Meshed Imposter

Using k-term and ignoring back faces (effectively, n->2n) the formula becomes
dfront=k/n Sin[θ]
and the entire formula will be
[(1-cos2θ) + (1+cos2θ)k/(2n) sinθ] 2Atn[r/d]

This approximation may fail with low number of polygons
and furthermore it is not the way meshed imposters are generated so
improvement is possible.

Image with depth

Frontface perfect, backface missing

IwDErr[n_,r_,d_,θ_,k_]:= (1-Cos[2θ]) 2 ArcTan[r/d];

(*LDI equals IwD: no backside. This is not completely fair because LDI can handle complex transparent objects better, but that's not
in our model...*)

LDIErr[n_,r_,d_,θ_,k_]:=IwDErr[n,r,d,θ,k]; (* LDI can also not be viewed from the side, mentioned values of 1.26 suggest no back face at all *)