[seqfan] Re: A063747 sign of power series expansion of the Gamma function

[Peter Luschny]

HP> It is also sometimes not obvious that floating-point arithmetic
HP> is being used in the background. And then one can hardly assume
HP> that everyone has half a century of experience in numerical computing.

Even that doesn't help.

If you want to read the story of how PARI bugs were searched
for at the Computing Center of the Polish Academy of Sciences
for half a year (August 2021 to December 2021), you can read it here.

The story also shows that increasing 'precision' does not
eliminate all sources of error. On the contrary, algorithms
that are correct using low precision become subtly wrong
when applying higher precision.

"In the end, it turned out that the cause of this error was
simply in the PARI implemented procedure for computing the
value of the zeta function which uses the classical Euler-Maclaurin
algorithm. Specifically, the values of the Bernoulli numbers
required to compute the zeta function were rounded unnecessarily.
It was due to double roundings occurring when caching Bernoulli
numbers, because of too frequent precision reductions."
>From the Appendix of  https://arxiv.org/pdf/2210.04609.pdf

Euler-Maclaurin-Bernoulli numbers! So not with any esoteric
calculations, but with the daily bread, so to speak.

Peter