when I add two floating point numbers the result is not correct above 262144

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Mark Ekblad
Mark Ekblad 2018-4-30
评论： John D'Errico 2018-4-30
When I execute the following I always get back 262144, it is like it is treating it as an integer
a=single(262144.000);
for i=1:1234,
a= a+single(0.01);
end;
display(a-floor(a));
sprintf('%10.6f',a)
single
0
ans =
'262144.000000'
or if I just do a = single(262144.000)+single(0.01) result is 262144

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采纳的回答

James Tursa
James Tursa 2018-4-30
编辑：James Tursa 2018-4-30
The amount you are adding, 0.01, is less than the eps of the number you are using. E.g.,
>> a = single(262144.000)
a =
262144
>> eps(a)
ans =
0.0313
>> a + 0.01
ans =
262144
So the result of the addition does not change the value of "a" because there is not enough precision in "a". I.e., the closest number to 262144.01 in IEEE single precision is in fact 262144. The next highest number is 262144.03125.

更多回答（2 个）

the cyclist
the cyclist 2018-4-30
It is not possible to exactly represent decimal numbers with a finite number of bits (in this case, 32 bits, because you are specifying single-precision).
So, you will get that
single(262144) + single(0.1)
is most closely representable by
2.6214409e5
but that
single(262144) + single(0.01)
is most closely represented as
262144
The numbers are not "being treated as integers", but as the closest representation in this 32 bit system as possible.
You would similarly see that
262144 + 1.e-10
is represented as
2.621440000000001e+05
but
262144 + 1.e-11
is represented (and displayed) as
262144
in double precision.
John D'Errico
John D'Errico 2018-4-30
Time to learn what eps means. After all, YOU were the one who chose to use single precision here. So that means you should also learn what the consequences of that decision will be.
a=single(262144.000)
a =
single
262144
eps(a)
ans =
single
0.03125
eps(a) is essentially the smallest number that can be added to a, and still get a different number.
a == a+eps(a)
ans =
logical
0
a == a+eps(a)/2
ans =
logical
1
Think of eps(a) as the size of the least significant bit in the number a.
  2 个评论
the cyclist
the cyclist 2018-4-30
As an addendum to help your understanding, note that
a = single(262144);
e = eps(a);
b = log2(e)
results in
b = -5
So, eps() returns the distance to the next-largest floating-point number (of the same precision), and in this case that distance is 2^(-5).
This illustrates the binary aspect of the representation.

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