Underflow

Occurs when the result of an arithmetic operation is smaller than the smallest normalized FP number which can be stored. In IEEE standard the result is a subnormal number ("gradual" underflow) or 0, if the result is small enough. Note that subnormal numbers have fewer bits of precision than normalized ones, so using them may lead to a loss of accuracy. For example, let

and compute . The mathematical result is . According to our roundoff error analysis, we expect the calculated to satisfy

where the bound on delta comes from the fact that we have two floating point multiplications, and (with exact rounding) each of them can introduce a roundoff error as large as the half the machine precision :

Since in our toy system , we expect the computed result to be in the range

However, the product underflows, and has to be represented by the subnormal number ; when multiplied by this gives , which means that the relative error is almost times larger than expected