Accurate Floating - Point Summation ∗

Given a vector of floating-point numbers with exact sum s, we present an algorithm for calculating a faithful rounding of s into the set of floating-point numbers, i.e. one of the immediate floating-point neighbors of s. If the s is a floating-point number, we prove that this is the result of our algorithm. The algorithm adapts to the condition number of the sum, i.e. it is very fast for mildly conditioned sums with slowly increasing computing time proportional to the condition number. All statements are also true in the presence of underflow. Furthermore algorithms with K-fold accuracy are derived, where in that case the result is stored in a vector of K floating-point numbers. We also present an algorithm for rounding the sum s to the nearest floating-point number. Our algorithms are fast in terms of measured computing time because they neither require special operations such as access to mantissa or exponent, they contain no branch in the inner loop, nor do they require extra precision: The only operations used are standard floating-point addition, subtraction and multiplication in one working precision, for example double precision. Moreover, in contrast to other approaches, the algorithms are ideally suited for parallelization. We also sketch dot product algorithms with similar properties.