1  /* @(#)s_cbrt.c 5.1 93/09/24 */
 2  /*
 3   * ====================================================
 4   * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
 5   *
 6   * Developed at SunPro, a Sun Microsystems, Inc. business.
 7   * Permission to use, copy, modify, and distribute this
 8   * software is freely granted, provided that this notice 
 9   * is preserved.
10   * ====================================================
11   */
12  
13  #if defined(LIBM_SCCS) && !defined(lint)
14  static char rcsid[] = "$NetBSD: s_cbrt.c,v 1.8 1995/05/10 20:46:49 jtc Exp $";
15  #endif
16  
17  #include "math.h"
18  #include "mathP.h"
19  
20  /* cbrt(x)
21   * Return cube root of x
22   */
23  #ifdef __STDC__
24  static const u_int32_t
25  #else
26  static u_int32_t
27  #endif
28  	B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
29  	B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
30  
31  #ifdef __STDC__
32  static const double
33  #else
34  static double
35  #endif
36  C =  5.42857142857142815906e-01, /* 19/35     = 0x3FE15F15, 0xF15F15F1 */
37  D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */
38  E =  1.41428571428571436819e+00, /* 99/70     = 0x3FF6A0EA, 0x0EA0EA0F */
39  F =  1.60714285714285720630e+00, /* 45/28     = 0x3FF9B6DB, 0x6DB6DB6E */
40  G =  3.57142857142857150787e-01; /* 5/14      = 0x3FD6DB6D, 0xB6DB6DB7 */
41  
42  #ifdef __STDC__
43  	double cbrt(double x) 
44  #else
45  	double cbrt(x) 
46  	double x;
47  #endif
48  {
49  	int32_t	hx;
50  	double r,s,t=0.0,w;
51  	u_int32_t sign;
52  	u_int32_t high,low;
53  
54  	GET_HIGH_WORD(hx,x);
55  	sign=hx&0x80000000; 		/* sign= sign(x) */
56  	hx  ^=sign;
57  	if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
58  	GET_LOW_WORD(low,x);
59  	if((hx|low)==0) 
60  	    return(x);		/* cbrt(0) is itself */
61  
62  	SET_HIGH_WORD(x,hx);	/* x <- |x| */
63      /* rough cbrt to 5 bits */
64  	if(hx<0x00100000) 		/* subnormal number */
65  	  {SET_HIGH_WORD(t,0x43500000);	/* set t= 2**54 */
66  	   t*=x; GET_HIGH_WORD(high,t); SET_HIGH_WORD(t,high/3+B2);
67  	  }
68  	else
69  	  SET_HIGH_WORD(t,hx/3+B1);
70  
71  
72      /* new cbrt to 23 bits, may be implemented in single precision */
73  	r=t*t/x;
74  	s=C+r*t;
75  	t*=G+F/(s+E+D/s);	
76  
77      /* chopped to 20 bits and make it larger than cbrt(x) */ 
78  	GET_HIGH_WORD(high,t);
79  	INSERT_WORDS(t,high+0x00000001,0);
80  
81  
82      /* one step newton iteration to 53 bits with error less than 0.667 ulps */
83  	s=t*t;		/* t*t is exact */
84  	r=x/s;
85  	w=t+t;
86  	r=(r-t)/(w+r);	/* r-s is exact */
87  	t=t+t*r;
88  
89      /* retore the sign bit */
90  	GET_HIGH_WORD(high,t);
91  	SET_HIGH_WORD(t,high|sign);
92  	return(t);
93  }