The log-volume change of Eq. 4, in the s → q * direction is ln (sin 2 (θ 0 ) sin (θ 1 )). When transforming in the opposite direction, the same expression is multiplied by -1. The singularities in this quantity are considered at points where sin (x) = 0 for any x, where the volume, and thus density, cannot be defined. For numerical stability of dealing with samples flowing near these points, the arguments of the log were clipped to be greater than a small but finite threshold (1e-8). Generally, clipping of this type can introduce errors during inference, which in the flow-based PGMs can be quantified by a mismatch in log volume changes associated with the forward and inverse transformations of the data.