This paper considers the Gaussian half-duplex relay channel (G-HD-RC): a channel model where a source transmits a message to a destination with the help of a relay that can not transmit and receive at the same time. It is shown that the cut-set upper bound on the capacity can be achieved to within a constant gap, regardless of the actual value of the channel parameters, by either Partial-Decode-and-Forward or Compress-and-Forward. The performance of these coding strategies is evaluated with both random and deterministic switch at the relay. Numerical evaluations show that the actual gap is less than what analytically obtained, and that random switch achieves higher rates than deterministic switch. As a result of this analysis, the generalized Degrees-of-Freedom of the G-HD-RC is exactly characterized for this channel. In order to get insights into practical schemes for the G-HD-RC that are less complex than Partial-Decode-and-Forward or Compress-and-Forward, the exact capacity of the Linear Deterministic Approximation (LDA) of the G-HD-RC at high-SNR is determined. It is shown that random switch and correlated non-uniform inputs bits are optimal for the LDA. It is then demonstrated that deterministic switch is to within one bit from the capacity. This latter scheme is translated into a coding strategy for the original G-HD-RC and its optimality to within a constant gap is proved. The gap attained by this scheme is larger than that of Partial-Decode-and-Forward, thereby pointing to an interesting practical tradeoff between gap to capacity and complexity.