# Type stability for higher derivatives in ForwardDiff

**URL:** <https://discourse.julialang.org/t/type-stability-for-higher-derivatives-in-forwarddiff/66686>\
**Category:** General Usage\
**Tags:** forwarddiff, type-stability\
**Created:** [August 19, 2021, 6:18pm UTC](https://discourse.julialang.org/t/type-stability-for-higher-derivatives-in-forwarddiff/66686 "2021-08-19T18:18:05Z")\
**Posts on this page:** 3\
**Page:** 1

<div class="post-metadata">

**Author:** ![JonasWickman](https://avatars.discourse-cdn.com/v4/letter/j/9de0a6/32.png) [@JonasWickman](https://discourse.julialang.org/u/JonasWickman)\
**Post date:** [August 19, 2021, 6:18pm UTC](https://discourse.julialang.org/t/type-stability-for-higher-derivatives-in-forwarddiff/66686/1 "2021-08-19T18:18:05Z")

</div>

I’m trying to compute all third partial derivatives of a function using ForwardDiff, but cannot get it to be type stable and I don’t understand why. Here is a minimal example:

```julia
using ForwardDiff
using StaticArrays

function tf( x )
    return ( exp( 0.1*x[1] + 0.2*x[2] ) )
end

function get_derivatives( f ) 
    
    ∇f(x) = ForwardDiff.gradient( f , x ) 
    ∇²f(x) = ForwardDiff.hessian( f , x ) 
    ∇³f(x) = ForwardDiff.jacobian( ∇²f , x )
    
    return ∇f, ∇²f, ∇³f

end

∇tf, ∇²tf, ∇³tf = get_derivatives( tf )

@code_warntype ∇³tf( @SVector [1.0,1.0] )

```

The output of `@code_warntype` is:

```julia
Variables
  #self#::Core.Const(var"#∇³f#329"{var"#∇²f#328"{typeof(tf)}}(var"#∇²f#328"{typeof(tf)}(tf)))
  x::SVector{2, Float64}

Body::Any
1 ─ %1 = ForwardDiff.jacobian::Core.Const(ForwardDiff.jacobian)
│ %2 = Core.getfield(#self#, :∇²f)::Core.Const(var"#∇²f#328"{typeof(tf)}(tf))
│ %3 = (%1)(%2, x)::Any
└── return %3

```

The gradient and hessian are both type stable. Searching around for related topics I read about the ForwardDiff chunk size, so I also tried implementing the following:

```julia
function get_derivatives_2( f ) 
    
    cfg_grad = ForwardDiff.GradientConfig( f , SVector{2}( [1.0,1.0] ) , ForwardDiff.Chunk{1}() )
    ∇f(x) = ForwardDiff.gradient( f , x , cfg_grad ) 

    cfg_jac_1 = ForwardDiff.JacobianConfig( ∇f , SVector{2}( [1.0,1.0] ) , ForwardDiff.Chunk{1}() )
    ∇²f(x) = ForwardDiff.jacobian( ∇f , x , cfg_jac_1 ) 

    cfg_jac_2 = ForwardDiff.JacobianConfig( ∇²f , SVector{2}( [1.0,1.0] ) , ForwardDiff.Chunk{1}() )
    ∇³f(x) = ForwardDiff.jacobian( ∇²f , x , cfg_jac_2 ) 
    
    return ∇f, ∇²f, ∇³f

end

```

However, this yielded the same result with the gradient and hessian being type stable, but the third derivative failing to be so.

Julia version: 1.6.2  
ForwardDiff version: v0.10.19

Any help with this would be much appreciated.

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<div class="post-metadata">

**Author:** ![longemen3000](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/longemen3000/32/7298_2.png) [@longemen3000](https://discourse.julialang.org/u/longemen3000)\
**Post date:** [August 19, 2021, 10:04pm UTC](https://discourse.julialang.org/t/type-stability-for-higher-derivatives-in-forwarddiff/66686/2 "2021-08-19T22:04:11Z")

</div>

yes, i found the same problem, but when needing the primal, first and second derivative. for what i understand, when using StaticArrays, the chunk is always the length of the SArray

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<div class="post-metadata">

**Author:** ![longemen3000](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/longemen3000/32/7298_2.png) [@longemen3000](https://discourse.julialang.org/u/longemen3000)\
**Post date:** [August 19, 2021, 10:07pm UTC](https://discourse.julialang.org/t/type-stability-for-higher-derivatives-in-forwarddiff/66686/3 "2021-08-19T22:07:56Z")

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related:

> [@Allocation on ForwardDiff + DiffResults + StaticArrays](https://discourse.julialang.org/t/allocation-on-forwarddiff-diffresults-staticarrays/46534/3):
>
> Cf [https://github.com/JuliaDiff/ForwardDiff.jl/pull/315](https://github.com/JuliaDiff/ForwardDiff.jl/pull/315) Something similar for Hessian may help resolve this.