Use autodiff in templated network rate evaluation

[yut23]

No description provided.

[yut23]

Timings for test_aprox_rates, with a loop added around the rate_* calls. RHS::rhs() calls the version without derivatives, while the analytic Jacobian calls the version with derivatives. The previous code always calculated the derivatives, hence the same timings on the left.

CPU (n_cell=64, loops=10):
- without derivatives:  5.34s ->  4.34s (-1.00s, -19%)
- with derivatives:     5.34s ->  5.71s (+0.37s,  +7%)

CUDA (n_cell=64, loops=500):
- without derivatives:  5.25s ->  4.42s (-0.83s, -16%)
- with derivatives:     5.25s ->  5.30s (+0.05s,  +1%)

[zingale]

test suite: http://groot.astro.sunysb.edu/Microphysics/test-suite/gfortran/2024-07-17-006/index.html

[zingale]

do you expect the diffs shown in the test suite?

[yut23]

No, I don't think the diffs should be that large. I'm even more confused by the diffs with the numerical Jacobian, as that shouldn't be using autodiff at all.

[zingale]

let me rerun the test

[yut23]

I found a few optimizations I made that were causing the differences with the numerical Jacobian, and I'm now pretty sure the other big diffs are just compounded from the roundoff diffs in the analytical Jacobian.

[zingale]

most test_rhs diffs are now roundoff, except for powerlaw, which is Inf
http://groot.astro.sunysb.edu/Microphysics/test-suite/gfortran/2024-07-23-001/index.html

[yut23]

This should be ready to merge. The Inf diffs for powerlaw are because we weren't calculating the temperature derivatives at all before.