math/log2.c

/*
 * Double-precision log2(x) function.
 *
 * Copyright (c) 2018, Arm Limited.
 * SPDX-License-Identifier: Apache-2.0
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *     http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

#include <math.h>
#include <stdint.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

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);

#if LOG2_POLY1_ORDER == 11
# define LO asuint64 (1.0 - 0x1.5b51p-5)
# define HI asuint64 (1.0 + 0x1.6ab2p-5)
#endif
  if (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 && unlikely (ix == asuint64 (1.0)))
	return 0;
      r = x - 1.0;
#if HAVE_FAST_FMA
      hi = r*InvLn2hi;
      lo = r*InvLn2lo + 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;
#if LOG2_POLY1_ORDER == 11
      /* 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;
#endif
      return y;
    }
  if (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).  */
#if HAVE_FAST_FMA
  /* rounding error: 0x1p-55/N.  */
  r = fma (z, invc, -1.0);
  t1 = r*InvLn2hi;
  t2 = r*InvLn2lo + 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;
#if LOG2_POLY_ORDER == 7
  /* 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;
#endif
  return y;
}