Fix issue that made order derivatives of besseli zero when

`isinteger(v)`.

Author: Chris Geoga

src/BesselFunctions/besseli.jl

@@ -414,7 +414,6 @@ _besseli(nu::Union{Int16, Float16}, x::Union{Int16, Float16}) = Float16(_besseli
 _besseli(nu::AbstractRange, x::T) where T = besseli!(zeros(T, length(nu)), nu, x)
 
 function _besseli(nu::T, x::T) where T <: Union{Float32, Float64}
-    isinteger(nu) && return _besseli(Int(nu), x)
     ~isfinite(x) && return x
     abs_nu = abs(nu)
     abs_x = abs(x)

test/besseli_test.jl

@@ -131,7 +131,8 @@ end
 (v, x) = 12.0, 3.2
 @test besseli(v,x) ≈ 7.1455266650203694069897133431e-7
 
-(v,x) = 13.0, -1.0
+# Negative arguments only allowed with v::Int!
+(v,x) = 13, -1.0
 @test besseli(v,x) ≈ -1.995631678207200756444e-14
 
 (v,x) = 12.6, -3.0
@@ -146,8 +147,7 @@ end
 (v, x) = -12.3, 8.2
 @test besseli(v,x) ≈ 0.267079696793126091886043602895
 
-(v, x) = -14.0, -9.9
+# Negative arguments only allowed with v::Int!
+(v, x) = -14, -9.9
 @test besseli(v,x) ≈ 0.2892290867115615816280234648
 
-(v, x) = -14.6, -10.6
-#@test besseli(v,x) ≈ -0.157582642056898598750175404443 - 0.484989503203097528858271270828*im

test/data/besseli/enzyme/README.md

@@ -1,22 +1,16 @@
 ```julia
-using ArbNumerics, DelimitedFiles, SpecialFunctions, Enzyme, Bessels
+using ArbNumerics, DelimitedFiles, SpecialFunctions
 
 # Because you can't just use FiniteDifferences due to the "numerical noise".
 simplefd(f,x,h=ArbReal(1e-40)) = (f(x+h)-f(x))/h
 
 ArbNumerics.setprecision(ArbReal, digits=100)
 arb_besseli(v,x)  = ArbNumerics.besseli(ArbReal(v), ArbReal(x))
 
-# TODO (cg 2023/05/30 11:44): temporarily, we make vgrid[end]=15.1 instead of
-# 15.0 or some other integer because the integer-order `besseli` derivatives
-# need to be fixed with dispatch, like `besselk`.
-vgrid = range(1e-3, 15.1,  length=20) 
+vgrid = range(1e-3, 15.0,  length=20) 
 xgrid = range(1e-3, 100.0, length=30)
 vx = vec(collect(Iterators.product(vgrid, xgrid)))
 
-our_dx(v,x) = autodiff(Forward, _x->Bessels.besseli(v, _x), Duplicated, Duplicated(x, 1.0))[2]
-our_dv(v,x) = autodiff(Forward, _v->Bessels.besseli(_v, x), Duplicated, Duplicated(v, 1.0))[2]
-
 if !isinteractive()
   ref_values = map(vx) do vxj
     (v,x) = vxj