What is the worst case error for (a - b) + b?

Question

When evaluating with IEEE 754 floating point numbers a and b, what is the worst case error in terms of the magnitude of a and b of the sum (a - b) + b? How close to a can I expect that to be?

Answer 1

100%. b may be so large that ab produces -b , and then (ab)+b produces zero.

For example, with IEEE-754 basic 64-bit binary, (1−2 ⁵⁴ )+2 ⁵⁴ yields 0, with round-to-nearest-ties-to-even. We can also have 100% in the other direction. If a is 1 and b is 2 ⁵³ +2, then (ab)+b produces 2.

Also, if b is infinity, (ab)+b produces a NaN.