Round-off error

In <a href="/facts/Computing/Kghtm9fX">computing</a>, a roundoff error, also called rounding error, is the difference between the result produced by a given <a href="/facts/Algorithm/fnl5NmRt">algorithm</a> using exact <a href="/facts/Arithmetic/hvHwkbMV">arithmetic</a> and the result produced by the same algorithm using finite-precision, <a href="/facts/Rounding/rCLYXN00">rounded</a> arithmetic. Rounding errors are due to inexactness in the representation of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> and the arithmetic operations done with them. This is a form of <a href="/facts/Quantization_error/fddS3BE2">quantization error</a>. When using approximation <a href="/facts/Equation/pxFzLAVR">equations</a> or algorithms, especially when using finitely many digits to represent real numbers (which in theory have infinitely many digits), one of the goals of <a href="/facts/Numerical_analysis/QT9cYa9z">numerical analysis</a> is to <a href="/facts/Error_analysis_(mathematics)/Mu8VHCQJ">estimate</a> computation errors. Computation errors, also called <a href="/facts/Numerical_error/5x9tkosI">numerical errors</a>, include both <a href="/facts/Truncation_error/WE4rqBTG">truncation errors</a> and roundoff errors.
When a sequence of calculations with an input involving any roundoff error are made, errors may accumulate, sometimes dominating the calculation. In <a href="/facts/Ill-conditioned/9Tjtk6wh">ill-conditioned</a> problems, significant error may accumulate.
In short, there are two major facets of roundoff errors involved in numerical calculations:

<ol><li>The ability of computers to represent both magnitude and precision of numbers is inherently limited.</li>
<li>Certain numerical manipulations are highly sensitive to roundoff errors. This can result from both mathematical considerations as well as from the way in which computers perform arithmetic operations.</li></ol>

Round-off error open-in-new

Round-off error