Question

Consider the IEEE-754 single precision floating point numbers $P=0xC1800000$ and $Q=0x3F5C2EF4$. Which one of the following corresponds to the product $P \times Q$, represented in IEEE-754 single precision format?

• A. 0x404C2EF4
• B. 0x405C2EF4
• C. 0xC15C2EF4
• D. 0xC14C2EF4

Hint:

To quickly verify IEEE-754 values: check the sign bit for positivity/negativity, compute exponent–bias for scaling, then approximate mantissa. Multiplication/division is faster by decoding exponents and mantissas.

Answer 1

The Correct Option is C

Step 1: Decode $P=0xC1800000$.
In IEEE-754 single precision: 1 sign bit, 8 exponent bits (bias 127), 23 fraction bits.
$0xC1800000 = 1100\;0001\;1000\;0000\;0000\;0000\;0000\;0000$.
- Sign = 1 (negative).
- Exponent = 1000 0011 = 131 $\Rightarrow$ actual exponent = 131–127 = 4.
- Mantissa = 1.0000000 (since fraction = 0).
Thus $P = -2^4 = -16$.
Step 2: Decode $Q=0x3F5C2EF4$.
$0x3F5C2EF4 = 0011\;1111\;0101\;1100\;0010\;1110\;1111\;0100$.
- Sign = 0 (positive).
- Exponent = 0111 1110 = 126 $\Rightarrow$ actual exponent = -1.
- Mantissa fraction ≈ 1.359375 (approx). More accurately, $Q \approx 0.86$.
Step 3: Multiply.
$P \times Q = (-16) \times (0.86) \approx -13.76$.
Step 4: Match with options.
Among the given hexadecimal IEEE-754 values:
- 0xC15C2EF4 corresponds to approximately -13.76.
Thus, correct answer = (C).
\[ \boxed{0xC15C2EF4} \]