Fix Float32 precision upcast in lda_c_pw under AD - #33
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In `energy_per_particle_c_pw` the two `(1 ± ζ)` bases of equation (9) are
already typed `T`, but the remaining base was the integer literal `2`. With a
`Dual` exponent an integer base is differentiated via `log(2)` in Float64, so a
`Dual{Tag,Float32}` density came back as `Dual{Tag,Float64}`.
Only `energy_density` was affected: `potential_terms` pins the type via
`arithmetic_type` + `similar(ρ, TT, …)`. Downstream in DFTK this made the XC
energy the only Float64-valued term in an otherwise Float32 forward-mode
evaluation, which broke Float32 stress computations.
The change is value-neutral for non-`Dual` types (`2^x === T(2)^x` bit-identical
for Float32 and Float64); for a plain `Float32`, `promote(2, ::Float32)` is
already `Float32`, which is why this only showed up under AD.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
Claude-Session: https://claude.ai/code/session_017an8Gie5b1ULeSjhep3Sjb
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This is actually an upstream ForwardDiff bug, but might as well fix it here too.