Rational{IntN}(pi) are poor approximations to Pi

[JeffreySarnoff]

These values are better approximations to Pi

Rational{Int32}(pi) = Int32(1068966896) // Int32(340262731)
Rational{Int64}(pi) = Int64(2646693125139304345) // Int64(842468587426513207)
Rational{Int128}(pi) = Int128(60728338969805745700507212595448411044) // Int128(19330430665609526556707216376512714945)

setprecision(BigFloat,500);
Q32pi =  Int32(1068966896) // Int32(340262731);
Q64pi = Int64(2646693125139304345) // Int64(842468587426513207);
Q128pi =  Int128(60728338969805745700507212595448411044) // Int128(19330430665609526556707216376512714945);

Float64(BigFloat(pi)-Q32), Float64(BigFloat(pi)-Rational{Int32}(pi)) # 3.1e-18, 1.2e-16
Float64(BigFloat(pi)-Q64), Float64(BigFloat(pi)-Rational{Int64}(pi)) # 1.4e-38, 1.2e-16
Float64(BigFloat(pi)-Q128) # -6.2e-76

[JeffreySarnoff]

And if Rational{64} means Int32//Int32, use Q32pi for Q64pi

[TotalVerb]

Is this operation meaningful? We have with the revised definitions

julia> 4Rational{Int64}(π)
ERROR: OverflowError()
 in *(::Rational{Int64}, ::Rational{Int64}) at ./rational.jl:196
 in *(::Int64, ::Rational{Int64}) at ./promotion.jl:191
 in eval(::Module, ::Any) at ./boot.jl:226

which is somewhat unfortunate. More precision is not necessarily a good thing.

Rational (from my understanding) is intended for exact arithmetic (approximate arithmetic is done much better using floats). Would it be better to disallow conversion of pi with an InexactError?

[JeffreySarnoff]

@TotalVerb How do you get that? Please show me what you changed, how you redefined things.

[TotalVerb]

@JeffreySarnoff What I did to change the definition was:

julia> Base.convert(::Type{Rational{Int64}}, c::Irrational{:π}) = Int64(2646693125139304345) // Int64(842468587426513207)

julia> 4Rational{Int64}(π)
ERROR: OverflowError()
 in *(::Rational{Int64}, ::Rational{Int64}) at ./rational.jl:196
 in *(::Int64, ::Rational{Int64}) at ./promotion.jl:191
 in eval(::Module, ::Any) at ./boot.jl:226

But you can reproduce it without changing anything with

julia> 4 * Int64(2646693125139304345) // Int64(842468587426513207)
ERROR: OverflowError()
 in *(::Rational{Int64}, ::Rational{Int64}) at ./rational.jl:196
 in *(::Int64, ::Rational{Int64}) at ./promotion.jl:191
 in eval(::Module, ::Any) at ./boot.jl:226

[JeffreySarnoff]

@TotalVerb Now I understand that.
Pick maximal constant K for which K * Rational{Int64}(pi) should not overflow,
and if K is 2^61, then the current definition is appropriate.
While K should be greater than 4, should it be greater than 2^31?

[JeffreySarnoff]

Apparently, not a pressing issue. I am retiring it.

[simonbyrne]

Good catch, I don't know how that happened!

The problem is this line:

julia/base/irrationals.jl

Line 16 in dcde84d

convert{T<:Integer}(::Type{Rational{T}}, x::Irrational) = convert(Rational{T}, Float64(x))

It would be better to call rationalize on a BigFloat, e.g. this gives the correct answer:

rationalize(Int32, big(pi), tol=0)

[JeffreySarnoff]

:)