Page 79 - Engineering Digital Design

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

can be represented many different ways in floating-point notation:

33
Planck's constant, h = 0.6625 x 10~
32
= 0.06625 x 10~
31
= 0.006625 x 10" .

All three adhere to the form of Eq. (2.17) and are, therefore, legitimate floating-point
numbers in radix 10. Thus, as the radix point floats to the left, the exponent is scaled
accordingly. The first form for h is said to be normalized because the most significant digit
(MSD) of M is nonzero, a means of standardizing the radix point position. Notice that the
sign for M is positive while that for E is negative.
In computers the FPN is represented in binary where the normalized representation
requires that the MSB for M always be 1. Thus, the range in M in decimal is

0.5 < M < 1.

Also, the fraction (mantissa) M is represented in sign-magnitude from. The normalized
format for a 32-bit floating-point number in binary, which agrees with the IEEE standard
[3], is shown in Fig. 2.1. Here, the sign bit (1 if negative or 0 if positive) is placed at bit
position 0 to indicate the sign of the fraction. Notice that the radix point is assumed to lie
between bit positions 8 and 9 to separate the E bit-field from the M bit-field.
Before two FPNs can be added or subtracted in a computer, the E fields must be compared
and equalized, and the M fields adjusted. The decision-making process can be simplified
if all exponents are converted to positive numbers by using the excess representation given
by Eq. (2.16). For a #-digit number in radix r, the exponent in Eq. (2.17) becomes

(i
l
E xs = E r+r ~ , (2.18)
q
where E is the actual exponent augmented by a bias of B = r ^. The range in the actual
exponent E r is usually taken to be
q l
-(r ~ - 1) < E r < +(/•*-' - 1).

Sign Exponent E Fraction F
Bit (radix 2, bias 127) (Mantissa)
Assumed
position of
radix point
FIGURE 2.1
IEEE standard bit format for 32-bit normalized floating-point representation.