Suppose we wished to apply the x86 CMP instruction to 32-bit operands that contained numbers in a floating-point format. For correct results, what requirements have to be met in the following areas?
a. The relative position of the significand, sign, and exponent fields.
b. The representation of the value zero.
c. The representation of the exponent.
d. Does the IEEE format meet these requirements? Explain.
The Correct Answer and Explanation is :
To correctly apply the x86 CMP instruction to 32-bit operands containing floating-point numbers, the requirements for floating-point format fields must align with the behavior of integer comparison. Here’s the analysis:
a. The relative position of the significand, sign, and exponent fields:
The floating-point format must arrange the fields such that the bitwise ordering of the numbers (when treated as unsigned integers) matches their numerical ordering. This is necessary for the CMP instruction, which performs a bitwise comparison. Specifically:
- The sign bit must be the most significant bit, allowing negative numbers to compare as less than positive numbers.
- The exponent should follow the sign bit, determining the scale of the number.
- The significand should be the least significant part.
This layout ensures the natural ordering of bits matches the numerical magnitude.
b. The representation of the value zero:
The representation of zero must ensure consistent comparison. A single, unique bit pattern (all zeros) should represent zero to avoid issues with ordering and equality checks. Representations with multiple zeros (e.g., +0 and -0) would require additional handling in comparison logic.
c. The representation of the exponent:
The exponent must be stored in biased form, where the minimum value maps to the lowest representable exponent and the maximum value maps to the highest. This ensures that the bitwise ordering of exponents aligns with their numerical ordering.
d. Does the IEEE format meet these requirements? Explain:
Yes, the IEEE 754 floating-point format meets these requirements:
- The sign bit is the most significant bit, followed by the biased exponent and the significand.
- Zero is represented uniquely as all zeros in the exponent and significand fields (with the sign bit optionally 0 or 1 for +0 or -0, though they compare equally).
- The exponent uses a bias (127 for single-precision), ensuring bitwise ordering corresponds to numerical ordering.
Thus, IEEE 754 format is compatible with the CMP instruction for correct floating-point comparisons.