diff --git a/base/math.jl b/base/math.jl
index c78816545a2eb..a9f259e854ee8 100644
--- a/base/math.jl
+++ b/base/math.jl
@@ -606,6 +606,10 @@ by Carlos F. Borges
 The article is available online at ArXiv at the link
   https://arxiv.org/abs/1904.09481
 
+    hypot(x...)
+
+Compute the hypotenuse ``\\sqrt{\\sum |x_i|^2}`` avoiding overflow and underflow.
+
 # Examples
 ```jldoctest; filter = r"Stacktrace:(\\n \\[[0-9]+\\].*)*"
 julia> a = Int64(10)^10;
@@ -621,85 +625,81 @@ Stacktrace:
 
 julia> hypot(3, 4im)
 5.0
+
+julia> hypot(-5.7)
+5.7
+
+julia> hypot(3, 4im, 12.0)
+13.0
 ```
 """
-hypot(x::Number, y::Number) = hypot(promote(x, y)...)
-hypot(x::Complex, y::Complex) = hypot(abs(x), abs(y))
-hypot(x::T, y::T) where {T<:Real} = hypot(float(x), float(y))
-function hypot(x::T, y::T) where {T<:Number}
-    if !iszero(x)
-        z = y/x
-        z2 = z*z
+hypot(x::Number) = abs(float(x))
+hypot(x::Number, y::Number, xs::Number...) = _hypot(float.(promote(x, y, xs...))...)
+function _hypot(x, y)
+    # preserves unit
+    axu = abs(x)
+    ayu = abs(y)
 
-        abs(x) * sqrt(oneunit(z2) + z2)
-    else
-        abs(y)
-    end
-end
+    # unitless
+    ax = axu / oneunit(axu)
+    ay = ayu / oneunit(ayu)
 
-function hypot(x::T, y::T) where T<:AbstractFloat
     # Return Inf if either or both inputs is Inf (Compliance with IEEE754)
-    if isinf(x) || isinf(y)
-        return T(Inf)
+    if isinf(ax) || isinf(ay)
+        return oftype(axu, Inf)
     end
 
     # Order the operands
-    ax,ay = abs(x), abs(y)
     if ay > ax
-        ax,ay = ay,ax
+        axu, ayu = ayu, axu
+        ax, ay = ay, ax
     end
 
     # Widely varying operands
-    if ay <= ax*sqrt(eps(T)/2)  #Note: This also gets ay == 0
-        return ax
+    if ay <= ax*sqrt(eps(typeof(ax))/2)  #Note: This also gets ay == 0
+        return axu
     end
 
     # Operands do not vary widely
-    scale = eps(T)*sqrt(floatmin(T))  #Rescaling constant
-    if ax > sqrt(floatmax(T)/2)
+    scale = eps(typeof(ax))*sqrt(floatmin(ax))  #Rescaling constant
+    if ax > sqrt(floatmax(ax)/2)
         ax = ax*scale
         ay = ay*scale
         scale = inv(scale)
-    elseif ay < sqrt(floatmin(T))
+    elseif ay < sqrt(floatmin(ax))
         ax = ax/scale
         ay = ay/scale
     else
-        scale = one(scale)
+        scale = oneunit(scale)
     end
-    h = sqrt(muladd(ax,ax,ay*ay))
+    h = sqrt(muladd(ax, ax, ay*ay))
     # This branch is correctly rounded but requires a native hardware fma.
     if Base.Math.FMA_NATIVE
         hsquared = h*h
         axsquared = ax*ax
-        h -= (fma(-ay,ay,hsquared-axsquared) + fma(h,h,-hsquared) - fma(ax,ax,-axsquared))/(2*h)
+        h -= (fma(-ay, ay, hsquared-axsquared) + fma(h, h,-hsquared) - fma(ax, ax, -axsquared))/(2*h)
     # This branch is within one ulp of correctly rounded.
     else
         if h <= 2*ay
             delta = h-ay
-            h -= muladd(delta,delta-2*(ax-ay),ax*(2*delta - ax))/(2*h)
+            h -= muladd(delta, delta-2*(ax-ay), ax*(2*delta - ax))/(2*h)
         else
             delta = h-ax
-            h -= muladd(delta,delta,muladd(ay,(4*delta-ay),2*delta*(ax-2*ay)))/(2*h)
+            h -= muladd(delta, delta, muladd(ay, (4*delta - ay), 2*delta*(ax - 2*ay)))/(2*h)
         end
     end
-    return h*scale
+    return h*scale*oneunit(axu)
+end
+function _hypot(x...)
+    maxabs = maximum(abs, x)
+    if isnan(maxabs) && any(isinf, x)
+        return oftype(maxabs, Inf)
+    elseif (iszero(maxabs) || isinf(maxabs))
+        return maxabs
+    else
+        return maxabs * sqrt(sum(y -> abs2(y / maxabs), x))
+    end
 end
-
-"""
-    hypot(x...)
-
-Compute the hypotenuse ``\\sqrt{\\sum |x_i|^2}`` avoiding overflow and underflow.
-
-# Examples
-```jldoctest
-julia> hypot(-5.7)
-5.7
-
-julia> hypot(3, 4im, 12.0)
-13.0
-```
-"""
-hypot(x::Number...) = sqrt(sum(abs2(y) for y in x))
 
