Of course, the 'crank' here is looking for a cipher with an easy-to-understand keystream; but my own goal was always to efficiently compute the digits of number

Of course, the 'crank' here is looking for a cipher with an easy-to-understand keystream; but my own goal was always to efficiently compute the digits of numbers for their particular algebraic properties, to an arbitrary distance. Though it looks like there really might not be any alternative to Newton's method (or similar) for many of them.