Skip to content

Avoid catastrophic cancellation in the derivatives of tanh and inverse trigonometric/hyperbolic functions #122

New issue

Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.

Sign up for GitHub

By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.

Already on GitHub? Sign in to your account

Merged
devmotion merged 3 commits into from
Nov 10, 2026
Merged

Avoid catastrophic cancellation in the derivatives of tanh and inverse trigonometric/hyperbolic functions #122

Show file tree
Hide file tree
Changes from all commits
Commits
Show all changes
3 commits
Select commit Hold shift + click to select a range
7a6fa44
Avoid catastrophic cancellation in tanh and inverse trig/hyperbolic d…
devmotion Nov 10, 2026
798d17f
Avoid overflow in the derivative of NaNMath.acosh
devmotion Nov 10, 2026
cf65762
Bump version to 1.17.0
devmotion Nov 10, 2026
File filter

Filter by extension

Filter by extension
Conversations
Failed to load comments.
Loading
Jump to
Jump to file
Failed to load files.
Loading
Diff view
Diff view
2 changes: 1 addition & 1 deletion Project.toml
View file
Open in desktop
Original file line number Diff line number Diff line change
@@ -1,6 +1,6 @@
name = "DiffRules"
uuid = "b552c78f-8df3-52c6-915a-8e097449b14b"
version = "1.17.0-dev"
version = "1.17.0"

[deps]
IrrationalConstants = "92d709cd-6900-40b7-9082-c6be49f344b6"
Expand Down
38 changes: 19 additions & 19 deletions src/rules.jl
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -12,8 +12,8 @@
@define_diffrule Base.abs2(x) = :( $x + $x )
@define_diffrule Base.inv(x) = :( -abs2(inv($x)) )
@define_diffrule Base.log(x) = :( inv($x) )
@define_diffrule Base.log10(x) = :( inv($x) / $logten )
@define_diffrule Base.log2(x) = :( inv($x) / $logtwo )
@define_diffrule Base.log10(x) = :( inv($logten * $x) )
@define_diffrule Base.log2(x) = :( inv($logtwo * $x) )
@define_diffrule Base.log1p(x) = :( inv($x + 1) )
@define_diffrule Base.exp(x) = :( exp($x) )
@define_diffrule Base.exp2(x) = :( exp2($x) * $logtwo )
Expand All @@ -33,30 +33,30 @@
@define_diffrule Base.cotd(x) = :( - deg2rad(1 + cotd($x)^2) )
@define_diffrule Base.sinpi(x) = :( π * cospi($x) )
@define_diffrule Base.cospi(x) = :( -(π * sinpi($x)) )
@define_diffrule Base.asin(x) = :( inv(sqrt(1 - $x^2)) )
@define_diffrule Base.acos(x) = :( -inv(sqrt(1 - $x^2)) )
@define_diffrule Base.asin(x) = :( inv(sqrt((1 - $x) * (1 + $x))) )
@define_diffrule Base.acos(x) = :( -inv(sqrt((1 - $x) * (1 + $x))) )
@define_diffrule Base.atan(x) = :( inv(1 + $x^2) )
@define_diffrule Base.asec(x) = :( inv(abs($x) * sqrt($x^2 - 1)) )
@define_diffrule Base.acsc(x) = :( -inv(abs($x) * sqrt($x^2 - 1)) )
@define_diffrule Base.asec(x) = :( inv(abs($x) * sqrt(($x - 1) * ($x + 1))) )
@define_diffrule Base.acsc(x) = :( -inv(abs($x) * sqrt(($x - 1) * ($x + 1))) )
@define_diffrule Base.acot(x) = :( -inv(1 + $x^2) )
@define_diffrule Base.asind(x) = :( inv(deg2rad(sqrt(1 - $x^2))) )
@define_diffrule Base.acosd(x) = :( -inv(deg2rad(sqrt(1 - $x^2))) )
@define_diffrule Base.asind(x) = :( inv(deg2rad(sqrt((1 - $x) * (1 + $x)))) )
@define_diffrule Base.acosd(x) = :( -inv(deg2rad(sqrt((1 - $x) * (1 + $x)))) )
@define_diffrule Base.atand(x) = :( inv(deg2rad(1 + $x^2)) )
@define_diffrule Base.asecd(x) = :( inv(deg2rad(abs($x) * sqrt($x^2 - 1))) )
@define_diffrule Base.acscd(x) = :( -inv(deg2rad(abs($x) * sqrt($x^2 - 1))) )
@define_diffrule Base.asecd(x) = :( inv(deg2rad(abs($x) * sqrt(($x - 1) * ($x + 1)))) )
@define_diffrule Base.acscd(x) = :( -inv(deg2rad(abs($x) * sqrt(($x - 1) * ($x + 1)))) )
@define_diffrule Base.acotd(x) = :( -inv(deg2rad(1 + $x^2)) )
@define_diffrule Base.sinh(x) = :( cosh($x) )
@define_diffrule Base.cosh(x) = :( sinh($x) )
@define_diffrule Base.tanh(x) = :( 1 - tanh($x)^2 )
@define_diffrule Base.tanh(x) = :( sech($x)^2 )
@define_diffrule Base.sech(x) = :( -tanh($x) * sech($x) )
@define_diffrule Base.csch(x) = :( -coth($x) * csch($x) )
@define_diffrule Base.coth(x) = :( -(csch($x)^2) )
@define_diffrule Base.asinh(x) = :( inv(sqrt($x^2 + 1)) )
@define_diffrule Base.acosh(x) = :( inv(sqrt($x^2 - 1)) )
@define_diffrule Base.atanh(x) = :( inv(1 - $x^2) )
@define_diffrule Base.asech(x) = :( -inv($x * sqrt(1 - $x^2)) )
@define_diffrule Base.acosh(x) = :( inv(sqrt($x - 1) * sqrt($x + 1)) )
@define_diffrule Base.atanh(x) = :( inv((1 - $x) * (1 + $x)) )
@define_diffrule Base.asech(x) = :( -inv($x * sqrt((1 - $x) * (1 + $x))) )
@define_diffrule Base.acsch(x) = :( -inv(abs($x) * sqrt(1 + $x^2)) )
@define_diffrule Base.acoth(x) = :( inv(1 - $x^2) )
@define_diffrule Base.acoth(x) = :( inv((1 - $x) * (1 + $x)) )
@define_diffrule Base.sinc(x) = :( cosc($x) )
@define_diffrule Base.deg2rad(x) = :( deg2rad(one($x)) )
@define_diffrule Base.mod2pi(x) = :( isinteger($x / $twoπ) ? oftype(float($x), NaN) : one(float($x)) )
Expand Down Expand Up @@ -265,10 +265,10 @@ _abs_deriv(x) = signbit(x) ? -one(x) : one(x)
@define_diffrule NaNMath.sin(x) = :( NaNMath.cos($x) )
@define_diffrule NaNMath.cos(x) = :( -NaNMath.sin($x) )
@define_diffrule NaNMath.tan(x) = :( 1 + NaNMath.pow(NaNMath.tan($x), 2) )
@define_diffrule NaNMath.asin(x) = :( inv(NaNMath.sqrt(1 - NaNMath.pow($x, 2))) )
@define_diffrule NaNMath.acos(x) = :( -inv(NaNMath.sqrt(1 - NaNMath.pow($x, 2))) )
@define_diffrule NaNMath.acosh(x) = :( inv(NaNMath.sqrt(NaNMath.pow($x, 2) - 1)) )
@define_diffrule NaNMath.atanh(x) = :( inv(1 - NaNMath.pow($x, 2)) )
@define_diffrule NaNMath.asin(x) = :( inv(NaNMath.sqrt((1 - $x) * (1 + $x))) )
@define_diffrule NaNMath.acos(x) = :( -inv(NaNMath.sqrt((1 - $x) * (1 + $x))) )
@define_diffrule NaNMath.acosh(x) = :( inv(NaNMath.sqrt($x - 1) * NaNMath.sqrt($x + 1)) )
@define_diffrule NaNMath.atanh(x) = :( inv((1 - $x) * (1 + $x)) )
@define_diffrule NaNMath.log(x) = :( inv($x) )
@define_diffrule NaNMath.log2(x) = :( inv($logtwo * $x) )
@define_diffrule NaNMath.log10(x) = :( inv($logten * $x) )
Expand Down
39 changes: 39 additions & 0 deletions test/runtests.jl
View file
Open in desktop
Original file line number Diff line number Diff line change
Expand Up @@ -186,6 +186,45 @@ for xytype in [:Float32, :Float64, :BigFloat]
end
end

