Chapter 1. Introduction
12
1.5.2
The exponent
The next 11 bits are used to represent the exponent. It is crucial to mention here that
the actual value of the exponent is not the value that is stored. For E the value stored
the true value of the exponent is given by equation
, E
bias
is the number 1023. In
order to represent negative as well as positive numbers the number 1023 is subtracted
from the number to get the real value. With 11 bits the largest number that can be
represented is 2047 and the lowest 0, after the bias subtraction we get 1024 as the highest
value and -1023 as the lowest that can be represented.
e = E − E
bias
(1.2)
1.5.3
The significand
The significand of the floating point number is represented by a string of 52 bits. However
the actual number is not just these 52 bits. The fraction f stored in these 52 bits is in
the range [0-1). The significand S is calculated with the equation
. A more detailed
explanation of the leading ’1’ digit follows in the next section.
S = 1.f
(1.3)
1.5.4
Floating point normalization
As mentioned earlier the actual bits of the significand part of the number are 53 and
not 52. This happens because the MSB is always assumed to be one and therefore
does not need to be stored which leads to storing area reduction. The final result of the
calculation of any two floating point numbers, must have the MSB of the significand part
equal to 1. However after a calculation it is often that the MSB will not be equal to 1, in
which case the number must be normalised so as to change the MSB to 1. If the number
is not normalized there are two possibilities, one the MSB is zero and second an overflow
has occurred and a extra bit has been concatenated to the left of the significand. If the
second has occurred no actual normalization has to take place regarding the significand
part of the number however the exponent should be right shifted by one(A more detailed