Are there span tables published for repairing a cut roof joist?
There is no mention where the cut out is but this should take care of it in any respect, as long as the mid span support is a series of posts and not a purlin...
Can you get a 2X6 in there that will give close to a full span? if the material is clean or can be scraped clean, a healthy bead of construction adhesive zig-zagged on the face of the old 2X and screw the 2X6 starting 4 inches from the ends with a pair of screws and 12+/- inches apart in the remainder in pairs also. Use 3 inch drywall screws or 2 3/4 inch SDS screws through the 2X6 into the 2X4. Make sure the screws draw tight the material, pilot if needed with a 3/16" bit.

If you have a mid span support that goes to good bearing through to the foundation, you can use a shorter repair joist, but a single joist that goes over multiple spans is stronger than a single span joist.

FWIW, it looks like a 2X4 will suffice from what I seen the table at page 396 of the 2012 building code (table R802.5.1).