Catastrophic cancellation

Summary

Catastrophic cancellation is the large loss of relative accuracy when one subtracts two nearly equal numbers that are only known approximately. The leading digits cancel, and what is left is dominated by the errors of the inputs. The subtraction itself is often exact in floating-point arithmetic; the damage comes from earlier round-off errors. It is the most common cause of numerically unstable formulas, and it is avoided by rewriting formulas, for example $\sqrt{x+1} - \sqrt{x} = 1/(\sqrt{x+1} + \sqrt{x})$.