OK, but how does one apply this advice in practice?
The radiative exchange between two surfaces goes as x^4-y^4 where x and y are the surface temperatures. I can worry about whether that's ill-conditioned all I want, but at the end of the day I'm further ahead evaluating (x-y)(x+y)(x^2+y^2) than the original expression.
Similarly, if I need to find x^2-1, what recourse do I have? Maybe there's some reduction that makes the computation more stable. But normally I would expect to discover such a reduction by turning x^2-1 into (x-1)(x+1), and then seeing if one of those factors cancels out somewhere. So it still seems like a good first step.
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.
Another way to do the same type of error analysis of numerical algorithms is to use a parameter sensitivity library. In the end, numerical stability can be thought of as the sensitivity of the final answer to separate independent changes in each intermediate value obtained in the computation of the answer. This is sometimes overlooked, I think, and it's quite easy to implement as well.
I have one quibble. They say "Use (x-y)(x+y) instead of x^2-y^2". This is true, but (in my opinion) misleading: the function (x,y) -> x^2-y^2 is itself ill-conditioned when x is near y, and using a more numerically stable algorithm to evaluate an ill-conditioned function will not improve the function's condition. If x and y are close to each other, the inaccuracy introduced by using x^2-y^2 is close to the inaccuracy due to using approximate values for x and y. So it you have x^2-y^2, I think the first thing you should worry about is whether your computation is ill-conditioned, and only then worry whether it's numerically stable. I say this because I sometimes see people try to evaluate x^2-1, notice it is inaccurate for x near 1, and try to replace x^2-1 with (x-1)*(x+1), which is still inaccurate.