The Diophantine Equation $x^4 + 2 y^4 = z^4 + 4 w^4$---a number of improvements

Abstract: The quadruple $$(1\,484\,801, 1\,203\,120, 1\,169\,407, 1\,157\,520)$$ already known is essentially the only non-trivial solution of the Diophantine equation $$x⁴ + 2 y⁴ = z⁴ + 4 w<sup>4$$ for $|x|$, $|y|$, $|z|$, and $|w|$ up to one hundred million. We describe the algorithm we used in order to establish this result, thereby explaining a number of improvements to our original approach.