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123 lines
4.5 KiB
C
123 lines
4.5 KiB
C
/* Single-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_f32.h"
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const static struct data
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{
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float32_t poly[4];
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float32_t table[5];
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float32_t one_third, two_thirds;
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} data = {
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/* Very rough approximation of cbrt(x) in [0.5, 1], generated with FPMinimax.
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*/
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.poly = { 0x1.c14e96p-2, 0x1.dd2d3p-1, -0x1.08e81ap-1,
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0x1.2c74c2p-3, },
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/* table[i] = 2^((i - 2) / 3). */
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.table = { 0x1.428a3p-1, 0x1.965feap-1, 0x1p0, 0x1.428a3p0, 0x1.965feap0 },
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.one_third = 0x1.555556p-2f,
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.two_thirds = 0x1.555556p-1f,
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};
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#define SmallestNormal 0x00800000
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#define Thresh 0x7f000000 /* asuint(INFINITY) - SmallestNormal. */
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#define MantissaMask 0x007fffff
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#define HalfExp 0x3f000000
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static svfloat32_t NOINLINE
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special_case (svfloat32_t x, svfloat32_t y, svbool_t special)
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{
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return sv_call_f32 (cbrtf, x, y, special);
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}
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static inline svfloat32_t
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shifted_lookup (const svbool_t pg, const float32_t *table, svint32_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 vector single-precision cbrt(x) using Newton iteration
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with initial guess obtained by a low-order polynomial. Greatest error
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is 1.64 ULP. This is observed for every value where the mantissa is
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0x1.85a2aa and the exponent is a multiple of 3, for example:
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_ZGVsMxv_cbrtf (0x1.85a2aap+3) got 0x1.267936p+1
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want 0x1.267932p+1. */
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svfloat32_t SV_NAME_F1 (cbrt) (svfloat32_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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svfloat32_t ax = svabs_x (pg, x);
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svuint32_t iax = svreinterpret_u32 (ax);
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svuint32_t sign = sveor_x (pg, svreinterpret_u32 (x), iax);
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/* Subnormal, +/-0 and special values. */
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svbool_t special = svcmpge (pg, svsub_x (pg, iax, SmallestNormal), 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 frexpf, which gets subnormal values wrong - these have to be
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special-cased as a result. */
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svfloat32_t m = svreinterpret_f32 (svorr_x (
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pg, svand_x (pg, svreinterpret_u32 (x), MantissaMask), HalfExp));
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svint32_t e = svsub_x (pg, svreinterpret_s32 (svlsr_x (pg, iax, 23)), 126);
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/* p is a rough approximation for cbrt(m) in [0.5, 1.0]. The better this is,
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the less accurate the next stage of the algorithm needs to be. An order-4
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polynomial is enough for one Newton iteration. */
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svfloat32_t p
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= sv_pairwise_poly_3_f32_x (pg, m, svmul_x (pg, m, m), d->poly);
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/* One iteration of Newton's method for iteratively approximating cbrt. */
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svfloat32_t m_by_3 = svmul_x (pg, m, d->one_third);
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svfloat32_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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/* 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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svfloat32_t ef = svmul_x (pg, svcvt_f32_x (pg, e), d->one_third);
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svint32_t ey = svcvt_s32_x (pg, ef);
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svint32_t em3 = svmls_x (pg, e, ey, 3);
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svfloat32_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 ldexpf. */
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svfloat32_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_f32 (svorr_x (pg, svreinterpret_u32 (y), sign)),
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special);
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/* Copy sign. */
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return svreinterpret_f32 (svorr_x (pg, svreinterpret_u32 (y), sign));
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}
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