That is just an unfortunately propogated bit of "internet wisdom." The two contain virtually the same amount of energy. The difference is that you can run a more aggressive fuel or spark tune with higher octane, to extract more useful energy out of the fuel, instead of dumping it out the radiator as waste heat.
I still wouldn't expect to see a 7%+ increase by switching to premium, and I especially wouldn't expect an ounce of magic elixir make an entire tank of E10 give the same mileage as pure gas.
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