Catastrophic cancellation

Remarks

Example: $x = 1.2341$, $y = 1.2259$ rounded to three significant digits give $\tilde x = \tilde y = 1.23$, so $\tilde x - \tilde y = 0$ while $x - y = 0.0082$; the relative error is $\abs{\delta} = 1$, although $\abs{\delta_1}, \abs{\delta_2} < 0.004$. Here the factor $(\abs{x} + \abs{y})/\abs{x - y}$ is about $300$. Non-example: for $x = 1.2341$ and $y = -1.2259$ the factor is $1$, so the relative error of $\tilde x - \tilde y$ is at most the larger input error; nothing cancels. The factor $(\abs{x} + \abs{y})/\abs{x - y}$ is the relative condition number of subtraction. By the Sterbenz lemma, if $x$ and $y$ are floating-point numbers with $y/2 \le x \le 2y$, then $x - y$ is a floating-point number, so the subtraction itself adds no error.