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For example, the following equation can be used to generate a Z table, where the mean $\mu$ is 0 and the standard deviation $\sigma$ is 1.

$$ \frac{1 + \operatorname{erf}\left(\frac{z - \mu}{\sigma\sqrt{2}}\right)}{2} $$

Ideally I am looking for the respective cumulative distribution functions expressed in a way that is easy to convert to source code.

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The analogous results for the given distributions can be found:

t: https://en.wikipedia.org/wiki/Student%27s_t-distribution

$\chi^2$: https://en.wikipedia.org/wiki/Chi-squared_distribution

F: https://en.wikipedia.org/wiki/F-distribution

What you want from those pages is the CDF (that is, Cumulative Density Function). The rest is a matter of determining what parameters to use.

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