Module autodiff 

Automatic differentiation using dual numbers.

A dual number a + b·ε where ε² = 0 carries both the value and the derivative of a function at a point. Evaluating f(Dual(x, 1)) yields Dual(f(x), f'(x)) in a single forward pass — no symbolic differentiation needed, and it works for arbitrary compositions.

This module provides:

• Dual — a 1st-order dual number type with full arithmetic
• eval — evaluate an Expr AST with dual numbers
• derivative — compute f'(x) at a point for any parseable expression
• gradient — compute the gradient of a multivariate expression
• jacobian — compute the Jacobian of a system of expressions

Structs

Dual

A first-order dual number: value + deriv * ε where ε² = 0.

Functions

derivative

Compute the derivative of f(var) at x using automatic differentiation.

eval

Evaluate an Expr AST using dual numbers, returning the value and derivative.

gradient

Compute the gradient of a multivariate expression at a point.

jacobian

Compute the Jacobian matrix of a system of expressions.