The study of computational problems in the ideal theory of local rings led to the notion of standard bases. These are a version of Gröbner bases for orderings which are not well orderings. In most cases appearing in applications, the tangent cone algorithm computes a standard basis of a given ideal [\textit{T. Mora}, \textit{G. Pfister} and \textit{C. Traverso}, ``An introduction to the tangent cone algorithm'', In: C. M. Hoffman (ed.), Issues in robotics and nonlinear geometry, JAI Press (1992)]. Roughly speaking, this is a variant of the Buchberger algorithm with suitable modifications to avoid infinitely long reductions. The complexity of the tangent cone algorithm is unknown. In the present paper the authors introduce a ``very lazy'' version of the algorithm which allows early interruptions based on a priori knowledge, for example, of the upper bound of the degrees of elements in a standard basis. In this case the algorithm is shown to have the same complexity as the Buchberger one.