TIL about floating point math:
float w=a*b;
float e=fma(-a,b,w);
w is the closest float possible to the real answer of a*b, but the difference between that and the actual answer can be significant.
e is that difference, and apparently the difference is exactly representable as a float.
Aug 14, 2026, 2:17 AM@demofox Difference will fail to be representable if underflow (or overflow) happens. But for reasonably well-scaled a and b, yes!
This algorithm is called "TwoProduct", and you can also do it pretty efficiently without FMA, using a technique that Dekker discovered surprisingly early.
Aug 14, 2026, 2:19 AM@demofox The difference due to rounding is also always representable for addition and subtraction in the default round-to-nearest mode, but not necessarily when using directed rounding. But multiplication works in every rounding mode.
@steve neat!
@steve thanks yeah, I found that paper, and a 2013 one evaluating kahan's formula for a 2d determinant ab-cd. Really neat stuff.