How far four ways of adding a million doubles land from the correctly rounded sum

Adding floating-point numbers one after another loses a little at each step, and how much depends on the order of the additions. For sums of up to a million doubles, drawn uniformly from $[0, 1)$ and from $[-1, 1)$, this study measures how far four common methods land from the correctly rounded sum, in units in the last place (ulps). Adding left to right, the error averages 196.86 ulps for a million positive values and grows with their number as about with fitted = , close to the square root that independent rounding errors predict. Pairwise summation stays within ulps at every length. Kahan's and Neumaier's compensated sums return the correctly rounded sum in all draws, including mixed-sign sums that nearly cancel. Every value is reproducible exactly, on any machine and Python version.