/* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.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.
 * ====================================================
 *
 * Optimized by Bruce D. Evans.
 */
/* cbrt(x)
 * Return cube root of x
 */

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

static const uint32_t
    B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
    B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */

/* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
static const double P0 = 1.87595182427177009643, /* 0x3ffe03e6, 0x0f61e692 */
    P1 = -1.88497979543377169875,                /* 0xbffe28e0, 0x92f02420 */
    P2 = 1.621429720105354466140,                /* 0x3ff9f160, 0x4a49d6c2 */
    P3 = -0.758397934778766047437,               /* 0xbfe844cb, 0xbee751d9 */
    P4 = 0.145996192886612446982;                /* 0x3fc2b000, 0xd4e4edd7 */

double cbrt(double x) {
  union {
    double f;
    uint64_t i;
  } u = {x};
  double_t r, s, t, w;
  uint32_t hx = u.i >> 32 & 0x7fffffff;

  if (hx >= 0x7ff00000) /* cbrt(NaN,INF) is itself */
    return x + x;

  /*
   * Rough cbrt to 5 bits:
   *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
   * where e is integral and >= 0, m is real and in [0, 1), and "/" and
   * "%" are integer division and modulus with rounding towards minus
   * infinity.  The RHS is always >= the LHS and has a maximum relative
   * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
   * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
   * floating point representation, for finite positive normal values,
   * ordinary integer divison of the value in bits magically gives
   * almost exactly the RHS of the above provided we first subtract the
   * exponent bias (1023 for doubles) and later add it back.  We do the
   * subtraction virtually to keep e >= 0 so that ordinary integer
   * division rounds towards minus infinity; this is also efficient.
   */
  if (hx < 0x00100000) { /* zero or subnormal? */
    u.f = x * 0x1p54;
    hx = u.i >> 32 & 0x7fffffff;
    if (hx == 0)
      return x; /* cbrt(0) is itself */
    hx = hx / 3 + B2;
  } else
    hx = hx / 3 + B1;
  u.i &= 1ULL << 63;
  u.i |= (uint64_t)hx << 32;
  t = u.f;

  /*
   * New cbrt to 23 bits:
   *    cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
   * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
   * to within 2**-23.5 when |r - 1| < 1/10.  The rough approximation
   * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
   * gives us bounds for r = t**3/x.
   *
   * Try to optimize for parallel evaluation as in __tanf.c.
   */
  r = (t * t) * (t / x);
  t = t * ((P0 + r * (P1 + r * P2)) + ((r * r) * r) * (P3 + r * P4));

  /*
   * Round t away from zero to 23 bits (sloppily except for ensuring that
   * the result is larger in magnitude than cbrt(x) but not much more than
   * 2 23-bit ulps larger).  With rounding towards zero, the error bound
   * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
   * in the rounded t, the infinite-precision error in the Newton
   * approximation barely affects third digit in the final error
   * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
   * before the final error is larger than 0.667 ulps.
   */
  u.f = t;
  u.i = (u.i + 0x80000000) & 0xffffffffc0000000ULL;
  t = u.f;

  /* one step Newton iteration to 53 bits with error < 0.667 ulps */
  s = t * t;             /* t*t is exact */
  r = x / s;             /* error <= 0.5 ulps; |r| < |t| */
  w = t + t;             /* t+t is exact */
  r = (r - t) / (w + r); /* r-t is exact; w+r ~= 3*t */
  t = t + t * r;         /* error <= 0.5 + 0.5/3 + epsilon */
  return t;
}