# The Machine Epsilon

Robert M. Corless

Tue Nov 28 18:19:22 PST 1995

The definition of the `machine epsilon' u given in the paper `Chaos and Continued Fractions' is the standard one used in numerical analysis. However, it is a bit tricky to understand for people not used to floating-point arithmetic, and is especially tricky in Maple because the value of u depends on the setting of Digits and also on whether or not hardware floating point (`evalhf`) is being used.

The definition is `the smallest number u which when added to 1 gives something different than 1'. Of course, in the real number system, this could only be u = 0, and indeed the machine epsilon acts something like zero in some cases in floating point systems. But in floating point, u is not zero, because of rounding. Moreover, u is not the smallest floating point number we can represent, as we will see.

Let us suppose we are working in Maple to 6 Digits. Then if we add `5.0*10^(-Digits)` to 1, we get as our result. Maple internally computed and then rounded to 6 Digits, to get that answer. If now we add `4.99999*10^(-Digits)` to 1, we get , which really is 1. Maple internally computed the result to a higher precision, and then (correctly) rounded the result. So here, u is . Things get more complicated when `evalhf` is involved.

But of course we can represent smaller numbers: , for example, is a perfectly representable number in Maple to 6 Digits. The difference is that the exponent (here 200) is independent of the number of digits in the `significand' (older works called this the `mantissa' but the current, more rational, practice is to use the word `significand').

The importance of the machine epsilon is that it measures the effects of rounding errors made when adding, subtracting, multiplying, or dividing two numbers. No matter how carefully you do any of these operations, you must make an error at least this big (in a relative sense) when you round your answer to the number of Digits you are using.