Mathematica, 17 14 bytes / 12 characters

Round[#/√2]& 

Try it online

-3 bytes because Mathematica accepts the char √, which I copied from this MathGolf answer.

@Mark S. suggested

⌊#/√2⌋& 

to save 1 byte and 5 characters, but this no longer rounds to the nearest integer, as it always rounds down. A similar program can be made to always round up.

Mathematica, 17 14 13 bytes / 12 7 characters

⌊#/√2⌋& 

Try it online

-3 bytes because Mathematica accepts the char √, which I copied from this MathGolf answer.

-1 byte, -5 characters, as per @Mark S. suggestion, by using ⌊⌋.

For just one more byte (but 5 more characters) I can always round to the nearest integer with

Round[#/√2]& 

Mathematica, 17 14 bytes / 12 characters

Round[#/√2]& 

Try it online

-3 bytes because Mathematica accepts the char √, which I copied from this MathGolf answer.

Mathematica, 17 14 bytes / 12 characters

Round[#/√2]& 

Try it online

-3 bytes because Mathematica accepts the char √, which I copied from this MathGolf answer.

@Mark S. suggested

⌊#/√2⌋& 

to save 1 byte and 5 characters, but this no longer rounds to the nearest integer, as it always rounds down. A similar program can be made to always round up.

Mathematica, 17 14 bytes (but 12 chars :'( )

Round[#/√2]& 

Try it online

-3 bytes because Mathematica accepts the char √, which I copied from this MathGolf answer.

Mathematica, 17 14 bytes / 12 characters

Round[#/√2]& 

Try it online

-3 bytes because Mathematica accepts the char √, which I copied from this MathGolf answer.