import

test/erf.jl

@@ -0,0 +1,137 @@
+function compute_f1(p::Vector{Float64}; x::Float64=0.0)
+  L, μ, σ = p
+
+  sq2σ = sqrt(2) * σ
+  f1 = (x - (μ - 0.5L)) / sq2σ
+
+  return f1
+end
+
+function compute_J_f1(p::Vector{Float64}; x::Float64=0.0)
+  L, μ, σ = p
+
+  sq2σ = sqrt(2) * σ
+  J_f1 = [0.5 -1 -(x - (μ - 0.5L)) / σ] / sq2σ
+  return J_f1
+end
+
+function compute_H_f1(p::Vector{Float64}; x::Float64=0.0)
+  L, μ, σ = p
+
+  sq2σ = sqrt(2) * σ
+
+  J_f1 = [0.5 -1 -(x - (μ - 0.5L)) / σ] / sq2σ
+
+  H_f1 = zeros(3, 3)
+  H_f1[1, 3] = H_f1[3, 1] = -0.5 / (σ * sq2σ)
+  H_f1[2, 3] = H_f1[3, 2] = 1 / (σ * sq2σ)
+  H_f1[3, 3] = -2 * J_f1[3] / σ
+
+  return H_f1
+end
+
+function compute_f2(p::Vector{Float64}; x::Float64=0.0)
+  L, μ, σ = p
+
+  sq2σ = sqrt(2) * σ
+  f2 = (x - (μ + 0.5L)) / sq2σ
+
+  return f2
+end
+
+function compute_J_f2(p::Vector{Float64}; x::Float64=0.0)
+  L, μ, σ = p
+
+  sq2σ = sqrt(2) * σ
+  J_f2 = [-0.5 -1 -(x - (μ + 0.5L)) / σ] / sq2σ
+  return J_f2
+end
+
+function compute_H_f2(p::Vector{Float64}; x::Float64=0.0)
+  L, μ, σ = p
+
+  sq2σ = sqrt(2) * σ
+
+  J_f2 = [-0.5 -1 -(x - (μ + 0.5L)) / σ] / sq2σ
+
+  H_f2 = zeros(3, 3)
+  H_f2[1, 3] = H_f2[3, 1] = 0.5 / (σ * sq2σ)
+  H_f2[2, 3] = H_f2[3, 2] = 1 / (σ * sq2σ)
+  H_f2[3, 3] = -2 * J_f2[3] / σ
+
+  return H_f2
+end
+
+@testset "Erf aux functions" begin
+  L = 10 * (rand() - 0.5)
+  μ = 10 * (rand() - 0.5)
+  σ = 10rand()
+
+  x = σ * (2 * rand() - 0.5) + μ
+  p = [L, μ, σ]
+
+  @test TaylorTest.check(compute_f1, compute_J_f1, p; x=x)
+  @test TaylorTest.check(compute_J_f1, compute_H_f1, p; x=x)
+  @test TaylorTest.check(compute_f2, compute_J_f2, p; x=x)
+  @test TaylorTest.check(compute_J_f2, compute_H_f2, p; x=x)
+end
+
+@testset "Difference of error functions" begin
+  function diff_of_erf(p::Vector{Float64}; x::Float64=0.0)
+    L, μ, σ = p
+
+    f1 = compute_f1(p; x=x)
+    f2 = compute_f2(p; x=x)
+
+    return 0.5 * erf(f2, f1)
+  end
+
+  function J_diff_of_erf(p::Vector{Float64}; x::Float64=0.0)
+    # d/dx erf(x)/2 = gauss([sqrt(2), 0, 1]; x=x)
+    L, μ, σ = p
+    sq2σ = sqrt(2) * σ
+
+    # d/dξ ( erf( f1(x) ) - erf( f2(x) ) )/2 = d/dξ f1(x) gauss([sqrt(2), 0, 1]; x=f1(x)) - d/dξ f2(x) gauss([sqrt(2), 0, 1]; x=f2(x))
+
+    f1 = compute_f1(p; x=x)
+    f2 = compute_f2(p; x=x)
+
+    J_f1 = compute_J_f1(p; x=x)
+    J_f2 = compute_J_f2(p; x=x)
+    return J_f1 .* gauss([sqrt(2), 0, 1]; x=f1) - J_f2 .* gauss([sqrt(2), 0, 1]; x=f2)
+  end
+
+  function H_diff_of_erf(p::Vector{Float64}; x::Float64=0.0)
+    # d/dx erf(x)/2 = gauss([sqrt(2), 0, 1]; x=x)
+    L, μ, σ = p
+    sq2σ = sqrt(2) * σ
+
+    # d²/dξ² ( erf( f1(x) ) - erf( f2(x) ) )/2 =   d²/dξ² f1(x) gauss([sqrt(2), 0, 1]; x=f1(x)) - d²/dξ² f2(x) gauss([sqrt(2), 0, 1]; x=f2(x))
+    # + (d/dξ f1(x))^2 gauss([sqrt(2), 0, 1]; x=f1(x)) - d²/dξ² f2(x) gauss([sqrt(2), 0, 1]; x=f2(x))
+
+    f1 = compute_f1(p; x=x)
+    f2 = compute_f2(p; x=x)
+
+    J_f1 = compute_J_f1(p; x=x)
+    J_f2 = compute_J_f2(p; x=x)
+
+    H_f1 = compute_H_f1(p; x=x)
+    H_f2 = compute_H_f2(p; x=x)
+
+    return (H_f1 .* gauss([sqrt(2), 0, 1]; x=f1) - H_f2 .* gauss([sqrt(2), 0, 1]; x=f2)
+            -
+            J_f1' * J_f1 .* f1 .* gauss([2 * sqrt(2), 0, 1]; x=f1) + J_f2' * J_f2 .* f2 .* gauss([2 * sqrt(2), 0, 1]; x=f2))
+  end
+
+  L = 10 * (rand() - 0.5)
+  μ = 10 * (rand() - 0.5)
+  σ = 10rand()
+
+  x = σ * (2 * rand() - 0.5) + μ
+  p = [L, μ, σ]
+
+  @test TaylorTest.check(diff_of_erf, J_diff_of_erf, p; x=x)
+  @test TaylorTest.check(J_diff_of_erf, H_diff_of_erf, p, [2, 3]; x=x)
+end
+
+