# Rules must not suffer from catastrophic cancellation, e.g., `1 - x^2` near `|x| = 1` or
# `1 - tanh(x)^2` for large `|x|` (#102): compare with the rule evaluated in high precision
below_one(T) = (1 - 4 * eps(T), -(1 - 64 * eps(T))) # 1 ± x for |x| < 1 near 1
above_one(T) = (1 + 4 * eps(T), -(1 + 64 * eps(T))) # x ± 1 for |x| > 1 near 1
acosh_xs(T) = (abs.(above_one(T))..., 2 * sqrt(floatmax(T))) # x^2 overflows for the last
@testset "accuracy: ($M, $f)" for (M, f, xs) in [
(:Base, :asin, below_one),
(:Base, :acos, below_one),
(:Base, :asind, below_one),
(:Base, :acosd, below_one),
(:Base, :atanh, below_one),
(:Base, :asech, T -> abs.(below_one(T))),
(:Base, :acoth, above_one),
(:Base, :acosh, acosh_xs),
(:Base, :asec, above_one),
(:Base, :acsc, above_one),
(:Base, :asecd, above_one),
(:Base, :acscd, above_one),
(:Base, :tanh, T -> T.((0.5, -5, 10, -20, 30, 300))),
(:NaNMath, :asin, below_one),
(:NaNMath, :acos, below_one),
(:NaNMath, :atanh, below_one),
(:NaNMath, :acosh, acosh_xs),
]
deriv = DiffRules.diffrule(M, f, :x)
@eval let
for T in (Float32, Float64), x in $xs(T)
# `x` is exact in `BigFloat`, so this is the derivative at `x` up to ~1e-70
expected = setprecision(BigFloat, 256) do
let x = big(x)
T($deriv)
end
end
@test $deriv isa T
@test $deriv ≈ expected rtol = 8 * eps(T)
end
end
end

end

@testset "diffrules" begin
Expand Down
Loading