The exponent field of a normal encoding can be neither all 0's nor all 1's, and for an explicit format, the integer bit must be 1:

*(a)
;*

*(b) If is explicit, then
.*

*(a) is explicit ;*

*(a)
;*

*(b)
.*

Let be a normal encoding for a format with and . The significand field is interpreted as a -exact value in the interval , i.e., with an implied a radix point following the leading bit, which is 1, either explicitly or implicitly. The value encoded by is the signed product of this value and a power of 2 determined by the exponent field. Since it is desirable for the range of exponents to be centered at 0, this field is interpreted with a bias of , i.e., the value of the exponent represented is

The following is trivial consequence of Definition 5.2.4.

*(a)
;*

*(b)
;*

*(c)
.*

*(a) ;*

*(b)
;*

*(c) is
-exact.*

The normal encoding of a representable value is derived as follows.

The next two lemmas establish an inverse relation between the encoding and decoding functions, from which it follows that the numbers that admit normal encodings are precisely those that satify Definition 5.2.5.

PROOF: Let , , , and . It is clear from Definition 5.2.4 that . By Lemma 5.2.1,

It also clear from Definition 5.2.4 that

Suppose is implicit. Then by Definitions 5.2.6 and 5.1.4 and Lemmas 2.4.9 and 2.2.5,

The explicit case is similar.

PROOF: We give the proof for the implicit case; the explicit case is similar.

Let , , , and . By Lemma 2.4.1, is a -bit vector and by Lemma 2.4.7,

Since is -exact,

We shall have occasion to refer to the smallest and largest positive numbers that admit normal representations.

If is a format, then

*(a)
;*

*(b)
is representable as a normal in ;*

*(c) is representable as a normal in , then
.*

PROOF: It is clear that is positive and satisfies Definition 5.2.5. Moreover, if and is representable as a normal in , then since , by Lemma 4.1.3.

If is a format, then

*(a)
;*

*(b)
is representable as a normal in ;*

*(c) If is representable as a normal in , then
.*

PROOF: It is clear that is positive and satisfies Definition 5.2.5. Let and . If is representable as a normal in , then by Definition 5.2.5, and is -exact. Thus, and by Lemma 4.2.21,

David Russinoff 2016-09-27