Accurate Parallel Floating-Point Accumulation

Abstract

Abstract-Using parallel associative reduction, iterative refinement, and conservative termination detection, we show how to use tree reduce parallelism to compute correctly rounded floating-point sums in O(log N ) depth at arbitrary throughput. Our parallel solution shows how we can continue to exploit Moore's Law scaling in transistor count to accelerate floatingpoint performance even when clock rates remain flat. Empirical evidence suggests our iterative algorithm only requires two tree reduce passes to converge to the accurate sum in virtually all cases. Furthermore, we develop the hardware implementation of a 250 MHz pipelined, native, residue-preserving IEEE-754 double-precision, floating-point adder on a Virtex 6 FPGA that requires only 48% more area than a standard adder without residue. Finally, we show how this module can be used as the base of a streaming accurate floating-point accumulation unit that can be tuned to consume m summands every cycle.

Parallel Algorithms for Summing Floating-Point Numbers

“…6 Indeed, their methods can involve as many as O(n) passes over the data. Kadric et al [22] provide a parallel pipelined method that takes a similar approach to the algorithm of Leuprecht and Oberaigner, while improving its convergence in practice, but their method nevertheless depends on inherently sequential pipelining and iterative refinement operations. Recently, Demmel and Nguyen [11] present a parallel floatingpoint summation method based on using a superaccumulator, but, like the previous sequential superaccumulator methods cited above, their method does not utilize a carry-free intermediate representation; hence, it has an inherently sequential carry-propagation step as a part of its "inner loop" computation.…”

How Can Applications of Blockchain and Artificial Intelligence Improve Performance of Internet of Things? -- A Survey

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Secure and Accurate Summation of Many Floating-Point Numbers

“…In addition to these solutions, there are a number of adaptive methods for exactly summing 𝑛 floating point numbers using various other data structures for representing intermediate results, which do not consider the security or privacy of the data. Further, these methods, which include ExBLAS [17] and algorithms by Zhu and Hayes [52,53], Demmel and Hida [21,22], Rump et al [46], Priest [43], Malcolm [39], Leuprecht and Oberaigner [38], Kadric et al [35], and Demmel and Nguyen [23], are not amenable to conversion to secure protocols with few rounds.…”