24
$\begingroup$

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!

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  • 4
    $\begingroup$ Internal`$EqualTolerance and Internal`$SameQTolerance. You can search the site for explanations $\endgroup$ – Michael E2 Mar 10 '15 at 0:17
  • 1
    $\begingroup$ @MichaelE2 Yep, just found those in the related questions. Should this one be closed? $\endgroup$ – 2012rcampion Mar 10 '15 at 0:18
  • $\begingroup$ That's a good question. I didn't find an exact duplicate in my cursory search. But I guess it's in the "Details" of the docs -- e.g., from Equal: "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)." I would consider that "easily found in the documentation," I guess. Others might think this is a nice clear question about an idiosyncratic behavior of Mathematica. Your choice, or you could wait and see if the community closes it. $\endgroup$ – Michael E2 Mar 10 '15 at 0:22
  • 1
    $\begingroup$ Yeah, there are nearly infinitely many corners of Mathematica and not enough time to explore them all. (Different users have different notions of "easily found", but still, I think if the answer is clearly explained in the details of the docs for the function, I consider it easily found, even if it's a function I never heard of. But I won't vote to close. I think the issue is important and common enough, that a little extra duplication of the documentation is ok.) $\endgroup$ – Michael E2 Mar 10 '15 at 0:33
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    $\begingroup$ Related:stackoverflow.com/questions/4983885/… $\endgroup$ – Michael E2 Mar 10 '15 at 0:40
14
$\begingroup$

You can define your own "precise equal" using Congruent () (entered as Esc===Esc or \[Congruent]):

Congruent[x_, y_] := Equal @@ SetPrecision[{x, y}, Infinity]

Now

x ≡ y
False

for your example.

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  • $\begingroup$ I usually use Rationalize[_, 0], but SetPrecision[_, Infinity] actually makes more sense. $\endgroup$ – 2012rcampion Mar 10 '15 at 0:37
  • 3
    $\begingroup$ @2012rcampion I would use SetPrecision, too. Rationalize does something different. See the answers and discussion in the comments by Mr.Wizard and me here. $\endgroup$ – Michael E2 Mar 10 '15 at 0:44
  • $\begingroup$ @MichaelE2 Yikes, thanks for clarifying this. You weren't kidding about "nearly infinitely many corners." $\endgroup$ – 2012rcampion Mar 10 '15 at 0:49
  • $\begingroup$ @2012rcampion I liked your syntax style use in your answer more. Hope you don't mind that I adopted it. $\endgroup$ – Karsten 7. Mar 10 '15 at 1:02
  • $\begingroup$ Sure, that's what this site's for! And nice job spotting the impetus for my question. =) $\endgroup$ – 2012rcampion Mar 10 '15 at 1:17
13
$\begingroup$

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.

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