The ultimate sanity check

[Gottfried]

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My ultimate sanity test is to prove symbolically that using the Taylor's series for \( f^n(z) \) that \( f^{a+b}(z)-f^{a}(f^{b}(z))=\mathcal{O}(z^k) \). For my check I was able to get to \( \mathcal{O}(z^{29}) \) where the Lyapunov multiplier \( \lambda \) is neither zero or a root of unity and the origin is set to a fixed point not infinity. I used no floating point in my calculations, only rational numbers, so I could obtain an exact answer.
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No one provided an answer to my question, so I will come at it from another direction. What software that members have written that is based on rational numbers and not floating point?

Hi Daniel -

 I had (have?) difficulties to understand what you're really asking. Just to try one road, detecting the point "rational numbers": I've described the coefficients of the powerseries of the iteration of \( f_t(z) = t^z -1 \) using \(u = \log(t) \) in terms of polynomials of rational coefficients with keeping \( u \) symbolic/replacable by any numerical value ( real or complex).       
By the usual fix-point shift, for \( b= t^{1/t} = \exp( u \exp (-u)) \) , I think this extends to the common tetration, which is then interpolated along the Schroeder-style.   
The most detailed description of those polynomials (in \( u \) and \( u^h \) where \( h \) is the iteration height) is in coefficients on my webpages (btw. I've not seen that details anywhere else so far).

Gottfried