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// N.B. only appropriate for IEEE doubles.
// Cube root implementation obtained from code with the following notice:
/* @(#)s_cbrt.c 1.3 95/01/18 */
/*
* ====================================================
* 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.
* ====================================================
*
*/
/* Sometimes it's necessary to define __LITTLE_ENDIAN explicitly
but these catch some common cases. */
#if defined(i386) || defined(i486) || \
defined(intel) || defined(x86) || defined(i86pc) || \
defined(__alpha) || defined(__osf__)
#define __LITTLE_ENDIAN
#endif
#ifdef __LITTLE_ENDIAN
#define __HI(x) *(1+(int*)&x)
#define __LO(x) *(int*)&x
#define __HIp(x) *(1+(int*)x)
#define __LOp(x) *(int*)x
#else
#define __HI(x) *(int*)&x
#define __LO(x) *(1+(int*)&x)
#define __HIp(x) *(int*)x
#define __LOp(x) *(1+(int*)x)
#endif
/* cbrt(x)
* Return cube root of x
*/
inline double cbrt(double x) {
static const unsigned
B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
static const double
C = 5.42857142857142815906e-01, /* 19/35 = 0x3FE15F15, 0xF15F15F1 */
D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */
E = 1.41428571428571436819e+00, /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */
F = 1.60714285714285720630e+00, /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */
G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */
int hx;
double r,s,t=0.0,w;
unsigned sign;
hx = __HI(x); /* high word of x */
sign=hx&0x80000000; /* sign= sign(x) */
hx ^=sign;
if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
if((hx|__LO(x))==0)
return(x); /* cbrt(0) is itself */
__HI(x) = hx; /* x <- |x| */
/* rough cbrt to 5 bits */
if(hx<0x00100000) /* subnormal number */
{__HI(t)=0x43500000; /* set t= 2**54 */
t*=x; __HI(t)=__HI(t)/3+B2;
}
else
__HI(t)=hx/3+B1;
/* new cbrt to 23 bits, may be implemented in single precision */
r=t*t/x;
s=C+r*t;
t*=G+F/(s+E+D/s);
/* chopped to 20 bits and make it larger than cbrt(x) */
__LO(t)=0; __HI(t)+=0x00000001;
/* one step newton iteration to 53 bits with error less than 0.667 ulps */
s=t*t; /* t*t is exact */
r=x/s;
w=t+t;
r=(r-t)/(w+r); /* r-s is exact */
t=t+t*r;
/* retore the sign bit */
__HI(t) |= sign;
return(t);
}
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