Improve floating-point Euclidean division for Float16 and Float32 (#49637)

Author: simonbyrne

base/div.jl

@@ -381,3 +381,9 @@ end
 # NOTE: C89 fmod() and x87 FPREM implicitly provide truncating float division,
 # so it is used here as the basis of float div().
 div(x::T, y::T, r::RoundingMode) where {T<:AbstractFloat} = convert(T, round((x - rem(x, y, r)) / y))
+
+# Vincent Lefèvre: "The Euclidean Division Implemented with a Floating-Point Division and a Floor"
+# https://inria.hal.science/inria-00070403
+# Theorem 1 implies that the following are exact if eps(x/y) <= 1
+div(x::Float32, y::Float32, r::RoundingMode) = Float32(round(Float64(x) / Float64(y), r))
+div(x::Float16, y::Float16, r::RoundingMode) = Float16(round(Float32(x) / Float32(y), r))

test/numbers.jl

@@ -1770,6 +1770,27 @@ end
         @test cld(-1.1, 0.1) == div(-1.1, 0.1, RoundUp)   ==  ceil(big(-1.1)/big(0.1)) == -11.0
         @test fld(-1.1, 0.1) == div(-1.1, 0.1, RoundDown) == floor(big(-1.1)/big(0.1)) == -12.0
     end
+    @testset "issue  #49450" begin
+        @test div(514, Float16(0.75)) === Float16(685)
+        @test fld(514, Float16(0.75)) === Float16(685)
+        @test cld(515, Float16(0.75)) === Float16(687)
+
+        @test cld(1, Float16(0.000999)) === Float16(1001)
+        @test cld(2, Float16(0.001999)) === Float16(1001)
+        @test cld(3, Float16(0.002934)) === Float16(1023)
+        @test cld(4, Float16(0.003998)) === Float16(1001)
+        @test fld(5, Float16(0.004925)) === Float16(1015)
+
+        @test div(4_194_307, Float32(0.75)) === Float32(5_592_409)
+        @test fld(4_194_307, Float32(0.75)) === Float32(5_592_409)
+        @test cld(4_194_308, Float32(0.75)) === Float32(5_592_411)
+
+        @test fld(5, Float32(6.556511e-7)) === Float32(7_626_007)
+        @test fld(10, Float32(1.3113022e-6)) === Float32(7_626_007)
+        @test fld(11, Float32(1.4305115e-6)) === Float32(7_689_557)
+        @test cld(16, Float32(2.8014183e-6)) === Float32(5_711_393)
+        @test cld(17, Float32(2.2053719e-6)) === Float32(7_708_451)
+    end
 end
 @testset "return types" begin
     for T in (Int8,Int16,Int32,Int64,Int128, UInt8,UInt16,UInt32,UInt64,UInt128)