On Fri, 3 Oct 2003, Russell McMahon wrote:

> > If anyone's interested, I optimized the (already optimized) 16-bit square
> > root routine that I wrote for the 18F family. It now takes between 55 and
> > 68 cycles to compute the square root of a 16-bit number. The new version
> > unrolls the loop and takes advantage of the invariant states. At the cost
> > of nearly doubling the code size, 7 more execution cycles could be saved.
> > However, one can only stoop so low to shave cycles...
>
> FWIW, and you all may have better ways of doing this, square root can be
> used directly to build a log function using otherwise only shift and add. It
> may be that this information is more useful to people with a 4 function plus
> sqrt calculators who want a log function :-)
>
> eg using said calculator
>
>     log10(X) ~= (sqrt.sqrt.sqrt.sqrt.sqrt.sqrt.sqrt.sqrt.(X) - 1 ) x 889

This can be re-written as:

   log10(X) ~=  (X^(1/2^11) - 1) * 889

(11 instead of 10 because of a subsequent RM post).

I don't see any 'obvious' analytical way to prove this. However, it might
be instructive to look at series expansions of log10 and X^Y and see if
there's some way to correlate the coefficients of the terms in the
expansions.

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