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In computing, a roundoff error, [1] also called rounding error, [2] is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. [3]
Precision is often the source of rounding errors in computation. The number of bits used to store a number will often cause some loss of accuracy. An example would be to store "sin(0.1)" in IEEE single precision floating point standard.
Accuracy is also used as a statistical measure of how well a binary classification test correctly identifies or excludes a condition. That is, the accuracy is the proportion of correct predictions (both true positives and true negatives) among the total number of cases examined. [10]
This type of rounding, which is also named rounding to a logarithmic scale, is a variant of rounding to a specified power. Rounding on a logarithmic scale is accomplished by taking the log of the amount and doing normal rounding to the nearest value on the log scale. For example, resistors are supplied with preferred numbers on a logarithmic scale.
This alternative definition is significantly more widespread: machine epsilon is the difference between 1 and the next larger floating point number.This definition is used in language constants in Ada, C, C++, Fortran, MATLAB, Mathematica, Octave, Pascal, Python and Rust etc., and defined in textbooks like «Numerical Recipes» by Press et al.
If the correction is also computed to ±1/2000 bit of accuracy (which does not require extra floating-point precision as long as the correction is less than 1/2000 the magnitude of the stored value f(x), and the computed correction is more than ±1/1000 of a bit away from exactly half a bit (the difficult rounding case), then it is known ...
Subtracting nearby numbers in floating-point arithmetic does not always cause catastrophic cancellation, or even any error—by the Sterbenz lemma, if the numbers are close enough the floating-point difference is exact. But cancellation may amplify errors in the inputs that arose from rounding in other floating-point arithmetic.
If this cannot be eliminated, potentially by resetting the instrument immediately before the experiment then it needs to be allowed by subtracting its (possibly time-varying) value from the readings, and by taking it into account while assessing the accuracy of the measurement.