# Padé approximation - usability in iterative algorithms

Firstly, I have to say that I don't understand Padé approximation well.
But I discovered that, it is more precise than Taylor series.
I have to create approximation for these functions: Log(x) and Tanh. And I have to create iterative algorithms (I must compute result with variable precision).

So my questions are:
Is Padé approximation usable (and more efficient than simple Taylor series) for this task?
If no, is there any better way to approximate these functions?

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What you can do that is equivalent to using a Padé approximant (SFAICT, there's no simple method for generating the coefficients of the numerator and denominator polynomials for the two functions you have, except by solving the appropriate Toeplitz system) is to use continued fraction expansions, which $\ln(1+x)$ and $\tanh(x)$ have by their virtue of being expressible as hypergeometric functions.
Of course, for proper use, you have to perform appropriate argument reductions (e.g. for $\tanh$, compute $x^{\star}=\frac{x}{2^n}$ where $n$ is an appropriate integer such that $x^{\star}$ is "small" enough, evaluate the continued fraction at $x^{\star}$, and use the double-argument formula for $\tanh$ to undo your previous transformation).