So, the problem isn't bounded by simple probabilities. There are other properties involved that cause it to diverge from the "naive" probabilistic estimates. What those properties are we don't know yet.
The "rounding" problem is called the "Table Maker's Dilemma" and it descends into some pretty fundamental transcendental number theory.
This is not an "easy" problem. Proof that PI was transcendental is relatively modern (1885).
In my experience, low order bits of transcendental functions make terrible rng sources (even discounting their runtime). Your example of the logarithm is great -- for large inputs, you need large changes to flip even low order bits.
Elkies [0] computes a piecewise linear approximation to his (number-theoretic) function of interest and does a lattice reduction on each piece to filter out the vast majority of the space. The same approach, works almost everywhere on all of the transcendental functions I've tried, and I believe the CRlibm group uses it.