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  "body": "\n## Definition\n\nLet $x, y$ be real numbers with $x \\ne y$, and let $\\tilde x = x(1 + \\delta_1)$ and $\\tilde y = y(1 + \\delta_2)$ be approximations of them with relative errors $\\delta_1, \\delta_2$, for example [[round-off errors|CPN6SD3BTV]].\nThen $\\tilde x - \\tilde y = (x - y)(1 + \\delta)$ with\n$$\\delta = \\frac{x \\delta_1 - y \\delta_2}{x - y}, \\qquad \\abs{\\delta} \\le \\frac{\\abs{x} + \\abs{y}}{\\abs{x - y}} \\max(\\abs{\\delta_1}, \\abs{\\delta_2}).$$\n*Catastrophic cancellation* is the case $\\abs{x - y} \\ll \\abs{x} + \\abs{y}$.\nThen the relative error $\\delta$ of the computed difference can be much larger than the relative errors of the inputs.\n\n## Remarks\n\nExample: $x = 1.2341$, $y = 1.2259$ 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 $\\abs{\\delta_1}, \\abs{\\delta_2} \u003c 0.004$.\nHere the factor $(\\abs{x} + \\abs{y})/\\abs{x - y}$ is about $300$.\nNon-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.\nThe factor $(\\abs{x} + \\abs{y})/\\abs{x - y}$ is the relative [[condition number|AB3S6MHJ8X]] of subtraction.\nBy 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.\n",
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