Little side-note: algorithm as given is scalar; however, its branch-free, and defined entirely in the header file. So, compilers will typically be able to vectorize it, and thus achieve speed up directly based on the vector size. I see potential [but architecture-dependent] optimization using Estrin scheme for evaluating the polynomial.
Yes, aim was to be acurate down to 1 lsb while significantly faster. Feel free to drop terms from the polynomial if you can live with less accurate results!
The coefficients were generated by a package called Sollya, I've used it a few times to develop accurate chebyshev approximations for functions.
I have developed very fast, accurate, and vectorizable atan() and atan2() implementations, leveraging AVX/SSE capabilities.
You can find them here [warning: self-signed SSL-Cert].