Balanced pairwise summation with range-failure status

Input real floating-point values $x_1,\dots ,x_n$, $n\ge 1$. Set $d=\lceil {\log }_{2}n\rceil$, place these values in leaves $1,\dots ,n$ from left to right, and append $2^d-n$ exact zero leaves. At level $\mathrm{\ell }=1,\dots ,d$, replace each consecutive pair of current nodes $(a_{\mathrm{\ell }-1,2j-1},a_{\mathrm{\ell }-1,2j})$, in increasing $j=1,\dots ,2^{d-\mathrm{\ell }}$, by $a_{\mathrm{\ell },j}=\mathrm{fl}(a_{\mathrm{\ell }-1,2j-1}+a_{\mathrm{\ell }-1,2j})\mathrm{.}$ If an operation fails, return range-failure with the first failing level $\mathrm{\ell }$ and node $j$ in this level-by-level, left-to-right order. For $d=0$, return $x_1$ with no additions. Otherwise return the single node $a_{d,1}$.

For termination and the distinction between the successful full-tree addition count and a failing initial segment, apply Finite arithmetic schedules terminate, with early failure counted separately to the prescribed levels. The operationwise model supplies each addition's finite value-or-failure return. Accuracy bounds require successful arithmetic.