GATE CS 2021 Set 1 Q.34 (CS Q.24): Number Representations and Computer Arithmetic

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 __________

Your answer

Official answer and solution

Solution

Decode the IEEE 754 single-precision floating point fields using bias 127.

Given single-precision IEEE 754 fields:

Sign bit: $S = 1$ (indicates a negative number).
Exponent: $E = 10000001_2 = 128 + 1 = 129_{10}$ .
With a bias of 127, the true exponent is $e = E - 127 = 129 - 127 = 2$ .

Fraction: $F = 11110000000000000000000_2$ .
The fractional value is: $f = \frac{1}{2^1} + \frac{1}{2^2} + \frac{1}{2^3} + \frac{1}{2^4} = 0.5 + 0.25 + 0.125 + 0.0625 = 0.9375$ The normalized significand is $M = 1 + f = 1 + 0.9375 = 1.9375$ .

Computing the value:

\text{Value} = (-1)^S \times M \times 2^e = (-1)^1 \times 1.9375 \times 2^2 = -1.9375 \times 4 = -7.75

Final answer

-7.75

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