GATE CS 2023 Q35 — Number Representations (Fixed and Floating Point)

To find the product $P \times Q$ in IEEE-754 single-precision format, we decode both numbers:

1. Decode $P = \text{0xC1800000}$:

• Binary: 1100 0001 1000 0000 0000 0000 0000 0000
• Sign bit ($S$) = $1$ (negative)
• Biased Exponent ($E$) = $10000011_2 = 131_{10} \implies$ Unbiased exponent $e_P = 131 - 127 = 4$
• Fraction ($F$) = $0 \implies$ Mantissa $M_P = 1.0$
• Value: $P = -1.0 \times 2^4$

2. Decode $Q = \text{0x3F5C2EF4}$:

• Binary: 0011 1111 0101 1100 0010 1110 1111 0100
• Sign bit ($S$) = $0$ (positive)
• Biased Exponent ($E$) = $01111110_2 = 126_{10} \implies$ Unbiased exponent $e_Q = 126 - 127 = -1$
• Fraction ($F$) = $10111000010111011110100_2$
• Value: $Q = +1.F \times 2^{-1}$

3. Compute the Product $P \times Q$:

• Sign of product = Negative (since negative $\times$ positive is negative)
• Product value: $-(1.0 \times 2^4) \times (1.F \times 2^{-1}) = -1.F \times 2^3$
• The unbiased exponent of the product is $3$.
• The biased exponent of the product is $3 + 127 = 130 = 10000010_2$.
• The fraction remains unchanged: $F = 10111000010111011110100_2$.

4. Reassemble into IEEE-754 Single Precision format:

• Sign (1 bit): 1
• Exponent (8 bits): 1000 0010
• Fraction (23 bits): 101 1100 0010 1110 1111 0100
• Combined 32-bit binary: 1100 0001 0101 1100 0010 1110 1111 0100
• Hexadecimal representation: 0xC15C2EF4