lib/libm/src/s_cbrt.c

   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  * $FreeBSD: src/lib/msun/src/s_cbrt.c,v 1.14 2005/12/20 01:21:30 bde Exp $
  15  * $NetBSD: s_cbrt.c,v 1.11 2002/05/26 22:01:54 wiz Exp $
  16  * $DragonFly: src/lib/libm/src/s_cbrt.c,v 1.2 2007/07/03 04:54:07 pavalos Exp $
  17  */
  18
  19 #include <math.h>
  20 #include "math_private.h"
  21
  22 /* cbrt(x)
  23  * Return cube root of x
  24  */
  25 static const u_int32_t
  26         B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
  27         B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
  28
  29 /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
  30 static const double
  31 P0 =  1.87595182427177009643,           /* 0x3ffe03e6, 0x0f61e692 */
  32 P1 = -1.88497979543377169875,           /* 0xbffe28e0, 0x92f02420 */
  33 P2 =  1.621429720105354466140,          /* 0x3ff9f160, 0x4a49d6c2 */
  34 P3 = -0.758397934778766047437,          /* 0xbfe844cb, 0xbee751d9 */
  35 P4 =  0.145996192886612446982;          /* 0x3fc2b000, 0xd4e4edd7 */
  36
  37 double
  38 cbrt(double x)
  39 {
  40         int32_t hx;
  41         union {
  42             double value;
  43             uint64_t bits;
  44         } u;
  45         double r,s,t=0.0,w;
  46         u_int32_t sign;
  47         u_int32_t high,low;
  48
  49         EXTRACT_WORDS(hx,low,x);
  50         sign=hx&0x80000000;             /* sign= sign(x) */
  51         hx  ^=sign;
  52         if(hx>=0x7ff00000) return(x+x); /* cbrt(NaN,INF) is itself */
  53
  54     /*
  55      * Rough cbrt to 5 bits:
  56      *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
  57      * where e is integral and >= 0, m is real and in [0, 1), and "/" and
  58      * "%" are integer division and modulus with rounding towards minus
  59      * infinity.  The RHS is always >= the LHS and has a maximum relative
  60      * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
  61      * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
  62      * floating point representation, for finite positive normal values,
  63      * ordinary integer divison of the value in bits magically gives
  64      * almost exactly the RHS of the above provided we first subtract the
  65      * exponent bias (1023 for doubles) and later add it back.  We do the
  66      * subtraction virtually to keep e >= 0 so that ordinary integer
  67      * division rounds towards minus infinity; this is also efficient.
  68      */
  69         if(hx<0x00100000) {             /* zero or subnormal? */
  70             if((hx|low)==0)
  71                 return(x);              /* cbrt(0) is itself */
  72             SET_HIGH_WORD(t,0x43500000); /* set t= 2**54 */
  73             t*=x;
  74             GET_HIGH_WORD(high,t);
  75             INSERT_WORDS(t,sign|((high&0x7fffffff)/3+B2),0);
  76         } else
  77             INSERT_WORDS(t,sign|(hx/3+B1),0);
  78
  79     /*
  80      * New cbrt to 23 bits:
  81      *    cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
  82      * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
  83      * to within 2**-23.5 when |r - 1| < 1/10.  The rough approximation
  84      * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
  85      * gives us bounds for r = t**3/x.
  86      *
  87      * Try to optimize for parallel evaluation as in k_tanf.c.
  88      */
  89         r=(t*t)*(t/x);
  90         t=t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));
  91
  92     /*
  93      * Round t away from zero to 23 bits (sloppily except for ensuring that
  94      * the result is larger in magnitude than cbrt(x) but not much more than
  95      * 2 23-bit ulps larger).  With rounding towards zero, the error bound
  96      * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
  97      * in the rounded t, the infinite-precision error in the Newton
  98      * approximation barely affects third digit in the the final error
  99      * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
 100      * before the final error is larger than 0.667 ulps.
 101      */
 102         u.value=t;
 103         u.bits=(u.bits+0x80000000)&0xffffffffc0000000ULL;
 104         t=u.value;
 105
 106     /* one step Newton iteration to 53 bits with error < 0.667 ulps */
 107         s=t*t;                          /* t*t is exact */
 108         r=x/s;                          /* error <= 0.5 ulps; |r| < |t| */
 109         w=t+t;                          /* t+t is exact */
 110         r=(r-t)/(w+r);                  /* r-t is exact; w+r ~= 3*t */
 111         t=t+t*r;                        /* error <= 0.5 + 0.5/3 + epsilon */
 112
 113         return(t);
 114 }