BeginPackage["Schofield`"]

$Assumptions = {θc > 0, θc > 1, gC[_] ∈ Reals, B > 0, γ > 0, ξ0 > 0}

β[D_:3] := Piecewise[
  {
    {1/8, D == 2},
    {0.326419, D == 3},
    {1/2, D == 4},
    {β, True}
  }
]

δ[D_:3] := Piecewise[
  {
    {15, D == 2},
    {4.78984, D == 3},
    {3, D == 4},
    {δ, True}
  }
]

α[D_:3] := Piecewise[
  {
    {0, D == 2},
    {0.11008, D == 3},
    {0, D == 4},
    {α, True}
  }
]

Δ[D_:3] := β[D] δ[D]

t[θ_] := ((θ)^2 - 1)
h[n_][θ_] := (1 - (θ/θc)^2) Sum[gC[i]LegendreP[(2 * i + 1), θ/θc], {i, 0, n}]

RFLow[B_, θc_][θ_] := -(1/(12 \[Pi]))((
     E^(-(1/(B^2 (\[Theta] - \[Theta]c)^2)))
     ExpIntegralE[7/6, -(1/(B^2 (\[Theta] - \[Theta]c)^2))] Gamma[1/
       6])/(\[Theta] - \[Theta]c)^2 + (-((
         2 E^(-(1/(B^2 \[Theta]c^2)))
         ExpIntegralE[7/6, -(1/(B^2 \[Theta]c^2))])/\[Theta]c^2) + (
        E^(-(1/(B^2 (\[Theta] + \[Theta]c)^2)))
        ExpIntegralE[7/
          6, -(1/(B^2 (\[Theta] + \[Theta]c)^2))])/(\[Theta] + \
  \[Theta]c)^2) Gamma[1/6] + (
     4 B E^(-(1/(B^2 (\[Theta] - \[Theta]c)^2)))
     ExpIntegralE[5/3, -(1/(B^2 (\[Theta] - \[Theta]c)^2))] Gamma[2/
       3])/(\[Theta] - \[Theta]c) + ((
        8 B E^(-(1/(B^2 \[Theta]c^2)))
        ExpIntegralE[5/3, -(1/(B^2 \[Theta]c^2))])/\[Theta]c - (
        4 B E^(-(1/(B^2 (\[Theta] + \[Theta]c)^2)))
        ExpIntegralE[5/
          3, -(1/(B^2 (\[Theta] + \[Theta]c)^2))])/(\[Theta] + \
  \[Theta]c)) Gamma[2/3])
RFHigh[ξ0_][ξ_] := (ξ^2+ξ0^2)^(1+0.085)

RF[n_][θ_] := AL RFLow[B, θc][θ] + AH RFHigh[θ0][θ] + Sum[A[i] LegendreP[(2 i), θ/θc] , {i, 0, n}]
RFReg[n_][θ_] := -AL (1/(12 \[Pi]))((
     (-((
         2 E^(-(1/(B^2 \[Theta]c^2)))
         ExpIntegralE[7/6, -(1/(B^2 \[Theta]c^2))])/\[Theta]c^2) + (
        E^(-(1/(B^2 (\[Theta] + \[Theta]c)^2)))
        ExpIntegralE[7/
          6, -(1/(B^2 (\[Theta] + \[Theta]c)^2))])/(\[Theta] + \
  \[Theta]c)^2) Gamma[1/6] + \
      ((
        8 B E^(-(1/(B^2 \[Theta]c^2)))
        ExpIntegralE[5/3, -(1/(B^2 \[Theta]c^2))])/\[Theta]c - (
        4 B E^(-(1/(B^2 (\[Theta] + \[Theta]c)^2)))
        ExpIntegralE[5/
          3, -(1/(B^2 (\[Theta] + \[Theta]c)^2))])/(\[Theta] + \
  \[Theta]c)) Gamma[2/3]) + AH RFHigh[θ0][θ] + Sum[A[i] LegendreP[(2 i), θ/θc], {i, 0, n}]
dRFc[n_][m_] := AL m! Gamma[7/6 + m/2] B^(m + 2) / (2 π) + D[RFReg[n][θ], {θ, m}] /. θ -> θc

