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/* Double-precision log2(x) function.
Copyright (C) 2018-2019 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 <math.h>
#include <stdint.h>
#include <math-svid-compat.h>
#include <libm-alias-finite.h>
#include <libm-alias-double.h>
#include "math_config.h"
#define T __log2_data.tab
#define T2 __log2_data.tab2
#define B __log2_data.poly1
#define A __log2_data.poly
#define InvLn2hi __log2_data.invln2hi
#define InvLn2lo __log2_data.invln2lo
#define N (1 << LOG2_TABLE_BITS)
#define OFF 0x3fe6000000000000
/* Top 16 bits of a double. */
static inline uint32_t
top16 (double x)
{
return asuint64 (x) >> 48;
}
double
__log2 (double x)
{
/* double_t for better performance on targets with FLT_EVAL_METHOD==2. */
double_t z, r, r2, r4, y, invc, logc, kd, hi, lo, t1, t2, t3, p;
uint64_t ix, iz, tmp;
uint32_t top;
int k, i;
ix = asuint64 (x);
top = top16 (x);
#define LO asuint64 (1.0 - 0x1.5b51p-5)
#define HI asuint64 (1.0 + 0x1.6ab2p-5)
if (__glibc_unlikely (ix - LO < HI - LO))
{
/* Handle close to 1.0 inputs separately. */
/* Fix sign of zero with downward rounding when x==1. */
if (WANT_ROUNDING && __glibc_unlikely (ix == asuint64 (1.0)))
return 0;
r = x - 1.0;
#ifdef __FP_FAST_FMA
hi = r * InvLn2hi;
lo = r * InvLn2lo + __builtin_fma (r, InvLn2hi, -hi);
#else
double_t rhi, rlo;
rhi = asdouble (asuint64 (r) & -1ULL << 32);
rlo = r - rhi;
hi = rhi * InvLn2hi;
lo = rlo * InvLn2hi + r * InvLn2lo;
#endif
r2 = r * r; /* rounding error: 0x1p-62. */
r4 = r2 * r2;
/* Worst-case error is less than 0.54 ULP (0.55 ULP without fma). */
p = r2 * (B[0] + r * B[1]);
y = hi + p;
lo += hi - y + p;
lo += r4 * (B[2] + r * B[3] + r2 * (B[4] + r * B[5])
+ r4 * (B[6] + r * B[7] + r2 * (B[8] + r * B[9])));
y += lo;
return y;
}
if (__glibc_unlikely (top - 0x0010 >= 0x7ff0 - 0x0010))
{
/* x < 0x1p-1022 or inf or nan. */
if (ix * 2 == 0)
return __math_divzero (1);
if (ix == asuint64 (INFINITY)) /* log(inf) == inf. */
return x;
if ((top & 0x8000) || (top & 0x7ff0) == 0x7ff0)
return __math_invalid (x);
/* x is subnormal, normalize it. */
ix = asuint64 (x * 0x1p52);
ix -= 52ULL << 52;
}
/* x = 2^k z; where z is in range [OFF,2*OFF) and exact.
The range is split into N subintervals.
The ith subinterval contains z and c is near its center. */
tmp = ix - OFF;
i = (tmp >> (52 - LOG2_TABLE_BITS)) % N;
k = (int64_t) tmp >> 52; /* arithmetic shift */
iz = ix - (tmp & 0xfffULL << 52);
invc = T[i].invc;
logc = T[i].logc;
z = asdouble (iz);
kd = (double_t) k;
/* log2(x) = log2(z/c) + log2(c) + k. */
/* r ~= z/c - 1, |r| < 1/(2*N). */
#ifdef __FP_FAST_FMA
/* rounding error: 0x1p-55/N. */
r = __builtin_fma (z, invc, -1.0);
t1 = r * InvLn2hi;
t2 = r * InvLn2lo + __builtin_fma (r, InvLn2hi, -t1);
#else
double_t rhi, rlo;
/* rounding error: 0x1p-55/N + 0x1p-65. */
r = (z - T2[i].chi - T2[i].clo) * invc;
rhi = asdouble (asuint64 (r) & -1ULL << 32);
rlo = r - rhi;
t1 = rhi * InvLn2hi;
t2 = rlo * InvLn2hi + r * InvLn2lo;
#endif
/* hi + lo = r/ln2 + log2(c) + k. */
t3 = kd + logc;
hi = t3 + t1;
lo = t3 - hi + t1 + t2;
/* log2(r+1) = r/ln2 + r^2*poly(r). */
/* Evaluation is optimized assuming superscalar pipelined execution. */
r2 = r * r; /* rounding error: 0x1p-54/N^2. */
r4 = r2 * r2;
/* Worst-case error if |y| > 0x1p-4: 0.547 ULP (0.550 ULP without fma).
~ 0.5 + 2/N/ln2 + abs-poly-error*0x1p56 ULP (+ 0.003 ULP without fma). */
p = A[0] + r * A[1] + r2 * (A[2] + r * A[3]) + r4 * (A[4] + r * A[5]);
y = lo + r2 * p + hi;
return y;
}
#ifndef __log2
strong_alias (__log2, __ieee754_log2)
libm_alias_finite (__ieee754_log2, __log2)
# if LIBM_SVID_COMPAT
versioned_symbol (libm, __log2, log2, GLIBC_2_29);
libm_alias_double_other (__log2, log2)
# else
libm_alias_double (__log2, log2)
# endif
#endif
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