csit2860/notes_20130205_csit2860.txt at master · fiskr/csit2860 · GitHub

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February 05, 2013
CSIT 2260: Machine organization and architecture

Multiplication
--------------

Arithmetic shift-
        when multiplying by powers of 2, shift numerals
        to the left by however many powers of 2 you are
        multiplying
        e.g.
         0100 = 4
        x0010 = 2
        -----
         1000 = 8
        Notice the one in 0100 was multiplied by one power of
        2, hence the bit was moved once to the left- 1000

Otherwise, just add the multiplicand for each 1 bit in the
multiplier and shift the next addition to the left. For a 0
bit in the multiplier, just shift for the next addition.
e.g.
                  1011
                x 1110
                ------
                     0  (1011 x 0) just shift to the left
                 10110  (1011 x 1) shift to the left
                101100  (1011 x 1) shift to the left
             + 1011000  (1011 x 1) shift to the left
               -------
              10011010


Floating Point Representation
-----------------------------
floating point calculations can be performed with any integer format
floating point emulation is the calculation of floating point
numbers without actually storing them
Usually hardware, not special software, is used to do these
calculations.

floating point numbers
        allow arbitrary number of decimal places to the right

Scientific notation:
Sign    Mantissa        Exponent
+/-     x.xx       X    base^x

1 bit   precision       range

Mantissa = Significand

e.g.
0 00110 10000000 = 2^6 X 0.1 = 32

Because
0 00110 10000000 = 2^6 X 0.1 = 32
is the same as
0 00111 01000000 = 2^7 X 0.01 = 32

hence normalization- a rule stating the mantissa must always
start with 1.

biased exponent
        number midway between the range expressible by
        the exponent.
        With a 5 bit exponent we use 16 as our bias- called
        "excess- 16" representation.
        e.g.
        0 10110 10000000 = 2^(22-16) x 0.1 = 2^6 x 0.1


Class examples
-------------
+0.5 (base 10) = 0.1 (base 2) = 0.1 x 2^0 (i.e. 0.1 x 1)
0 10000 10000000 = 2^(16 - 16) x 0.1 = 2^0 x 0.1

-1.125 (base 10) = -1.001 (base 2) = 0.1001  x 2^1
1 10001 100100000 = - 2^(17 - 16) x 0.1001

3.576