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Here's a solution which is vectorizable: http://gallium.inria.fr/blog/fast-vectorizable-math-approx/

I've also read that Chebyshev polynomials are far better for approximating functions than Taylor series (eg https://en.wikipedia.org/wiki/Approximation_theory)

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This matters less once you put things through the horner scheme. And if you are not you are leaving performance on the table.

Horner does not conflict with vectorization in this case. If you read the code, it actually uses Horner — the vectorizability applies to when you call the online function in a loop.

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