diff --git a/Project.toml b/Project.toml
index dd614606..c73f4e7c 100644
--- a/Project.toml
+++ b/Project.toml
@@ -27,7 +27,6 @@ ArrayInterface = "7.1"
 ForwardDiff = "0.10.13"
 GPUArraysCore = "0.1, 0.2"
 GenericSchur = "0.5.3"
-JET = "0.9, 0.10, 0.11"
 LinearAlgebra = "1.10"
 Pkg = "1"
 PrecompileTools = "1"
@@ -42,7 +41,6 @@ julia = "1.10"
 [extras]
 Aqua = "4c88cf16-eb10-579e-8560-4a9242c79595"
 ForwardDiff = "f6369f11-7733-5829-9624-2563aa707210"
-JET = "c3a54625-cd67-489e-a8e7-0a5a0ff4e31b"
 Pkg = "44cfe95a-1eb2-52ea-b672-e2afdf69b78f"
 Random = "9a3f8284-a2c9-5f02-9a11-845980a1fd5c"
 SafeTestsets = "1bc83da4-3b8d-516f-aca4-4fe02f6d838f"
@@ -50,4 +48,4 @@ StaticArrays = "90137ffa-7385-5640-81b9-e52037218182"
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
 
 [targets]
-test = ["Aqua", "ForwardDiff", "JET", "Pkg", "Test", "SafeTestsets", "StaticArrays", "Random"]
+test = ["Aqua", "ForwardDiff", "Pkg", "Test", "SafeTestsets", "StaticArrays", "Random"]
diff --git a/src/exp_generic.jl b/src/exp_generic.jl
index b1837432..11d2bcc1 100644
--- a/src/exp_generic.jl
+++ b/src/exp_generic.jl
@@ -130,7 +130,9 @@ end
 
 """
     struct ExpMethodGeneric{T}
-    ExpMethodGeneric()=ExpMethodGeneric{Val{13}}();
+    ExpMethodGeneric()=ExpMethodGeneric{Val{13}}()
+    ExpMethodGeneric(k::Integer)=ExpMethodGeneric{Val{k}}()
+    ExpMethodGeneric(::Type{T}) where T = ExpMethodGeneric{Val{pade_order_for_type(T)}}()
 
 Generic exponential implementation of the method `ExpMethodHigham2005`,
 for any exp argument `x`  for which the functions
@@ -138,16 +140,73 @@ for any exp argument `x`  for which the functions
 UniformScaling objects) are defined. The type `T` is used to adjust the
 number of terms used in the Pade approximants at compile time.
 
+For high-precision types like `BigFloat`, the Padé order is automatically
+selected based on the precision to achieve machine-precision accuracy.
+You can also manually specify the order: `ExpMethodGeneric(k)` uses a
+`(k,k)` Padé approximant. To automatically select based on element type,
+use `ExpMethodGeneric(T)` where `T` is the element type.
+
 See "The Scaling and Squaring Method for the Matrix Exponential Revisited"
 by Higham, Nicholas J. in 2005 for algorithm details.
 """
 struct ExpMethodGeneric{T} end
-ExpMethodGeneric() = ExpMethodGeneric{Val(13)}();
+ExpMethodGeneric() = ExpMethodGeneric{Val(13)}()
+ExpMethodGeneric(k::Integer) = ExpMethodGeneric{Val{k}()}()
+
+"""
+    pade_order_for_type(::Type{T}) where T
+
+Compute the minimum Padé order k required for machine-precision accuracy
+for a given floating-point type T. The (k,k) Padé approximant for exp(x)
+has error bounded by (x/2)^(2k+1) / (2k+1)! for |x| ≤ 1.
+"""
+function pade_order_for_type(::Type{T}) where {T}
+    # Get precision in bits
+    p = _precision_bits(T)
+    # For standard Float64, use the optimized k=13
+    p <= 64 && return 13
+    # For higher precision, compute required k
+    # We need: (1/2)^(2k+1) / (2k+1)! < 2^(-p)
+    # Adding a small buffer for safety
+    target = big(1) // big(2)^(p + 10)
+    for k in 13:500
+        bound = (big(1) // 2)^(2k + 1) / factorial(big(2k + 1))
+        if bound < target
+            return k
+        end
+    end
+    return 500  # fallback for extremely high precision
+end
+
+_precision_bits(::Type{Float16}) = 11
+_precision_bits(::Type{Float32}) = 24
+_precision_bits(::Type{Float64}) = 53
+_precision_bits(::Type{BigFloat}) = precision(BigFloat)
+_precision_bits(::Type{Complex{T}}) where {T} = _precision_bits(T)
+_precision_bits(::Type{T}) where {T <: AbstractFloat} = precision(T)
+# Fallback for other numeric types (integers, etc.)
+_precision_bits(::Type{T}) where {T <: Number} = 53
+
+ExpMethodGeneric(::Type{T}) where {T} = ExpMethodGeneric{Val{pade_order_for_type(T)}()}()
+
+# Extract the element type from various input types
+_eltype(x::Number) = typeof(x)
+_eltype(x::AbstractArray{T}) where {T} = T
+
+# Determine if the default k=13 is sufficient for the given element type
+_needs_higher_order(::Type{T}) where {T} = _precision_bits(T) > 64
 
