In real world applications is there a concrete benefit when using O(log(log(n)) instead of O(log(n)) algorithms ?

This is the case when one use for instance van Emde Boas trees instead of more conventional binary search tree implementations. 
But for example, if we take n < 10^6 then in the best case the double logarithmic algorithm outperforms the logarithmic one by (approximately) a factor of 5. And also in general the implementation is more tricky and complex. 

Given that I personally prefer BST over VEB-trees, what do you think ?

*One could easily demonstrate that :*

![enter image description here][2]

 


  [1]: https://i.sstatic.net/GKHn0.png
  [2]: https://i.sstatic.net/Nk1eu.png