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# Denormal Representations

Of the two values of the exponent field that lie outside of the range of normal encodings, the upper extreme is reserved for the encoding of infinities and other non-numerical entities, which will not be discussed here, while an exponent field of 0 is used to encode numerical values that lie below the normal range.

If the exponent and significand fields of an encoding with respect to a format with implicit MSB are both 0, with exponent field 0, then the encoded value itself is 0. If the exponent field is 0 and the significand field is not, then if the significand field is also 0, then the encoded value itself the encoding is said to be denormal.

Definition 5.3.1   (dencodingp) Let and with and , and let be a -bit vector. Then

and

Theere are two differences between the decoding formulas for denormal and normal representations:

1. For a denormal encoding, the implicit MSB is taken to be 0 rather than 1, so that the value represented by the significand field is .

2. The power of 2 represented by the zero exponent field of a denormal encoding, which might be expected to be , is instead the same as the minimum value of the normal range, .

Definition 5.3.2   (ddecode) Let , , and with and . If , then

(sgn-ddecode, expo-ddecode, sig-ddecode) Let , , and with and . Assume that and let .

 (a) (b) (c) .

PROOF: (a) is trivial; (b) and (c) follow from Lemmas 4.1.11 and 4.1.12

The class of reals that are representable as denormal encodings is recognized by the following predicate.

Definition 5.3.3   (drepp) Let , , and with and . Then if and only if all of the following are true:

 (a) , (b) , and (c) is -exact.

If a number is so representable, then its encoding is constructed as follows.

Definition 5.3.4   (dencode) Let , , and with and . If , then

where

Next, we examine the relationship between the decoding and encoding functions.

(drepp-ddecode, dencode-ddecode) Let , , and with and . If , then

and

PROOF: Let . Since ,

and by Lemma 4.1.2,

which is equivalent to Definition 5.3.3(b). In order to prove (c), we must show, according to Definition 4.2.1, that

But

This establishes .

Now by Definition 5.3.2,

Therefore, by Definitions 5.3.1, 5.3.4, and 5.2.1 and Lemmas 2.4.9 and 2.2.5,

(dencodingp-dencode, ddecode-dencode) Let , , and with and . If , then

and

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

and

Since is -exact,

and since ,

by Lemma 4.1.7, which implies

Finally, according to Definition 5.3.2,

The smallest positive denormal is computed by the following function:

Definition 5.3.5   (spd) For all and , .

(positive-spd, drepp-spd, spd-smallest)
Let and with and .

 (a) (b) (c) If , , and , then .

PROOF: It is clear that is positive. To show that is -exact, we need only observe that

Finally, since

holds and moreover, is the smallest positive that satisfies

Every number with a denormal representation is a multiple of the smallest positive denormal.

(spd-mult) Let , , and with and . Then if and only if for some , .

PROOF: For , let . Then and

We shall show, by induction on , that for . First note that for all such ,

Suppose that for some , . Then is -exact, and by Lemma 4.2.16, so is . But since , it follows from Lemma 4.2.5 that is also -exact. Since

, i.e., , and hence, .

Now suppose that and . Let . Clearly, , and . It follows from Lemma 4.2.17 that , and consequently, is -exact. Thus, by Lemma 4.2.16,

Next: Rebiasing Exponents Up: Floating-Point Formats Previous: Representations with Implicit Leading   Contents
david.m.russinoff 2015-02-08