We present TeV gamma-ray observations of the Crab Nebula, the standard reference source in ground-based gamma-ray astronomy, using data from the High Altitude Water Cherenkov (HAWC) Gamma-Ray Observatory. In this analysis we use two independent energy-estimation methods that utilize extensive air shower variables such as the core position, shower angle, and shower lateral energy distribution. In contrast, the previously published HAWC energy spectrum roughly estimated the shower energy with only the number of photomultipliers triggered. This new methodology yields a much improved energy resolution over the previous analysis and extends HAWCs ability to accurately measure gamma-ray energies well beyond 100 TeV. The energy spectrum of the Crab Nebula is well fit to a log parabola shape $left(frac{dN}{dE} = phi_0 left(E/textrm{7 TeV}right)^{-alpha-betalnleft(E/textrm{7 TeV}right)}right)$ with emission up to at least 100 TeV. For the first estimator, a ground parameter that utilizes fits to the lateral distribution function to measure the charge density 40 meters from the shower axis, the best-fit values are $phi_o$=(2.35$pm$0.04$^{+0.20}_{-0.21}$)$times$10$^{-13}$ (TeV cm$^2$ s)$^{-1}$, $alpha$=2.79$pm$0.02$^{+0.01}_{-0.03}$, and $beta$=0.10$pm$0.01$^{+0.01}_{-0.03}$. For the second estimator, a neural network which uses the charge distribution in annuli around the core and other variables, these values are $phi_o$=(2.31$pm$0.02$^{+0.32}_{-0.17}$)$times$10$^{-13}$ (TeV cm$^2$ s)$^{-1}$, $alpha$=2.73$pm$0.02$^{+0.03}_{-0.02}$, and $beta$=0.06$pm$0.01$pm$0.02. The first set of uncertainties are statistical; the second set are systematic. Both methods yield compatible results. These measurements are the highest-energy observation of a gamma-ray source to date.
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