numerics - Different floating-point numbers equal?

Let's define two different numbers.

x = 1.

y = 1. + 2^-52 (* equivalently, 1 + $MachineEpsilon *)

Let's make sure they're different with FullForm:

x // FullForm (* 1.` *)
y // FullForm (* 1.0000000000000002` *)

Those look pretty close... let's make sure they're different. I'm not a wizard with the developer tools, but I can export them as IEEE double-precision floating point numbers (which I'd bet is their internal representation):

StringJoin @@ 
 IntegerString[Reverse@ToCharacterCode[ExportString[x, "Real64"]], 
  16, 2]

(* 3ff0000000000000 *)
StringJoin @@ 
 IntegerString[Reverse@ToCharacterCode[ExportString[y, "Real64"]], 
  16, 2]
(* 3ff0000000000001 *)

We can see that they are indeed different. They represent the two numbers:

$$ \begin{align} x &= (1.){\underbrace{000 \cdots 000}_\text{51 zeros}}0_2 \times 2^{01111111111_2 - 1023} \equiv 1 \\ y &= (1.){\underbrace{000 \cdots 000}_\text{51 zeros}}1_2 \times 2^{01111111111_2 - 1023} \equiv 1 + \frac{1}{2^{52}} \end{align} $$

That is, x is exactly one, and y is the smallest IEEE double greater than one. Ok, so they're different. Hey Mathematica, you know they're diff-

x == y (* True *)

Oh. What if we try-

x === y (* True *)

Hey Python, you use doubles, right? Are you seeing this?

>>> 1. == 1.0000000000000002
False

Maybe it's because you're using quads?

>>> 1. == 1.0000000000000001

True

Yeah, I didn't think so. Mathematica, are you sure? I mean, this doesn't seem right...

y - x (* 2.22045*10^-16 *)

Aha! I knew it! Now let's try this:

y - x == 0 (* False *)

Success! Now let's just double-check (pun intended):

1.0000000000000001 - 1. (* 0. *)

% == 0 (* True *)

So you are using double-precision...

My question is, Why do Equal and SameQ return True, even though these numbers are obviously different? SameQ ignores the last bit, and Equal ignores the last seven bits!

Answer

It seems I found my answer in OleksandrR's comment to this question. He says,

Bear in mind Equal applies an extra tolerance in Mathematica. The proper comparison is

Block[{Internal`$EqualTolerance = -Infinity}, 1 == 1 + $MachineEpsilon] (* False *)
Block[{Internal`$EqualTolerance = -Infinity}, 1 == 1 + $MachineEpsilon/2] (* True *)

In fact, the value of Internal`$EqualTolerance * Log2[10] is 7., meaning that it ignores the last seven bits, just as I discovered!

(Analogously, Internal`$SameQTolerance * Log2[10] is 1., i.e. it drops the last bit.)

Note that this is mentioned in the documentation for Equal, under Details:

• Approximate numbers with machine precision or higher are considered equal if they differ in at most their last seven binary digits (roughly their last two decimal digits).


• For numbers below machine precision the required tolerance is reduced in proportion to the precision of the numbers.

However, I never thought to look at it, since (thought) I knew what == means! Lesson learned, always check the documentation, especially if you don't think you need to.