Fusible Numbers II

We define a canonical representation of a fusible number as a representation with only 0.

For instance, the shortest canonical representations of 21/16 are :

• (((0~0)~0)~(0~0))~(0~0)
• (((0~0)~0)~0)~((0~0)~0)
• (((0~0)~0)~0)~(0~(0~0))
• ((0~(0~0))~(0~0))~(0~0)
• ((0~(0~0))~0)~((0~0)~0)
• ((0~(0~0))~0)~(0~(0~0))
• ((0~0)~((0~0)~0))~(0~0)
• ((0~0)~(0~(0~0)))~(0~0)
• ((0~0)~0)~(((0~0)~0)~0)
• ((0~0)~0)~((0~(0~0))~0)
• ((0~0)~0)~(0~((0~0)~0))
• ((0~0)~0)~(0~(0~(0~0)))
• (0~((0~0)~0))~((0~0)~0)
• (0~((0~0)~0))~(0~(0~0))
• (0~(0~(0~0)))~((0~0)~0)
• (0~(0~(0~0)))~(0~(0~0))
• (0~(0~0))~(((0~0)~0)~0)
• (0~(0~0))~((0~(0~0))~0)
• (0~(0~0))~(0~((0~0)~0))
• (0~(0~0))~(0~(0~(0~0)))
• (0~0)~(((0~0)~0)~(0~0))
• (0~0)~((0~(0~0))~(0~0))
• (0~0)~((0~0)~((0~0)~0))
• (0~0)~((0~0)~(0~(0~0)))

and the last in alphabetic order is (0~(0~(0~0)))~(0~(0~0))

What is the last (in alphabetic order) shortest canonical representation of 111/64?
Hint : The number of 0s in a representation is 13.
[My timing: 4 sec]