[dmlorenzetti]

The first thing you should worry about is whether your computation is ill-conditioned, and only then worry whether it's numerically stable.

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.

[conistonwater]

I don't know about your radiative exchange example, but it is often possible to reparametrize the problem. For example, sometimes you can replace the parameters (x,y) with (s,t)=(x-y, y), which would get you further: x^2-y^2 = s (s+2t), which is stable at s=0, and if x>y>0, then it's stable everywhere. This can shift the region of instability to somewhere far from typical parameter values. Otherwise, yes, this isn't solvable.