gets slightly off value when summing

I found this problem

>> p1

p1 =

    0.0050

>> p2

p2 =

    0.0450

>> p1+p2 - 0.05

ans =

  -6.9389e-18
  

Obviously p1 + p2 is 0.05 but I think due to loss of data when summed the result is not exactly equal to 0.05. Is there a way to fix this (as in, how do i restore p1+p2 to 0.05)? the code above is not what I am using, it's only for reference. Thanks in advance.

Answer 1

[Eric Delgado]

It's float operation universe. :)

p1 = 0.005;
p2 = 0.0450;

% Instead of:
p1+p2 == 0.05

ans = logical
   0

% Try this:
abs(p1+p2-0.05) <= 1e-5 % You could use 1e-17 and still will receive a true logical value

ans = logical
   1

[Sharon Timotius]

My goal is to restore the p1+p2 to 0.05. Is there a way? thanks

[Eric Delgado]

Nop... but you can use round to force it.

p1 = 0.005;
p2 = 0.045;

p1+p2 == 0.05

ans = logical
   0

round(p1+p2, 2) == 0.05

ans = logical
   1

[John D'Errico]

That could in general be dangerous, that is, using round to try to FORCE it to sum correctly. This is because even the number 0.05 is not exactly 0.05. 0.05 is NOT representable using floating point arithmetic, because the number is stored in a binary form using 52 binary bits.

In binary, the number 0.05 would be represented as the infinitely repeating bit sequence

0.0000110011001100110011001100110011001100110011001100110011...

where the ones represent negative powers of 2.

format long g
x = sum(2.^[-5 -6 -9 -10 -13 -14 -17 -18 -21 -22 -25 -26 -29 -30 -33 -34 -37 -38 -41 -42 -45 -46 -49 -50 -53 -54 -56])

x = 
                      0.05

More accurately

sprintf('%0.55f',x)

ans = '0.0500000000000000027755575615628913510590791702270507812'

So while it LOOKS like 0.05, it is not, for the same reason that 1/3 is not finitely representable as a decimal. We can even do this test:

x == 0.05

ans = logical
   1

And MATLAB even THINKS that number is the same as 0.05. But we know it is not.

So rounding such a result could conceivably produce something other than 0.05. Do I need to come up with a counter-example?