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 perforned inK times the working precision, forK 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 theseK-times compensated algorithms are competitive forK up to 4, i.e., up to 212 mantissa bits.

中文翻译：

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用于准确、有效和快速多项式评估的算法

我们调查了一类算法来评估具有浮点系数的多项式以及使用 IEEE-754 浮点算法执行的计算。其原理是使用霍纳算法一次性或递归地应用多项式评估的无错误变换，并准确地对最终分解求和。这些补偿算法与在 K 倍工作精度中执行的 Horner 算法一样准确，对于 K 是任意正整数。我们通过先验误差分析证明了这种精度属性。我们还提供经过验证的动态边界，并应用这些结果来计算忠实的四舍五入评估。这些补偿算法很快。我们通过对重要环境的数值实验来说明它们的实际效率。