Setting aside all other variables for a moment, "racing around" is something ill-defined and subjective, but it is quite entirely plausible that something like hard acceleration, while not in the short term, can get you better mileage in the long run.
To understand why this might be the case, you need to understand the power band. An engine's performance is typically rated by torque and horsepower. The torque specification will usually be measured as the maximum torque an engine can produce at a given speed (RPM), while horsepower will measure the maximum amount of power also at a rated speed. Thus the power band is generally defined as the operating speed in between the values of which peak torque and horsepower operate.
In this illustration, take the "flexible" engine. It has a peak torque rated say at about 245 ft-lbs at 2500 RPM with horsepower peaking at about 225 at 5500 RPM. The power band would be somewhere between 2500-5500 RPM. Rarely are these graphs nicely curved as pictured here. The torque would usually start drop immediately after its peak.
When the engine is operating within the power band, it is producing power the most efficiently. Under ideal operating conditions (maximum engine load and throttle), an engine will produce power the most efficiently at peak torque RPM.
But these conditions almost never happen on the road for a sustained period of time, so you will never have the maximal efficiency and so it's not guaranteed that it will be at that speed in typical driving conditions. In-fact the torque and power curves will typically be different depending on the engine load and throttle.
To understand these conditions, you would have to look at a BSFC or SFC (Brake / Specific Fuel Consumption) map for your vehicle if one exists.
So it's very possible that when you're "racing around" you're operating in the power range and so are increasing fuel economy in the long term. Generally when trying to maximize fuel economy, it's often recommended you accelerate briskly. Of course if you're not looking ahead you might have to brake and consume more fuel to accelerate.