We present a compensated algorithm to evaluate the first derivative of B'ezier tensor product surface in floating arithmetic. The algorithm based on error-free transformation is fast in terms of measured computing time, and the computed results are as accurate as the classic de Casteljau tensor product algorithm in twice the working precision. We also propose the relative forward error bound of this compensated algorithm. Our numerical experiments illustrate these results. The algorithm is performed at a given working double precision and portable for any arithmetic obey to the IEEE-754 standard.
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