Strange result in subtraction operation

I have made the following operation in Matlab: 4.238257316854621e+00 - 4.238257316854609e+00

The obvoius result is 1.12e-14, but the answer returned was 1.154631945610163e-14.

I know that math operations are affected by the limited precision of calculations (floating point representation) but this result is really surprising to me.

Can anyone provide some explanation? Is there any way to improve the accuracy of this calculation?

Answer 1

[Les Beckham]

The "obvious" result is 1.2e-14 not 1.12e-14 (the different last two digits in your original numbers at the 14 decimal place give 2.1 - 0.9 = 1.2). So 1.154631945610163e-14 is within 4.5e-16 of the "obvious" result which is consistent with the expected floating point precision.

Answer 2

[Jan]

format long g
a = 4.238257316854621;
b = 4.238257316854609;

wanted = 1.12e-14;
got    = a - b

got = 
      1.15463194561016e-14

drift = max(eps(a), eps(b))  % Expected error:

drift = 
      8.88178419700125e-16

got - wanted

ans = 
      3.46319456101627e-16

So the found result is inside the expected error range for IEEE754 doubles. Remember, that you cannot store 16 digits reliably:

digits(20)
a = vpa(4.238257316854621)

a = 

4.2382573168546207043

You see, the double value is not store accurately already. To do so, you need a char vector as input:

aa = vpa('4.238257316854621')

aa = 

4.238257316854621

No, there is no way to improve the accuracy except for increasing the precision, e.g. by using SYM objects or quadrupel precision.

bb = vpa('4.238257316854609')

bb = 

4.238257316854609

aa - bb

ans = 

0.000000000000011999999999999544971

[Marcio Coelho de Mattos]

I understood. Thanks for the detailed explanation.