From b0259dc9726c5bb8bc87ab240b0d14c42b5bc032 Mon Sep 17 00:00:00 2001
From: Lyndon White <lyndon.white@invenialabs.co.uk>
Date: Sat, 28 Mar 2020 14:20:10 +0000
Subject: [PATCH 1/4] Compute coeffs using Rationals

---
 src/methods.jl | 8 +++++---
 1 file changed, 5 insertions(+), 3 deletions(-)

diff --git a/src/methods.jl b/src/methods.jl
index 20b56ea9..1400134f 100644
--- a/src/methods.jl
+++ b/src/methods.jl
@@ -157,10 +157,12 @@ end
 
 # Compute coefficients for the method
 function _coefs(grid::AbstractVector{<:Real}, p::Integer, q::Integer)
-    C = [g^i for i in 0:(p - 1), g in grid]
-    x = zeros(Int, p)
+    # For high precision on the \ we use Rational, and to prevent overfloats we use Int128
+    # At the end we go to Float64 for fast floating point math (rather than rational math)
+    C = [Rational{Int128}(g^i) for i in 0:(p - 1), g in grid]
+    x = zeros(Rational{Int128}, p)
     x[q + 1] = factorial(q)
-    return C \ x
+    return Float64.(C \ x)
 end
 
 # Estimate the bound on the function value and its derivatives at a point

From d6e9ddbab645a90aaa38dd39333f598ba07dc32c Mon Sep 17 00:00:00 2001
From: Lyndon White <lyndon.white@invenialabs.co.uk>
Date: Sat, 4 Apr 2020 22:23:19 +0100
Subject: [PATCH 2/4] add tests of accuracy at high orders

---
 test/methods.jl | 11 +++++++++++
 1 file changed, 11 insertions(+)

diff --git a/test/methods.jl b/test/methods.jl
index 4b46d33e..fe886626 100644
--- a/test/methods.jl
+++ b/test/methods.jl
@@ -37,6 +37,17 @@ using FiniteDifferences: Forward, Backward, Central, Nonstandard
         @test central_fdm(5, 1)(abs, 0.001) ≈ 1.0
     end
 
+    @testset "Accuracy at high orders, with high adapt" begin
+        # Regression test against issues with precision during computation of _coeffs
+        # see https://github.com/JuliaDiff/FiniteDifferences.jl/issues/64
+
+        @test fdm(central_fdm(9, 5), exp, 1.0, adapt=4) ≈ exp(1) atol=1e-7
+
+        poly(x) = 4x^3 + 3x^2 + 2x + 1
+        @test fdm(central_fdm(9, 3), poly, 1.0, adapt=4) ≈ 24 atol=1e-11
+    end
+
+
     @testset "Printing FiniteDifferenceMethods" begin
         @test sprint(show, central_fdm(2, 1)) == """
             FiniteDifferenceMethod:

From 2a92c613776250bc82befccb18030fe1117e14ee Mon Sep 17 00:00:00 2001
From: Lyndon White <oxinabox@ucc.asn.au>
Date: Mon, 6 Apr 2020 13:02:07 +0100
Subject: [PATCH 3/4] Update src/methods.jl

Co-Authored-By: willtebbutt <wt0881@my.bristol.ac.uk>
---
 src/methods.jl | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/methods.jl b/src/methods.jl
index 1400134f..41485561 100644
--- a/src/methods.jl
+++ b/src/methods.jl
@@ -157,7 +157,7 @@ end
 
 # Compute coefficients for the method
 function _coefs(grid::AbstractVector{<:Real}, p::Integer, q::Integer)
-    # For high precision on the \ we use Rational, and to prevent overfloats we use Int128
+    # For high precision on the \ we use Rational, and to prevent overflows we use Int128
     # At the end we go to Float64 for fast floating point math (rather than rational math)
     C = [Rational{Int128}(g^i) for i in 0:(p - 1), g in grid]
     x = zeros(Rational{Int128}, p)

From 7eed57f280d0f6da96d19e22ba4945f4dc530d0d Mon Sep 17 00:00:00 2001
From: Lyndon White <oxinabox@ucc.asn.au>
Date: Mon, 6 Apr 2020 13:02:47 +0100
Subject: [PATCH 4/4] update version number

---
 Project.toml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/Project.toml b/Project.toml
index 7388759d..5b1df0e1 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,6 +1,6 @@
 name = "FiniteDifferences"
 uuid = "26cc04aa-876d-5657-8c51-4c34ba976000"
-version = "0.9.3"
+version = "0.9.4"
 
 [deps]
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"