[stephencanon]

The correct way to evaluate x^2 - 1 is by using fma(x,x,-1). Now that Intel and AMD have finally made FMA available in hardware on commodity parts (better late than never!), it's realistic to start using this much more freely.

[conistonwater]

That won't do much for x near 1, because the function x^2-1 is itself ill-conditioned. In other words, it is relevant that the floating-point value of x is itself only an approximation to some true value of x. So computing x^2-1 exactly for a given floating-point value of x does not give a good approximation to the true value of x^2-1. This is a mathematical property of the function x^2-1, and cannot be fixed with any algorithm. This is basically why I consider the example x^2-1 => (x-y)(x+y) misleading.

[stephencanon]

There are multiple ways to analyze computations; condition number is one of them. It is relevant when inputs are assumed to be approximations to some hidden "correct" value. However, this assumption is not always warranted when dealing with floating-point numbers; sometimes, we would instead like to analyze errors under the assumption that the inputs are exact values. When this is the case, fma(x,x,-1) produces a correctly rounded result, whereas (x-1)(x+1) does not, and the ill-conditioning of the function x^2 - 1 does not enter into the analysis anywhere.

If you are used to working with complex computations (most of what is usually referred to as numerical analysis, for example), you likely do not find yourself in the latter situation very often. However, for those of us who work on low-level details of floating-point on a regular basis, the latter analysis is often critical.

[conistonwater]

Okay, I agree, that is completely reasonable. I do usually work with fairly complex algorithms, so I care about individual operations less than about the final result.

[acqq]

The expression won't turn to be good conditioned only by using fma. You actually missed the point of conistonwater's post.

[stephencanon]

Condition number only matters if you assume that the inputs are perturbed by some error. Under the assumption that inputs are exact (which is appropriate to make for some uses), it does not enter into the analysis.