(Adds TPG declining to comment) By Anirban Sen and Milana Vinn NEW YORK, Aug 3 (Reuters) - Buyout firms Francisco Partners and TPG saved hundreds of millions of dollars in their $6.5 billion deal to buy New Relic by walking away from negotiations with the business software company in May and then returning two months later with a lower offer, people familiar with the matter said. The two private equity firms offered more than $90 per share in cash to acquire New Relic in May, a bid that the...