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James Demmel wrote:My point was that even a single hardware register is implicitly sorting, bucket sorting by exponents, because it needs to be able to add and possibly cancel operands with overlapping mantissas.Ulrich's point was that a hardware implemention simply uses a subset of the exponent bits to index into the correct bucket, so to speak. Carries may of course propagate into higher-exponent buckets, and in order to reduce carry propagation, one would use two accumuluators in practice, summing positive and negative terms separately, and doing one big subtraction at the end. In a parallell environment one might use several accumulator pairs.
Malcolm's approach{\sc M.~Malcolm}, {\em {On accurate floating-point summation}}, Comm. ACM, 14
(1971), pp.~731--736. reduces carries to almost not existent. Best wishes, Siegfried -- ===================================================== Prof. Dr. Siegfried M. Rump Institute for Reliable Computing Hamburg University of Technology Schwarzenbergstr. 95 21071 Hamburg Germany phone +49 40 42878 3027 fax +49 40 42878 2489 http://www.ti3.tu-harburg.de and Visiting Professor at Waseda University Faculty of Science and Engineering Shinjuku Lambdax Bldg. 902 2-4-12 Okubo, Shinjuku-ku Tokyo 169-0072 Japan phone/fax in Japan +81 3 5286 3414