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When expanding $g(f)$ or $f^{-1}$ to high order, one could try to quickly detect if $f$ is a rational or algebraic function of low degree and handle this specially (using binomial transform, etc).
When expanding$g(f)$ or $f^{-1}$ to high order, one could try to quickly detect if $f$ is a rational or algebraic function of low degree and handle this specially (using binomial transform, etc).