# 1.3  Chop

The following function, which relates to the fixed-point registers defined in Part V, truncates a real number to a specifed numebr of fractional bits:

Definition 1.3.1   (chop) For and ,

Notation: We shall abbreviate as .

PROOF: This is an instance of Definition 1.2.1

PROOF:

and

PROOF:

PROOF:

PROOF: Since , , which implies

and

PROOF:

Notation: We shall abbreviate as .

If , then is the largest multiple of that does not exceed . The following lemmas have found application in the analysis of floating-point adders.

PROOF:

(a) By Lemma 1.1.5,

(b) By Lemmas 1.2.1, 1.2.3, and 1.1.5,

(chop-int-neg) Let , , , and . . If and , then

PROOF: We simplify the expression on the left using Lemmas 1.1.5 and 1.1.8:

For the expression on the right, we apply the same two lemmas and Lemma 1.1.7:

David Russinoff 2017-08-01