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/* Double-precision AdvSIMD log1p
Copyright (C) 2023 Free Software Foundation, Inc.
This file is part of the GNU C Library.
The GNU C Library is free software; you can redistribute it and/or
modify it under the terms of the GNU Lesser General Public
License as published by the Free Software Foundation; either
version 2.1 of the License, or (at your option) any later version.
The GNU C Library is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
Lesser General Public License for more details.
You should have received a copy of the GNU Lesser General Public
License along with the GNU C Library; if not, see
<https://www.gnu.org/licenses/>. */
#include "v_math.h"
#include "poly_advsimd_f64.h"
const static struct data
{
float64x2_t poly[19], ln2[2];
uint64x2_t hf_rt2_top, one_m_hf_rt2_top, umask, inf, minus_one;
int64x2_t one_top;
} data = {
/* Generated using Remez, deg=20, in [sqrt(2)/2-1, sqrt(2)-1]. */
.poly = { V2 (-0x1.ffffffffffffbp-2), V2 (0x1.55555555551a9p-2),
V2 (-0x1.00000000008e3p-2), V2 (0x1.9999999a32797p-3),
V2 (-0x1.555555552fecfp-3), V2 (0x1.249248e071e5ap-3),
V2 (-0x1.ffffff8bf8482p-4), V2 (0x1.c71c8f07da57ap-4),
V2 (-0x1.9999ca4ccb617p-4), V2 (0x1.7459ad2e1dfa3p-4),
V2 (-0x1.554d2680a3ff2p-4), V2 (0x1.3b4c54d487455p-4),
V2 (-0x1.2548a9ffe80e6p-4), V2 (0x1.0f389a24b2e07p-4),
V2 (-0x1.eee4db15db335p-5), V2 (0x1.e95b494d4a5ddp-5),
V2 (-0x1.15fdf07cb7c73p-4), V2 (0x1.0310b70800fcfp-4),
V2 (-0x1.cfa7385bdb37ep-6) },
.ln2 = { V2 (0x1.62e42fefa3800p-1), V2 (0x1.ef35793c76730p-45) },
/* top32(asuint64(sqrt(2)/2)) << 32. */
.hf_rt2_top = V2 (0x3fe6a09e00000000),
/* (top32(asuint64(1)) - top32(asuint64(sqrt(2)/2))) << 32. */
.one_m_hf_rt2_top = V2 (0x00095f6200000000),
.umask = V2 (0x000fffff00000000),
.one_top = V2 (0x3ff),
.inf = V2 (0x7ff0000000000000),
.minus_one = V2 (0xbff0000000000000)
};
#define BottomMask v_u64 (0xffffffff)
static float64x2_t VPCS_ATTR NOINLINE
special_case (float64x2_t x, float64x2_t y, uint64x2_t special)
{
return v_call_f64 (log1p, x, y, special);
}
/* Vector log1p approximation using polynomial on reduced interval. Routine is
a modification of the algorithm used in scalar log1p, with no shortcut for
k=0 and no narrowing for f and k. Maximum observed error is 2.45 ULP:
_ZGVnN2v_log1p(0x1.658f7035c4014p+11) got 0x1.fd61d0727429dp+2
want 0x1.fd61d0727429fp+2 . */
VPCS_ATTR float64x2_t V_NAME_D1 (log1p) (float64x2_t x)
{
const struct data *d = ptr_barrier (&data);
uint64x2_t ix = vreinterpretq_u64_f64 (x);
uint64x2_t ia = vreinterpretq_u64_f64 (vabsq_f64 (x));
uint64x2_t special = vcgeq_u64 (ia, d->inf);
#if WANT_SIMD_EXCEPT
special = vorrq_u64 (special,
vcgeq_u64 (ix, vreinterpretq_u64_f64 (v_f64 (-1))));
if (__glibc_unlikely (v_any_u64 (special)))
x = v_zerofy_f64 (x, special);
#else
special = vorrq_u64 (special, vcleq_f64 (x, v_f64 (-1)));
#endif
/* With x + 1 = t * 2^k (where t = f + 1 and k is chosen such that f
is in [sqrt(2)/2, sqrt(2)]):
log1p(x) = k*log(2) + log1p(f).
f may not be representable exactly, so we need a correction term:
let m = round(1 + x), c = (1 + x) - m.
c << m: at very small x, log1p(x) ~ x, hence:
log(1+x) - log(m) ~ c/m.
We therefore calculate log1p(x) by k*log2 + log1p(f) + c/m. */
/* Obtain correctly scaled k by manipulation in the exponent.
The scalar algorithm casts down to 32-bit at this point to calculate k and
u_red. We stay in double-width to obtain f and k, using the same constants
as the scalar algorithm but shifted left by 32. */
float64x2_t m = vaddq_f64 (x, v_f64 (1));
uint64x2_t mi = vreinterpretq_u64_f64 (m);
uint64x2_t u = vaddq_u64 (mi, d->one_m_hf_rt2_top);
int64x2_t ki
= vsubq_s64 (vreinterpretq_s64_u64 (vshrq_n_u64 (u, 52)), d->one_top);
float64x2_t k = vcvtq_f64_s64 (ki);
/* Reduce x to f in [sqrt(2)/2, sqrt(2)]. */
uint64x2_t utop = vaddq_u64 (vandq_u64 (u, d->umask), d->hf_rt2_top);
uint64x2_t u_red = vorrq_u64 (utop, vandq_u64 (mi, BottomMask));
float64x2_t f = vsubq_f64 (vreinterpretq_f64_u64 (u_red), v_f64 (1));
/* Correction term c/m. */
float64x2_t cm = vdivq_f64 (vsubq_f64 (x, vsubq_f64 (m, v_f64 (1))), m);
/* Approximate log1p(x) on the reduced input using a polynomial. Because
log1p(0)=0 we choose an approximation of the form:
x + C0*x^2 + C1*x^3 + C2x^4 + ...
Hence approximation has the form f + f^2 * P(f)
where P(x) = C0 + C1*x + C2x^2 + ...
Assembling this all correctly is dealt with at the final step. */
float64x2_t f2 = vmulq_f64 (f, f);
float64x2_t p = v_pw_horner_18_f64 (f, f2, d->poly);
float64x2_t ylo = vfmaq_f64 (cm, k, d->ln2[1]);
float64x2_t yhi = vfmaq_f64 (f, k, d->ln2[0]);
float64x2_t y = vaddq_f64 (ylo, yhi);
if (__glibc_unlikely (v_any_u64 (special)))
return special_case (vreinterpretq_f64_u64 (ix), vfmaq_f64 (y, f2, p),
special);
return vfmaq_f64 (y, f2, p);
}
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