Is there a formula to find the p-value from a given z-score?

Question

I know that you can use a z-table to find p-values given a z-score, however I am curious if you can calculate it without the z-table or a calculator. Is it possible? If so, how do you do it? I've tried searching, but the only thing that came up was a post on the Actuarial Outpost, and none of the answers really addressed the question.

Note: I am curious because I want to write a Python program that will do this, and writing an entire z-table, and then searching the table for a given value seems incredibly tedious and inefficient. I am aware that the scipy library can do this, however it seems kinda lazy, and on top of that I'm just curious.

edit: The answer was actually really simple, just take the integral of the normal distribution and approximating it. However, this doesn't seem satisfying or tangible enough for me, mostly because I can't convert it to code-form (sorry for the trouble by the way). Are there any other ways?

Answer 1

There is no other way. You need a numerical method - approximating the integral f the normal, or using a program in a library or a $z$-table - which in turn depend on numerical methods.

Answer 2

You can create Taylor series approximation of the normal PDF and integrate that. There will be some error, but you can choose the number of expansion terms to minimize the error to an arbitrary threshold.