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Comment by flatline | Hacker News Reader

It is 2023 and our tooling still encourages the same mistakes people were making 40 years ago. Can we really not have equality operators that do a comparison with a 1% tolerance or something as a sensible default equivalence instead of blatantly wrong bitwise comparison?

I would even be happy with a default set of compiler warnings or errors, which I don’t believe I have ever seen.

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From a user POV, I think == and === on floats should simply be undefined in any language. It should be a compile time or runtime error.

There can be a separate function `float_equals()` with explicit args that does what people want

The only reason to use the same syntax == is for POLYMORPHIC code that is actually correct.

But it's not going to be correct with floats, because they don't obey the same algebraic laws ... So the syntax should be different!

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I believe the Erlang compiler optimization presented as justification for this change is a good example

The compiler wants to reason about code, independent of types

But that reasoning about the =:= operator is wrong for the float case.

what people want varies, but often what people doing numerical programming want is to write their monomorphic code in infix syntax so they can more easily see bugs in it

they're already familiar with rounding errors

from that perspective you're suggesting taking a step backwards from fortran i toward assembly language

the erlang compiler bug presented is not an example of what you're talking about because you're talking about arithmetic, and the buggy comparison operator it's using is not an arithmetic comparison operator

Fortran would be an interesting case, because say C++ is polymorphic, and so are all dynamic programming languages with more than 1 type (x == y could be a string or float comparison)

But I would still say that floating point equality is a vanishingly rare operation

I can't think of any real use case for it, other than maybe some (bad, incomplete) unit tests that assert 1.0 == x and assert 1.0 == y

And that use case is perfectly served by float_equals(), and arguably served better if it has some options. Although probably abs(x - y) < eps is just as good, and you don't even need float_equals()

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Can you show some real use cases for floating point equality in good, production code? (honest question)

It's similar to hashing floats, which Go and Python do allow. I made an honest request for examples of float hashing:

https://lobste.rs/s/9e8qsh/go_1_21_may_have_clear_x_builtin#...

https://old.reddit.com/r/ProgrammingLanguages/comments/10bm2...

I didn't get any answers that appeared realistic. Some people said you might want to create a histogram of floats -- but that's obviously better served by bucketing floats, so you're hashing integers.

Another person said the same thing for quantizing to mesh.

One person suggested fraud detection for made-up values, but that was clearly addressed by converting the float to a bit pattern, and hashing the bit pattern.

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This isn't a theoretical question since we're working on this for https://www.oilshell.org.

For that case it seems pretty clearly OK to omit === on floats, since I use awk and R for simple stats and percentages, and that's likely what people would do with a shell.

Though as always I'm open to concrete counterexamples.

i gave some examples in https://news.ycombinator.com/item?id=35883963, though arguably those are not very concrete. also i pointed out there why unit tests that use exact comparison on floats are not necessarily bad: ieee-784 specifies bit-exact results for the five fundamental arithmetic operations, and if your compiler is introducing rounding errors, you want your tests to fail. (or introducing nonerrors, in the case where it's introducing fmas.)

even hashing is useful for memoization

i think probably you're asking in the wrong places

the lapack source might be a better place to look; we can probably suppose that if something is in lapack it's realistic and also not a bug, and numerical code doesn't get more 'good, production' than lapack

in dgelsx for example i find

          ELSE IF( ANRM.EQ.ZERO ) THEN
    *
    *        Matrix all zero. Return zero solution.
    *
             CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )

and also (fairly dubious, at least to my uneducated eye)

    *     Determine RANK using incremental condition estimation
    *
          WORK( ISMIN ) = ONE
          WORK( ISMAX ) = ONE
          SMAX = ABS( A( 1, 1 ) )
          SMIN = SMAX
          IF( ABS( A( 1, 1 ) ).EQ.ZERO ) THEN
             RANK = 0
             CALL DLASET( 'F', MAX( M, N ), NRHS, ZERO, ZERO, B, LDB )
             GO TO 100
          ELSE
             RANK = 1
          END IF

as well as some sentinel-value checks

and in dlar1v:

                WORK( INDS+I ) = S*WORK( INDLPL+I )*L( I )
                IF( WORK( INDLPL+I ).EQ.ZERO )
         $                      WORK( INDS+I ) = LLD( I )

and also

                TMP = D( I ) / DMINUS
                IF(DMINUS.LT.ZERO) NEG2 = NEG2 + 1
                WORK( INDUMN+I ) = L( I )*TMP
                WORK( INDP+I-1 ) = WORK( INDP+I )*TMP - LAMBDA
                IF( TMP.EQ.ZERO )
         $           WORK( INDP+I-1 ) = D( I ) - LAMBDA

and also

          IF( ABS(MINGMA).EQ.ZERO )
         $   MINGMA = EPS*WORK( INDS+R1-1 )

of course float comparisons for ordering are much more common than float comparisons for equality, but there are numerous realistic examples of float comparisons for equality in lapack and, i think, in applied mathematics in general

but if you ask an audience of type theorists, sysadmins, and web developers, they won't know that; they aren't numerical analysts and probably haven't inverted a submatrix since sophomore math class, if ever

