Probably not completely half no.
I am a SAHM. I get up with the children in the week (we have 2 daughters, DD1 is 5 years old and DD2 is 1 years old). My partner will either be in bed as been working late shift/nights or be at work if doing early shift.
So I do school runs. Most laundry/ironing/putting away (though he will chuck a load in if he sees it and will hang out washing sometimes. I do general day to day sweeping, polishing, hoovering etc. But if he is working earlies so home in afternoon he will sweep through at night while I am putting the children to bed.
He usually baths them about twice a week.
I cook, and either of us will wash up depending on who gets there first.
He will hoover the stairs/landing a couple of times a week (I hate that job!)
I tend to keep the toilet clean.
Every weekend he has a major sweep downstairs (moving sofa out etc) and complete mop through.
He does all gardening and takes bins out on bin day.
I get up in night if either children wake up. We both have a lie in on weekends (take a day each).
I tend to take charge of medicine etc as can get confusing if others start giving it as well in case they forget to say. If I am going out I put the times down when medicine has been had, when can have next etc. If he does need to give it he will put a tick and write the time. But 9/10 times it is me.
When he is at home either of us will change nappies, depending on who notices it first
So he is pretty good really