GATE CS 2022 Q.41: ALU, Datapath and Control Unit

Consider three floating point numbers A, B and C stored in registers RA, RB and RC, respectively as per IEEE-754 single precision floating point format. The 32-bit content stored in these registers (in hexadecimal form) are as follows.

\text{RA} = \text{0xC1400000}, \quad \text{RB} = \text{0x42100000}, \quad \text{RC} = \text{0x41400000}

Which one of the following is FALSE?

$A + C = 0$

$C = A + B$

$B = 3C$

$(B - C) > 0$

Official answer and solution

Official answer: B
From: Official GATE 2022 answer key

Solution

Decode the IEEE-754 single precision floating point hexadecimal representations to their real values and evaluate each relationship.

In IEEE-754 single precision: bit 31 is sign $s$ , bits 30..23 are biased exponent $e$ , and bits 22..0 are fraction $f$ .

$\text{RA} = \text{0xC1400000} = 1100\ 0001\ 0100\ \dots_2$ :
- $s = 1$ (negative)
- $e = 1000\ 0010_2 = 130$ . Unbiased exponent: $130 - 127 = 3$ .
- $f = .100\dots_2 = 0.5$ . Value: $A = -1 \times (1 + 0.5) \times 2^3 = -1.5 \times 8 = -12.0$ .
$\text{RC} = \text{0x41400000} = 0100\ 0001\ 0100\ \dots_2$ :
- $s = 0$ (positive), same magnitude as $A$ . Value: $C = +12.0$ .
$\text{RB} = \text{0x42100000} = 0100\ 0010\ 0001\ \dots_2$ :
- $s = 0$ (positive)
- $e = 1000\ 0100_2 = 132$ . Unbiased exponent: $132 - 127 = 5$ .
- $f = .0010\dots_2 = \frac{1}{8} = 0.125$ . Value: $B = (1 + 0.125) \times 2^5 = 1.125 \times 32 = 36.0$ .

Testing the options:

(A) $A + C = -12.0 + 12.0 = 0$ . (TRUE)
(B) $C = A + B \implies 12.0 = -12.0 + 36.0 = 24.0$ , which is FALSE.
(C) $B = 3C \implies 36.0 = 3 \times 12.0 = 36.0$ . (TRUE)
(D) $(B - C) = 36.0 - 12.0 = 24.0 > 0$ . (TRUE)

Thus, Option (B) is the FALSE statement.

Final answer

Option (B)

Common trap

Incorrectly calculating the fraction value for RB ( $1.125 \times 32 = 36$ ).

The answer is the one in the official GATE answer key. The solution is this site's own explanation.

Source

Paper: GATE 2022 Computer Science and Information Technology (CS)
Question: 41 of 65
Section: Computer Science and Information Technology

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