We used to do a lot of these kinds of things when I was studying for my Applied Mathematics degree (which I received with Distinction). From what I can remember, if the plots were complex you would break it down into parts and use different techniques for each - the end result was some big complex formula.
Then they invented libraries to do much of that manual leg work - e.g. MatLab.
These days compute resources are so cheap that if you don't need super high precision then a lookup table will work just as well and probably even be quicker. They are also easier to understand and maintain - unless you enjoy a challenge.