Choose a derivative path and measure it

Start with the operator your residual needs, then choose the smallest suitable representation.

• One activation derivative: a supported activation fast path.
• A few network directions: directional jets.
• All mixed partials through a small order: a mixed jet with an explicit coefficient budget.
• A deep Laplacian: direct contraction.
• A repeated deep Laplacian: inspect the exact-support or estimator report.

Measure the actual training loop in addition to derivative evaluation. Include warmup, compilation, memory, numerical error, and parameter-gradient checks. Large high-order coefficients can cause cancellation even when the algebra is correct.

Follow the benchmark methodology and compare with the published evidence.