All computers designed from 1985 use so called IEEE floating point arithmetics which means that there is a machine independent standard of the of floating point number treatment. This means that the floating point numbers are expressed in the form, where is normalized integer mantisa represented by 52 bits and e is another integer within the interval related to the number bits reserved for exponents representation. It is the finiteness of exponent which limits the interval of real numbers that can be represented by floating point numbers. The smallest floating-point number is is the underflow limit and can be viewed as the computational threshold. The maximum floating point number, pointing to the overflow limit, is These two limits should be distinguished from another important quantity associated with representation of floating point numbers, namely a unity round-off error, also called machine epsilon, corresponding to the distance from 1.0 to the next larger floating point number. Its value is and it is closely associated with the build up of roundoff errors. The number of decimal digits corresponding to 52 binary digits is approximately 16. It can be determined from, which gives.
unit_roundoff = u, where 1 + u is different from 1 machine-epsilon = a – 1 ; where a is smallest representable number greater than 1 machine_epsilon = 2*u