src/math/cbrt.c

   1 /* origin: FreeBSD /usr/src/lib/msun/src/s_cbrt.c */
   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 /* cbrt(x)
  15  * Return cube root of x
  16  */
  17
  18 #include <math.h>
  19 #include <stdint.h>
  20
  21 static const uint32_t
  22 B1 = 715094163, /* B1 = (1023-1023/3-0.03306235651)*2**20 */
  23 B2 = 696219795; /* B2 = (1023-1023/3-54/3-0.03306235651)*2**20 */
  24
  25 /* |1/cbrt(x) - p(x)| < 2**-23.5 (~[-7.93e-8, 7.929e-8]). */
  26 static const double
  27 P0 =  1.87595182427177009643,  /* 0x3ffe03e6, 0x0f61e692 */
  28 P1 = -1.88497979543377169875,  /* 0xbffe28e0, 0x92f02420 */
  29 P2 =  1.621429720105354466140, /* 0x3ff9f160, 0x4a49d6c2 */
  30 P3 = -0.758397934778766047437, /* 0xbfe844cb, 0xbee751d9 */
  31 P4 =  0.145996192886612446982; /* 0x3fc2b000, 0xd4e4edd7 */
  32
  33 double cbrt(double x)
  34 {
  35         union {double f; uint64_t i;} u = {x};
  36         double_t r,s,t,w;
  37         uint32_t hx = u.i>>32 & 0x7fffffff;
  38
  39         if (hx >= 0x7ff00000)  /* cbrt(NaN,INF) is itself */
  40                 return x+x;
  41
  42         /*
  43          * Rough cbrt to 5 bits:
  44          *    cbrt(2**e*(1+m) ~= 2**(e/3)*(1+(e%3+m)/3)
  45          * where e is integral and >= 0, m is real and in [0, 1), and "/" and
  46          * "%" are integer division and modulus with rounding towards minus
  47          * infinity.  The RHS is always >= the LHS and has a maximum relative
  48          * error of about 1 in 16.  Adding a bias of -0.03306235651 to the
  49          * (e%3+m)/3 term reduces the error to about 1 in 32. With the IEEE
  50          * floating point representation, for finite positive normal values,
  51          * ordinary integer divison of the value in bits magically gives
  52          * almost exactly the RHS of the above provided we first subtract the
  53          * exponent bias (1023 for doubles) and later add it back.  We do the
  54          * subtraction virtually to keep e >= 0 so that ordinary integer
  55          * division rounds towards minus infinity; this is also efficient.
  56          */
  57         if (hx < 0x00100000) { /* zero or subnormal? */
  58                 u.f = x*0x1p54;
  59                 hx = u.i>>32 & 0x7fffffff;
  60                 if (hx == 0)
  61                         return x;  /* cbrt(0) is itself */
  62                 hx = hx/3 + B2;
  63         } else
  64                 hx = hx/3 + B1;
  65         u.i &= 1ULL<<63;
  66         u.i |= (uint64_t)hx << 32;
  67         t = u.f;
  68
  69         /*
  70          * New cbrt to 23 bits:
  71          *    cbrt(x) = t*cbrt(x/t**3) ~= t*P(t**3/x)
  72          * where P(r) is a polynomial of degree 4 that approximates 1/cbrt(r)
  73          * to within 2**-23.5 when |r - 1| < 1/10.  The rough approximation
  74          * has produced t such than |t/cbrt(x) - 1| ~< 1/32, and cubing this
  75          * gives us bounds for r = t**3/x.
  76          *
  77          * Try to optimize for parallel evaluation as in __tanf.c.
  78          */
  79         r = (t*t)*(t/x);
  80         t = t*((P0+r*(P1+r*P2))+((r*r)*r)*(P3+r*P4));
  81
  82         /*
  83          * Round t away from zero to 23 bits (sloppily except for ensuring that
  84          * the result is larger in magnitude than cbrt(x) but not much more than
  85          * 2 23-bit ulps larger).  With rounding towards zero, the error bound
  86          * would be ~5/6 instead of ~4/6.  With a maximum error of 2 23-bit ulps
  87          * in the rounded t, the infinite-precision error in the Newton
  88          * approximation barely affects third digit in the final error
  89          * 0.667; the error in the rounded t can be up to about 3 23-bit ulps
  90          * before the final error is larger than 0.667 ulps.
  91          */
  92         u.f = t;
  93         u.i = (u.i + 0x80000000) & 0xffffffffc0000000ULL;
  94         t = u.f;
  95
  96         /* one step Newton iteration to 53 bits with error < 0.667 ulps */
  97         s = t*t;         /* t*t is exact */
  98         r = x/s;         /* error <= 0.5 ulps; |r| < |t| */
  99         w = t+t;         /* t+t is exact */
 100         r = (r-t)/(w+r); /* r-t is exact; w+r ~= 3*t */
 101         t = t+t*r;       /* error <= 0.5 + 0.5/3 + epsilon */
 102         return t;
 103 }