Let $u$ 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 $\gamma_k:=k u / (1-k u)$ can be improved into $k u$, 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 $\beta \ge 2$. The derived error bounds are valid under a mandatory restriction of $k$. Moreover, we prove a similar bound for Horner's polynomial evaluation scheme.