I think actually you could cut budget much more than in half, and go to the Moon.

It's surprising how - comparatively - easy is - and was - to go to the Moon. Do you realize that for today technologies it's a walk in the park? It was done half a century ago. But that's not enough - even then in many aspects it was quite doable. Remember, the Moon race started as a calculated idea to win - all crucial requirements were approximately achievable, erring on "more likely than not". For example, F-1 first stage engine had a rather modest characteristics, except the size (which already provided a plenty of problems). The whole architecture was decided in favor of a single bug launcher - because docking was considered "risky", even though one docking (after liftoff from the Moon) and even one-half more docking (after translunar injection, when Apollo CM detached, turned over and docked with LM) eventually remained. Problems were being solved with size, not with smarter decisions - because the latter required time and were riskier.

So... Rocket science isn't a rocket science. Yes, Moon landing is one of the greatest technological achievements up to now. But it was done then, and for that much.