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Consider two polynomials and with multiple precision floating point coefficients. Although the product can in principle be computed efficiently using FFT multiplication, this algorithm is numerically stable only if the coefficients of are of the same order of magnitude and similarly for . In this paper, we present a new asymptotically fast multiplication algorithm which has a “better” numerically stability. We also provide some theoretical support for what we feel to be “better”, by introducing the concept of “relative Newton error”.

**Keywords: **polynomial, floating point number,
multiplication, algorithm, FFT

**A.M.S. subject classification: **65T50, 42-04, 68W30

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**Errata.** Unfortunately, an annoying error slipped into
the preprint: the bound (3) is wrong. Consequently, the complexity
estimates in the paper only work in one direction. Nevertheless, we have
implemented a first version of the algorithm in