We offer a more formal justification for the successes of our recently communicated "directed Heller-Herman-Kluk-Kay" (DHK) time propagator by examining its performance in one-dimensional bound systems which exhibit at least quasi-periodic motion. DHK is distinguished by its single one-dimensional integral--a vast simplification over the usual 2N-dimensional integral in full Heller-Herman-Kluk-Kay (for an N-dimensional system). We find that DHK accurately captures particular coherent state autocorrelations when its single integral is chosen to lie along these states' fastest growing manifold, as long as it is not perpendicular to their action gradient. Moreover, the larger the action gradient, the better DHK will perform. We numerically examine DHK's accuracy in a one-dimensional quartic oscillator and illustrate that these conditions are frequently satisfied such that the method performs well. This lends some explanation for why DHK frequently seems to work so well and suggests that it may be applicable to systems exhibiting quite strong anharmonicity.