Line data    Source code

       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             :  * Optimized by Bruce D. Evans.
      13             :  */
      14             : 
      15             : //#include <sys/cdefs.h>
      16             : //__FBSDID("$FreeBSD$");
      17             : 
      18             : #include "math_private.h"
      19             : 
      20             : /* cbrt(x)
      21             :  * Return cube root of x
      22             :  */
      23             : static const u_int32_t
      24             :         B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
      25             :         B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
      26             : 
      27             : /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
      28             : static const double
      29             : P0 =  1.87595182427177009643,           /* 0x3ffe03e6, 0x0f61e692 */
      30             : P1 = -1.88497979543377169875,           /* 0xbffe28e0, 0x92f02420 */
      31             : P2 =  1.621429720105354466140,          /* 0x3ff9f160, 0x4a49d6c2 */
      32             : P3 = -0.758397934778766047437,          /* 0xbfe844cb, 0xbee751d9 */
      33             : P4 =  0.145996192886612446982;          /* 0x3fc2b000, 0xd4e4edd7 */
      34             : 
      35             : double
      36           0 : cbrt(double x)
      37             : {
      38             :         int32_t hx;
      39             :         union {
      40             :             double value;
      41             :             uint64_t bits;
      42             :         } u;
      43           0 :         double r,s,t=0.0,w;
      44             :         u_int32_t sign;
      45             :         u_int32_t high,low;
      46             : 
      47           0 :         EXTRACT_WORDS(hx,low,x);
      48           0 :         sign=hx&0x80000000;                 /* sign= sign(x) */
      49           0 :         hx  ^=sign;
      50           0 :         if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
      51             : 
      52             :     /*
      53             :      * Rough cbrt to 5 bits:
      54             :      *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
      55             :      * where e is integral and >= 0, m is real and in [0, 1), and "/" and
      56             :      * "%" are integer division and modulus with rounding towards minus
      57             :      * infinity.  The RHS is always >= the LHS and has a maximum relative
      58             :      * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
      59             :      * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
      60             :      * floating point representation, for finite positive normal values,
      61             :      * ordinary integer division of the value in bits magically gives
      62             :      * almost exactly the RHS of the above provided we first subtract the
      63             :      * exponent bias (1023 for doubles) and later add it back.  We do the
      64             :      * subtraction virtually to keep e >= 0 so that ordinary integer
      65             :      * division rounds towards minus infinity; this is also efficient.
      66             :      */
      67           0 :         if(hx<0x00100000) {          /* zero or subnormal? */
      68           0 :             if((hx|low)==0)
      69           0 :                 return(x);              /* cbrt(0) is itself */
      70           0 :             SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
      71           0 :             t*=x;
      72           0 :             GET_HIGH_WORD(high,t);
      73           0 :             INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0);
      74             :         } else
      75           0 :             INSERT_WORDS(t,sign|(hx/3+B1),0);
      76             : 
      77             :     /*
      78             :      * New cbrt to 23 bits:
      79             :      *    cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
      80             :      * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
      81             :      * to within 2**-23.5 when |r - 1| < 1/10.  The rough approximation
      82             :      * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
      83             :      * gives us bounds for r = t**3/x.
      84             :      *
      85             :      * Try to optimize for parallel evaluation as in k_tanf.c.
      86             :      */
      87           0 :         r=(t*t)*(t/x);
      88           0 :         t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));
      89             : 
      90             :     /*
      91             :      * Round t away from zero to 23 bits (sloppily except for ensuring that
      92             :      * the result is larger in magnitude than cbrt(x) but not much more than
      93             :      * 2 23-bit ulps larger).  With rounding towards zero, the error bound
      94             :      * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
      95             :      * in the rounded t, the infinite-precision error in the Newton
      96             :      * approximation barely affects third digit in the final error
      97             :      * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
      98             :      * before the final error is larger than 0.667 ulps.
      99             :      */
     100           0 :         u.value=t;
     101           0 :         u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL;
     102           0 :         t=u.value;
     103             : 
     104             :     /* one step Newton iteration to 53 bits with error < 0.667 ulps */
     105           0 :         s=t*t;                          /* t*t is exact */
     106           0 :         r=x/s;                          /* error <= 0.5 ulps; |r| < |t| */
     107           0 :         w=t+t;                          /* t+t is exact */
     108           0 :         r=(r-t)/(w+r);                  /* r-t is exact; w+r ~= 3*t */
     109           0 :         t=t+t*r;                        /* error <= 0.5 + 0.5/3 + epsilon */
     110             : 
     111           0 :         return(t);
     112             : }