Floating-point arithmetic is based on the observation that every nonzero real number admits a representation of the form

The decomposition property is immediate.

The definition of may be restated as follows:

PROOF:

(a) If , then , which implies

(b) If , then , which implies

PROOF: since , the lemma follows from Lemma 4.1.3(b).

The width of a bit vector is determined by its exponent.

PROOF: This is just a restatement of the second inequality of Definition 4.1.1.

PROOF: This is an instance of Lemma 1.2.16.

:

We have the following bounds on .

PROOF: Definition 4.1.1 yields

PROOF: Since , , and hence .

PROOF: Since , , where . It follows from Definition 4.1.1 that , and therefore .

Changing the sign of a number does not affect its exponent or significand.

A shift does not affect the sign or significand.

PROOF:

(a) .

(b) .

(c) .

We have the following formulas for the components of a product.

PROOF:

(a) .

(b) Since
and
,
we have

If , then

On the other hand, if , then similarly,

(c) If
, then

Otherwise,

David Russinoff 2017-08-01