diff --git a/Project.toml b/Project.toml
index 181190c..f524ed8 100644
--- a/Project.toml
+++ b/Project.toml
@@ -3,6 +3,9 @@ uuid = "c1ae055f-0cd5-4b69-90a6-9a35b1a98df9"
 authors = ["David Widmann"]
 version = "0.1.0"
 
+[deps]
+LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+
 [compat]
 julia = "1"
 
diff --git a/README.md b/README.md
index c2f89a2..383c148 100644
--- a/README.md
+++ b/README.md
@@ -4,3 +4,19 @@
 [![Coverage](https://codecov.io/gh/devmotion/RealDot.jl/branch/main/graph/badge.svg)](https://codecov.io/gh/devmotion/RealDot.jl)
 [![Coverage](https://coveralls.io/repos/github/devmotion/RealDot.jl/badge.svg?branch=main)](https://coveralls.io/github/devmotion/RealDot.jl?branch=main)
 [![Code Style: Blue](https://img.shields.io/badge/code%20style-blue-4495d1.svg)](https://github.com/invenia/BlueStyle)
+
+This package only contains and exports a single function `realdot(x, y)`.
+It computes `real(LinearAlgebra.dot(x, y))` while avoiding computing the imaginary part of `LinearAlgebra.dot(x, y)` if possible.
+
+The real dot product is useful when one treats complex numbers as embedded in a real vector space.
+For example, take two complex arrays `x` and `y`.
+Their real dot product is `real(dot(x, y)) == dot(real(x), real(y)) + dot(imag(x), imag(y))`.
+This is the same result one would get by reinterpreting the arrays as real arrays:
+```julia
+xreal = reinterpret(real(eltype(x)), x)
+yreal = reinterpret(real(eltype(y)), y)
+real(dot(x, y)) == dot(xreal, yreal)
+```
+
+In particular, this function can be useful if you define pullbacks for non-holomorphic functions (see e.g. [this discussion in the ChainRulesCore.jl repo](https://github.com/JuliaDiff/ChainRulesCore.jl/pull/474)).
+It was implemented initially in [ChainRules.jl](https://github.com/JuliaDiff/ChainRules.jl) in [this PR](https://github.com/JuliaDiff/ChainRules.jl/pull/216) as `_realconjtimes`.
diff --git a/src/RealDot.jl b/src/RealDot.jl
index d9d83db..6f286d9 100644
--- a/src/RealDot.jl
+++ b/src/RealDot.jl
@@ -1,5 +1,22 @@
 module RealDot
 
-# Write your package code here.
+using LinearAlgebra: LinearAlgebra
+
+export realdot
+
+"""
+    realdot(x, y)
+
+Compute `real(dot(x, y))` while avoiding computing the imaginary part if possible.
+
+This function can be useful if you work with derivatives of functions on complex
+numbers. In particular, this computation shows up in pullbacks for non-holomorphic
+functions.
+"""
+@inline realdot(x, y) = real(LinearAlgebra.dot(x, y))
+@inline realdot(x::Complex, y::Complex) = muladd(real(x), real(y), imag(x) * imag(y))
+@inline realdot(x::Real, y::Number) = x * real(y)
+@inline realdot(x::Number, y::Real) = real(x) * y
+@inline realdot(x::Real, y::Real) = x * y
 
 end
diff --git a/test/runtests.jl b/test/runtests.jl
index cc75a0c..7cdb6d5 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -1,6 +1,63 @@
 using RealDot
+using LinearAlgebra
 using Test
 
+# struct need to be defined outside of tests for julia 1.0 compat
+# custom complex number (tests fallback definition)
+struct CustomComplex{T} <: Number
+    re::T
+    im::T
+end
+
+Base.real(x::CustomComplex) = x.re
+Base.imag(x::CustomComplex) = x.im
+
+Base.conj(x::CustomComplex) = CustomComplex(x.re, -x.im)
+
+function Base.:*(x::CustomComplex, y::Union{Real,Complex})
+    return CustomComplex(reim(Complex(reim(x)...) * y)...)
+end
+Base.:*(x::Union{Real,Complex}, y::CustomComplex) = y * x
+function Base.:*(x::CustomComplex, y::CustomComplex)
+    return CustomComplex(reim(Complex(reim(x)...) * Complex(reim(y)...))...)
+end
+
+# custom quaternion to test definition for hypercomplex numbers
+# adapted from Quaternions.jl
+struct Quaternion{T<:Real} <: Number
+    s::T
+    v1::T
+    v2::T
+    v3::T
+end
+
+Base.real(q::Quaternion) = q.s
+Base.conj(q::Quaternion) = Quaternion(q.s, -q.v1, -q.v2, -q.v3)
+
+function Base.:*(q::Quaternion, w::Quaternion)
+    return Quaternion(
+        q.s * w.s - q.v1 * w.v1 - q.v2 * w.v2 - q.v3 * w.v3,
+        q.s * w.v1 + q.v1 * w.s + q.v2 * w.v3 - q.v3 * w.v2,
+        q.s * w.v2 - q.v1 * w.v3 + q.v2 * w.s + q.v3 * w.v1,
+        q.s * w.v3 + q.v1 * w.v2 - q.v2 * w.v1 + q.v3 * w.s,
+    )
+end
+
+function Base.:*(q::Quaternion, w::Union{Real,Complex,CustomComplex})
+    a, b = reim(w)
+    return q * Quaternion(a, b, zero(a), zero(a))
+end
+Base.:*(w::Union{Real,Complex,CustomComplex}, q::Quaternion) = conj(conj(q) * conj(w))
+
 @testset "RealDot.jl" begin
-    # Write your tests here.
+    scalars = (
+        randn(), randn(ComplexF64), CustomComplex(randn(2)...), Quaternion(randn(4)...)
+    )
+    arrays = (randn(10), randn(ComplexF64, 10))
+
+    for inputs in (scalars, arrays)
+        for x in inputs, y in inputs
+            @test realdot(x, y) == real(dot(x, y))
+        end
+    end
 end