* Mike Stay <metaweta@gmail.com> [Apr 15. 2015 17:38]:
Addition isn't even associative in floating point: http://en.wikipedia.org/wiki/Associative_property#Nonassociativity_of_floati... so floating point numbers don't even form a monoid under addition. Is addition commutative?
I don't think so: Take one value L so large that adding 1.0 does not change it at all. Also take very many 1.0's (indeed take L of them). Now 1.0 + 1.0 + 1.0 + ... + 1.0 + L ==> 2.0 * L but L + 1.0 + 1.0 + 1.0 + ... + 1.0 ==> L
Is there some concise characterization of what kind of structure they do form, like "Addition and multiplication are magmas with an absorbing element NaN such that ..."?
That's gonna be hard to answer (even when ignoring quiet vs. signaling NaNs, negative zero, and denormalized underflow). Suggest to check out (no idea how helpful as I never asked myself the question above): David Goldberg: What Every Computer Scientist Should Know about Floating-Point Arithmetic, ACM Computing Surveys, vol.23, no.1, pp.5-48, \bdate{March-1991} http://www.validlab.com/goldberg/paper.ps <--= edited reprint Jean-Michel Muller, Nicolas Brisebarre, Florent de Dinechin, Claude-Pierre Jeannerod, Vincent Lef\`{e}vre, Guillaume Melquiond, Nathalie Revol, Damien Stehl\'{e}, Serge Torres: Handbook of Floating-Point Arithmetic, Birkh\"{a}user, \bdate{2010} If I'd needed to know in a hurry, I'd ask Paul Zimmermann. If you find out, kindly tell us. Best regards, jj
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com
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