  In appropriate frameworks, automatic differentiation is transparent to the
user at the cost of being a significant computational burden when the number of
operations is large. For iterative algorithms, implicit differentiation
alleviates this issue but requires custom implementation of Jacobian
evaluation. In this paper, we study one-step differentiation, also known as
Jacobian-free backpropagation, a method as easy as automatic differentiation
and as performant as implicit differentiation for fast algorithms (e.g.,
superlinear optimization methods). We provide a complete theoretical
approximation analysis with specific examples (Newton's method, gradient
descent) along with its consequences in bilevel optimization. Several numerical
examples illustrate the well-foundness of the one-step estimator.