(there are 6550 files in lapack 3.9.0 of which i looked through 15 to find these examples, suggesting that there are on the order of 1000 realistic examples in there; eventually i'd probably find one that wasn't even comparing to zero)

plausibly instead of giving them floating point by default you should give them fixed point with, say, nine digits after the decimal point; that was the approach knuth took in tex, for example, to guarantee reproducibility, but it also gives human-predictable rounding. and in most cases it should be perfectly adequate for simple stats and percentages. as a bonus you get 64 bits of precision instead of 53

see also https://news.ycombinator.com/item?id=35902861 for more examples

Thanks for actually going and looking! I would say that this supports the idea of float_equals(x, 0.0) or even float_equals_zero(x) :)

It wouldn't be too surprising to me if 90% of float equalities in real code are comparisons to 0.0 or 1.0 or -1.0 (that's what I meant in my original post, there was a typo)

The point about x >= 0 and x <= 0 is interesting. I suppose to be consistent, you would also remove those operators on floats, although for something awk-like, I don't think it matters either way.

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The other point is that it looks like the Fortran code isn't actually using == or x.eq.zero polymorphically!

So the first point is that I would want to understand conceptually where float equality would be equal. You mentioned fixed points, although there I also suspect that abs(x - y) < eps is a better test in most situations, though certainly there could be counterexamples.

The second point is -- would any of those situations be polymorphic with int and float, or string and float? It depends on the language, but it seems vanishingly unlikely in most languages, and in particular in say in C++

I would probably go look at the equivalent Eigen code if it mattered with respect to Oil, but it probably doesn't

yeah, i also wrote a cube root routine using newton's method yesterday and came to the conclusion that |x - x'| < threshold was a better approach in that case. maybe in every case, maybe not

i agree that prefix notation is not as bad for these as i had thought before looking

i think there are plausible cases where your float might be polymorphic with double, complex, a vector, a matrix, or an element of a galois field, but i think it would only be polymorphic with int in cases like stopping change propagation when an observable value gets recomputed to the same value from new inputs, and in those cases what you need is exact bitwise equality, not arithmetic equality

an awk-like thing might be better with fixed-point, like tex and metafont; you could call the data type 'btc` and represent it as a 64-bit integer that's multiplied by 10*-8 for output

The tolerance you need depends on how you calculated the number, so there's no way to have a sensible default.

The problem with defining an “epsilon” (1% in your case) is that there is no value that would please everyone. For some, 1% will be fine, but others may need 0.0001%.

The solution is to either use decimal floats if they suit your need (and eat the performance penalty), or to use a linter that flags float comparisons by equality.

And then there's the question of how you use the epsilon - i.e. whether 1.2e-16 and 1e-20 are "within 1%". Sometimes those are very much different sizes, but if you're comparing sin(π) (i.e. the 1.2e-16) with some other almost-zero computation, they're very much within 1%.

That’s bad enough, but it is not the problem with defining an “epsilon”, it’s a problem.

Another one is that you would lose transitivity on equality. With a 1% epsilon, you would have

   1.000 == 1.005
   1.005 == 1.010
   1.010 == 1.015

but

   1.000 != 1.015

You also would have to be careful on how to define that 1% error range. the naive "x is equal to all numbers between 0.99x and 1.01x" would mean 1.0 would be considered equal to 0.99, but 0.99 would not be considered equal to 1.0 (1.01 times 0.99 is less than 1.0)

You also lose that, if a == b and c == d it follows that a + c == b + d.

The behavior around zero also will go against intuition. If you consider 0.99 and 1.0 to be equal, do you really want 1.0E-100 and -1.0E-100 be different?

> Can we really not have equality operators that do a comparison with a 1% tolerance or something as a sensible default equivalence instead of blatantly wrong bitwise comparison?

Then it wouldn't really be an equality comparison any more. For instance according to your rule 1.0 and 1.01 are the same number, but they clearly are not. But if you want to do this regularly you could create a special type and overload the equality operator (if your language supports it, otherwise you'll have to call the function directly) that points to 'boolean about_equal(float f1, float f2, float epsilon)' or something equivalent where epsilon is the amount that you would allow f1 and f2 to deviate from each other to still call them equal. And you could define epsilon as ((abs(f1) + abs(f2)) / 100) .

The bitwise comparison isn't 'blatantly wrong', it's the expected behavior in just about every programming language. I'm not aware of any language that has a native float operator that does this but as you can see you can usually simply add one yourself if you really need it. But this need rarely comes up and usually indicates that you are doing something wrong.

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