Out-of-Control Rounding Errors
J, 84 bytes

Starting from a 1 element list the function keeps all the possible intermediate numbers in the list by evaling the next expression and adding it's up and down rounded copies.

Will golf further and add explanation tomorrow.
Can't find obvious ways to golf it more.

f=.3 :'/:~~.&lt;.&gt;((,&gt;.,&lt;.)@".@(":@],''x'',;@[))&amp;.&gt;/|.(&gt;@{.;_2&lt;\}.);:y rplc''/'';''%'''


Passes all the tests.

Usage:

   f '1/3*2*2*2*2*2*2'
0 16 20 21 22 24 32 64
   f '1/3*9'
0 3 9


Try it here.