Estimate maximum relative error of concatenated floating-point operations

Question

According to this very elaborate answer I would estimate the maximum relative error δ_res,max of the following computation like this:

// Pseudo code    
float a, b, c; // Prefilled IEEE 754 floats with double precision    
res = a / b * c;

res = a * (1 + δ_a) / ( b * (1 + δ_b) ) * (1 + δ_a/b) * c * (1 + δ_c) * (1 + δ_a/b*c)

= a / b * c * (1 + δ_a) / (1 + δ_b) * (1 + δ_a/b) * (1 + δ_c) * (1 + δ_a/b*c)

= a / b * c * (1 + δ_res)

=> δ_res = (1 + δ_a) / (1 + δ_b) * (1 + δ_a/b) * (1 + δ_c) * (1 + δ_a/b*c) - 1

All δs are within the bounds of ± ε / 2, where ε is 2^-52.

=> δ_res,max = (1 + ε / 2)^4 / (1 - ε / 2) - 1 ≈ 2.5 * ε

Is this a valid approach for error estimation that can be used for every combination of basic floating-point operations?

PS:

Yes, I read "What Every Computer Scientist Should Know About Floating-Point Arithmetic". ;)

Answer 1

Well, it's probably a valid approach. I'm not sure how you've jockeyed that last line, but your conclusion is basically correct (though note that, since the theoretical error can exceed 2.5e, in practice the error bound is 3e).

And yes, this is a valid approach which will work for any floating-point expression of this form. However, the results won't always be as clean. Once you have addition/subtraction in the mix, rather than just multiplication and division, you won't usually be able to cleanly separate the exact expression from an error multiplier. Instead, you'll see input terms and error terms getting multiplied directly together, rather than the pleasantly relatively constant bound here.

As a useful example, try deriving the maximum relative error for (a+b)-a (assuming a and b are exact).