First of all, we had to make sure that the roots we computed were accurate to at least 10,000 (resp. 20,000) digits. We computed these roots using Maple as well as Mathematica, having them compute the roots to an accuracy of 10,010 digits. We then did two checks on the computed approximation to . First, we tested that by checking that . Second, we tested that the 10,000th through 10,005th digits were not all zeros or nines. This ensures that we actually computed the first 10,000 digits of the decimal expansion of . (We note that Maple initially did not give us an accuracy of 10,000 digits for all of the cube roots, so that we had to increase the precision here.) We then had to make sure that we computed the statistics and probability values accurately---or at least to a reasonable precision, since we used asymptotic formulas anyway. We did this by implementing them both in Maple and in Mathematica and comparing the results. We detected no significant discrepancy.

We claim that these measures reasonably ensure the reliability of our experimental results.