The statisticians have a bunch of tricks to transform the formulas into more-easily computable forms, e.g. calculate both the average and the standard deviation in a single pass through the data instead of one pass to calculate the average and a second to calculate the standard deviation. Converting the math in here to efficient code isn't very easy.

You mean Welford's algorithm. Since code was requested:

https://jonisalonen.com/2013/deriving-welfords-method-for-co...

Someone should write a „math notation for programmers“ article. Certainly would help me anyway.

And also it's something programmers need to be skilled at.

Yes. But also, probably just about every language will have a module with these functions. Python's NumPy and SciPy should have all of these built-in.

...except it depends on knowledge of the t distribution but has no information on how to approximate it.

It is a good frequentist's toolbox, but it is not immediately translatable to code, no.