I thought the OP’s code was fine. My main point was that the top-level poster’s argument about what counted as “functional” seemed silly, and that if he really wanted to go all the way he could write a more completely generic function which could also be made faster by getting data flowing through the CPU more efficiently or using a GPU. (I shouldn’t have phrased this as “the proper thing to do” without clarification, but I can’t edit it now.)

> if for some reason a program is really doing this calculation over and over

Most of the time when you want to evaluate sin(x) for one number x, you want to also evaluate it for a million other numbers. For example, you might be updating the velocity vectors of all the rockets in a space game, finding the map coordinates of a polyline representing the coastline of England, converting color space coordinates for every pixel in an image from CIELAB to CIELCh, or whatever.

In cases where you are just calling the sine function as a one-off, performance is pretty well irrelevant.

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Since sin(x) has a few steps before and after the polynomial evaluation, you are right that it usually makes sense to hard-code the whole function with domain normalization etc. (ideally still vectorized), though those other steps could also be genericized if necessary, since you could also use them for various other special functions.

The nice thing about just storing the polynomial coefficients as data is that you can then do more things with them than evaluation alone. For inspiration, look at the Chebfun project, http://www.chebfun.org/ https://people.maths.ox.ac.uk/trefethen/cacm.pdf

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One last thing: I recommend trying to re-frame trigonometry problems as vector problems whenever possible, and thereby not needing the sine function at all. Code which avoids transcendental function evaluations altogether is usually simpler to reason about and faster.