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Make abs, abs_imag, inv, and / resistant to under/overflow (#122)
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* Use hypot in abs and abs_imag

* Use approximate check

* Add tests against abs/exp under/overflow

* Increment patch number

* Ensure tests pass on 1.0

* Backport _hypot code for older Julia versions

* Mark test as non broken

* Implement abs without hypot

* Implement abs_imag efficiently

* Handle over/underflow for inv and div

* Make inv implementation more like Complex's

* Test abs_imag for all Julia versions

* Add tests for inv under/overflow

* Test more abs/abs_imag cases

* Simplify tests with isequal

* Use isequal

* Add isequal

* Add under/overflow tests for div

* Add missing abs_imag tests

* Apply suggestions from code review

Co-authored-by: Yuto Horikawa <[email protected]>

* Update test/Quaternion.jl

Co-authored-by: Yuto Horikawa <[email protected]>

* Update test/Quaternion.jl

Co-authored-by: Yuto Horikawa <[email protected]>

* Increment patch number

Co-authored-by: Yuto Horikawa <[email protected]>
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@sethaxen @hyrodium
sethaxen and hyrodium authored Jan 9, 2023
1 parent 7e0efc6 commit dc7f5a1
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Showing 3 changed files with 128 additions and 6 deletions.
2 changes: 1 addition & 1 deletion Project.toml
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Original file line number Diff line number Diff line change
@@ -1,6 +1,6 @@
name = "Quaternions"
uuid = "94ee1d12-ae83-5a48-8b1c-48b8ff168ae0"
version = "0.7.3"
version = "0.7.4"

