implement a fast atan approximation function

[zhichen3]

implement a fast atan function using an approximate function for x in [-1, 1], for |x| > 1, the identity arctan(x) = sign(pi/2, x) - arctan(1/x) (Thanks to Eric) is used.

Two versions of the approximate functions are included.

1. Efficient Approximations for the Arctangent Function by Rajan 2006: Equation 9: Max error ~0.0015 rad
2. https://stackoverflow.com/questions/42537957/fast-accurate-atan-arctan-approximation-algorithm: Max error ~0.00063 rad

Generally I find the form from stackoverflow is more accurate and also faster. 1) is also discussed in 2).

In general, from testing, atanf is roughly more than 2 times faster than std::atan.

[zhichen3]

Not sure why hip failed to compile. Also I kept the volatile qualifier for testing because runtime wouldn't increase with iteration otherwise.

[zingale]

I suspect the HIP failure is due to changes in ROCm and there is a namespace clash now.

[zingale]

can you try not including <AMReX_GpuUtility.H> and instead directly include just what you need?

[zingale]

or alternately, include <AMReX_Gpu.H>, since that seems to do some namespace magic.

[yut23]

It's probably this line:

Microphysics/networks/iso7/actual_network.H

Line 10 in 30469dd

using namespace amrex;

[yut23]

atanf() may conflict with the definition from the standard library. I think fast_atan() works fine.

[zingale]

indeed, we should get rid of those using namespace amrex
I'll try that in a separate PR

[zhichen3]

I figured checking nan is unnecessary, so I removed it and also removed AMReX_GpuUtility.H since I don't need isnan anymore. I guess this also fixed hip compilation error.

[yut23]

I've been looking at the screening code a bunch for the autodiff stuff, and I think we may want to include the third-order term in the Taylor series, since we do sqrt_gamma - atan(sqrt_gamma) which cancels to zero at first-order. Here's a comparison between the first and third order series: the relative error of the first-order series goes up to around 32% and levels off. There's also a discontinuity when it switches from fast_atan_1 to the Taylor series.

[zhichen3]

sounds good to me