Three floating point numbers X, Y, and Z are stored in three registers Rx,Ry,Rx​,Ry​, and RzRz​, respectively in IEEE 754 single precision format as given below in hexadecimal:
Rx=0xC1100000,Ry=0x40C00000 and Rz=0x41400000
Which of the following option(s) is/are CORRECT?

The correct options are:
1. 4(X + Y) + Z = 0
2. 2Y − Z = 0
3. 4X + 3Z = 0

Step-by-Step Explanation:

To determine which options are correct, we first need to decode the values of X, Y, and Z stored in registers Rx, Ry, and Rz using the IEEE 754 single-precision floating-point format.

Recall that the standard format for IEEE 754 single precision (32-bit) representation consists of:
- Sign bit (S): 1 bit (determines if the number is positive or negative).
- Exponent (E): 8 bits biased by 127.
- Fraction / Mantissa (F): 23 bits.

The decimal value is given by:
Value = $(-1{)}^S\times (1+F)\times 2^{E-127}$

1. Decoding Rx = 0xC1100000 (Value X)
Converting the hexadecimal representation to binary:
C1100000 = 1100 0001 0001 0000 0000 0000 0000 0000
Now we split the bits according to the format:
- Sign bit (S): 1 (Negative)
- Biased Exponent (E): 1000 0010 (binary) = 130 (decimal)
- Fraction (F): 001 0000 0000 0000 0000 0000 (binary) = $2^{-3}=0.125$

Let's calculate the value of X:
Exponent factor: $2^{130-127}=2^3=8$
Significand (1 + F): 1 + 0.125 = 1.125
Value of X = $-1\times 1.125\times 8=-9$

2. Decoding Ry = 0x40C00000 (Value Y)
Converting the hexadecimal representation to binary:
40C00000 = 0100 0000 1100 0000 0000 0000 0000 0000
Splitting the bits:
- Sign bit (S): 0 (Positive)
- Biased Exponent (E): 1000 0001 (binary) = 129 (decimal)
- Fraction (F): 100 0000 0000 0000 0000 0000 (binary) = $2^{-1}=0.5$

Let's calculate the value of Y:
Exponent factor: $2^{129-127}=2^2=4$
Significand (1 + F): 1 + 0.5 = 1.5
Value of Y = $1\times 1.5\times 4=6$

3. Decoding Rz = 0x41400000 (Value Z)
Converting the hexadecimal representation to binary:
41400000 = 0100 0001 0100 0000 0000 0000 0000 0000
Splitting the bits:
- Sign bit (S): 0 (Positive)
- Biased Exponent (E): 1000 0010 (binary) = 130 (decimal)
- Fraction (F): 100 0000 0000 0000 0000 0000 (binary) = $2^{-1}=0.5$

Let's calculate the value of Z:
Exponent factor: $2^{130-127}=2^3=8$
Significand (1 + F): 1 + 0.5 = 1.5
Value of Z = $1\times 1.5\times 8=12$

Verification of Options:
We have: X = -9, Y = 6, Z = 12.

- Option 1: 4(X + Y) + Z = 0
$4(-9+6)+12=4(-3)+12=-12+12=0$ (CORRECT)

- Option 2: 2Y − Z = 0
$2\left(6\right)-12=12-12=0$ (CORRECT)

- Option 3: 4X + 3Z = 0
$4(-9)+3\left(12\right)=-36+36=0$ (CORRECT)

- Option 4: X + Y + Z = 0
$-9+6+12=9\ne 0$ (INCORRECT)