/* origin: FreeBSD /usr/src/lib/msun/src/e_log2.c */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunSoft, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */
/*
 * Return the base 2 logarithm of x.  See log.c for most comments.
 *
 * Reduce x to 2^k (1+f) and calculate r = log(1+f) - f + f*f/2
 * as in log.c, then combine and scale in extra precision:
 *    log2(x) = (f - f*f/2 + r)/log(2) + k
 */

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

static const double ivln2hi =
                        1.44269504072144627571e+00, /* 0x3ff71547, 0x65200000 */
    ivln2lo = 1.67517131648865118353e-10,           /* 0x3de705fc, 0x2eefa200 */
    Lg1 = 6.666666666666735130e-01,                 /* 3FE55555 55555593 */
    Lg2 = 3.999999999940941908e-01,                 /* 3FD99999 9997FA04 */
    Lg3 = 2.857142874366239149e-01,                 /* 3FD24924 94229359 */
    Lg4 = 2.222219843214978396e-01,                 /* 3FCC71C5 1D8E78AF */
    Lg5 = 1.818357216161805012e-01,                 /* 3FC74664 96CB03DE */
    Lg6 = 1.531383769920937332e-01,                 /* 3FC39A09 D078C69F */
    Lg7 = 1.479819860511658591e-01;                 /* 3FC2F112 DF3E5244 */

double log2(double x) {
  union {
    double f;
    uint64_t i;
  } u = {x};
  double_t hfsq, f, s, z, R, w, t1, t2, y, hi, lo, val_hi, val_lo;
  uint32_t hx;
  int k;

  hx = u.i >> 32;
  k = 0;
  if (hx < 0x00100000 || hx >> 31) {
    if (u.i << 1 == 0)
      return -1 / (x * x); /* log(+-0)=-inf */
    if (hx >> 31)
      return (x - x) / 0.0; /* log(-#) = NaN */
    /* subnormal number, scale x up */
    k -= 54;
    x *= 0x1p54;
    u.f = x;
    hx = u.i >> 32;
  } else if (hx >= 0x7ff00000) {
    return x;
  } else if (hx == 0x3ff00000 && u.i << 32 == 0)
    return 0;

  /* reduce x into [sqrt(2)/2, sqrt(2)] */
  hx += 0x3ff00000 - 0x3fe6a09e;
  k += (int)(hx >> 20) - 0x3ff;
  hx = (hx & 0x000fffff) + 0x3fe6a09e;
  u.i = (uint64_t)hx << 32 | (u.i & 0xffffffff);
  x = u.f;

  f = x - 1.0;
  hfsq = 0.5 * f * f;
  s = f / (2.0 + f);
  z = s * s;
  w = z * z;
  t1 = w * (Lg2 + w * (Lg4 + w * Lg6));
  t2 = z * (Lg1 + w * (Lg3 + w * (Lg5 + w * Lg7)));
  R = t2 + t1;

  /*
   * f-hfsq must (for args near 1) be evaluated in extra precision
   * to avoid a large cancellation when x is near sqrt(2) or 1/sqrt(2).
   * This is fairly efficient since f-hfsq only depends on f, so can
   * be evaluated in parallel with R.  Not combining hfsq with R also
   * keeps R small (though not as small as a true `lo' term would be),
   * so that extra precision is not needed for terms involving R.
   *
   * Compiler bugs involving extra precision used to break Dekker's
   * theorem for spitting f-hfsq as hi+lo, unless double_t was used
   * or the multi-precision calculations were avoided when double_t
   * has extra precision.  These problems are now automatically
   * avoided as a side effect of the optimization of combining the
   * Dekker splitting step with the clear-low-bits step.
   *
   * y must (for args near sqrt(2) and 1/sqrt(2)) be added in extra
   * precision to avoid a very large cancellation when x is very near
   * these values.  Unlike the above cancellations, this problem is
   * specific to base 2.  It is strange that adding +-1 is so much
   * harder than adding +-ln2 or +-log10_2.
   *
   * This uses Dekker's theorem to normalize y+val_hi, so the
   * compiler bugs are back in some configurations, sigh.  And I
   * don't want to used double_t to avoid them, since that gives a
   * pessimization and the support for avoiding the pessimization
   * is not yet available.
   *
   * The multi-precision calculations for the multiplications are
   * routine.
   */

  /* hi+lo = f - hfsq + s*(hfsq+R) ~ log(1+f) */
  hi = f - hfsq;
  u.f = hi;
  u.i &= (uint64_t)-1 << 32;
  hi = u.f;
  lo = f - hi - hfsq + s * (hfsq + R);

  val_hi = hi * ivln2hi;
  val_lo = (lo + hi) * ivln2lo + lo * ivln2hi;

  /* spadd(val_hi, val_lo, y), except for not using double_t: */
  y = k;
  w = y + val_hi;
  val_lo += (y - w) + val_hi;
  val_hi = w;

  return val_lo + val_hi;
}