It's amazing to think that just after Rolls-Royce Holdings (LSE:RR) completed its life-saving rights issue in November 2020, it shares were changing hands for just over £1. The share price was given another boost on 30 July when the group released its results for the first six months of 2026. Delving deeperCompared to the first half of 2025, its operating profit increased by 46% and its operating margin improved from 19.1% to 22.5% (both on an underlying basis). As a result, it now expects an underlying operating profit for 2026 of £4.7bn-£4.9bn (previously £4bn-£4.2bn) and free cash flow of £3.8bn-£4bn (previously £3.6bn-£3.8bn). Seeing into the futureAs a result of its first-half performance, it says it has "further confidence" in its mid-term (2028) guidance of £4.9bn-£5.2bn of operating profit.