base/math/s_cbrt.c

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/* @(#)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.
 * ====================================================
 */

#if defined(LIBM_SCCS) && !defined(lint)
static char rcsid[] = "$NetBSD: s_cbrt.c,v 1.8 1995/05/10 20:46:49 jtc Exp $";
#endif

#include "math.h"
#include "mathP.h"

/* cbrt(x)
 * Return cube root of x
 */
#ifdef __STDC__
static const u_int32_t
#else
static u_int32_t
#endif
    B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
    B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */

#ifdef __STDC__
static const double
#else
static double
#endif
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 */

#ifdef __STDC__
    double cbrt(double x) 
#else
    double cbrt(x) 
    double x;
#endif
{
    int32_t hx;
    double r,s,t=0.0,w;
    u_int32_t sign;
    u_int32_t high,low;

    GET_HIGH_WORD(hx,x);
    sign=hx&0x80000000;         /* sign= sign(x) */
    hx  ^=sign;
    if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
    GET_LOW_WORD(low,x);
    if((hx|low)==0) 
        return(x);      /* cbrt(0) is itself */

    SET_HIGH_WORD(x,hx);    /* x <- |x| */
    /* rough cbrt to 5 bits */
    if(hx<0x00100000)       /* subnormal number */
      {SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
       t*=x; GET_HIGH_WORD(high,t); SET_HIGH_WORD(t,high/3+B2);
      }
    else
      SET_HIGH_WORD(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) */ 
    GET_HIGH_WORD(high,t);
    INSERT_WORDS(t,high+0x00000001,0);


    /* 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 */
    GET_HIGH_WORD(high,t);
    SET_HIGH_WORD(t,high|sign);
    return(t);
}

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