3 ms·

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 comp

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."

tgb 6y ago
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.
zozbot234 6y ago
> "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.)