float-rational == is not transitive

[StefanKarpinski]

Example:

julia> (x,y,z) = (1//10, 0.1, 3602879701896397//36028797018963968);

julia> x == y
true

julia> y == z
true

julia> x == z
false

Originally raised here. To fix this, we would have to make comparisons between rationals and floats do something more conservative, such as converting the float to the exact rational value that it represents. For example, this conversion would do it for many Float64s:

function float2rat(x::Float64)
  ux = reinterpret(Uint64,x)
  int(sign(x))*(1+int(ux & (2^52-1))//2^52)*(2//1)^int((((ux>>52)&(2^11-1))-1023))
end

However, this does have some problems in that it overflows for floats with large exponents, so something slightly cleverer has to be done to make this work. One also doesn't actually need to construct the rational number to do this comparison – cross multiplication and an equality check will also work.

[mlubin]

I think it's reasonable to restrict == to mean exact equality. isapprox should be used in other cases.

[StefanKarpinski]

I agree. We should ensure that == is transitive for all types in Base or we may end up like JavaScript and PHP (where equality isn't even reflexive). It's kind of amazing how difficult it is to make == transitive. There are several intrinsics for int-float comparisons that were fiendishly difficult to concoct.

[StefanKarpinski]

I also kind of like the idea of being able to demonstrate that 1/10 cannot be exactly represented as a float like this:

julia> 1/10
0.1

julia> ans == 1//10
false

[StefanKarpinski]

Still todo: inequalities. These are always even more of a pain.

[stevengj]

As I mentioned on the mailing list, I think promotion of (float,rational) to rational is a mistake; it makes the Rational type positively dangerous in a floating-point/numerical context, because it means any rational will propagate through the computation, destroying accuracy and performance of whatever it touches. Transitivity of == in mixed rational/float comparison is a small price to pay to avoid this.