from e / 3, which
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is an integer in [-2, 2], and can be looked up in the table T. Hence the
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result is assembled as:
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cbrt(x) = cbrt(m) * t * 2 ^ round(e / 3) * sign. */
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svfloat64_t eb3f = svmul_x (pg, svcvt_f64_x (pg, e), d->one_third);
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svint64_t ey = svcvt_s64_x (pg, eb3f);
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svint64_t em3 = svmls_x (pg, e, ey, 3);
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svfloat64_t my = shifted_lookup (pg, d->table, em3);
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my = svmul_x (pg, my, a);
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/* Vector version of ldexp. */
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svfloat64_t y = svscale_x (pg, my, ey);
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if (__glibc_unlikely (svptest_any (pg, special)))
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return special_case (
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x, svreinterpret_f64 (svorr_x (pg, svreinterpret_u64 (y), sign)),
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special);
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/* Copy sign. */
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return svreinterpret_f64 (svorr_x (pg, svreinterpret_u64 (y), sign));
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}
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