Lets consider anlaytical function power series expansion.
$$f(x)=\sum_{i=0}^{\infty}a_ix^i$$
Assume that this function convergent in any point of complex plane.
But for big values of $x$ this function will require more calculations to get to some approximate value, and accordingly for small $x$, the approximation would be reached much quicker. For example: In order to calculate the $e^5$ with $$\sum_{i=0}^{12}\frac{5^i}{i!}=148.114$$ While $$e^5=148.413$$ So we reached 3-digit accuracy.
Now if we do the same for $e^{1/5}$
$$\sum_{i=0}^{12}\frac{1}{5^i i!}=1.22140275816017$$
16 digits of which are accurate.
Now the question is if there is a known method of accelerating calculations of function in some point (for example $e^5$) using some transformation to the unit disk (using calculations of $e^x$ for $x<1$)?
Here is a useful link, unfortunatley the methods described here does not work for my case. http://numbers.computation.free.fr/Constants/Miscellaneous/seriesacceleration.html#Cohen