fix spurious overflow for Float16(::Rational) (#52395)

Fixes #52394.

Also fixes `Float32` for `UInt128`, since currently
`Float32((typemax(UInt128)-0x01) // typemax(UInt128))` gives `Nan32`.

base/rational.jl

@@ -145,6 +145,14 @@ function (::Type{T})(x::Rational{S}) where T<:AbstractFloat where S
  P = promote_type(T,S)
  convert(T, convert(P,x.num)/convert(P,x.den))::T
 end
+ # avoid spurious overflow (#52394). (Needed for UInt16 or larger;
+ # we also include Int16 for consistency of accuracy.)
+Float16(x::Rational{<:Union{Int16,Int32,Int64,UInt16,UInt32,UInt64}}) =
+ Float16(Float32(x))
+Float16(x::Rational{<:Union{Int128,UInt128}}) =
+ Float16(Float64(x)) # UInt128 overflows Float32, include Int128 for consistency
+Float32(x::Rational{<:Union{Int128,UInt128}}) =
+ Float32(Float64(x)) # UInt128 overflows Float32, include Int128 for consistency
 
 function Rational{T}(x::AbstractFloat) where T<:Integer
  r = rationalize(T, x, tol=0)

test/float16.jl

@@ -203,6 +203,11 @@ const minsubf16_32 = Float32(minsubf16)
 # issues #33076
 @test Float16(1f5) == Inf16
 
+# issue #52394
+@test Float16(10^8 // (10^9 + 1)) == convert(Float16, 10^8 // (10^9 + 1)) == Float16(0.1)
+@test Float16((typemax(UInt128)-0x01) // typemax(UInt128)) == Float16(1.0)
+@test Float32((typemax(UInt128)-0x01) // typemax(UInt128)) == Float32(1.0)
+
 @testset "conversion to Float16 from" begin
  for T in (Float32, Float64, BigFloat)
  @testset "conversion from $T" begin