Final answer:
To calculate the product of the given numbers in half-precision format, we must use binary scientific notation, adjust for mantissa and exponent, and implement rounding procedures. The result will be less accurate than a calculator due to the limited precision of 16 bits.
Step-by-step explanation:
To calculate the product of −8.0546875 × 10⁰ and −1.79931640625 × 10⁻¹ by hand using 16-bit half precision format:
First, we convert both numbers into binary scientific notation to fit into the half-precision format. We separate the sign bit, exponent, and mantissa, taking note of the implicit leading 1 for the mantissa in IEEE 754 representation. In this format, we have 1 bit for sign, 5 bits for exponent, and 10 bits for the mantissa.
Second, we multiply the mantissas and add the exponents, being careful to adjust for the bias of the exponent in this format (which is 15 for half precision).
Third, we apply the guard, round, and sticky bits concept while rounding to the nearest even value as required, and correct the exponent if necessary due to mantissa normalization.
Finally, we compare the result with the non-rounded calculation you might get using a calculator. Since calculators don't consider the precision limits of half-precision floating-point format, the result from a calculator will be more precise. We might expect a result that is slightly less accurate due to rounding and the limited precision of 16-bits. The use of guard, round, and sticky bits helps to mitigate errors during the rounding process.
For example, when a calculator multiplies 2.48 times 6.3, the result of 15.687 is more precise than what we can represent in half precision, which ignores the limits on significant figures.