C - floating point rounding

Question

I'm trying to understand how floating point numbers work.

I think I'd like to test out what I know / need to learn by evaluating the following: I would like to find the smallest x such that x + 1 = x, where x is a floating point number.

As I understand it, this would happen in the case where x is large enough so that x + 1 is closer to x than the next number higher than x representable by floating point. So intuitively it seems it would be the case where I don't have enough digits in the significand. Would this number x then be the number where the significand is all 1's. But then I can't seem to figure out what the exponent would have to be. Obviously it would have to be big (relative to 10^0, anyway).

Answer 1

You just need an expression for the value of the LS bit in the mantissa in terms of the exponent. When this is > 1 then you have met your condition. For a single precision float the LS bit has a value of 2^-24*2^exp, so the condition would me met when exp is > 24, i.e. 25. The smallest (normalized) number where this condition would be satisfied would therefore be 1.0 * 2^25 = 33554432.0f.

I haven't checked this, so my maths may be off somewhere (e.g. by a factor of 2) and it's also possible that the FP unit does rounding beyond the 24th bit, so there may be a further factor of 2 needed to account for this, but you get the general idea...