Apologies if the question is too elementary here.

For a certain computational application I need to approximate Gaussian distribution $e^{-x^2}$ with specific absolute precision (within $10^{-7}$ over $\mathbb{R}$), preferably with rational functions.

Alas, I'm not familiar with approximation theory. Google pointed me toward Pade approximation as the way to go. Alas, I still don't know how to derive Pade approximation for a given function, much less how to ensure the approximation would fit to the prescribed precision. Could you point me towards the relevant information?

EDIT:

Here is some background. For some engineering computations I need analytical expressions for certain integrals; in this particular case for $e^{-x^2} \operatorname{erf}(x+a)$, where $a$ is a parameter that changes throughout the computation. Mathematica was unable to provide an analytical expression of $\int e^{-x^2}\operatorname{erf}(x+a)\,dx$ and neither was I.

That caused me thinking how can I approximate well known functions such as $e^{-x^2}$ and $\operatorname{erf}(x+a)$ (and possibly a few others I may need) so that their products would be analytically integrable?

It's very well known that one can obtain an analytic expression for an integral of a rational function, and that products of rational functions are rational. Thus the idea: approximate the relevant functions with rational ones using Pade approximation and integrate the products of Pade approximations. That would basically replace the integral I want with an expression containing a bunch of rational functions and $\log$s and $\arctan$s.

I've got to tightly control the precision of the above approximations though to have the model perform as expected. Thus the question.

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