 atan(y::Real, x::Real) = atan(promote(float(y),float(x))...)
 atan(y::T, x::T) where {T<:AbstractFloat} = Base.no_op_err("atan", T)
@@ -1146,12 +1146,7 @@ for func in (:sin,:cos,:tan,:asin,:acos,:atan,:sinh,:cosh,:tanh,:asinh,:acosh,
     end
 end
 
-for func in (:atan,:hypot)
-    @eval begin
-        $func(a::Float16,b::Float16) = Float16($func(Float32(a),Float32(b)))
-    end
-end
-
+atan(a::Float16,b::Float16) = Float16(atan(Float32(a),Float32(b)))
 cbrt(a::Float16) = Float16(cbrt(Float32(a)))
 sincos(a::Float16) = Float16.(sincos(Float32(a)))
 
diff --git a/test/math.jl b/test/math.jl
index 35e341801a73b..eaf782334c213 100644
--- a/test/math.jl
+++ b/test/math.jl
@@ -1107,8 +1107,68 @@ end
 
     isdefined(Main, :Furlongs) || @eval Main include("testhelpers/Furlongs.jl")
     using .Main.Furlongs
-    @test hypot(Furlong(0), Furlong(0)) == Furlong(0.0)
-    @test hypot(Furlong(3), Furlong(4)) == Furlong(5.0)
-    @test hypot(Complex(3), Complex(4)) === 5.0
-    @test hypot(Complex(6, 8), Complex(8, 6)) === 10.0*sqrt(2)
+    @test (@inferred hypot(Furlong(0), Furlong(0))) == Furlong(0.0)
+    @test (@inferred hypot(Furlong(3), Furlong(4))) == Furlong(5.0)
+    @test (@inferred hypot(Furlong(NaN), Furlong(Inf))) == Furlong(Inf)
+    @test (@inferred hypot(Furlong(Inf), Furlong(NaN))) == Furlong(Inf)
+    @test (@inferred hypot(Furlong(0), Furlong(0), Furlong(0))) == Furlong(0.0)
+    @test (@inferred hypot(Furlong(Inf), Furlong(Inf))) == Furlong(Inf)
+    @test (@inferred hypot(Furlong(1), Furlong(1), Furlong(1))) == Furlong(sqrt(3))
+    @test (@inferred hypot(Furlong(Inf), Furlong(NaN), Furlong(0))) == Furlong(Inf)
+    @test (@inferred hypot(Furlong(Inf), Furlong(Inf), Furlong(Inf))) == Furlong(Inf)
+    @test isnan(hypot(Furlong(NaN), Furlong(0), Furlong(1)))
+    ex = @test_throws ErrorException hypot(Furlong(1), 1)
+    @test startswith(ex.value.msg, "promotion of types ")
+
+    @test_throws MethodError hypot()
+    @test (@inferred hypot(floatmax())) == floatmax()
+    @test (@inferred hypot(floatmax(), floatmax())) == Inf
+    @test (@inferred hypot(floatmin(), floatmin())) == √2floatmin()
+    @test (@inferred hypot(floatmin(), floatmin(), floatmin())) == √3floatmin()
+    @test (@inferred hypot(1e-162)) ≈ 1e-162
+    @test (@inferred hypot(2e-162, 1e-162, 1e-162)) ≈ hypot(2, 1, 1)*1e-162
+    @test (@inferred hypot(1e162)) ≈ 1e162
+    @test hypot(-2) === 2.0
+    @test hypot(-2, 0) === 2.0
+    let i = typemax(Int)
+        @test (@inferred hypot(i, i)) ≈ i * √2
+        @test (@inferred hypot(i, i, i)) ≈ i * √3
+        @test (@inferred hypot(i, i, i, i)) ≈ 2.0i
+        @test (@inferred hypot(i//1, 1//i, 1//i)) ≈ i
+    end
+    let i = typemin(Int)
+        @test (@inferred hypot(i, i)) ≈ -√2i
