We analyze the forward error in the floating point summation of real numbers, for computations in low precision or extreme-scale problem dimensions that push the limits of the precision. We present a systematic recurrence for a martingale on a computational tree, which leads to explicit and interpretable bounds with nonlinear terms controlled explicitly rather than by big-O terms. Two probability parameters strengthen the precision-awareness of our bounds: one parameter controls the first order terms in the summation error, while the second one is designed for controlling higher order terms in low precision or extreme-scale problem dimensions. Our systematic approach yields new deterministic and probabilistic error bounds for three classes of mono-precision algorithms: general summation, shifted general summation, and compensated (sequential) summation. Extension of our systematic error analysis to mixed-precision summation algorithms that allow any number of precisions yields the first probabilistic bounds for the mixed-precision FABsum algorithm. Numerical experiments illustrate that the probabilistic bounds are accurate, and that among the three classes of mono-precision algorithms, compensated summation is generally the most accurate. As for mixed precision algorithms, our recommendation is to minimize the magnitude of intermediate partial sums relative to the precision in which they are computed.

我们分析实数浮点求和中的前向误差，以进行低精度或极端规模问题维度的计算，从而突破精度的极限。我们提出了计算树上鞅的系统递归，这导致了显式且可解释的边界，非线性项由显式控制，而不是由大 O 项控制。两个概率参数增强了边界的精度意识：一个参数控制求和误差中的一阶项，而第二个参数旨在控制低精度或极端规模问题维度中的高阶项。我们的系统方法为三类单精度算法产生新的确定性和概率误差界限：一般求和、移位一般求和、和补偿（顺序）求和。将我们的系统误差分析扩展到允许任意数量精度的混合精度求和算法，产生混合精度的第一个概率界限FABsum算法。数值实验表明，概率界限是准确的，并且在三类单精度算法中，补偿求和通常是最准确的。对于混合精度算法，我们的建议是最小化中间部分和相对于计算精度的大小。