# Numpy corrcoef giving different result from manual calculation

I'm trying to calculate a partial correlation matrix for a high dimensional problem. I'm using this paper as a guide. I'm also referencing this function from Pingouin.

Starting from the inverse covariance matrix of the variables:

inv_cov = np.linalg.pinv(np.cov(var_mat))


Then using equation 2.5 from the paper, calculating partial correlation between variable 0 and 1:

-inv_cov[0,1] / np.sqrt(inv_cov[0,0] * inv_cov[1,1])
>>> 0.8683796638219119


That formula seems to be the same as numpy's corrcoef function, however, when I use that:

-np.corrcoef(inv_cov)[0,1]
>>> 0.7114911879489708


Using the Pingouin function returns the same as the manual calculation:

inv_diag = np.diag(np.sqrt(1 / np.diag(inv_cov)))
partial_corr = -1 * (inv_diag @ inv_cov @ inv_diag)
partial_corr[0,1]
>>> 0.8683796638219119


What's the difference in Numpy's corrcoef that causes it to have a different value? In the linked paper they use R's cov2cor function as an example of how to calculate the matrix, does corrcoef not do the same thing?