Added comments about the apparently-magic rational function used in

the second step of approximating cbrt(x).  It turns out to be neither
very magic not nor very good.  It is just the (2,2) Pade approximation
to 1/cbrt(r) at r = 1, arranged in a strange way to use fewer operations
at a cost of replacing 4 multiplications by 1 division, which is an
especially bad tradeoff on machines where some of the multiplications
can be done in parallel.  A Remez rational approximation would give
at least 2 more bits of accuracy, but the (2,2) Pade approximation
already gives 6 more bits than needed.  (Changed the comment which
essentially says that it gives 3 more bits.)

Lower order Pade approximations are not quite accurate enough for
double precision but are plenty for float precision.  A lower order
Remez rational approximation might be enough for double precision too.
However, rational approximations inherently require an extra division,
and polynomial approximations work well for 1/cbrt(r) at r = 1, so I
plan to switch to using the latter.  There are some technical
complications that tend to cost a division in another way.

1 files changed, 15 insertions, 1 deletions

diff --git a/lib/msun/src/s_cbrt.c b/lib/msun/src/s_cbrt.c
index 4e7eaa8..6d90078 100644
--- a/lib/msun/src/s_cbrt.c
+++ b/lib/msun/src/s_cbrt.c
@@ -72,7 +72,21 @@ cbrt(double x)
 	} else
 	    SET_HIGH_WORD(t,sign|(hx/3+B1));
 
-    /* new cbrt to 23 bits; may be implemented in single precision */
+    /*
+     * New cbrt to 26 bits; may be implemented in single precision:
+     *    cbrt(x) = t*cbrt(x/t**3) ~= t*R(x/t**3)
+     * where R(r) = (14*r**2 + 35*r + 5)/(5*r**2 + 35*r + 14) is the
+     * (2,2) Pade approximation to cbrt(r) at r = 1.  We replace
+     * r = x/t**3 by 1/r = t**3/x since the latter can be evaluated
+     * more efficiently, and rearrange the expression for R(r) to use
+     * 4 additions and 2 divisions instead of the 4 additions, 4
+     * multiplications and 1 division that would be required using
+     * Horner's rule on the numerator and denominator.  t being good
+     * to 32 bits means that |t/cbrt(x)-1| < 1/32, so |x/t**3-1| < 0.1
+     * and for R(r) we can use any approximation to cbrt(r) that is good
+     * to 20 bits on [0.9, 1.1].  The (2,2) Pade approximation is not an
+     * especially good choice.
+     */
 	r=t*t/x;
 	s=C+r*t;
 	t*=G+F/(s+E+D/s);