 function exponential!(
         x, method::ExpMethodGeneric{Vk},
         cache = alloc_mem(x, method)
     ) where {Vk}
+    # For high-precision types with default k=13, automatically use higher order
+    T = _eltype(x)
+    if Vk === Val{13}() && _needs_higher_order(T)
+        k = pade_order_for_type(T)
+        return exponential!(x, ExpMethodGeneric{Val{k}()}(), cache)
+    end
+
     nx = opnorm(x, 1)
     if isnan(nx) || nx > 4611686018427387904 # i.e. 2^62 since it would cause overflow in 2^s
         # This should (hopefully) ensure that the result is Inf or NaN depending on
diff --git a/test/basictests.jl b/test/basictests.jl
index c1ae2f88..0212e4eb 100644
--- a/test/basictests.jl
+++ b/test/basictests.jl
@@ -126,6 +126,34 @@ end
     @test vec(res) ≈ exp.(ts)
 end
 
+@testset "Issue 44 - BigFloat precision" begin
+    using ExponentialUtilities: pade_order_for_type
+
+    # Test that pade_order_for_type returns reasonable values
+    @test pade_order_for_type(Float64) == 13
+    @test pade_order_for_type(BigFloat) >= 24  # depends on precision
+
+    # Test scalar BigFloat accuracy
+    _x = range(-10, stop = 10, length = 50)
+    bigfloat_x = big.(_x)
+    max_err = maximum(
+        abs,
+        (x -> exp_generic(x) / exp(x) - 1).(bigfloat_x)
+    ) / eps(BigFloat)
+    @test max_err < 100  # Should be within ~100 eps
+
+    # Test with different precisions
+    for bits in [128, 256, 512]
+        setprecision(bits) do
+            x = big"0.5"
+            result = exp_generic(x)
+            exact = exp(x)
+            rel_err = abs(result - exact) / exact
+            @test rel_err < 100 * eps(BigFloat)
+        end
+    end
+end
+
 @testset "naive_matmul" begin
     A = Matrix(reshape((1.0:(23.0^2)) ./ 700, (23, 23)))
     @test exp_generic(A) ≈ exp(A)
diff --git a/test/jet/Project.toml b/test/jet/Project.toml
new file mode 100644
index 00000000..0cf8f6e9
--- /dev/null
+++ b/test/jet/Project.toml
@@ -0,0 +1,6 @@
+[deps]
+JET = "c3a54625-cd67-489e-a8e7-0a5a0ff4e31b"
+Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
+
+[compat]
+JET = "0.9, 0.10, 0.11"
diff --git a/test/jet/jet.jl b/test/jet/jet.jl
new file mode 100644
index 00000000..fcc3a428
--- /dev/null
+++ b/test/jet/jet.jl
@@ -0,0 +1,41 @@
+using ExponentialUtilities, JET, Test
+
+@testset "JET static analysis" begin
+    # Test key entry points for type stability and correctness
+    # Using report_call to check for runtime errors
+
+    @testset "expv" begin
+        rep = JET.report_call(expv, (Float64, Matrix{Float64}, Vector{Float64}))
+        @test length(JET.get_reports(rep)) == 0
+    end
+
+    @testset "arnoldi" begin
+        rep = JET.report_call(arnoldi, (Matrix{Float64}, Vector{Float64}))
+        @test length(JET.get_reports(rep)) == 0
+    end
+
+    @testset "phi" begin
+        rep = JET.report_call(phi, (Matrix{Float64}, Int))
+        @test length(JET.get_reports(rep)) == 0
+    end
+
+    @testset "exponential!" begin
+        rep = JET.report_call(ExponentialUtilities.exponential!, (Matrix{Float64},))
+        @test length(JET.get_reports(rep)) == 0
+    end
+
