Irrespective of what you contributed to the purchase price, if you purchased as Joint Tenants then you are entitled to half each, and if you purchased as Tenants in Common then you are entitled to whatever shares you agreed at the time of purchase.
In calculating what your partner is entitled to, do not forget that the only way you and they can ultimately get hold of the equity is to sell the property. Therefore if you buy out your ex partner you should bring into account the notional costs of sale (estate agents fees, legal fees, VAT etc) - say 3% of the value of the house.
To do the calculation take a realistic value of the property, deduct the notional costs of sale, take off the cost of redeeming the mortgage (remember to allow for any early redemption penalties) and divide what is left by two (or the appropriate proportions if Tenants in Common).
For negotiaing purposes you can offer whatever you like, but the above gives you the "fair" answer.