The Fundamental Axiom of Floating Point Arithmetic

Re:The Fundamental Axiom of Floating Point Arithmetic by vortico on Wednesday August 05, 2020 @04:06AM (#24058357)

I think you're misinterpretating the fundamental axiom because it's still true for `x-y`. What you're thinking is that the fundamental axiom applies to the composition of operations `closest_float(x) + closest_float(y)`, but the axiom only applies to each individual operation.

The axiom can be stated more rigorously "If f is a (possibly multivariate) function on the set of floats to the reals, the IEEE-compliant approximation of f must give the closest float for all float values of its arguments."

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Re:The Fundamental Axiom of Floating Point Arithmetic by zozbot234 on Wednesday August 05, 2020 @08:15AM (#24059952)

> "If f is a (possibly multivariate) function on the set of floats to the reals, the IEEE-compliant approximation of f must give the closest float for all float values of its arguments."

It should be noted that this is not exactly true for some IEEE functions, due to the table-maker's dilemma: computing the "closest float approximation" to an arbitrary function can be computationally infeasible if the true result is close to being halfway in-between two consecutive floats. (This applies to the default rounding mode, but can be extended by analogy to other rounding modes.)

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Re:The Fundamental Axiom of Floating Point Arithmetic by tgb on Wednesday August 05, 2020 @07:27AM (#24059558)

They're correct that the "In mathematical terms" version of it is wrong (even for addition, if one is negative). It's missing fl()'s around the other a's and b's. Your statement is greatly superior than the poor man's version presented in the article.

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