+    @testset "phiv" begin
+        rep = JET.report_call(phiv, (Float64, Matrix{Float64}, Vector{Float64}, Int))
+        @test length(JET.get_reports(rep)) == 0
+    end
+
+    @testset "kiops" begin
+        rep = JET.report_call(kiops, (Float64, Matrix{Float64}, Vector{Float64}))
+        @test length(JET.get_reports(rep)) == 0
+    end
+
+    @testset "expv_timestep" begin
+        rep = JET.report_call(expv_timestep, (Float64, Matrix{Float64}, Vector{Float64}))
+        @test length(JET.get_reports(rep)) == 0
+    end
+end
diff --git a/test/qa.jl b/test/qa.jl
index 6dba9775..cfa2c442 100644
--- a/test/qa.jl
+++ b/test/qa.jl
@@ -1,4 +1,4 @@
-using ExponentialUtilities, Aqua, JET
+using ExponentialUtilities, Aqua
 @testset "Aqua" begin
     Aqua.find_persistent_tasks_deps(ExponentialUtilities)
     Aqua.test_ambiguities(ExponentialUtilities, recursive = false)
@@ -12,43 +12,3 @@ using ExponentialUtilities, Aqua, JET
     Aqua.test_unbound_args(ExponentialUtilities)
     Aqua.test_undefined_exports(ExponentialUtilities)
 end
-
-@testset "JET static analysis" begin
-    # Test key entry points for type stability and correctness
-    # Using report_call to check for runtime errors
-
-    @testset "expv" begin
-        rep = JET.report_call(expv, (Float64, Matrix{Float64}, Vector{Float64}))
-        @test length(JET.get_reports(rep)) == 0
-    end
-
-    @testset "arnoldi" begin
-        rep = JET.report_call(arnoldi, (Matrix{Float64}, Vector{Float64}))
-        @test length(JET.get_reports(rep)) == 0
-    end
-
-    @testset "phi" begin
-        rep = JET.report_call(phi, (Matrix{Float64}, Int))
-        @test length(JET.get_reports(rep)) == 0
-    end
-
-    @testset "exponential!" begin
-        rep = JET.report_call(ExponentialUtilities.exponential!, (Matrix{Float64},))
-        @test length(JET.get_reports(rep)) == 0
-    end
-
-    @testset "phiv" begin
-        rep = JET.report_call(phiv, (Float64, Matrix{Float64}, Vector{Float64}, Int))
-        @test length(JET.get_reports(rep)) == 0
-    end
-
-    @testset "kiops" begin
-        rep = JET.report_call(kiops, (Float64, Matrix{Float64}, Vector{Float64}))
-        @test length(JET.get_reports(rep)) == 0
-    end
-
-    @testset "expv_timestep" begin
-        rep = JET.report_call(expv_timestep, (Float64, Matrix{Float64}, Vector{Float64}))
-        @test length(JET.get_reports(rep)) == 0
-    end
-end
diff --git a/test/runtests.jl b/test/runtests.jl
index ce7c7fd4..68fc7ea6 100644
--- a/test/runtests.jl
+++ b/test/runtests.jl
@@ -10,6 +10,12 @@ function activate_gpu_env()
     return Pkg.instantiate()
 end
 
+function activate_jet_env()
+    Pkg.activate("jet")
+    Pkg.develop(PackageSpec(path = dirname(@__DIR__)))
+    return Pkg.instantiate()
+end
+
 @time begin
     if GROUP == "All" || GROUP == "Core"
         @time @safetestset "Quality Assurance" include("qa.jl")
@@ -20,4 +26,9 @@ end
         activate_gpu_env()
         @time @safetestset "GPU Tests" include("gpu/gputests.jl")
     end
+
+    if GROUP == "JET"
+        activate_jet_env()
+        @time @safetestset "JET Tests" include("jet/jet.jl")
+    end
 end