Assume that a set of real numbers x belonging to the interval
is represented in such a way that the following specifications are satisfied:
d1 is the maximum distance between small exactly-represented non zero numbers;
d2 is the maximum distance between large exactly-represented numbers;
xmin is the maximum distance between 0 and the smallest exactly-represented numbers:
where the adjectives small and large refer to the absolute value of the corresponding numbers.
Every number x will be represented in the form ±s.be, with b ≥ 2, s being the significand and e the exponent.
In order to make the implementation of the arithmetic operations easier (Section 16.2), the two following conditions must be satisfied:
Thus x is expressed in the form
The values of p, emin, and emax are chosen in such a way that
Example 16.1 Define a floating-point representation system where
Choose B = 2. A straightforward solution of the system (16.2)–(16.5) is
The smallest nonzero exactly-represented positive number is 2−30; the distance between small exactly-represented numbers is
the largest exactly-represented positive number is
the distance between large exactly-represented numbers is