/* @(#)s_cbrt.c 5.1 93/09/24 */

/*

 * ====================================================

 * 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.

 */

#include "math_private.h"

/* cbrt(x)

 * Return cube root of x

 */

static const u_int32_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

s_cbrt(double x)

{

  int32_t hx;

  union {

      double value;

      u_int64_t bits;

  } u;

  double r,s,t=0.0,w;

  u_int32_t sign;

  u_int32_t high,low;

  EXTRACT_WORDS(hx,low,x);

  sign=hx&0x80000000;     /* sign= sign(x) */

  hx  ^=sign;

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

    /*

     * 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? */

      if((hx|low)==0)

    return(x);    /* cbrt(0) is itself */

      SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */

      t*=x;

      GET_HIGH_WORD(high,t);

      INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0);

  } else

      INSERT_WORDS(t,sign|(hx/3+B1),0);

    /*

     * 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 k_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.value=t;

  u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL;

  t=u.value;

    /* 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);

}