Effective de la Valle Poussin style bounds on the first Chebyshev function

In 1898 Charles Jean de la Valle Poussin, as part of his famed proof of the prime number theorem, developed an ineffective bound on the first Chebyshev function of the form: $$\mathrm{\mid }\theta (x)-x\mathrm{\mid }=\mathcal{O}(x\exp (-K\sqrt{\ln x}))\mathrm{.}$$ This bound holds for $x$ sufficiently large, $x\ge x_0$, and $K$ some unspecified positive constant. To the best of my knowledge this bound has never been made effective – I have never yet seen this bound made fully explicit, with precise values being given for $x_0$ and $K$. Herein, using a number of effective results established over the past 50 years, I shall develop two very simple explicit fully effective bounds of this type: $$\mathrm{\mid }\theta (x)-x\mathrm{\mid }<x\exp (-\frac{1}{4}\sqrt{\ln x});(x\ge 2)\mathrm{.}$$ $$\mathrm{\mid }\theta (x)-x\mathrm{\mid }<x\exp (-\frac{1}{3}\sqrt{\ln x});(x\ge 3)\mathrm{.}$$ Many other fully explicit bounds along these lines can easily be developed. For instance one can trade off stringency against range of validity: $$\mathrm{\mid }\theta (x)-x\mathrm{\mid }<\frac{1}{2}x\exp (-\frac{1}{4}\sqrt{\ln x});(x\ge 29),$$ $$\mathrm{\mid }\theta (x)-x\mathrm{\mid }<\frac{1}{2}x\exp (-\frac{1}{3}\sqrt{\ln x});(x\ge 41)\mathrm{.}$$ With hindsight, some of these effective bounds could have been established almost 50 years ago.