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[Jesús Muñoz]

Hi again:

I am glad to see how Shawul's UCT run and also new Adam Hair's estimate based in estimating BF with polynomials and logarithms suit in my interval. I have worked a little on my 'method' and I discovered that is a little better than I expected! It can also estimates Perft in even cases of n (at first I thought it only could estimate odd values of plies). I have adjust another polynomial for even values, because the 'odd - even effect' is brutal when I work with my alpha - beta couple of parameters. The idea is splitting odd and even values of n (I think I said that some weeks ago) and also Reinhard exposed this effect with his inverse interpolation.

First I worked with cubic functions of alpha and beta in both even and odd values of n; secondly, I switched to a quartic function in alpha and beta (odd case) when maintained cubic function for alpha and beta (in the case of even values of n). The results were incredibly similar (sure: less than 1% of difference between cases). I had no access to true known Perft values so I did the estimates in a clumsy way: I remembered all the Perft except Perft(10), which I remembered it was around 6.9352e+13, and I did the calculations (with a normal scientific calculator) assuming Perft(10) ~ 6.93525e+13 (trying to reduce the difference). I have rounded the final values because it is too messy giving all the numbers when they are not exact, because of the issue involving Perft(10). The estimates up to Perft(20) are:

Perft(13) ~ 1.980468096e+18 (I started from this value (my estimate), as it was the true one).
Perft(14) ~ 6.1861e+19
Perft(15) ~ 2.0134e+21
Perft(16) ~ 6.5013e+22
Perft(17) ~ 2.1729e+24
Perft(18) ~ 7.2067e+25
Perft(19) ~ 2.4604e+27
Perft(20) ~ 8.3348e+28

These estimates are like a copy of Labelle estimates, but all the math took me some hours (also adjusting by hand the polynomials, but it was fast) and the ways for reaching these numbers are completely independent.

http://wismuth.com/chess/statistics-games.html

It is amazing for me how similar are my values. BF's are almost the same as Labelle ones, and always my estimates are a little less than Labelle ones. These results are the arithmetic averages of the results given by alpha and beta estimates. They also have error bars that I have omitted, but there were smaller in even cases than in odd cases. Besides, error bars grow when n grows: Perft(14) to Perft(20) (only even values of n) gave error bars from ~ ± 0.0053% to ~ ± 0.0566%; Perft(15) to Perft(19) (only odd values of n) gave error bars from ~ ± 0.05% to ~ ± 0.17% (remember that error bar was ~ ± 0.01022% for n = 13).

Splitting polynomials is crucial: in a first momment I tried to estimate Perft(14) to Perft(20) with the help of an only polynomial and results were horrible! Althouh Perft(14) and Perft(15) were around 0.65% less than the final results I post, BF's from n = 16 to n = 20 were 26, 9(?), 18... and Perft(20) estimate was ~ 9.85912e+27 (horrible!).

Maybe it is better to increase one degree the polynomials ASAP it could be done (when new estimates are calculated). But it was simply too much work for me.

Fortunately, Mr. Edwards has posted again. His effort computing Perft(13) is very much appreciated here. Please keep the good work!

Regards from Spain.

Ajedrecista.
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