This paper speeds up Brent's algorithms for various high-precision computations in the power series ring C[[t]]. If it takes time 3 to compute a product then it takes time roughly 5:6 to compute a reciprocal; roughly 8:2 to compute a quotient or a logarithm; roughly 6:5 to compute a square root; roughly 9 to compute both a square root and a reciprocal square root; and roughly 10:4 to compute an exponential. The same ideas apply to approximate computations in R, Q p, etc.

This paper presents lower bounds and upper bounds on the distribution of smooth integers; builds an algebraic framework for the bounds; shows how the bounds can be computed at extremely high speed using FFT-based power-series exponentiation; explains how one can choose the parameters to achieve any desired level of accuracy; and discusses several generalizations.