Reliable computing: numerical and rounding errors

The arithmetic performed in a machine involves numbers with only a finite number of digits, with the results that many calculations are performed with approximate representations of the actual numbers. In typical computer, only a relatively small subset of the real number system is used for the representation of all real numbers. This subset contains only rational numbers, both positive and negative, and stores a fractional part, called the mantisa, together with an exponential part, called the characteristic. Floating point systems are specified by four parameters, Fl(B,p,L,U) , and elements of these systems are specified by three parameters, (s,m,e): • s is the sign of a floating-point number, • m is its (unsigned) mantissa, and • e is its exponent.