[clean-list] BigInt & GMP on Linux

[Ronny Wichers Schreur]

John van Groningen writes (to the Clean list):

> By the way, the example program that was included with ExtendedArith also
> computes pi, but in a different way using arctan and is more than 20 times
> faster for 2500 digits.

But it's not incremental.

> (the last digit is however computed incorrectly).


This is because of the accumulated rounding errors for the terms
in the Taylor series for arctan. There's a comment to that effect
in my version, but apparently it was removed in the ExtendedArith
distribution.

The number of terms that the algorithm computes to get to d digits
of pi is roughly

    0.5*(1/log10 18 + 1/log10 57 + 1/log10 239) * d = 0.89 * d.

The maximum rounding error for each term is one ULP (unit in the
last position). The terms for arctan alternate in sign, so the
total error is less than 0.89*d/2 ULPs. For d=2500 this is about
1117 ULPs. In practice it's much better, because many errors
from the negative and positive terms cancel out.


Cheers,

Ronny Wichers Schreur