We've all seen high level pianists play with immaculate polish on their notes. How is this done?
well, you know, lots of practice.
i'd say just record yourself. video and audio. break the song into 4 bar loops with a few beats' overlap. don't go to the next loop till the one you're on is perfect. and always end a loop's practice on a perfect take so your nerves get wired like THAT overnight.
but, in the end, nobody cares about perfection except the musician. you should care about it. but you really can't expect to play perfectly every dang time through the piece. your average take can be pretttty close to perfect, though.
A high level concert pianist will practise a minimum of 8 hours a day. You'd think at that level they wouldn't need to practise much at all. However, usually without the music in front of them, they have to be note perfect, and that's the best proven way to do it, as the other answers also say.
There is also the propensity for players at that level to be able to make mistakes that they can recover from. Mistakes for them won't often be wrong notes, so the audience won't hear things like that, but could be a different voicing in their piece, playing over someone else in the wrong place, or forgetting a repeat. They will have played the piece so much that it'll be an almost automatic correction that, unless the listener plays the piece themselves, or has heard it thousands of times, won't be spotted. And even if that's the case, it could be argued that that was the preferred way the performer wanted it!
You know what they say:
“Miss one day of practice, I notice; miss two, the orchestra notice; miss three, the audience notices”
You need lot of practice to achieve this levels. Actually, professional pianists sometimes play a lot of notes wrong in concerts.
"A musician only has to become skilled enough that their playing is more precise than the discernment of said listener".