In general, I find that minimax approximation is an underappreciated tool, especially the quite simple Remez algorithm to generate an optimal polynomial approximation [0]. With some modifications, you can adapt it to optimize for either absolute or relative error within an interval, or even come up with rational-function approximations. (Though unfortunately, many presentations of the algorithm use overly-simple forms of sample point selection that can break down on nontrivial input curves, especially if they contain small oscillations.)

[0] https://en.wikipedia.org/wiki/Remez_algorithm

▲

herf 6 hours ago | parent | next [-]

They teach a lot of Taylor/Maclaurin series in Math classes (and trig functions are sometimes called "CORDIC" which is an old method too) but these are not used much in actual FPUs and libraries. Maybe we should update the curricula so people know better ways.

	▲	bee_rider 4 hours ago \| parent [-]
		Taylor series makes a lot more sense in a math class, right? It is straightforward and (just for example), when you are thinking about whether or not a series converges in the limit, why care about the quality of the approximation after a set number of steps?

▲

jason_s 6 hours ago | parent | prev [-]

Not sure I would call Remez "simple"... it's all relative; I prefer Chebyshev approximation which is simpler than Remez.

	▲	LegionMammal978 6 hours ago \| parent \| next [-]
		Perhaps, but at least I find it very simple for the optimality properties it gives: there is no inherent need to say, "I know that better parameters likely exist, but the algorithm to find them would be hopelessly expensive," as is the case in many forms of minimax optimization.
	▲	stephencanon 6 hours ago \| parent \| prev [-]
		Ideally either one is just a library call to generate the coefficients. Remez can get into trouble near the endpoints of the interval for asin and require a little bit of manual intervention, however.