high performance computing on graphics processing units: hgpu.org

On modern multi-core, many-core, and heterogeneous architectures, floating-point computations, especially reductions, may become non-deterministic and thus non-reproducible mainly due to non-associativity of floating-point operations. We introduce a solution to compute deterministic sums of floating-point numbers efficiently and with the best possible accuracy. Our multi-level algorithm consists of two main stages: a filtering stage that uses fast vectorized floating-point expansions; an accumulation stage based on superaccumulators in a high-radix carry-save representation. We present implementations on recent Intel desktop and server processors, on Intel Xeon Phi accelerator, and on AMD and NVIDIA GPUs. We show that the numerical reproducibility and bit-perfect accuracy can be achieved at no additional cost for large sums that have dynamic ranges of up to 90 orders of magnitude by leveraging arithmetic units that are left underused by standard reduction algorithms.