Linesearch assumes f to be Enzyme compatible

**Describe the bug 🐞**

If you have a function `f` which is ForwardDiff compatible but not Enzyme compatible, the line search fails after `Enzyme` has been loaded (works before). It seems like internally its using Enzyme unconditionally, even though autodiff is specified as ForwardDiff specificially.

Ideally I would be able to fix the Enzyme problems by defining a trait like this:
```julia
enzyme_use_dupllicated_annotation(::ResidualFun) = true
```

**Minimal Reproducible Example 👇**

```julia
using Pkg
pkg"activate --temp"
pkg"add NonlinearSolve, PreallocationTools, ForwardDiff, Enzyme"

using NonlinearSolve, PreallocationTools, ForwardDiff

# first we construct a probme which is ForwardDiff comaptible but fails on Enzyme
struct ResidualFun
    cache::DiffCache{Vector{Float64}, Vector{Float64}}
end

ResidualFun(n::Int) = ResidualFun(DiffCache(zeros(n)))

function (f::ResidualFun)(res, u, p)
    tmp = get_tmp(f.cache, u)
    @. tmp = u^2 - p
    @. res = tmp - u .+ 1.0
    return nothing
end

n = 4
fun = ResidualFun(n)
u0 = ones(n)
p = fill(2.0, n)

prob = NonlinearLeastSquaresProblem(NonlinearFunction(fun, resid_prototype=zeros(n)), u0, p)

alg1 = GaussNewton(; linesearch=BackTracking(), autodiff=AutoForwardDiff())
solve(prob, alg1)

alg2 = TrustRegion(; autodiff=AutoForwardDiff(), radius_update_scheme=RadiusUpdateSchemes.Bastin)
solve(prob, alg2)

using Enzyme

solve(prob, alg1) # Enzyme Error
solve(prob, alg2) # Enzyme Error

