Question

3.5 (base 10) == 01000000011000000000000000000000 (base 2)
-2.0625 (base 10) == 11000000000001000000000000000000 (base 2)
-7.21875 (base 10) == 11000000111001110000000000000000 (base 2)

Calculate 3.5 * -2.0625 (= -7.21875) using the algorithm for floating point multiplication. You must show work for credit!

Answer 1

In single precison point format with 32 bit representation we can represent a number as

sign                    fraction/significand/mantissa (23 bits)
 |                      /                                     \
 |  exponent (8 bits)  /                                       \
 |   /           \    /                                         \
 0  1 0 0 0 0 0 0 0  1 0 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 0 1 1 0 1 1

Similarly a given number can be represented with the format

by using these two concepts we can rewrite the numbers as

01000000011000000000000000000000 = 0 10000000 11000000000000000000000   

(sign) 128 6291456

So this number could be represented like (-1)⁰     * 2^(128-127) * (1+6291456*2^-23)   = 1*2¹*1.75 = 2

11000000000001000000000000000000 = 1 10000000 00001000000000000000000

(sign) 128 262144

So this number could be represented like (-1)¹ * 2^(128-127) * (1+262144*2^-23)   = -1*2*1.03125 =2.0625

Their multiplication can do in separate.

That is sign part is multiplied first

So (-1)⁰ * (-1)¹ = (-1)¹ = -1 ( this represents the result is a negative number so we can use 1 as the first bit in the answer)

Now we are multiplying the exponential part

2¹  * 2¹ = 2^{2 ( So the multiplication result will be} 2^{2 )}

Now we need to multiply the mantissa part

Finally we will get the result as (-1)¹ * 2² * (1+6750208*2^-23) = (-1)¹ * 4 * 1.8046825 = -7.21875