libm/s_cbrt.c

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