Irrational Acceleration of a Continued Fraction of $\pi$

An application of (iterated) Bauer-Muir acceleration can give an Apéry-like continued fraction for $\pi$ with irrational coefficients, and much faster convergence. It can be considered a generalized continued fraction with the same matrix representation as the standard case, but the dimension increased due to irrationality. The construction is also given for $\ln (2)$ and $\sqrt[3]{2}$.