FLOATING POINT OPERATIONS
The scientific notation has a single digit to the left of the decimal point. A number in scientific notation that has no leading 0s is called a normalized number, which is the usual way to write it. Foating point - Computer arithmetic that represents numbers in which the binary point is not fixed. Floating-point numbers are usually a multiple of the size of a word.
The representation of a MIPS
floating-point number is shown below, where s is the sign of the floating-point
number (1 meaning negative), exponent is the value of the 8-bit exponent field
(including the sign of the exponent), and fraction is the 23-bit number. This
representation is called sign and magnitude, since the sign has a separate bit
from the rest of the number.
A standard scientific notation for
reals in normalized form offers three advantages. · It simplifies exchange of data that includes
floating-point
numbers; It simplifies the floating-point
arithmetic algorithms to know that numbers will always be in this form;
·
It increases the accuracy of the numbers that can
be stored in a word, since the unnecessary leading 0s are replaced by real
digits to the right of the binary point.
Fig. 2.14
Scientific notation
Floating point addition
Step 1. To be able to add these
numbers properly, align the decimal point of the number that has the smaller
exponent. Hence, we need a form of the smaller number, 1.610ten x10–1, that
matches the larger exponent. We obtain this by observing that there are
multiple representations of an unnormalized floating-point number in scientific
notation:
Step 1.1.610ten x10–1 =
0.1610ten x100 = 0.01610ten x101
Step 2. Next comes the addition
of the significands: 9.999ten+ 0.016ten The sum is 10.015ten x101.
Step 3. This sum is not in
normalized scientific notation, so we need to adjust it: 10.015ten x101
= 1.0015ten x102
Thus, after the addition we may
have to shift the sum to put it into normalized form, adjusting the exponent
appropriately.
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