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IEEE representation

by Maylis
Tags: ieee, representation
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Maylis
#19
Aug11-14, 12:46 PM
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P: 575
Between any powers of 2, are there 2^52 representable numbers? I just guessed on that one. Is the number of representable numbers between 2^3 and 2^4 2^52 as well?? Or how do you determine that

log2(eps(1))

ans =

   -52
so the distance between 1 and the next representable number is 2^ -52, now I see that one. I think I discovered a pattern for determing the value of eps(x).

So you just express x as a power of 2, if its a whole number, then you do 2^x-52, and that is eps(x). If yoy have something that is not expressed as a power of 2, then you just go back to the lower number that can be, so eps(5) = eps(4)

So eps(4), 4 is 2^2, so eps(4) is 2^(2-52) = 2^-50.

So now, I can say with some confidence I understand F.1, F.3, and F.4. Now where I am stuck is F.2 and F.5.

Edit: I might be able to justify F.5 now, let's give this a whirl

So I know eps(2^52) is 1, because 2^(52-52) is 0, hence 2^0 = 1. So, everything in between there also has an eps(x) = 1. That means you can have ##2^{52}, 1+2^{52}, 2+2^{52},3+2^{52} ....2^{53}##. So there are 2^52 numbers in between 2^52 and 2^53.

So I guess what you should do to find the number of representable numbers is subtract the highest number from the lowerst number in the interval, then divide by eps(lower interval)??

##2^{53} - 2^{52} = 2^{52}/1 = 2^{52}## representable numbers

So to create another example, the number of representable numbers between [4 8).

eps(4) = 2^(2-52) = 2^-50 spacing between.

##8-4 = 4##, so ##4/2^{-50} = 4*2^{50}## representable numbers between 4 and 8
D H
#20
Aug12-14, 11:59 AM
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There are 252 representable numbers in the interval [2n,2n+1) if 2n has a normalized representation. For example, there are none in the interval [21024,21025) because 21024 is out of range.

What about the denormalized numbers? There is only one representable number in the interval [2-1074,2-1073), 2-1074 itself. The next largest representable number after 2-1074 is 2-1073. There are two representable numbers in the interval [2-1073,2-1072), three in [2-1072,2-1071), and so on, until you get to the interval [2-1022,2-1021), which contains 252 representable numbers. Every power of 2 interval from that one to [21023,21024) does contain 252 representable numbers.
Maylis
#21
Aug12-14, 09:19 PM
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P: 575
Quote Quote by D H View Post
There are 252 representable numbers in the interval [2n,2n+1)
Does that mean the relative spacing between all numbers is the same for normalized numbers?
D H
#22
Aug12-14, 09:20 PM
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Of course not. It means the spacing is the same for all representable numbers between 2n and 2n+1.
Maylis
#23
Aug12-14, 10:30 PM
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P: 575
EDIT: nevermind, I get it now.

I still don't understand the thing about the smallest representable number. Someone said it was 1+2^53, that is very big to be a smallest representable number, how do you do the analysis to determine that?
D H
#24
Aug12-14, 11:37 PM
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P: 15,201
Your question F.2 asked for the smallest non-representable integer. That obviously needs to be a biggish number.


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