```

```julia
ERROR: 
If you are using Enzyme by selecting the `AutoEnzyme` object from ADTypes, you may want to try setting the `function_annotation` option as follows:

        AutoEnzyme(; function_annotation=Enzyme.Duplicated)

This hint appears because DifferentiationInterface and Enzyme are both loaded. It does not necessarily imply that Enzyme is being called through DifferentiationInterface.

EnzymeMutabilityException: Function argument passed to autodiff cannot be proven readonly.
If the the function argument cannot contain derivative data, instead call autodiff(Mode, Const(f), ...)
See https://enzyme.mit.edu/index.fcgi/julia/stable/faq/#Activity-of-temporary-storage for more information.
The potentially writing call is   store ptr addrspace(10) %.fca.0.0.1.extract, ptr %6, align 8, !dbg !33, !tbaa !27, !alias.scope !35, !noalias !38, using   %6 = getelementptr inbounds i8, ptr %4, i64 8, !dbg !33

Stacktrace:
  [1] NonlinearFunction
    @ ~/.julia/packages/SciMLBase/obTYa/src/scimlfunctions.jl:2656 [inlined]
  [2] NonlinearFunction
    @ ~/.julia/packages/SciMLBase/obTYa/src/scimlfunctions.jl:0 [inlined]
  [3] diffejulia_NonlinearFunction_218064_inner_1wrap
    @ ~/.julia/packages/SciMLBase/obTYa/src/scimlfunctions.jl:0
  [4] macro expansion
    @ ~/.julia/packages/Enzyme/S3nC6/src/compiler.jl:6672 [inlined]
  [5] enzyme_call
    @ ~/.julia/packages/Enzyme/S3nC6/src/compiler.jl:6151 [inlined]
  [6] CombinedAdjointThunk
    @ ~/.julia/packages/Enzyme/S3nC6/src/compiler.jl:6035 [inlined]
  [7] autodiff
    @ ~/.julia/packages/Enzyme/S3nC6/src/Enzyme.jl:528 [inlined]
  [8] autodiff
    @ ~/.julia/packages/Enzyme/S3nC6/src/Enzyme.jl:549 [inlined]
  [9] value_and_pullback!(f!::NonlinearFunction{…}, y::Vector{…}, tx::Tuple{…}, prep::DifferentiationInterfaceEnzymeExt.EnzymeReverseTwoArgPullbackPrep{…}, backend::AutoEnzyme{…}, x::Vector{…}, ty::Tuple{…}, contexts::DifferentiationInterface.Constant{…})
    @ DifferentiationInterfaceEnzymeExt ~/.julia/packages/DifferentiationInterface/M8gIf/ext/DifferentiationInterfaceEnzymeExt/reverse_twoarg.jl:142
 [10] pullback!
    @ ~/.julia/packages/DifferentiationInterface/M8gIf/src/first_order/pullback.jl:580 [inlined]
 [11] #15
    @ ~/.julia/packages/SciMLJacobianOperators/k05eI/src/SciMLJacobianOperators.jl:317 [inlined]
 [12] JacobianOperator
    @ ~/.julia/packages/SciMLJacobianOperators/k05eI/src/SciMLJacobianOperators.jl:161 [inlined]
 [13] mul!
    @ ~/.julia/packages/SciMLJacobianOperators/k05eI/src/SciMLJacobianOperators.jl:229 [inlined]
 [14] __mul!
    @ ~/.julia/packages/MaybeInplace/1tMVs/src/MaybeInplace.jl:307 [inlined]
 [15] macro expansion
    @ ~/.julia/packages/MaybeInplace/1tMVs/src/MaybeInplace.jl:171 [inlined]
 [16] solve!(cache::NonlinearSolveFirstOrder.GenericTrustRegionSchemeCache{…}, J::Matrix{…}, fu::Vector{…}, u::Vector{…}, δu::Vector{…}, descent_stats::@NamedTuple{…})
    @ NonlinearSolveFirstOrder ~/.julia/packages/NonlinearSolveFirstOrder/ih1YC/src/trust_region.jl:478
 [17] step!(cache::NonlinearSolveFirstOrder.GeneralizedFirstOrderAlgorithmCache{…}; recompute_jacobian::Nothing)
    @ NonlinearSolveFirstOrder ~/.julia/packages/NonlinearSolveFirstOrder/ih1YC/src/solve.jl:344
 [18] step!
    @ ~/.julia/packages/NonlinearSolveFirstOrder/ih1YC/src/solve.jl:269 [inlined]
 [19] #step!#179
    @ ~/.julia/packages/NonlinearSolveBase/1vIgs/src/solve.jl:609 [inlined]
 [20] step!
    @ ~/.julia/packages/NonlinearSolveBase/1vIgs/src/solve.jl:603 [inlined]
 [21] solve!(cache::NonlinearSolveFirstOrder.GeneralizedFirstOrderAlgorithmCache{…})
    @ NonlinearSolveBase ~/.julia/packages/NonlinearSolveBase/1vIgs/src/solve.jl:304
 [22] __solve(::NonlinearLeastSquaresProblem{…}, ::GeneralizedFirstOrderAlgorithm{…}; kwargs::@Kwargs{…})
    @ NonlinearSolveBase ~/.julia/packages/NonlinearSolveBase/1vIgs/src/solve.jl:290
 [23] __solve
    @ ~/.julia/packages/NonlinearSolveBase/1vIgs/src/solve.jl:286 [inlined]
 [24] #solve_call#149
    @ ~/.julia/packages/NonlinearSolveBase/1vIgs/src/solve.jl:179 [inlined]
 [25] solve_call
    @ ~/.julia/packages/NonlinearSolveBase/1vIgs/src/solve.jl:137 [inlined]
 [26] solve_up(prob::NonlinearLeastSquaresProblem{…}, sensealg::Nothing, u0::Vector{…}, p::Vector{…}, args::GeneralizedFirstOrderAlgorithm{…}; originator::SciMLBase.ChainRulesOriginator, kwargs::@Kwargs{…})
    @ NonlinearSolveBase ~/.julia/packages/NonlinearSolveBase/1vIgs/src/solve.jl:132
 [27] solve_up
    @ ~/.julia/packages/NonlinearSolveBase/1vIgs/src/solve.jl:117 [inlined]
 [28] #solve#147
    @ ~/.julia/packages/NonlinearSolveBase/1vIgs/src/solve.jl:89 [inlined]
 [29] solve(prob::NonlinearLeastSquaresProblem{…}, args::GeneralizedFirstOrderAlgorithm{…})
    @ NonlinearSolveBase ~/.julia/packages/NonlinearSolveBase/1vIgs/src/solve.jl:47
 [30] top-level scope
    @ REPL[20]:1
 [31] top-level scope
    @ REPL:1
Some type information was truncated. Use `show(err)` to see complete types.

