Question

Assume we are using the simple model for the floating-point representation given in this book (the representation uses a 14-bit format, 5 bits for the exponent with a bias of 15, a zero-normalized mantissa of 8 bits, and a single sign bit). What is the real number represented by 01110010000101? Show how the computer would add this number with 238.5 using the floating-point addition algorithm. Verify your results by adding them in decimal, what is the relative error in the representation of the output sum. Note from student: Attempting to convert 01110010000101 to decimal resulted in 12,448. I'm not sure if this is correct and I cannot find a converter online. Any help would be appreciated!

Answer 1

Solution for the problem is provided below, please comment if any doubts:

Here the 14-bit floating point representation is used with:

            5 bit exponent, 8 bit zero-normalized mantissa and a sign bit.

Part I:

The floating point number given is: 01110010000101

The sign bit = 0, Positive

The 5 bit exponent = 11100 = 28, but it is a biased value by 15, thus the original exponent value =28-15=13

The mantissa part is: 10000101, since zero normalized, the mantissa will be =

= 0.10000101

Now shift the point to 13 bits right and place 0’s if place is not there

0.10000101*2¹³ = 1000010100000 = 4256

Thus the real number equivalent of 01110010000101 = 4256

Part II:

To add with 238.5

First convert 238.5 to floating point format

First convert to binary, 238.5 = 11101110.1

Now convert to zero normalized mantissa and compute the exponent as the shifted bits of point.

11101110.1=0.111011101*2^8

Now perform the addition of , 0.10000101*2¹³ and 0.111011101*2⁸

For that first make both the exponents same

0.111011101*2⁸ = 0.00000111011101*2¹³

Now perform the addition of equal exponent numbers

0.1000010100000*2¹³ + 0.00000111011101*2¹³

=0.10001001111101*2¹³

Now convert back to floating point, eliminate bits after 8 bits in result

Add 15 to exponent, 13+15=28= 11100

Mantissa: 10001001

Sign: 0

Result= 01110010001001

Convert back to decimal

= 0. 10001001*2¹³

⁼1000100100000

=4384

The actual result has to be = 4256+238.5 =4494.5

The relative error = (4494.5-4384)/ 4494.5 = 0.02 or 2%