Forward-mode automatic differentiation on a finite computational graph

Input a graph from Finite straight-line computational programs and real-arithmetic tapes, an admissible point $x$, and a direction $v\in {\mathbb{R}}^n$. Require every primitive to be $C^1$ on its declared open domain or supplied as a piecewise-$C^1$ primitive with the branch rules of the graph definition. Evaluate all primal node values in topological order and store the tape. Set each input tangent to ${\dot{z}}_{{\mathrm{i}\mathrm{n}}_i}={\dot{x}}_i=v_i$, using the distinct input labels of the graph. For each operation $j=1,\dots ,T$, compute ${\dot{z}}_j=D{\varphi }_j(z_{p_{j,1}},\dots ,z_{p_{j,r_j}})({\dot{z}}_{p_{j,1}},\dots ,{\dot{z}}_{p_{j,r_j}})\mathrm{.}$ Return the primal scalar output $f(x)$ and its tangent: return $(z_o,{\dot{z}}_o)$ if $o$ is an operation node, and $(x_i,v_i)$ if $o$ is input coordinate $i$. If the input is outside the admissible domain, return DOMAIN_ERROR; if a piecewise primitive is at a branch boundary or has no unique strict active branch, return NONDIFFERENTIABLE and no tangent. All derivative evaluations in this definition are exact real-arithmetic operations on the specified primitive maps.