```
**Not Working Environment (please complete the following information):**

  - Output of `using Pkg; Pkg.status()`

```julia
(jl_oInbUj) pkg> st
Status `/tmp/jl_oInbUj/Project.toml`
  [7da242da] Enzyme v0.13.129
  [f6369f11] ForwardDiff v1.3.2
⌃ [8913a72c] NonlinearSolve v4.15.0
⌃ [d236fae5] PreallocationTools v0.4.34
Info Packages marked with ⌃ have new versions available and may be upgradable.
```

  - Output of `using Pkg; Pkg.status(; mode = PKGMODE_MANIFEST)`
<details>

```julia
(jl_oInbUj) pkg> st -m
Status `/tmp/jl_oInbUj/Manifest.toml`
  [47edcb42] ADTypes v1.21.0
  [7d9f7c33] Accessors v0.1.43
  [79e6a3ab] Adapt v4.4.0
  [4fba245c] ArrayInterface v7.22.0
  [70df07ce] BracketingNonlinearSolve v1.7.1
  [fa961155] CEnum v0.5.0
  [d360d2e6] ChainRulesCore v1.26.0
  [38540f10] CommonSolve v0.2.6
  [bbf7d656] CommonSubexpressions v0.3.1
  [34da2185] Compat v4.18.1
  [a33af91c] CompositionsBase v0.1.2
  [2569d6c7] ConcreteStructs v0.2.3
  [187b0558] ConstructionBase v1.6.0
  [163ba53b] DiffResults v1.1.0
  [b552c78f] DiffRules v1.15.1
⌃ [a0c0ee7d] DifferentiationInterface v0.7.15
  [ffbed154] DocStringExtensions v0.9.5
  [4e289a0a] EnumX v1.0.6
  [7da242da] Enzyme v0.13.129
  [f151be2c] EnzymeCore v0.8.18
  [e2ba6199] ExprTools v0.1.10
  [55351af7] ExproniconLite v0.10.14
  [9aa1b823] FastClosures v0.3.2
  [6a86dc24] FiniteDiff v2.29.0
  [f6369f11] ForwardDiff v1.3.2
  [069b7b12] FunctionWrappers v1.1.3
  [77dc65aa] FunctionWrappersWrappers v0.1.3
  [46192b85] GPUArraysCore v0.2.0
  [61eb1bfa] GPUCompiler v1.8.2
  [3587e190] InverseFunctions v0.1.17
  [92d709cd] IrrationalConstants v0.2.6
  [82899510] IteratorInterfaceExtensions v1.0.0
  [692b3bcd] JLLWrappers v1.7.1
  [ae98c720] Jieko v0.2.1
  [ba0b0d4f] Krylov v0.10.5
  [929cbde3] LLVM v9.4.6
  [87fe0de2] LineSearch v0.1.6
  [7ed4a6bd] LinearSolve v3.58.0
  [2ab3a3ac] LogExpFunctions v0.3.29
  [e6f89c97] LoggingExtras v1.2.0
  [1914dd2f] MacroTools v0.5.16
  [bb5d69b7] MaybeInplace v0.1.4
  [2e0e35c7] Moshi v0.3.7
  [77ba4419] NaNMath v1.1.3
⌃ [8913a72c] NonlinearSolve v4.15.0
⌃ [be0214bd] NonlinearSolveBase v2.11.2
⌅ [5959db7a] NonlinearSolveFirstOrder v1.11.1
  [9a2c21bd] NonlinearSolveQuasiNewton v1.12.0
  [26075421] NonlinearSolveSpectralMethods v1.6.0
  [d8793406] ObjectFile v0.5.0
⌃ [d236fae5] PreallocationTools v0.4.34
  [aea7be01] PrecompileTools v1.3.3
  [21216c6a] Preferences v1.5.1
  [3cdcf5f2] RecipesBase v1.3.4
  [731186ca] RecursiveArrayTools v3.48.0
  [189a3867] Reexport v1.2.2
  [ae029012] Requires v1.3.1
  [7e49a35a] RuntimeGeneratedFunctions v0.5.17
⌃ [0bca4576] SciMLBase v2.138.1
  [19f34311] SciMLJacobianOperators v0.1.12
⌃ [a6db7da4] SciMLLogging v1.8.0
  [c0aeaf25] SciMLOperators v1.15.1
  [431bcebd] SciMLPublic v1.0.1
  [53ae85a6] SciMLStructures v1.10.0
  [6c6a2e73] Scratch v1.3.0
  [efcf1570] Setfield v1.1.2