+        @test (@inferred hypot(i, i, i)) ≈ -√3i
+        @test (@inferred hypot(i, i, i, i)) ≈ -2.0i
+    end
+    @testset "$T" for T in (Float32, Float64)
+        @test (@inferred hypot(T(Inf), T(NaN))) == T(Inf) # IEEE754 says so
+        @test (@inferred hypot(T(Inf), T(3//2), T(NaN))) == T(Inf)
+        @test (@inferred hypot(T(1e10), T(1e10), T(1e10), T(1e10))) ≈ 2e10
+        @test isnan_type(T, hypot(T(3), T(3//4), T(NaN)))
+        @test hypot(T(1), T(0)) === T(1)
+	    @test hypot(T(1), T(0), T(0)) === T(1)
+        @test (@inferred hypot(T(Inf), T(Inf), T(Inf))) == T(Inf)
+        for s in (zero(T), floatmin(T)*1e3, floatmax(T)*1e-3, T(Inf))
+            @test hypot(1s, 2s)     ≈ s * hypot(1, 2)   rtol=8eps(T)
+            @test hypot(1s, 2s, 3s) ≈ s * hypot(1, 2, 3) rtol=8eps(T)
+        end
+    end
+    @testset "$T" for T in (Float16, Float32, Float64, BigFloat)
+        let x = 1.1sqrt(floatmin(T))
+            @test (@inferred hypot(x, x/4)) ≈ x * sqrt(17/BigFloat(16))
+            @test (@inferred hypot(x, x/4, x/4)) ≈ x * sqrt(9/BigFloat(8))
+        end
+        let x = 2sqrt(nextfloat(zero(T)))
+            @test (@inferred hypot(x, x/4)) ≈ x * sqrt(17/BigFloat(16))
+            @test (@inferred hypot(x, x/4, x/4)) ≈ x * sqrt(9/BigFloat(8))
+        end
+        let x = sqrt(nextfloat(zero(T))/eps(T))/8, f = sqrt(4eps(T))
+            @test hypot(x, x*f) ≈ x * hypot(one(f), f) rtol=eps(T)
+            @test hypot(x, x*f, x*f) ≈ x * hypot(one(f), f, f) rtol=eps(T)
+        end
+    end
+    # hypot on Complex returns Real
+    @test (@inferred hypot(3, 4im)) === 5.0
+    @test (@inferred hypot(3, 4im, 12)) === 13.0
 end
diff --git a/test/testhelpers/Furlongs.jl b/test/testhelpers/Furlongs.jl
index 2f43834674e8e..8abe583925d93 100644
--- a/test/testhelpers/Furlongs.jl
+++ b/test/testhelpers/Furlongs.jl
@@ -29,6 +29,13 @@ Base.oneunit(x::Type{Furlong{p,T}}) where {p,T} = Furlong{p,T}(one(T))
 Base.zero(x::Furlong{p,T}) where {p,T} = Furlong{p,T}(zero(T))
 Base.zero(::Type{Furlong{p,T}}) where {p,T} = Furlong{p,T}(zero(T))
 Base.iszero(x::Furlong) = iszero(x.val)
+Base.float(x::Furlong{p}) where {p} = Furlong{p}(float(x.val))
+Base.eps(::Type{Furlong{p,T}}) where {p,T<:AbstractFloat} = Furlong{p}(eps(T))
+Base.eps(::Furlong{p,T}) where {p,T<:AbstractFloat} = eps(Furlong{p,T})
+Base.floatmin(::Type{Furlong{p,T}}) where {p,T<:AbstractFloat} = Furlong{p}(floatmin(T))
+Base.floatmin(::Furlong{p,T}) where {p,T<:AbstractFloat} = floatmin(Furlong{p,T})
+Base.floatmax(::Type{Furlong{p,T}}) where {p,T<:AbstractFloat} = Furlong{p}(floatmax(T))
+Base.floatmax(::Furlong{p,T}) where {p,T<:AbstractFloat} = floatmax(Furlong{p,T})
 
 # convert Furlong exponent p to a canonical form.  This
 # is not type stable, but it doesn't matter since it is used