Computer Arithematic

I want to subtract 1 from 1+0.5*10^-15. When I enter 1+0.5*10^-15 in MATLAB command prompt, I get 1.000000000000000e+000 and not 1.0000000000000005e+000. I understand it is because of the displaying limitation of format long e. But when I subtract 1 from this number (1.000000000000000e+000), I get 4.440892098500626e-016 which is very close to 5.000000000000000e-016 but not exactly equal to it. Now this is not a displaying limitation of format long e. Why is this happening?

Answer 1

[James Tursa]

The limitations of floating point arithmetic. IEEE double precision can't store your numbers exactly, so the closest number in the floating point model is picked. E.g., here are the exact decimal equivalents of your numbers:

>> num2strexact(0.5*10^-15)

ans =

5.00000000000000038852699938330539619153592800597507572746280857245437800884246826171875e-16

>> num2strexact(1+0.5*10^-15)

ans =

1.000000000000000444089209850062616169452667236328125

>> num2strexact((1+0.5*10^-15)-1)

ans =

4.44089209850062616169452667236328125e-16

For example, the nearest numbers in the IEEE double precision floating point model to the one picked for your 1+0.5*10^-15 are:

>> num2strexact((1+0.5*10^-15)+eps(1+0.5*10^-15))

ans =

1.0000000000000006661338147750939242541790008544921875

>> num2strexact((1+0.5*10^-15)-eps(1+0.5*10^-15))

ans =

1.0000000000000002220446049250313080847263336181640625

You can find the num2strexact utility on the FEX here:

http://www.mathworks.com/matlabcentral/fileexchange/22239-num2strexact-exact-version-of-num2str

[Ashish Bhatt]

Thank you. You said closed number is picked. Then why is num2strexact(1+0.5*10^-15) not equal to either of the nearest numbers (num2strexact((1+0.5*10^-15)+eps(1+0.5*10^-15)) or num2strexact((1+0.5*10^-15)-eps(1+0.5*10^-15))) in your reply above?

[James Tursa]

The three numbers in the IEEE floating point model nearest 1+0.5*10^-15 in order are:

1.0000000000000002220446049250313080847263336181640625

1.000000000000000444089209850062616169452667236328125

1.0000000000000006661338147750939242541790008544921875

The one that is closest to 1+0.5*10^-15 is the middle one in the above list. All of these are shown in my post.