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 $e^{-x^2}\text{erf}(x+a)$, where $a$ is a parameter that changes throughout the computation. Mathematica was unable to provide an analytical expression of $\int e^{-x^2}\text{erf}(x+a)dx$ and neither was I.

That caused me thinking how can I approximate well known functions such as $e^{-x^2}$ and $\text{erf}(x+a)$ (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.

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.

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?

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 $e^{-x^2}\text{erf}(x+a)$, where $a$ is a parameter that changes throughout the computation. Mathematica was unable to provide an analytical expression of $\int e^{-x^2}\text{erf}(x+a)dx$ and neither was I.

That caused me thinking how can I approximate well known functions such as $e^{-x^2}$ and $\text{erf}(x+a)$ (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.