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Mathematics-Online lexicon:

Loss of significant digits

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The absolute relative error of a floating point addition can be estimated by

$\displaystyle \frac{\vert\text{R}(\text{R}x+\text{R}y)-(x+y)\vert}{\vert x+y\ve... ...}+\frac{\vert x\vert+\vert y\vert}{\vert x+y\vert}(\text{eps}+\text{eps} ^2). $

For summands of the same sign, the right-hand side is $ \leq 2$eps$ +O($eps$ ^2)$. On the other hand, if $ y\approx -x$, the quotient $ (\vert x\vert+\vert y\vert)/\vert x+y\vert$ becomes very large. If the first $ s$ digits of $ x$ and $ y$ with respect to the base $ \beta$ coincide, the relative error can be

$\displaystyle > 2\beta^{s-1}$eps$\displaystyle . $

The leading digits cancel when forming the difference, potentially causing a large error, since the difference is represented by fewer accurate digits.

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  automatically generated 7/25/2011