I have a question needing to find the largest representable integer using (6 exponent and 9 mantissa)
I know that there is a split of 5 bits for the exponent and 10 bits for the mantissa with 1 sign bit.
I know how to find the low and high digits for the mantissa. I use the function (k/(2^(mantissa))). But how do I find the high and low digits for the exponent? Does it have something to do with the amount of bits (16)?
I am looking at examples that say the high and low for (5 exponent is -16 and 15). But how they got to there is where I am confused.
Thanks
If your floating-point format follows the pattern set by IEEE 754, then the encoded exponent is biased by half its maximum value, rounded down. Thus, 5 exponent bits can hold codes from 0 to 31. Half 31 rounded down is 15. Thus an exponent code of 1 represents a mathematical exponent of 1−15 = −14, and an exponent code of, say, 27, represents a mathematical exponent of 27−15 = 12.
Additionally, in IEEE 754 binary floating-point, the maximum exponent code is reserved to represent infinities and NaNs. So the maximum exponent code for finite values in your case would be 30, representing a mathematical exponent of 30−15 = 15.
However, there is no law stating anybody must use IEEE 754. So the mere fact that your format has 1 sign bit, 5 exponent bits, and 10 significand bits does not tell us what the actual mathematical exponent values are. Somebody could choose to bias the exponent code by another value or to use the maximum value for regular numbers, not infinities and NaNs. And, given your information that the mathematical exponent range is from −15 to 16, it seems like the specification might be that all exponent codes represent numbers, and there are no infinities or NaNs. This suggests the exponent is biased by 15, and the maximum exponent code of 31 represents a mathematical exponent of 16.
It also suggests there are no subnormal numbers, as subnormal numbers would usually be encoded by an exponent code of 0, which would mean that the implicit leading bit of the significand is 0 (instead of 1 for normal numbers) and the mathematical exponent would clamp at −14 instead of decreasing to −15. The fact that your information says the minimum exponent is −15 suggests this is not occurring, so there are no subnormal numbers in this format.
Again, though, there is no law about what floating-point formats have to be. Somebody could make other choices. There should be a specification that describes this floating-point format, and that is where the necessary information should come from.