Nth digit of PI

[Kent]

Clive Tooth <clive at pisquaredoversix.force9.co.uk> wrote in message
news:8hqhhi$396$1 at mail.pl.unisys.com...
> Robert Israel wrote in message <8houlu$j9s$1 at nntp.itservices.ubc.ca>...
>
> >[...]
> >There are two different algorithms here, for two different problems.
> >
> >The BBP algorithm can find a hexadecimal (or more generally, base
> >2^k) digit of pi, faster than any known algorithm that would find
> >all the digits up to this one.  This is the exciting one.
>
> Although BBP do mention that "This algorithm is, by a factor of
> log(log(log(n))), asymptotically slower than the fastest known algorithms
> for generating the n-th digit by generating all of the first n digits of
> log(2) or pi."

I'm not getting this. It's "faster than any known algorithm that would find
all the digits up to this one", but it's "asymptotically slower than the
fastest known algorithms
 for generating the n-th digit by generating all of the first n digits of
 log(2) or pi". How can it be both faster and asymptotically slower? Does
this mean "faster for small enough values of n, but slower for large enough
values"? If so, about how big are "small enough" and "large enough"?

too lazy to follow the links and try to figure it out for himself,
Kent.