Attach a cheap risk score to each neural-network prediction and skip an expensive verifier, ensemble, diffusion refinement, retrieval call, or human review when the score is below a calibrated threshold. Independently audit a random subset of skipped examples using the expensive ground-truth procedure, and select the largest skip threshold whose exact confidence bound keeps the violation rate below a target budget.
Replace repeated evaluations of a Gaussian tail or Mills ratio in a neural loss or sampler with a short shifted-Hermite expansion. Choose a positive reference threshold x and represent the actual threshold as x+t; the same expansion then handles a whole batch of different shifts t using recursively generated Hermite coefficients.
Implement \(\log(1+u)\) through the bounded residual \(\rho(u)\), refining the argument by repeated square roots and propagating the residual backward with a rational recurrence. Use this table-free approximation in softplus and log-sum-exp kernels, where millions of logarithms are evaluated and predictable arithmetic can outperform general-purpose library calls on vectorized hardware.