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fix hypot with multiple arguments #38158

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Oct 30, 2020
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fix hypot with multiple arguments #38158

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95 changes: 45 additions & 50 deletions base/math.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -610,6 +610,10 @@ 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;
Expand All @@ -625,85 +629,81 @@ 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)
Expand Down Expand Up @@ -1150,12 +1150,7 @@ 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)))
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Shouldn't we still use this for hypot of `Float16?

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@vtjnash vtjnash Oct 26, 2020 •
edited
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We could, though as the PR was written, it would have been the wrong function to overload (since we want to always give it once chance to call float, then give it once chance to dispatch on the result, even if the type of the answer is the unchaged). It would be possible to define:

_hypot(a::Float16, b::Float16) = Float16(_hypot(Float32(a), Float32(b)))
_hypot(xs::Float16...) = Float16(_hypot(Float32.(xs)...))

But this also fall out of the generic definition _hypot.

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This indeed seems to have broken things #38311 (comment)

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)))

Expand Down
68 changes: 64 additions & 4 deletions test/math.jl
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Expand Up @@ -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)))
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Can there be a test hypot(Furlong(1), 1) etc. to check that an error is thrown? I think this might fail currently.

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This is what it does currently (master and PR):

julia> hypot(Furlongs.Furlong(1), 1)
ERROR: promotion of types Main.Furlongs.Furlong{1, Int64} and Int64 failed to change any arguments
Stacktrace:
 [1] error(::String, ::String, ::String)
   @ Base ./error.jl:42
 [2] sametype_error(input::Tuple{Main.Furlongs.Furlong{1, Int64}, Int64})
   @ Base ./promotion.jl:316
 [3] not_sametype(x::Tuple{Main.Furlongs.Furlong{1, Int64}, Int64}, y::Tuple{Main.Furlongs.Furlong{1, Int64}, Int64})
   @ Base ./promotion.jl:310
 [4] promote
   @ ./promotion.jl:293 [inlined]
 [5] hypot(::Main.Furlongs.Furlong{1, Int64}, ::Int64)
   @ Base.Math ./math.jl:641
 [6] top-level scope
   @ REPL[5]:1

And I added a test for it also to make sure that continues to operate the same. Is that what you were intending?

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Thanks for adding the test and for explaining that this PR doesn't change the error behavior of hypot(x, y) for disparate dimensions. I think it would be better for a user if hypot throws an error that has to do with a failure to compute on disparate dimensions instead of a promotion error, but that's not the point of this PR I suppose.

I still think it's the case that _hypot(Furlongs.Furlong(1), 1) will produce a meaningless result instead of an error. Perhaps that should be documented, even though this function should be not used externally.

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Looks like that also still fails, since we need to find the maximum value:

julia> Base.Math._hypot(Furlongs.Furlong(1), 1)
ERROR: MethodError: no method matching isless(::Main.Furlongs.Furlong{0, Float64}, ::Float64)
Closest candidates are:
  isless(::Main.Furlongs.Furlong{p, T} where T<:Number, ::Main.Furlongs.Furlong{p, T} where T<:Number) where p at /Users/jameson/julia/test/testhelpers/Furlongs.jl:64
  isless(::Any, ::Missing) at missing.jl:88
  isless(::Missing, ::Any) at missing.jl:87
  ...
Stacktrace:
 [1] <(x::Main.Furlongs.Furlong{0, Float64}, y::Float64)
   @ Base ./operators.jl:279
 [2] >(x::Float64, y::Main.Furlongs.Furlong{0, Float64})
   @ Base ./operators.jl:305
 [3] _hypot(x::Main.Furlongs.Furlong{1, Int64}, y::Int64)
   @ Base.Math ./math.jl:657
 [4] top-level scope
   @ REPL[2]:1

I can require them to have the same type via the method signature, but I'm more inclined to let the user call Math._hypot directly if they want to, and leave the error checking to the public hypot function.

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
7 changes: 7 additions & 0 deletions test/testhelpers/Furlongs.jl
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Expand Up @@ -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
Expand Down