/* origin: FreeBSD /usr/src/lib/msun/src/s_log1p.c */
/*
 * ====================================================
 * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 *
 * Developed at SunPro, a Sun Microsystems, Inc. business.
 * Permission to use, copy, modify, and distribute this
 * software is freely granted, provided that this notice
 * is preserved.
 * ====================================================
 */
/* double log1p(double x)
 * Return the natural logarithm of 1+x.
 *
 * Method :
 *   1. Argument Reduction: find k and f such that
 *                      1+x = 2^k * (1+f),
 *         where  sqrt(2)/2 < 1+f < sqrt(2) .
 *
 *      Note. If k=0, then f=x is exact. However, if k!=0, then f
 *      may not be representable exactly. In that case, a correction
 *      term is need. Let u=1+x rounded. Let c = (1+x)-u, then
 *      log(1+x) - log(u) ~ c/u. Thus, we proceed to compute log(u),
 *      and add back the correction term c/u.
 *      (Note: when x > 2**53, one can simply return log(x))
 *
 *   2. Approximation of log(1+f): See log.c
 *
 *   3. Finally, log1p(x) = k*ln2 + log(1+f) + c/u. See log.c
 *
 * Special cases:
 *      log1p(x) is NaN with signal if x < -1 (including -INF) ;
 *      log1p(+INF) is +INF; log1p(-1) is -INF with signal;
 *      log1p(NaN) is that NaN with no signal.
 *
 * Accuracy:
 *      according to an error analysis, the error is always less than
 *      1 ulp (unit in the last place).
 *
 * Constants:
 * The hexadecimal values are the intended ones for the following
 * constants. The decimal values may be used, provided that the
 * compiler will convert from decimal to binary accurately enough
 * to produce the hexadecimal values shown.
 *
 * Note: Assuming log() return accurate answer, the following
 *       algorithm can be used to compute log1p(x) to within a few ULP:
 *
 *              u = 1+x;
 *              if(u==1.0) return x ; else
 *                         return log(u)*(x/(u-1.0));
 *
 *       See HP-15C Advanced Functions Handbook, p.193.
 */

#include "libm.h"

static const double ln2_hi = 6.93147180369123816490e-01, /* 3fe62e42 fee00000 */
    ln2_lo = 1.90821492927058770002e-10,                 /* 3dea39ef 35793c76 */
    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 log1p(double x) {
  union {
    double f;
    uint64_t i;
  } u = {x};
  double_t hfsq, f, c, s, z, R, w, t1, t2, dk;
  uint32_t hx, hu;
  int k;

  hx = u.i >> 32;
  k = 1;
  if (hx < 0x3fda827a || hx >> 31) { /* 1+x < sqrt(2)+ */
    if (hx >= 0xbff00000) {          /* x <= -1.0 */
      if (x == -1)
        return x / 0.0;     /* log1p(-1) = -inf */
      return (x - x) / 0.0; /* log1p(x<-1) = NaN */
    }
    if (hx << 1 < 0x3ca00000 << 1) { /* |x| < 2**-53 */
      /* underflow if subnormal */
      if ((hx & 0x7ff00000) == 0)
        FORCE_EVAL((float)x);
      return x;
    }
    if (hx <= 0xbfd2bec4) { /* sqrt(2)/2- <= 1+x < sqrt(2)+ */
      k = 0;
      c = 0;
      f = x;
    }
  } else if (hx >= 0x7ff00000)
    return x;
  if (k) {
    u.f = 1 + x;
    hu = u.i >> 32;
    hu += 0x3ff00000 - 0x3fe6a09e;
    k = (int)(hu >> 20) - 0x3ff;
    /* correction term ~ log(1+x)-log(u), avoid underflow in c/u */
    if (k < 54) {
      c = k >= 2 ? 1 - (u.f - x) : x - (u.f - 1);
      c /= u.f;
    } else
      c = 0;
    /* reduce u into [sqrt(2)/2, sqrt(2)] */
    hu = (hu & 0x000fffff) + 0x3fe6a09e;
    u.i = (uint64_t)hu << 32 | (u.i & 0xffffffff);
    f = u.f - 1;
  }
  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;
  dk = k;
  return s * (hfsq + R) + (dk * ln2_lo + c) - hfsq + f + dk * ln2_hi;
}