We survey a class of algorithms to evaluate polynomials with floating point coefficients and for computation performed with IEEE-754 floating point arithmetic. The principle is to apply, once or recursively, an error-free transformation of the polynomial evaluation with the Horner algorithm and to accurately sum the final decomposition. These compensated algorithms are as accurate as the Horner algorithm performed in K times the working precision, for K an arbitrary positive integer. We prove this accuracy property with an a priori error analysis. We also provide validated dynamic bounds and apply these results to compute a faithfully rounded evaluation. These compensated algorithms are fast. We illustrate their practical efficiency with numerical experiments on significant environments. Comparing to existing alternatives these K-times compensated algorithms are competitive for K up to 4, i.e., up to 212 mantissa bits.
Abstract. Numerical reproducibility failures appear in massively parallel floating-point computations. One way to guarantee the numerical reproducibility is to extend the IEEE-754 correct rounding to larger computing sequences, as for instance for the BLAS libraries. Is the overcost for numerical reproducibility acceptable in practice? We present solutions and experiments for the level 1 BLAS and we conclude about the efficiency of these reproducible routines.
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