Question

Write a program that converts a given floating point binary number with a 24-bit normalized mantissa and an 8-bit exponent to its decimal (i.e. base 10) equivalent. For the mantissa, use the representation that has a hidden bit, and for the exponent use a bias of 127 instead of a sign bit. Of course, you need to take care of negative numbers in the mantissa also. Use your program to answer the following questions: (a) Mantissa: 11110010 11000101 01101010, exponent: 01011000. What is the base-10 number? (b) What is the largest number (in base 10) the system can represent? (c) What is the smallest non-zero positive base-10 number the system can represent? (d) What is the smallest difference between two such numbers? Give your answer in base 10. (e) How many significant base-10 digits can we trust using such a representation?

Mention: Matlab

Answer 1

Solution for the above question is provided below. If any doubt please comment below.

a)

%Given inputs
expnt='01011000';
mant='111100101100010101101010';

% Signbit
signBit=str2num(mant(1));

%Find mantissa
manti=mant(2:24);

% compute exponent
exponent=0;
dj=length(expnt);
for di=1:length(expnt)
   exponent=str2num(expnt(di))*2^(dj-1)+exponent;
    dj=dj-1;
end

% Find bias
ex=-127+exponent;
decim=0;

%compute the value
for di=1:length(manti)
   decim=str2num(manti(di))*2^(-di)+decim;
end

format long

% Result
res=(-1)^signBit*(1+decim)*2^(ex)

Output:

b)

To find it make exponent as all 1's and also make mantissa all as 1 except first bit in the above program

%Given inputs

expnt= '11111111';

mant='011111111111111111111111';

Output:

c)

%Given inputs

expnt= '00000000';

mant= '000000000000000000000000' ;

Output:

d)

First run the program using below mantissa and exponent

expnt='00000000';
mant='011111111111111111111111';

Second run the program using below mantissa and exponent

expnt='00000000';
mant='011111111111111111111110';

Find the difference of both result

Output:

e)

Number of significant base-10 digits = 2^-23