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s_cbrt.c
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1 
2 /* @(#)s_cbrt.c 1.3 95/01/18 */
3 /*
4  * ====================================================
5  * Copyright (C) 1993 by Sun Microsystems, Inc. All rights reserved.
6  *
7  * Developed at SunSoft, a Sun Microsystems, Inc. business.
8  * Permission to use, copy, modify, and distribute this
9  * software is freely granted, provided that this notice
10  * is preserved.
11  * ====================================================
12  *
13  */
14 
15 #include "fdlibm.h"
16 
17 /* cbrt(x)
18  * Return cube root of x
19  */
20 #ifdef __STDC__
21 static const unsigned
22 #else
23 static unsigned
24 #endif
25  B1 = 715094163, /* B1 = (682-0.03306235651)*2**20 */
26  B2 = 696219795; /* B2 = (664-0.03306235651)*2**20 */
27 
28 #ifdef __STDC__
29 static const double
30 #else
31 static double
32 #endif
33 C = 5.42857142857142815906e-01, /* 19/35 = 0x3FE15F15, 0xF15F15F1 */
34 D = -7.05306122448979611050e-01, /* -864/1225 = 0xBFE691DE, 0x2532C834 */
35 E = 1.41428571428571436819e+00, /* 99/70 = 0x3FF6A0EA, 0x0EA0EA0F */
36 F = 1.60714285714285720630e+00, /* 45/28 = 0x3FF9B6DB, 0x6DB6DB6E */
37 G = 3.57142857142857150787e-01; /* 5/14 = 0x3FD6DB6D, 0xB6DB6DB7 */
38 
39 #ifdef __STDC__
40  double cbrt(double x)
41 #else
42  double cbrt(x)
43  double x;
44 #endif
45 {
46  int hx;
47  double r,s,t=0.0,w;
48  unsigned sign;
49 
50 
51  hx = __HI(x); /* high word of x */
52  sign=hx&0x80000000; /* sign= sign(x) */
53  hx ^=sign;
54  if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
55  if((hx|__LO(x))==0)
56  return(x); /* cbrt(0) is itself */
57 
58  __HI(x) = hx; /* x <- |x| */
59  /* rough cbrt to 5 bits */
60  if(hx<0x00100000) /* subnormal number */
61  {__HI(t)=0x43500000; /* set t= 2**54 */
62  t*=x; __HI(t)=__HI(t)/3+B2;
63  }
64  else
65  __HI(t)=hx/3+B1;
66 
67 
68  /* new cbrt to 23 bits, may be implemented in single precision */
69  r=t*t/x;
70  s=C+r*t;
71  t*=G+F/(s+E+D/s);
72 
73  /* chopped to 20 bits and make it larger than cbrt(x) */
74  __LO(t)=0; __HI(t)+=0x00000001;
75 
76 
77  /* one step newton iteration to 53 bits with error less than 0.667 ulps */
78  s=t*t; /* t*t is exact */
79  r=x/s;
80  w=t+t;
81  r=(r-t)/(w+r); /* r-s is exact */
82  t=t+t*r;
83 
84  /* retore the sign bit */
85  __HI(t) |= sign;
86  return(t);
87 }