RFC[n_][θ_] := RF[n][θ] + AL I Sign[Im[θ]] ((θ-θc)Exp[-1/(B(θ-θc))]-(-θ-θc)Exp[-1/(B(-θ-θc))])

ddξ[h_][f_] := D[f, θ] / D[h[θ] / RealAbs[t[θ]]^Δ[3], θ]
ddη[h_][f_] := D[f, θ] / D[t[θ] / h[θ]^(1 / Δ[3]), θ]
dFdξLow[n_, h_][m_] := Module[{ff, hh}, Nest[ddξ[hh], ff[θ] / t[θ]^(3 * ν[3]), m] /. θ -> θc /. Map[Derivative[#][ff][θc] -> dRFc[n][#] &, Range[0, m]] /. hh -> h]
dFdξHigh[n_, h_][m_] := Module[{ff, hh}, Nest[ddξ[hh], ff[θ] / t[θ]^(3 * ν[3]), m] /. θ -> 0 /. Map[Derivative[#][ff][0] -> eqHighRHS[RF[n]][#] &, Range[0, m]] /. hh -> h]
dFdη[n_, h_][m_][tt_] := Module[{ff, hh}, Nest[ddη[hh], h[θ]^(-3 * ν[3] / Δ[3]) ff[θ], m] /. θ -> tt /. Map[Derivative[#][ff][tt] -> Derivative[#][RF[n]][tt] &, Range[0, m]] /. hh -> h]
dFdξLowList[n_, h_][m_] := Module[{ff, hh}, NestList[ddξ[hh], ff[θ] / t[θ]^(3 * ν[3]), m] /. θ -> θc /. Map[Derivative[#][ff][θc] -> dRFc[n][#] &, Range[0, m]] /. Map[Derivative[#][hh][θc] -> Derivative[#][h][θc] &, Range[0, m]]]
dFdξHighList[n_, h_][m_] := Module[{ff, hh}, NestList[ddξ[hh], ff[θ] / t[θ]^(3 * ν[3]), m] /. θ -> 0 /. Map[Derivative[#][ff][0] -> eqHighRHS[RF[n]][#] &, Range[0, m]] /. hh -> h]
dFdηList[n_, h_][m_][tt_] := Module[{ff, hh}, NestList[ddη[hh], h[θ]^(-3 * ν[3] / Δ[3]) (ff[θ]), m] /. θ -> tt /. Map[Derivative[#][ff][tt] -> Derivative[#][RF[n]][tt] &, Range[0, m]] /. hh -> h]

ruleB[g_] := B - 0.0464597
ruleθ0[g_] := Simplify[g[I θ0]/(-t[I θ0])^Δ[3]/I] - 
ruleAL[g_] := AL + t[θc]^2 OverBar[s] / (2 π) * (- g'[θc] / t[θc]^Δ[2])

eqLowRHSReg[n_][m_] := dRFc[n][m]

eqLowLHS[h_][m_] :=D[
  t[θ]^2 (Gl[h[θ] t[θ]^-Δ[2]] + Log[t[θ]^2]/(8 π)),
  {θ, m} ] /. θ -> θc

eqLow[n_, h_][m_] := (eqLowRHSReg[n][m] - eqLowLHS[h][m]) / m!

eqHighRHS[F_][m_] := D[F[θ], {θ, m} ] /. θ -> 0

eqHighLHS[h_][m_] := D[(-t[θ])^2 (Gh[h[θ] (-t[θ])^-Δ[2]] + Log[(-t[θ])^2]/(8 π)), {θ, m} ] /. θ -> 0

eqHigh[n_, h_][m_] := (eqHighRHS[RF[n]][m] - eqHighLHS[h][m]) / m!

