Define function for rounding to precision

with the second argument too, although if you want to enforce always rounding up it likely isn't worth it. I don't know why equivalent versions of ceil and floor don't exist. It was added relatively recently to round rather than in the original implementation, but I would have thought all that family of functions should have been updated the same way.

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John D'Errico 2019-8-14

编辑：John D'Errico 2019-8-14

Adam makes a good point, in that if round was extended as it was, then why did ceil and floor not get the same treatment? My guess is for the same reason I never thought to look to see if ceil and floor had the new capability. It is just something I've never had the slightest desire to do. Yes, I'll suppose a round to n digits has been at times useful, though I think it has been more when I answer a question about exactly that. Personally, I tend not to use it at all.

So why not? Why do I rarely ever use the capability to round a number to tenths, hundreths, etc? I tend not to do so, because I know that it is a fallacy in double precision floating point arithmetic to trust that a number is exactly rounded/truncated/ceiled to the nearest 0.1, or almost any fraction of an integer. (I do trust that Adam knows all this.)

format long g
X = rand(1,3)
X =
         0.925425280986515       0.00558112226984153         0.186388406230158
Y = floor(X*10)/10
Y =
                       0.9                         0                       0.1

But are they exactly 0.9. 0, and 0.1? Only in the second case are they so.

sprintf('%0.55f',Y(1))
ans =
    '0.9000000000000000222044604925031308084726333618164062500'
sprintf('%0.55f',Y(2))
ans =
    '0.0000000000000000000000000000000000000000000000000000000'
sprintf('%0.55f',Y(3))
ans =
    '0.1000000000000000055511151231257827021181583404541015625'

You can round/ceil/floor a number exactly to a fraction that is a pure power of 2 though. So to round/ceil/floor to the nearest multiple of 1/8, thus a pure power of 2, even though it is a negative power of 2?

Y8 = floor(X*8)/8
Y8 =
                     0.875                         0                     0.125
sprintf('%0.55f',Y8(1))
ans =
    '0.8750000000000000000000000000000000000000000000000000000'

We can see it is stored as exactly 0.875 too, using my num2bin utility.

num2bin(Y8(1))
ans = 
  struct with fields:
              Class: 'double'
               Sign: 1
           Exponent: -1
           Mantissa: '11100000000000000000000000000000000000000000000000000'
    BinaryExpansion: [-1 -2 -3]
              BiSci: '1.1100000000000000000000000000000000000000000000000000 B-1'
            BiCimal: '0.11100000000000000000000000000000000000000000000000000'

So Y8(1) is EXACTLY represented internally as 1/2 + 1/4 + 1/8 = 0.875.

So, while I accept that floor and ceil arguably should have been extended as was done to round, most of the time, you really are not getting what you asked for anyway.

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