Question

Consider the following representation of a number in IEEE 754 single-precision floating point format with a bias of 127.
\[ S : 1 E : 10000001 F : 11110000000000000000000 \] Here \( S \), \( E \), and \( F \) denote the sign, exponent, and fraction components of the floating-point representation.
The decimal value corresponding to the above representation (rounded to 2 decimal places) is \(\underline{\hspace{2cm}}\).

Hint:

In IEEE 754 format, the value is computed as \( (-1)^S \times (1.F)_2 \times 2^{(E - \text{bias})} \).

Answer 1

Correct Answer: -7.75

Step 1: Determine the sign.
The sign bit is \( S = 1 \), which indicates that the number is negative.

Step 2: Decode the exponent.
The exponent bits are \( E = 10000001_2 \). Converting to decimal:
\[ 10000001_2 = 129 \] The actual exponent is:
\[ 129 - 127 = 2 \]

Step 3: Determine the mantissa.
The fraction bits are \( F = 11110000000000000000000 \). The normalized mantissa is:
\[ 1.1111_2 \]

Step 4: Convert mantissa to decimal.
\[ 1.1111_2 = 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} = 1.9375 \]

Step 5: Compute the final value.
\[ \text{Value} = - (1.9375 \times 2^2) = -7.75 \] % Final Answer

Final Answer: \[ \boxed{-7.75} \]