Performance: Midpoint Spin Push for Thin Kicks

[ax3l]

Evaluate the Thomas-BMT generator once, at the momentum halfway through the kick, instead of applying two half rotations around the phase space kick. This is the structure already used by TaperedPL.

Reversibility is preserved exactly. The reversed element maps p_out back to p_in, so it sees the same midpoint, and the negated strength gives the inverse rotation. Roundtrip residual stays at round-off for all four elements (≤ 2.7e-16 either way).

Accuracy improves by 2x. Both forms are second order; they differ at third order, where the midpoint rule has half the error constant of the trapezoidal rule that two half kicks amount to. Error against a converged reference:

element	half kicks	midpoint	ratio
Kicker	3.215e-03	1.597e-03	2.01x
Buncher	5.694e-12	2.753e-12	2.07x
NonlinearLens	2.260e-07	1.130e-07	2.00x
ThinDipole	2.430e-07	1.005e-07	2.42x

The reference applies each element N times at strength/N. These kicks depend only on coordinates the kick leaves unchanged, so the N-fold product reproduces the identical phase space result (|dp| < 9e-16) while the spin resolves N substeps, making N->inf an exact reference for the same impulse. Converged at N=1e4. ThinDipole sits above 2x because its +h frame-rotation term is momentum-independent and contributes no error to either scheme.

It is also cheaper, at 1 tbmt_precession_vector + 1 rotate_spin per particle instead of 2 of each:

element	half kicks	midpoint	spin push	spin-only part
Kicker	7.77 ms	4.91 ms	-37%	-43%
Buncher	7.54 ms	5.05 ms	-33%	-43%
NonlinearLens	11.90 ms	9.21 ms	-23%	-37%
ThinDipole	3.98 ms	2.86 ms	-28%	-43%

"spin-only part" subtracts the [nospin] time of the same element. NonlinearLens gains least in total because its F'(zeta) evaluation dominates and is unaffected.

Applies to Kicker, Buncher, NonlinearLens and ThinDipole, whose fields depend only on coordinates the kick leaves unchanged. Buncher also changes the energy deviation, so pt is averaged along with the transverse momenta. ShortRF keeps the half kick form: it changes the reference energy, so px/py/pt are renormalized across the kick and a plain average would mix two normalizations.

Measured locally (GCC, OMP, SIMD, double, 1M particles, min of 3, ~4% run-to-run spread), not CodSpeed. Error numbers are at 100x the benchmark-file element strengths, chosen so the difference is resolvable — at nominal strengths it falls below the reference round-off floor and is irrelevant in practice.

[cemitch99]

Nice, thanks.