Representing numbers with IEEE754 with Round to Nearest Even

Question

I'm currently learning about IEEE754 standard and rounding, and I have an exercise which is the following:

Add -325.875 to 0.546875 in IEEE754, but with 3 bits dedicated to the mantissa instead of 23.

I'm having a lot of trouble doing this, especially representing the intermediary values, and the guard/round/sticky bits. Can someone give me a step-by-step solution, to the problem?

My biggest problem is that obviously I can't represent 0.546875 as 0.100011 as that would have more precision than the system has. So how would that be represented?

Apologies if the wording is confusing.

Answer 1

Preliminaries

The preferred term for the fraction portion of a floating-point number is “significand,” not “mantissa.” “Mantissa” is an old word for the fraction portion of a logarithm. Mantissas are logarithmic; adding to the mantissa multiplies the number represented. Significands are linear; adding to the significand adds to the number represented (as scaled by the exponent).

When working with a significand, use its mathematical precision, not the number of bits in the storage format. The IEEE-754 binary32 format has 23 bits in its primary field for the encoding of a significand, but another bit is encoded via the exponent field. Mathematically, numbers in the binary32 format behave as if they have 24 bits in their significands.

So, the task is to work with numbers with four bits in their significands, not three.

Work

In binary, −325.875 is −101000101.111₂•2. In scientific notation, that is −1.01000101111₂•2⁸. Rounding it to four bits in the significand gives −1.010₂•2⁸.

In binary, 0.546875 is .100011₂. In scientific notation, that is 1.00011₂•2⁻¹. Rounding it to four bits in the significand gives 1.001₂•2⁻¹. Note that the first four bits are 1000, but they are immediately followed by 11, so we round up. 1.00011 is closer to 1.001 than it is to 1.000.

So, in a floating-point format with four-bit significands, we want to add −1.010₂•2⁸ and 1.001₂•2⁻¹. If we adjust the latter number to have the same exponent as the former, we have −1.010₂•2⁸ and 0.000000001001₂•2⁸. To add those, we note the signs are different, so we want to subtract the magnitudes. It may help to line up the digits as we were taught in elementary school:

1.010000000000
0.000000001001
——————————————
1.001111110111 

Thus, the mathematical result would be −1.001111110111₂•2⁸. However, we need to round the significand to four bits. The first four bits are 1001, but they are followed by 11, so we round up, producing 1010. So the final result is −1.010₂•2⁸.

−1.010₂•2⁸ is −1.25•2⁸ = −320.