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.

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Step 2: Decode the exponent.
The exponent bits are:

E = 10000001₂

Converting to decimal:

10000001₂ = 129

Using the bias (127) for single-precision IEEE 754:

Actual exponent = 129 − 127 = 2

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Step 3: Determine the mantissa.
The fraction bits are:

F = 11110000000000000000000

Adding the implicit leading 1 for a normalized number, the mantissa becomes:

1.1111₂

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Step 4: Convert the mantissa to decimal.

1.1111₂ = 1 + 1/2 + 1/4 + 1/8 + 1/16

= 1.9375

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Step 5: Compute the final value.

Value = −(1.9375 × 2²)

= −7.75

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Final Answer:

−7.75