[dpsanders]

Summary

Stacked on #84 (forward-mode AD), which is stacked on #83 (derivative hardening). Adds hessian(ex, vars) as the third AD entry point, reusing the same scalar rule table (unary_rule / binary_rule) as gradient and tangent.

How it works

Forward-over-reverse:

1. gradient_code(ex, vars) — one reverse-mode pass. Yields an SSA whose variables.gradient = [g₁, …, gₙ] are symbolic handles into the code.
2. binarize_ssa — decompose any n-ary muls the adjoint left behind (e.g. 2 * _b̄ * x) so the tangent pass can consume them.
3. tangent_code — attach a forward-mode tangent pass on top. Produces tangent assignments for every SSA variable, including the gᵢs.
4. Compiled (inputs, direction) → (value, gradient, gradient_tangent). Invoking it with direction eᵢ gives row i of the Hessian. The user-facing hessian closure loops over i = 1..n and assembles the n × n matrix.

API

@variables x y z
H = hessian(x^2 * y + y^3 + z^3, [x, y, z])
val, grad, M = H((2.0, 3.0, 1.0))
# val :: Float64
# grad :: Tuple{Float64, Float64, Float64}
# M :: Matrix{Float64}  (3×3, Schwarz-symmetric for C² inputs)

Also in this PR

tangent_eq is hardened to accept non-call rhs's. Reverse-mode SSA contains _c̄ := 1 (constant seed) and identity copies like _b̄ := _c̄; the previous tangent_eq assumed every rhs was a function call and errored the moment gradient SSA was used as forward-mode input. Tangent of a bare constant is 0, tangent of a bare variable is the variable's own tangent — both fall out of dotted dispatching on Real vs BasicSymbolic.

Test plan

• 95/95 tests pass locally (68 inherited, +27 new for Hessian).
• Closed-form check against x² + y² (diagonal [2, 2]).
• Cross-term x·y (off-diagonal 1s).
• x²y + y³, sin(xy) — matches hand-derived Hessian.
• 3-variable case x² + y² + z² + xyz — dense 3×3.
• Schwarz symmetry check on three non-trivial expressions.

Bump to v0.8.0.

Performance note

The runtime cost is n tangent-pass invocations per hessian(…)(inputs) call. Fine for the small-n symbolic regime the package targets; denser cases can compile all n tangent passes into a single SSA later.

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