Weird result, why I can't find -0.809 and 0.309 in -1:0.001:1???

why I can't find -0.809 and 0.309 in -1:0.001:1???

[Stephen23]

"why I can't find -0.809 and 0.309 in -1:0.001:1???"

Because they are not there.

In fact, those exact values don't even exist as binary floating point numbers.

"Weird result"

This is a completely expected result with binary floating point numbers:

https://www.mathworks.com/matlabcentral/answers/57444-faq-why-is-0-3-0-2-0-1-not-equal-to-zero

https://www.mathworks.com/matlabcentral/answers/316889-why-two-equal-numbers-are-not-equal

http://www.mathworks.com/matlabcentral/answers/102419-how-do-i-determine-if-the-error-in-my-answer-is-the-result-of-round-off-error-or-a-bug

http://www.mathworks.com/matlabcentral/answers/140656-about-numerical-difficulty-of-equality-comparison-operation-between-two-double-real-numbers-in-matlab

This is worth reading as well:

https://docs.oracle.com/cd/E19957-01/806-3568/ncg_goldberg.html

http://faculty.tarleton.edu/agapie/documents/cs_343_arch/papers/1991_Goldberg_FloatingPoint.pdf

Answer 1

[David Goodmanson]

Hello JX,

you are looking for exact equality with floating point numbers, and a exact equality does not always occur.

x = -1:.001:1;
format long
x(1810)

ans = 0.809000000000000 % looks good

x(1810) - .809

ans = -1.110223024625157e-16 % isn't good

Since there are only a finite number of bits (64 for double precision numbers like these), most floating point numbers cannot be stored exactly in memory. Here there is disagreement in the 16th decimal place, which is to be expected in some (not all) of the 2001 cases for the x array.

If you take a look with format hex, the way these numbers are actually stored,

>> format hex

x(1810)

ans = 3fe9e353f7ced916

809

ans = 3fe9e353f7ced917

you can see that first, .809 is stored as well as possible with 64 bits but not exactly (as would be the case with a bunch of trailing hex zeros), and second that the two numbers differ by 1 in the last bit. The equality check fails.

[Jingli Xie]

How can I find -0.809 and 0.309 in -1:0.001:1?

[Stephen23]

"How can I find -0.809 and 0.309 in -1:0.001:1?"

Never compare binary floating point values for exact equality.

Always compare the absolute difference against a tolerance, e.g.:

tol = 1e-5;
idx = abs(A-B)<tol

[Steven Lord]

Another way to compare with a tolerance is to use ismembertol.

>> ismembertol(-0.809, -1:0.001:1, eps)
ans =
  logical
   1

If I'd used a tolerance of 0 as the third input, to only detect cases where -0.809 is an exact down-to-the-last-bit match for an element in the vector, the answer would have been false (logical 0.)