In this paper we establish a forward error analysis of the generalized complete Horner scheme for a polynomial $$p = \sum {a_j X^{n - j} } $$ with pivotal pointsz 1, ...,z n . The error analysis is based on the linearization method whose fundamental tools are systems of linear error equations and associated condition numbers which yield optimal bounds of the possible errors under data perturbations and rounding errors in floating point arithmetic. For Horner's scheme the bounds may be calculated by simple recurrences. The ordinary complete Horner scheme is characterized byz=z 1=...=z n . In contrast to the hitherto known error estimates for this special case our new optimal bounds for the polynomialp atz differ from those for the polynomial $$p_a = \sum {\left| {a_j } \right|X^{n - j} } $$ at |z| and thus take into account the possible partial cancellation of terms. The error estimates are illustrated by a series of numerical examples.