Page 89 - Engineering Digital Design

60 CHAPTER 2 / NUMBER SYSTEMS, BINARY ARITHMETIC, AND CODES

the divisor must be represented in the decimal range 0.5 < operand < 1. If both operands
are so represented, the direct quadratic convergence method is ideally suited for use with
the mantissas in normalized FPN notation described in Section 2.8.
For this approach the quotient is given by

_ DIVIDEND D D
DIVISOR D s
D D'kQ'k l'k 2-" Q
(2.26)
D s-ko-ki'k 2"- ^ 1
The process begins with D s = 1 — a, where a = 1 — D s. But since D s < 1, it follows that
a < 1 . Next choose

fco = 1 + a

giving the updated product

which is closer to 1 than DS- Now set

giving the updated product

4
2
2
D s • k 0 • ki = (1 - <x )(l + a ) = 1 - a .
Continuing, set

so that the updated product becomes

4
8
4
D s • k 0 • ki • k 2 = (1 - c* )(l + a ) = 1 - or , etc.
This process continues until the desired number of iterations has been reached or until the
updated product D s • k 0 • k\ • k 2 • • • = 1.
Notice that each kj is 1 plus the radix complement of the product of D s and all the k
factors to that point. This can be generalized mathematically as

(2-27)
D s
rC