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124 lines
4.6 KiB
C
124 lines
4.6 KiB
C
/* Single-precision vector (AdvSIMD) 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 "v_math.h"
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#include "poly_advsimd_f32.h"
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const static struct data
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{
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float32x4_t poly[4], one_third;
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float table[5];
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} data = {
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.poly = { /* Very rough approximation of cbrt(x) in [0.5, 1], generated with
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FPMinimax. */
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V4 (0x1.c14e96p-2), V4 (0x1.dd2d3p-1), V4 (-0x1.08e81ap-1),
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V4 (0x1.2c74c2p-3) },
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.table = { /* table[i] = 2^((i - 2) / 3). */
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0x1.428a3p-1, 0x1.965feap-1, 0x1p0, 0x1.428a3p0, 0x1.965feap0 },
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.one_third = V4 (0x1.555556p-2f),
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};
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#define SignMask v_u32 (0x80000000)
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#define SmallestNormal v_u32 (0x00800000)
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#define Thresh vdup_n_u16 (0x7f00) /* asuint(INFINITY) - SmallestNormal. */
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#define MantissaMask v_u32 (0x007fffff)
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#define HalfExp v_u32 (0x3f000000)
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static float32x4_t VPCS_ATTR NOINLINE
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special_case (float32x4_t x, float32x4_t y, uint16x4_t special)
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{
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return v_call_f32 (cbrtf, x, y, vmovl_u16 (special));
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}
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static inline float32x4_t
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shifted_lookup (const float *table, int32x4_t i)
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{
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return (float32x4_t){ table[i[0] + 2], table[i[1] + 2], table[i[2] + 2],
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table[i[3] + 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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_ZGVnN4v_cbrtf(0x1.85a2aap+3) got 0x1.267936p+1
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want 0x1.267932p+1. */
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VPCS_ATTR float32x4_t V_NAME_F1 (cbrt) (float32x4_t x)
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{
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const struct data *d = ptr_barrier (&data);
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uint32x4_t iax = vreinterpretq_u32_f32 (vabsq_f32 (x));
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/* Subnormal, +/-0 and special values. */
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uint16x4_t special = vcge_u16 (vsubhn_u32 (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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float32x4_t m = vbslq_f32 (MantissaMask, x, v_f32 (0.5));
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int32x4_t e
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= vsubq_s32 (vreinterpretq_s32_u32 (vshrq_n_u32 (iax, 23)), v_s32 (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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float32x4_t p = v_pairwise_poly_3_f32 (m, vmulq_f32 (m, m), d->poly);
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float32x4_t one_third = d->one_third;
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float32x4_t two_thirds = vaddq_f32 (one_third, one_third);
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/* One iteration of Newton's method for iteratively approximating cbrt. */
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float32x4_t m_by_3 = vmulq_f32 (m, one_third);
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float32x4_t a
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= vfmaq_f32 (vdivq_f32 (m_by_3, vmulq_f32 (p, p)), two_thirds, p);
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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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float32x4_t ef = vmulq_f32 (vcvtq_f32_s32 (e), one_third);
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int32x4_t ey = vcvtq_s32_f32 (ef);
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int32x4_t em3 = vsubq_s32 (e, vmulq_s32 (ey, v_s32 (3)));
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float32x4_t my = shifted_lookup (d->table, em3);
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my = vmulq_f32 (my, a);
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/* Vector version of ldexpf. */
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float32x4_t y
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= vreinterpretq_f32_s32 (vshlq_n_s32 (vaddq_s32 (ey, v_s32 (127)), 23));
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y = vmulq_f32 (y, my);
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if (__glibc_unlikely (v_any_u16h (special)))
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return special_case (x, vbslq_f32 (SignMask, x, y), special);
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/* Copy sign. */
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return vbslq_f32 (SignMask, x, y);
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}
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libmvec_hidden_def (V_NAME_F1 (cbrt))
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HALF_WIDTH_ALIAS_F1 (cbrt)
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