Introduction

omnibias is a small mathematical layer that lets you call σⁿ(z) — the n-th derivative of a base activation — in closed form, for arbitrary n, with bit-stable accuracy and a single σ evaluation regardless of order. The same primitive runs on PyTorch, JAX, and Keras 3; the polynomial coefficients come from one shared pure-Python module, so the closed-form activation / derivative math is bit-identical across backends by construction (end-to-end layer numerics still follow the chosen backend).

The whole pitch in one line:

omnibias gives you the n-th derivative of an activation in one forward pass at machine precision — where nested autodiff grows exponentially in cost and accumulates round-off, and finite differences lose roughly n digits by the n-th derivative.

Why it exists

Most deep-learning frameworks differentiate through nested layers via automatic differentiation (AD). That is the right answer for first derivatives. It is the wrong answer when you need:

• the Laplacian for a physics-informed network, a continuous normalizing flow, or a Stein operator — nested AD grows as you iterate Δᵏ;
• the Hessian for second-order optimization (KFAC, natural gradient) or Bayesian-PINN Fisher information;
• σⁿ for n ≥ 3 — 4th-order biharmonic / strain-gradient PDEs, 6th-order plate/shell models, score-based diffusion with high-order Stein operators.

For the Riccati class of activations the derivative tower has a closed form, and omnibias delivers it.

How it fits together

Every backend imports the same coefficients from omnibias-core, which is why a given (activation, order) pair is float64-ULP-equal across frameworks.

The math, in one paragraph

The core identity is Riccati: sigmoid'(z) = s(1 − s) and tanh'(z) = 1 − t². Differentiating these identities repeatedly yields polynomial recurrences (Eulerian for sigmoid, a Legendre-style recurrence for tanh, the probabilist's Hermite identity for the Gaussian). Each higher derivative is a polynomial in the single value σ(z) you already computed — so the whole tower costs one activation evaluation. See Closed-form derivatives for the full derivation.

What to read next

• New here? Go straight to the Quickstart.
• Want the conceptual model? Read Closed-form derivatives and Operator-typed layers.
• Care about correctness boundaries? Read Exactness & scope — the canonical page for what is closed-form, what is autodiff-exact, and what is numerical.
• Building something specific? Jump to the Tutorials.
• Need to clear legal review? Read Licensing — most of omnibias is Apache-2.0, and commercial terms for the certified-decision packages come from info@derivon.ai.

Honest by design

omnibias is deliberate about scope. We never blur closed-form, autodiff-exact, and numerical results — each carries an explicit label. That honesty is what makes the certified register worth trusting.