Yes, I have flexi - schooled a G&T child, and have home-educated another through to GCSEs.
Flexi-schooling is where parents come to a voluntary agreement with the school to home-educate for part of the week, and it's different from the reduced timetable for children with health problems which some people have mentioned. The difficulty with flexi-schooling is that the government guidance on it was changed in 2013 to say that the school should mark the child as having 'authorised absence', Code C, on the register, when the child is not in school. Prior to 2013 schools could mark the child as 'Educated offsite', and this did not count as an absence. In my area, several under-subscribed primary schools were happy to take home-educated children part-time under a flexi-school arrangement, until the guidance changed. The school received full-time funding even though the child attended part-time.
It's still permissible for state school heads to agree to a flexi-schooling arrangement, but the absences will impact on the school's absence figures. The only way round this is for the school to 'take ownership' of what the child is doing on the days out of school. If the school is prepared to take responsibility for the days out of school then some still use Code B, 'Educated off-site', although there is no explicit government guidance about this.
The most comprehensive sources of information on flexi-schooling are:
EdYourself on Flexi-schooling and
Flexischooling families UK facebook group.
Personally I think you would be better off at this stage sending your child to school full-time and concentrating on extension activities out of school hours. If your child enjoys maths, look at the UK Mathematics Trust - they have great challenges for mathsy kids, competitions, run the UK Maths Olympiad, and a mentorship scheme where able children can get online coaching.
If your child is good at music then that would be an area where you will find many opportunities to encourage them. Is there a conservatoire near you?
If languages appeal - see how he gets on with www.duolingo.com as a starter.
Have you considered talking to independent schools about scholarships?
Independent schools are more likely to consider flexi schooling than state schools as they don't have to worry about the absence figures.
Home education worked well for my family for a long time, but it does take a lot of effort going to all the activities in order to keep the social contact up, and this eats into the time you have available to do other things. I think there is a lot to be said for the range of skills that they learn in school, working with different teachers, understanding that different people have different expectations, and so on. Some schools are brilliant at providing differentiation activities for able children. Try not to make assumptions about what schools could and couldn't cope with; each one will be a bit different. Perhaps you could ask for meetings with teachers at a few different schools in your area and just ask them what they could offer your child?
When my oldest went to sixth form, at a maths specialist selective school, he met other kids who were interested in the same things. Some of these children had achieved amazing things; they had not been held back. Instead, because they'd ended up at 'geek school', they found like-minded friends who could push each other on. The school had some pupils who'd done Open University maths courses whilst at school because they'd taken A-level maths early, or done it concurrently with their maths A-levels. Do find out what's out there, before assuming that school won't have anything to offer.