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There is a need to discuss about the evaluation of the polynomial with the floating-point coefficients and the computing which is performed through IEE 754 floating points. Here, the principles are based on handling the recursive actions with the action free transformation for the polynomial evaluation. This is done through the use of Horner algorithm which is considered to be the sum for the final decomposition. The compensated algorithm includes the Horner algorithm where the performance needs to be mapped with working on the precisions of the K which is the arbitrary positive integer. The accuracy property is mapped with the priori error analysis and then providing the validation of the dynamic bounds with applying the results to compute the rounded evaluation. Here, the compensated algorithm is fast, and the practical efficiency is mapped with the numerical experiences for the significant standards

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