[deps]
LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
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58 changes: 54 additions & 4 deletions src/Quaternion.jl
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Expand Up @@ -136,11 +136,42 @@ Quaternion{Int64}(1, -2, -3, -4)
```
"""
Base.conj(q::Quaternion) = Quaternion(q.s, -q.v1, -q.v2, -q.v3)
Base.abs(q::Quaternion) = sqrt(abs2(q))
function Base.abs(q::Quaternion)
a = max(abs(q.s), abs(q.v1), abs(q.v2), abs(q.v3))
if isnan(a) && isinf(q)
return typeof(a)(Inf)
elseif iszero(a) || isinf(a)
return a
else
return sqrt(abs2(q / a)) * a
end
end
Base.float(q::Quaternion{T}) where T = convert(Quaternion{float(T)}, q)
abs_imag(q::Quaternion) = sqrt(q.v2 * q.v2 + (q.v1 * q.v1 + q.v3 * q.v3)) # ordered to match abs2
function abs_imag(q::Quaternion)
a = max(abs(q.v1), abs(q.v2), abs(q.v3))
if isnan(a) && (isinf(q.v1) | isinf(q.v2) | isinf(q.v3))
return oftype(a, Inf)
elseif iszero(a) || isinf(a)
return a
else
return sqrt((q.v1 / a)^2 + (q.v2 / a)^2 + (q.v3 / a)^2) * a
end
end
Base.abs2(q::Quaternion) = RealDot.realdot(q,q)
Base.inv(q::Quaternion) = conj(q) / abs2(q)
function Base.inv(q::Quaternion)
if isinf(q)
return quat(
copysign(zero(q.s), q.s),
flipsign(-zero(q.v1), q.v1),
flipsign(-zero(q.v2), q.v2),
flipsign(-zero(q.v3), q.v3),
)
end
a = max(abs(q.s), abs(q.v1), abs(q.v2), abs(q.v3))
p = q / a
iq = conj(p) / (a * abs2(p))
return iq
end

Base.isreal(q::Quaternion) = iszero(q.v1) & iszero(q.v2) & iszero(q.v3)
Base.isfinite(q::Quaternion) = isfinite(q.s) & isfinite(q.v1) & isfinite(q.v2) & isfinite(q.v3)
Expand Down Expand Up @@ -182,9 +213,28 @@ function Base.:*(q::Quaternion, w::Quaternion)
return Quaternion(s, v1, v2, v3)
end

Base.:/(q::Quaternion, w::Quaternion) = q * inv(w)
function Base.:/(q::Quaternion{T}, w::Quaternion{T}) where T
# handle over/underflow while matching the behavior of /(a::Complex, b::Complex)
a = max(abs(w.s), abs(w.v1), abs(w.v2), abs(w.v3))
if isinf(w)
if isfinite(q)
return quat(
zero(T)*sign(q.s)*sign(w.s),
-zero(T)*sign(q.v1)*sign(w.v1),
-zero(T)*sign(q.v2)*sign(w.v2),
-zero(T)*sign(q.v3)*sign(w.v3),
)
end
return quat(T(NaN), T(NaN), T(NaN), T(NaN))
end
p = w / a
return (q * conj(p)) / RealDot.realdot(w, p)
end

Base.:(==)(q::Quaternion, w::Quaternion) = (q.s == w.s) & (q.v1 == w.v1) & (q.v2 == w.v2) & (q.v3 == w.v3)
function Base.isequal(q::Quaternion, w::Quaternion)
isequal(q.s, w.s) & isequal(q.v1, w.v1) & isequal(q.v2, w.v2) & isequal(q.v3, w.v3)
end

"""
extend_analytic(f, q::Quaternion)
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74 changes: 73 additions & 1 deletion test/Quaternion.jl
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Expand Up @@ -59,6 +59,13 @@ end
@test Quaternion(1, 2, 3, 4) != Quaternion(1, 2, 3, 5)
end

@testset "isequal" begin
@test isequal(Quaternion(1, 2, 3, 4), Quaternion(1.0, 2.0, 3.0, 4.0))
@test !isequal(Quaternion(1, 2, 3, 4), Quaternion(5, 2, 3, 4))
@test isequal(Quaternion(NaN, -0.0, Inf, -Inf), Quaternion(NaN, -0.0, Inf, -Inf))
@test !isequal(Quaternion(NaN, 0.0, Inf, -Inf), Quaternion(NaN, -0.0, Inf, -Inf))
end

@testset "convert" begin
@test convert(Quaternion{Float64}, 1) === Quaternion(1.0)
@test convert(Quaternion{Float64}, Quaternion(1, 2, 3, 4)) ===
Expand Down Expand Up @@ -153,7 +160,31 @@ end
@test conj(conj(q)) === q
@test conj(conj(qnorm)) === qnorm
@test float(Quaternion(1, 2, 3, 4)) === float(Quaternion(1.0, 2.0, 3.0, 4.0))
@test Quaternions.abs_imag(q) == abs(Quaternion(0, q.v1, q.v2, q.v3))
@test Quaternions.abs_imag(q) ≈ abs(Quaternion(0, q.v1, q.v2, q.v3))
end

@testset "abs/abs_imag don't over/underflow" begin
for x in [1e-300, 1e300, -1e-300, -1e300]
