Three floating point numbers $X, Y,$ and $Z$ are stored in three registers $\text{Rx}, \text{R}_Y,$ and $\text{R}_Z$, respectively in IEEE 754 single precision format as given below in hexadecimal: $\text{Rx} = \text{0xC1100000}, \text{R}_Y = \text{0x40C00000},$ and $\text{R}_Z = \text{0x41400000}$ Which of the following option(s) is/are CORRECT?

In IEEE 754 single-precision floating-point format (32-bit):

• Sign bit ($S$): 1 bit

• Biased exponent ($E$): 8 bits (bias = 127)

• Fraction / Mantissa ($M$): 23 bits with an implicit leading 1.

The real value is given by $(-1)^S \times (1.M)_2 \times 2^{E - 127}$.

1. For register $\text{Rx} = \text{0xC1100000}$:

   • Binary: 1 10000010 00100000000000000000000

   • $S = 1$

   • $E = 10000010_2 = 130 \implies \text{Actual exponent} = 130 - 127 = 3$

   • Mantissa $= 1.001_2 = 1 + \frac{1}{8} = 1.125 = \frac{9}{8}$

   • $X = -1 \times \frac{9}{8} \times 2^3 = -9.0$

2. For register $\text{R}_Y = \text{0x40C00000}$:

   • Binary: 0 10000001 10000000000000000000000

   • $S = 0$

   • $E = 10000001_2 = 129 \implies \text{Actual exponent} = 129 - 127 = 2$

   • Mantissa $= 1.1_2 = 1.5 = \frac{3}{2}$

   • $Y = +1 \times 1.5 \times 2^2 = 6.0$

3. For register $\text{R}_Z = \text{0x41400000}$:

   • Binary: 0 10000010 10000000000000000000000

   • $S = 0$

   • $E = 10000010_2 = 130 \implies \text{Actual exponent} = 130 - 127 = 3$

   • Mantissa $= 1.1_2 = 1.5 = \frac{3}{2}$

   • $Z = +1 \times 1.5 \times 2^3 = 12.0$

Now evaluate each option:

• (A) $4(X + Y) + Z = 4(-9 + 6) + 12 = 4(-3) + 12 = -12 + 12 = 0$ (Correct)

• (B) $2Y - Z = 2(6) - 12 = 12 - 12 = 0$ (Correct)

• (C) $4X + 3Z = 4(-9) + 3(12) = -36 + 36 = 0$ (Correct)

• (D) $X + Y + Z = -9 + 6 + 12 = 9 \neq 0$ (Incorrect)