eqMid[F_, h_][m_] := D[
  F[θ] - t[θ]^2 Log[h[θ]^2]/(8 Δ[2]π) - h[θ]^((2-α[2])/Δ[2]) Φ[η]
    /. η -> t[θ] / h[θ]^(1 / Δ[2]),
  {θ, m} ] / m! /. θ -> 1

δ0 = 10^(-14);

Φs = {
  -1.197733383797993,
  -0.318810124891,
  0.110886196683,
  0.01642689465,
  -2.639978 10^-4,
  -5.140526 10^-4,
  2.08856 10^-4,
  -4.4819 10^-5,
  3.16 10^-7,
  4.31 10^-6,
  -1.99 10^-6
}

Gls = {
  Around[0, δ0],
  Around[-OverBar[s], δ0],
  Around[−0.048953289720, 2 10^(-12)],
  Around[ 0.0388639290, 1 10^(-10)],
  Around[-0.068362121, 1 10^(-9)],
  Around[ 0.18388371, 1 10^(-8)],
  Around[-0.659170, 1 10^(-6)],
  Around[ 2.937665, 3 10^(-6)],
  Around[-15.61, 10^(-2)],
  96.76,
  -679,
  5340
}

Ghs = {
  Around[0, δ0],
  Around[0, δ0],
  Around[ -1.84522807823, 10^(-11)],
  Around[0, δ0],
  Around[  8.3337117508, 10^(-10)],
  Around[0, δ0],
  Around[-95.16897, 10^(-5)],
  Around[0, δ0],
  Around[1457.62, 3 10^(-2)],
  0,
  Around[-25891, 2],
  0,
  502000
}

dRule[sym_][f_, i_] := Derivative[i[[1]] - 1][sym][0] -> f (i[[1]] - 1)!

ΦRules = MapIndexed[dRule[Φ], Φs];
GlRules = MapIndexed[dRule[Gl], Gls];
GhRules = MapIndexed[dRule[Gh], Ghs];

ClearAll[gC]
rules := Join[ΦRules, GlRules, GhRules]
(*ξ0 := 0.18930*)
(*gC[0] := 1*)
tC[0] := 1
(*gC[0] := 1*)

eq[n_, g_][m_, p_, q_] := Flatten[Join[{ruleB[g], ruleθ0[g], g'[0] - 1}, eqLow[n, g][#] & /@ Range[0, m],eqMid[RF[n], g][#] & /@ Range[0, p], eqHigh[n, g] /@ Range[0, q, 2]]] //. rules /. Around[x_, _] :> x

 (* *)
chiSquaredLow[n_, g_][m_] := Total[(((#[[1]] /. rules)["Value"] - #[[2]])^2 / (#[[1]] /. rules)["Uncertainty"]^2)& /@ ({Gls[[#+1]], dFdξLow[n, g][#] / #!} & /@ Range[0, m])]
chiSquaredHigh[n_, g_][m_] := Total[(((#[[1]] /. rules)["Value"] - #[[2]])^2 / (#[[1]] /. rules)["Uncertainty"]^2)& /@ ({Ghs[[#+1]], dFdξHigh[n, g][#] / #!} & /@ Range[0, m])]
chiSquared[F_, g_][m_] := chiSquaredLow[F, g][m] + chiSquaredHigh[F, g][m] + ruleB[g]^2 / δ0^2 + ruleθ0[g]^2 / 0.00005^2

newSol[eqs_, oldSol_, newVars_, δ_:0, γ_:0, opts___] := FindRoot[
  eqs,
  Join[
    {#1, #2 + γ * RandomVariate[NormalDistribution[]]} & @@@ (oldSol /. Rule -> List),
    Thread[{newVars, δ * RandomVariate[NormalDistribution[], Length[newVars]]}]
  ],
  MaxIterations -> 50000,
  opts
]

EndPackage[]