Overview

We saw in the previous chapter that the finite IEEE floating point numbers can all be expressed in the form

where p is the precision of the floating point system with, for normalized numbers, b ₀ = 1 and E _min ? E ? E _max and, for subnormal numbers and zero, b ⁰ = 0 and E = E _min. We denoted the largest normalized number by N _max and the smallest positive normalized number by N _min. There are also two infinite floating point numbers, ?.

We now introduce a new definition. We say that a real number x is in the normalized range of the floating point system if

The numbers 0 and ? and the subnormal numbers are not in the normalized range of the floating point system, although they are all valid floating point numbers.

Suppose that a real number x is not a floating point number. Then at least one (and perhaps both) of the following must be true:

• x is outside the normalized range (its absolute value is greater than N _max or less than N _min). For example, the numbers 2 ¹³⁰ and 2 ^?130 are both outside the normalized range of the single format.

• The binary expansion of x requires more than p bits to specify the number exactly; equivalently, the floating point precision p is too...