We present the latest version of pinocchio , a code that generates catalogues of DM haloes in an approximate but fast way with respect to an N–body simulation .
This code version extends the computation of particle and halo displacements up to 3rd-order Lagrangian Perturbation Theory ( LPT ) , in contrast with previous versions that used Zeldovich approximation ( ZA ) .

We run pinocchio on the same initial configuration of a reference N–body simulation , so that the comparison extends to the object–by–object level .
We consider haloes at redshifts 0 and 1 , using different LPT orders either for halo construction - where displacements are needed to decide particle accretion onto a halo or halo merging - or to compute halo final positions .

We compare the clustering properties of pinocchio haloes with those from the simulation by computing the power spectrum and 2-point correlation function ( 2PCF ) in real and redshift space ( monopole and quadrupole ) , the bispectrum and the phase difference of halo distributions .
We find that 2LPT and 3LPT give noticeable improvement .
3LPT provides the best agreement with N–body when it is used to displace haloes , while 2LPT gives better results for constructing haloes .
At the highest orders , linear bias is typically recovered at a few per cent level .

In Fourier space and using 3LPT for halo displacements , the halo power spectrum is recovered to within 10 per cent up to k _ { max } \sim 0.5 h / Mpc .
The results presented in this paper have interesting implications for the generation of large ensemble of mock surveys aimed at accurately compute covariance matrices for clustering statistics .