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:
(a) If , then ;
(b) If , then .
(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.
We have the following formulas for the components of a product.
On the other hand, if , then similarly,
David Russinoff 2017-08-01