Programatically add a product to cart with custom price

This is relatively straightforward but after days of research I'm still at a loss as to how to achieve it.

If I have a custom module which presents a customer with a form where they fill out various details, my code then calculates a price based on these inputs.

So my module has a calculated price, we just need to add a generic pre-defined product to the cart and modify the price...done. Can anyone provide a couple of snippets on how to do this? I've looked at various hooks and cant find any clear information on how to do this.