Fix conversion Irrational => Rational

Fixes #16513.

base/irrationals.jl

@@ -13,11 +13,30 @@ promote_rule{s,T<:Number}(::Type{Irrational{s}}, ::Type{T}) = promote_type(Float
 convert(::Type{AbstractFloat}, x::Irrational) = Float64(x)
 convert(::Type{Float16}, x::Irrational) = Float16(Float32(x))
 convert{T<:Real}(::Type{Complex{T}}, x::Irrational) = convert(Complex{T}, convert(T,x))
-convert{T<:Integer}(::Type{Rational{T}}, x::Irrational) = convert(Rational{T}, Float64(x))
+
+# NOTE: We use a @generated functions to compute the constants.
+# Hopefully in future we can replace this with @pure.
+# These return values which can be spliced directly into the AST.
+@generated function convert{T<:Integer}(::Type{Rational{T}}, x::Irrational)
+    o = precision(BigFloat)
+    p = 256
+    while true
+        setprecision(BigFloat, p)
+        bx = BigFloat(x())
+        r = rationalize(T, bx, tol=0)
+        if abs(BigFloat(r) - bx) > eps(bx)
+            setprecision(BigFloat, o)
+            return r
+        end
+        p += 32
+    end
+end
+convert(::Type{Rational{BigInt}}, x::Irrational) = throw(ArgumentError("Cannot convert an Irrational to a Rational{BigInt}: use rationalize(Rational{BigInt}, x) instead"))
 
 @generated function (t::Type{T}){T<:Union{Float32,Float64},s}(c::Irrational{s},r::RoundingMode)
-    f = T(big(c()),r())
-    :($f)
+    setprecision(BigFloat, 256) do
+        T(big(c()),r())
+    end
 end
 
 =={s}(::Irrational{s}, ::Irrational{s}) = true

test/numbers.jl

@@ -2086,6 +2086,12 @@ for f in (trunc, round, floor, ceil)
 
 @test isa(convert(Float64, big(1)//2), Float64)
 
+# issue 16513
+@test convert(Rational{Int32}, pi) == 1068966896 // 340262731
+@test convert(Rational{Int64}, pi) == 2646693125139304345 // 842468587426513207
+@test convert(Rational{Int128}, pi) == 60728338969805745700507212595448411044 // 19330430665609526556707216376512714945
+@test_throws ArgumentError convert(Rational{BigInt}, pi)
+
 # issue 5935
 @test rationalize(Int8,  nextfloat(0.1)) == 1//10
 @test rationalize(Int64, nextfloat(0.1)) == 300239975158034//3002399751580339