Forward-over-reverse Hessian-vector products

Input a $C^2$ graph, a point $x$, and direction $v$. Run Forward-mode automatic differentiation on a finite computational graph and retain its primal tape and all input and operation tangents. If that call returns DOMAIN_ERROR or NONDIFFERENTIABLE, return the same status and stop without a Hessian-vector product. Otherwise initialize every operation and input cotangent and tangent-cotangent accumulator to zero. Seed the output by setting its cotangent/tangent-cotangent pair to $(1,0)$: if $o$ is an operation, set $({\bar{z}}_o,{\dot{\bar{z}}}_o)=(1,0)$; if $o$ is input coordinate $i$, set $({\bar{x}}_i,{\dot{\bar{x}}}_i)=(1,0)$. In reverse topological order, for each node $j$ and parent slot $k$, let $A_{j,k}=D_k{\varphi }_j(z_{p_{j,1}},\dots ,z_{p_{j,r_j}})$, compute $c_{j,k}=A_{j,k}^T{\bar{z}}_j,{\dot{c}}_{j,k}=A_{j,k}^T{\dot{\bar{z}}}_j+(D{A}_{j,k}(z_{p_{j,1}},\dots ,z_{p_{j,r_j}})[{\dot{z}}_{p_{j,1}},\dots ,{\dot{z}}_{p_{j,r_j}}]{)}^T{\bar{z}}_j,$ and add $c_{j,k}$ and ${\dot{c}}_{j,k}$ to the parent's cotangent and tangent-cotangent accumulators. Use $({\bar{x}}_i,{\dot{\bar{x}}}_i)$ for input coordinate $i$ and $({\bar{z}}_{\mathrm{\ell }},{\dot{\bar{z}}}_{\mathrm{\ell }})$ for operation node $\mathrm{\ell }$. If the same parent occurs in multiple slots, add every slot contribution. Return $({\dot{\bar{x}}}_1,\dots ,{\dot{\bar{x}}}_n)$. Repeated parent slots receive separate additions. The admissible case requires every primitive evaluation to be in its declared domain and, for a piecewise primitive, on a unique strict active branch; a branch boundary or ambiguous selection returns NONDIFFERENTIABLE through the initial call. The transpose is the real Euclidean transpose, not a complex adjoint.