Final answer:
To sum A and B according to floating-point formatting rules, align their exponents, sum the mantissas, apply guard, round, and sticky bits, and then round to the nearest even. After these steps, assuming we maintain five significant figures, the calculated sum might be approximately 3.02754.
Step-by-step explanation:
To calculate the sum of A and B with the given floating-point formatting rules, we need to align the exponents first, align the mantissas (the part of a floating-point number that represents the significant digits), apply the rules for guard, round, and sticky bits, and then round to the nearest even. The values A=2.6125×10¹ and B=4.150390625×10⁻¹.
First, since A's exponent is 1 and B's exponent is -1, we must adjust B's exponent to match A by moving the decimal point in B's mantissa two places to the right, resulting in B=0.4150390625×10¹.
The aligned mantissa will then be A=2.6125 and B=0.4150390625. When these are added, the sum is 3.0275390625.
Then, apply the rules for guard, round, and sticky bits. If a bit is dropped in the process and the remaining number is already even, we leave it unchanged. If it's odd, we round up.
In this case, following these rules might lead to a result of 3.02754 (assuming we are utilizing a precision that maintains five significant figures after the rounding process).
Round to the nearest even is typically used in floating-point arithmetic to reduce rounding errors over multiple calculations. The guard bit helps in determining how to round by looking at the next bit (the round bit). If the dropped digits result in a tie, the sticky bit captures whether any further bits were non-zero.