at all vs impossibly

at all

adv
  • Indicating degree, quantity or frequency greater than zero: to the slightest degree, in any way, at any time or in any circumstance. 

impossibly

adv
  • Contrary to what had been thought possible. 

  • Not possibly; in an impossible manner. 

  • To the point of impossibility. 

How often have the words at all and impossibly occurred in a corpus of books? (source: Google Ngram Viewer )