GATE CS 2000 Q2.14 — Number Representations (Fixed and Floating Point)

In a 32-bit single-precision floating-point representation (such as IEEE 754), the significand has 24 bits of precision, which offers approximately 7 decimal digits of precision.

For $X$:

1. X := A + B: Since $A$ and $B$ have equal magnitudes but opposite signs, they cancel out exactly: $$X = 2.0 \times 10^{30} + (-2.0 \times 10^{30}) = 0.0$$
2. X := X + C: Adding $C$ gives: $$X = 0.0 + 1.0 = 1.0$$

For $Y$:

1. Y := A + C: We attempt to add $2.0 \times 10^{30}$ and $1.0$. Because their magnitudes differ by 30 orders of decimal magnitude (which vastly exceeds the 7-digit precision limit of 32-bit floats), the smaller value $1.0$ is rounded off to zero during alignment. Thus: $$Y = 2.0 \times 10^{30} + 1.0 \approx 2.0 \times 10^{30}$$
2. Y := Y + B: Adding $B$ gives: $$Y = 2.0 \times 10^{30} + (-2.0 \times 10^{30}) = 0.0$$

Therefore, the final values are $X = 1.0$ and $Y = 0.0$, which corresponds to Option B.