Improved error bounds for floating-point products and Horner’s scheme

Let $$\mathbf{u}$$ denote the relative rounding error of some floating-point format. Recently it has been shown that for a number of standard Wilkinson-type bounds the typical factors $$\gamma _k:=k\mathbf{u}/(1-k\mathbf{u})$$ can be improved into $$k\mathbf{u}$$ , and that the bounds are valid without restriction on $$k$$ . Problems include summation, dot products and thus matrix multiplication, residual bounds for $$LU$$ - and Cholesky-decomposition, and triangular system solving by substitution. In this note we show a similar result for the product $$\prod _{i=0}^k{x_i}$$ of real and/or floating-point numbers $$x_i$$ , for computation in any order, and for any base $$\beta \geqslant 2$$ . The derived error bounds are valid under a mandatory restriction of $$k$$ . Moreover, we prove a similar bound for Horner’s polynomial evaluation scheme.