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Can we compute the first $n$ digits of $\pi$ in $F(n)$ time?
It follows that if one can compute $2^{ 42n} \pi$ to within an error of at most $1$, one can compute the first $n$ digits of $\pi$, since the only way the knowledge of $2^{42n}\pi$ to within an error of … So running an algorithm "assuming there are no long carries" to compute the first $42n$ digits will suffice unconditionally to compute the first $n$ digits.
(1): K. Mahler. On the approximation of π. …