Computer Arithmetic - Fundamentals of Computer Organization and Architecture - page 75

Figure 4.11 Representation of a floating-point number

normally represented as a signed binary fraction, e represents the exponent, and b
represents the base (radix) of the exponent.

Example Figure 4.11 is a representation of a floating-point number having

m ¼ 1 bit. If the value stored in S is 0, the

number is positive and if the value stored in S is 1, the number is negative.
The exponent in the above example, can only represent positive numbers 0
through 255. To represent both positive and negative exponents, a fixed value,
called a bias, is subtracted from the exponent field to obtain the true exponent.
Assume that in the above example a bias

23 bits, e ¼ 8 bits, and S (sign bit)

¼

¼ 128 is used, then true exponents in

þ

the range 127 (stored as 255 in the

exponent field) can be represented. Based on this representation, the exponent

2 128 (stored as 0 in the exponent field) to

þ 4

can be represented by storing 132 in the exponent field, while the exponent 12

2

can be represented by storing 116 in the exponent field.
Assuming that b ¼ 2, then an FP number such as 1.75 can be represented in any

of the forms shown in Figure 4.12.
To simplify performing operations on FP numbers and to increase their precision,
they are always represented in what is called normalized forms. An FP number is
said to be normalized if the leftmost bit of the mantissa is 1. Therefore, among
the three above possible representations for 1.75, the first representation is normal-
ized and should be used.
Since the most significant bit (MSB) in a normalized FP number is always 1, then
this bit is often not stored and is assumed to be a hidden bit to the left of the radix
point, that is, the stored mantissa is 1.m. Therefore, a nonzero normalized number
represents the value ( 1) s * (1 : m) * 2 e 128 .

Floating-Point Arithmetic Addition / Subtraction The difficulty in adding

two FP numbers stems from the fact that they may have different exponents.
Therefore, before adding two FP numbers, their exponents must be equalized, that
is, the mantissa of the number that has smaller magnitude of exponent must be
aligned.

0.111*2 1 +

0 10000001 11100000000000000000000

+0.0111*2 2 0 10000010 01110000000000000000000

+0.00000000000000000000111*2 21

0 10010101 00000000000000000000111

Figure 4.12 Different representation of an FP number