 Scale-aware stopping criterion for Newton iteration

Near a well-conditioned simple root, a numerically stable Newton root-finding iteration cannot usually resolve steps far below the local spacing of floating-point numbers. That spacing has relative scale comparable to machine epsilon $ϵ$, motivating the displayed position-based criterion with a modest constant $C$. It combines an absolute scale near zero with a relative scale for large inputs. A residual-only test is changed by rescaling the equation, whereas the root and exact Newton step are unchanged. A tiny step alone is not a universal accuracy certificate: derivative errors, stagnation and cycles require separate consideration.