  We analyse the forward error in the floating point summation of real numbers,
from algorithms that do not require recourse to higher precision or better
hardware. We derive informative explicit expressions, and new deterministic and
probabilistic bounds for errors in three classes of algorithms: general
summation,shifted general summation, and compensated (sequential) summation.
Our probabilistic bounds for general and shifted general summation hold to all
orders. For compensated summation, we also present deterministic and
probabilistic first and second order bounds, with a first order bound that
differs from existing ones. Numerical experiments illustrate that the bounds
are informative and that among the three algorithm classes, compensated
summation is generally the most accurate method.