@test abs(quat(x, 0, 0, 0)) == abs(x)
@test abs(quat(0, x, 0, 0)) == abs(x)
@test abs(quat(0, 0, x, 0)) == abs(x)
@test abs(quat(0, 0, 0, x)) == abs(x)
@test Quaternions.abs_imag(quat(0, x, 0, 0)) == abs(x)
@test Quaternions.abs_imag(quat(0, 0, x, 0)) == abs(x)
@test Quaternions.abs_imag(quat(0, 0, 0, x)) == abs(x)
end
@test isnan(abs(quat(NaN, NaN, NaN, NaN)))
@test abs(quat(NaN, Inf, NaN, NaN)) == Inf
@test abs(quat(-Inf, NaN, NaN, NaN)) == Inf
@test abs(quat(0.0)) == 0.0
@test abs(quat(Inf)) == Inf
@test abs(quat(1, -Inf, 2, 3)) == Inf
@test isnan(Quaternions.abs_imag(quat(0, NaN, NaN, NaN)))
@test Quaternions.abs_imag(quat(0, Inf, NaN, NaN)) == Inf
@test Quaternions.abs_imag(quat(0, NaN, -Inf, NaN)) == Inf
@test Quaternions.abs_imag(quat(0.0)) == 0.0
@test Quaternions.abs_imag(quat(0.0, 0.0, Inf, 0.0)) == Inf
@test Quaternions.abs_imag(quat(0, 1, -Inf, 2)) == Inf
end

@testset "algebraic properties" begin
Expand All @@ -171,6 +202,21 @@ end
end
end

@testset "inv does not under/overflow" begin
x = 1e-300
y = inv(x)
@test isequal(inv(quat(x, 0.0, 0.0, 0.0)), quat(y, -0.0, -0.0, -0.0))
@test isequal(inv(quat(0.0, x, 0.0, 0.0)), quat(0.0, -y, -0.0, -0.0))
@test isequal(inv(quat(0.0, 0.0, x, 0.0)), quat(0.0, -0.0, -y, -0.0))
@test isequal(inv(quat(0.0, 0.0, 0.0, x)), quat(0.0, -0.0, -0.0, -y))
@test isequal(inv(quat(y, 0.0, 0.0, 0.0)), quat(x, -0.0, -0.0, -0.0))
@test isequal(inv(quat(0.0, y, 0.0, 0.0)), quat(0.0, -x, -0.0, -0.0))
@test isequal(inv(quat(0.0, 0.0, y, 0.0)), quat(0.0, -0.0, -x, -0.0))
@test isequal(inv(quat(0.0, 0.0, 0.0, y)), quat(0.0, -0.0, -0.0, -x))
@test isequal(inv(quat(-Inf, 1, -2, 3)), quat(-0.0, -0.0, 0.0, -0.0))
@test isequal(inv(quat(1, -2, Inf, 3)), quat(0.0, 0.0, -0.0, -0.0))
end

@testset "isreal" begin
@test isreal(Quaternion(1, 0, 0, 0))
@test !isreal(Quaternion(2, 1, 0, 0))
Expand Down Expand Up @@ -275,6 +321,32 @@ end
@test q2 \ q ≈ inv(q2) * q
@test q / x ≈ x \ q ≈ inv(x) * q
end
@testset "no overflow/underflow" begin
@testset for x in [1e-300, 1e300, -1e-300, -1e300]
@test quat(x) / quat(x) == quat(1)
@test quat(x) / quat(0, x, 0, 0) == quat(0, -1, 0, 0)
@test quat(x) / quat(0, 0, x, 0) == quat(0, 0, -1, 0)
@test quat(x) / quat(0, 0, 0, x) == quat(0, 0, 0, -1)
@test quat(0, x, 0, 0) / quat(x, 0, 0, 0) == quat(0, 1, 0, 0)
@test quat(0, x, 0, 0) / quat(0, x, 0, 0) == quat(1, 0, 0, 0)
@test quat(0, x, 0, 0) / quat(0, 0, x, 0) == quat(0, 0, 0, -1)
@test quat(0, x, 0, 0) / quat(0, 0, 0, x) == quat(0, 0, 1, 0)
end
@testset for T in [Float32, Float64]
o = one(T)
z = zero(T)
inf = T(Inf)
nan = T(NaN)
@testset for s in [1, -1], t in [1, -1]
@test isequal(quat(o) / quat(s*inf), quat(s*z, -z, -z, -z))
@test isequal(quat(o) / quat(s*inf, t*o, z, t*z), quat(s*z, -t*z, -z, -t*z))
@test isequal(quat(o) / quat(s*inf, t*nan, t*z, z), quat(s*z, nan, -t*z, -z))
@test isequal(quat(o) / quat(s*inf, t*inf, t*z, z), quat(s*z, -t*z, -t*z, -z))
end
@test isequal(quat(inf) / quat(inf, 1, 2, 3), quat(nan, nan, nan, nan))
@test isequal(quat(inf) / quat(inf, 1, 2, -inf), quat(nan, nan, nan, nan))
end
end
end

@testset "^" begin
Expand Down

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Registration pull request created: JuliaRegistries/General/75386

After the above pull request is merged, it is recommended that a tag is created on this repository for the registered package version.

This will be done automatically if the Julia TagBot GitHub Action is installed, or can be done manually through the github interface, or via:

git tag -a v0.7.4 -m "<description of version>" dc7f5a1f8e8e107af3706b1af75f98d8ae68e956
git push origin v0.7.4

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