API REFERENCE/omnibias-torch

omnibias.torch.fastpath.legendre

0.5.0rc1Release candidatePython >=3.10Apache-2.0

PyTorch activation derivatives, trainable operator layers, network jets, and direct differential-operator contractions.

import omnibias.torch.fastpath.legendre

Module contract & mathematical conventions

Closed-form derivatives of tanh via the Legendre-style recursion.

Let t = tanh(z). Then t' = 1 - t^2 (a Riccati identity), and each higher derivative of tanh is a polynomial in t:

tanh^(n)(z) = T_n(t), T_0(t) = t, T_{n+1}(t) = (1 - t^2) * T_n'(t).

Each step of the recursion factors out (1 - t^2), the same weight appearing in the Legendre orthogonality on [-1, 1] -- hence the module name. The coefficient generator lives in the canonical omnibias.core.polynomials module shared by torch and JAX.

Shared formulas do not imply universal floating-point identity. Verify dtype, shape, derivative-order support, and the stated numerical assumptions for the path you use.

Module source ↗Source revision 9a2d73df0573Running reference examples →

Imports & aliases

These names share a definition with another public module. Follow a name for its full contract, parameters, and examples.

Import from this module	Canonical reference
tanh_polynomial_coeffs	omnibias.core.tanh_polynomial_coeffs

FUNCTION

tanh_nth_derivative

omnibias.torch.fastpath.legendre.tanh_nth_derivativeSource ↗

tanh_nth_derivative(z: Tensor, n: int) -> Tensor

tanh^(n)(z) evaluated in closed form.

Parameters

Name	Type & default	Contract
zpositional-or-keyword	TensorRequired	See the operation contract and implementation below.
npositional-or-keyword	intRequired	See the operation contract and implementation below.

ReturnsTensor

Explicit validation & failure cases

• When n < 0, raises ValueError(f'order n must be >= 0, got {n}').

These are explicit source guards. Backend operations may also raise for invalid shapes, dtypes, or unsupported devices.

Executed regression · cmbconv1d shape

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_cmbconv1d_shape() -> None:
    layer = cmbConv1d(2, 4, kernel_size=3, padding=1, op="grad", base="tanh")
    out = layer(torch.randn(1, 2, 16))
    assert out.shape == (1, 4, 16)

python -m pytest "packages/omnibias-torch/tests/test_blocks.py::test_cmbconv1d_shape" -q

Read the implementation

def tanh_nth_derivative(z: Tensor, n: int) -> Tensor:
    """``tanh^(n)(z)`` evaluated in closed form."""
    if n < 0:
        raise ValueError(f"order n must be >= 0, got {n}")
    t = torch.tanh(z)
    if n == 0:
        return t
    coeffs = tanh_polynomial_coeffs(n)
    return _horner(coeffs, t)

Related APIsomnibias.core.tanh_polynomial_coeffs↗

FUNCTION

tanh_tower

omnibias.torch.fastpath.legendre.tanh_towerSource ↗

tanh_tower(z: Tensor, max_order: int) -> Tensor

All derivatives through max_order from one native tanh evaluation.

Parameters

Name	Type & default	Contract
zpositional-or-keyword	TensorRequired	See the operation contract and implementation below.
max_orderpositional-or-keyword	intRequired	See the operation contract and implementation below.

ReturnsTensor

Explicit validation & failure cases

• When max_order < 0, raises ValueError('max_order must be nonnegative').

These are explicit source guards. Backend operations may also raise for invalid shapes, dtypes, or unsupported devices.

Executed regression · hessian requires order two

Recorded coverage confirms this test executed the API body. This example belongs to the pinned upstream test suite. It may use fixtures and imports from its full file. Run it from that checkout with its test dependencies installed.

Full file, imports & fixtures ↗

def test_hessian_requires_order_two() -> None:
    layers, x0 = _build_mlp(seed=6)
    jet = mlp_jet_mv(x0, layers, 1)
    with pytest.raises(ValueError, match="hessian needs order >= 2"):
        jet_hessian(jet, int(x0.shape[0]), 1)

python -m pytest "packages/omnibias-torch/tests/test_jet_mv.py::test_hessian_requires_order_two" -q

Read the implementation

def tanh_tower(z: Tensor, max_order: int) -> Tensor:
    """All derivatives through max_order from one native tanh evaluation."""
    if max_order < 0:
        raise ValueError("max_order must be nonnegative")
    value = torch.tanh(z)
    rows = [value] + [_horner(tanh_polynomial_coeffs(n), value) for n in range(1, max_order + 1)]
    return torch.stack(rows, dim=0)

Related APIsomnibias.core.tanh_polynomial_coeffs↗

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