  [727e6d20] SimpleNonlinearSolve v2.10.0
  [276daf66] SpecialFunctions v2.7.1
  [1e83bf80] StaticArraysCore v1.4.4
  [10745b16] Statistics v1.11.1
  [53d494c1] StructIO v0.3.1
  [2efcf032] SymbolicIndexingInterface v0.3.46
  [a759f4b9] TimerOutputs v0.5.29
  [e689c965] Tracy v0.1.6
  [7cc45869] Enzyme_jll v0.0.249+0
  [1d5cc7b8] IntelOpenMP_jll v2025.2.0+0
  [dad2f222] LLVMExtra_jll v0.0.38+0
  [ad6e5548] LibTracyClient_jll v0.13.1+0
  [856f044c] MKL_jll v2025.2.0+0
  [efe28fd5] OpenSpecFun_jll v0.5.6+0
  [1317d2d5] oneTBB_jll v2022.0.0+1
  [0dad84c5] ArgTools v1.1.2
  [56f22d72] Artifacts v1.11.0
  [2a0f44e3] Base64 v1.11.0
  [ade2ca70] Dates v1.11.0
  [8ba89e20] Distributed v1.11.0
  [f43a241f] Downloads v1.7.0
  [7b1f6079] FileWatching v1.11.0
  [9fa8497b] Future v1.11.0
  [b77e0a4c] InteractiveUtils v1.11.0
  [ac6e5ff7] JuliaSyntaxHighlighting v1.12.0
  [4af54fe1] LazyArtifacts v1.11.0
  [b27032c2] LibCURL v0.6.4
  [76f85450] LibGit2 v1.11.0
  [8f399da3] Libdl v1.11.0
  [37e2e46d] LinearAlgebra v1.12.0
  [56ddb016] Logging v1.11.0
  [d6f4376e] Markdown v1.11.0
  [ca575930] NetworkOptions v1.3.0
  [44cfe95a] Pkg v1.12.1
  [de0858da] Printf v1.11.0
  [9a3f8284] Random v1.11.0
  [ea8e919c] SHA v0.7.0
  [9e88b42a] Serialization v1.11.0
  [6462fe0b] Sockets v1.11.0
  [2f01184e] SparseArrays v1.12.0
  [f489334b] StyledStrings v1.11.0
  [fa267f1f] TOML v1.0.3
  [a4e569a6] Tar v1.10.0
  [cf7118a7] UUIDs v1.11.0
  [4ec0a83e] Unicode v1.11.0
  [e66e0078] CompilerSupportLibraries_jll v1.3.0+1
  [deac9b47] LibCURL_jll v8.15.0+0
  [e37daf67] LibGit2_jll v1.9.0+0
  [29816b5a] LibSSH2_jll v1.11.3+1
  [14a3606d] MozillaCACerts_jll v2025.11.4
  [4536629a] OpenBLAS_jll v0.3.29+0
  [05823500] OpenLibm_jll v0.8.7+0
  [458c3c95] OpenSSL_jll v3.5.4+0
  [bea87d4a] SuiteSparse_jll v7.8.3+2
  [83775a58] Zlib_jll v1.3.1+2
  [8e850b90] libblastrampoline_jll v5.15.0+0
  [8e850ede] nghttp2_jll v1.64.0+1
  [3f19e933] p7zip_jll v17.7.0+0
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`
```

</details>

  - Output of `versioninfo()`

```julia
Julia Version 1.12.4
Commit 01a2eadb047 (2026-01-06 16:56 UTC)
Build Info:
  Official https://julialang.org release
Platform Info:
  OS: Linux (x86_64-linux-gnu)
  CPU: 16 × AMD Ryzen 9 8945HS w/ Radeon 780M Graphics
  WORD_SIZE: 64
  LLVM: libLLVM-18.1.7 (ORCJIT, znver3)
  GC: Built with stock GC
Threads: 16 default, 1 interactive, 16 GC (on 16 virtual cores)
Environment:
  JULIA_PKG_PRESERVE_TIERED_INSTALLED = true
  JULIA_PKG_PRECOMPILE_AUTO = 0
  JULIA_NUM_THREADS = auto
  JULIA_EDITOR = emacsclient
```

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Uh oh!

Linesearch assumes f to be Enzyme compatible #837

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Uh oh!

Linesearch assumes f to be Enzyme compatible #837

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