GATE Overflow Test Series | Digital Logic | Test 1 | Question: 12

Question

Which of the following $32-bit$ bits stream correctly represents the maximum possible normalized value represented in IEEE-754 single-precision representation?

• A. $\underbrace{0}_{\text{sign}}\underbrace{11111111}_{\text{exponent}}\underbrace{11111111111111111111111}_{\text{mantissa}}$
  
• B. $\underbrace{0}_{\text{sign}}\underbrace{11111111}_{\text{exponent}}\underbrace{11111111111111111111110}_{\text{mantissa}}$
  
• C. $\underbrace{0}_{\text{sign}}\underbrace{11111110}_{\text{exponent}}\underbrace{11111111111111111111110}_{\text{mantissa}}$
  
• D. $\underbrace{0}_{\text{sign}}\underbrace{11111110}_{\text{exponent}}\underbrace{11111111111111111111111}_{\text{mantissa}}$

Answer 1

For normalized value in IEEE-754 representation the exponent field cannot be all $1s.$ So, to get the maximum exponent we should make the exponent field $1111110$ which equals $254$ but with a bias of $127$ (used to have negative exponent in IEEE-754 representation), this equals $127.$
    
    Sign bit must be $0$ to make the number positive.
    
    Mantissa bits must be all $1s$ to maximize the number so that the represented number equals $1.\underbrace{111..1}_{\text{23 1s}} = 1+ (1 - 2^{-23}) = 2 - 2^{-23}.$
    
    So, correct option is D and the represented value $ = (2 - 2^{-23}) \times 2^{127}$