The classical Huaxin Lin's theorem shows that the distance from a matrix A to the set of normal matrices can be estimated in terms of its self-commutator [A,A*]. We obtain a quantitative version of this theorem, "optimal" with respect to the power of self-commutator. Under certain assumptions on A, our approach can be extended to the case of general bounded operators in Hilbert spaces and to elements of C*-algebras of real rank zero. The results are joint with Professor Yuri Safarov from King's College London.