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The goal of this paper is to analyze two polynomial evaluation schemes for mul-tiple precision floating point arithmetic. Polynomials are used extensively in nu-merical computations (Taylor series for mathematical functions, root finding) but a rigorous bound of the error on the final result is seldom provided. We provide such an estimate for the two schemes and find how to reduce the number of operations required at run-time by a dynamic error analysis. This work is useful for floating point polynomial arithmetic. Key words: polynomial evaluation, bounded error 1 Goal and motivations The goal is to compare two polynomial evaluation schemes. We want to com-pute the sum: P (x) = l∑ i=0 aix i and provide an error bound on the final result with respect to the number of summands l + 1 and the precision F used for the intermediate results (the ”internal ” precision)