Grumman said that the USN gave them a F4U and told them that the USN only wanted one fighter type, and to get the Hellcat performance up to that of the F4U. Grumman's tests showed that the F4U was faster down low because it did not have the provisions to prevent icing that they considered to be essential. Their high altitude tests showed that most of the speed advantage of the F4U at altitude was due to airspeed indicator error!
Let's see how long time Grumman test pilot Corky Meyer tells the story. I have trimmed the quotes here and there to tighten the focus.
"In the desperate climate of World War II, the Navy decided that the easiest, quickest and least costly way to tweak the utmost performance out of its fighter planes would be to let rival manufacturers test the latest versions of each other's products. So it was in the summer of 1943 when the Navy delivered into Grumman hands the newest Corsair (F4U-1D Buno 17781)...
"Grumman's specific orders from the Navy were to improve the Hellcat's speed by 20 knots and put better ailerons on it so that it would compare favorably to the incomparable Corsair...
"We were also pleased to learn that we had not been singled out for the harassment of our sterling product when we heard that Chance Vought, our friendly competitor from the other side of the Long Island Sound, was also sent an F6F-3 Hellcat and ordered to improve Corsair visibility, cockpit internal layout, stall characteristics, and also redesign the landing gear oleos...
"In my first flight I discovered that the Corsair did indeed indicate 20 knots faster and did have really smooth and powerful ailerons compared to our Hellcat...
"To simplify the evaluation and data reduction, we decided to test fly the Hellcat and the Corsair in close formation. Performance could then be compared directly at the critical altitudes of the main stage, high and low blower altitudes of the engine's superchargers, and from cruise to high speed level flight with water injection in actual comparison instead of complex calculation...
"Except for the Corsair having a 15 knot actual speed advantage over the Hellcat in the main sea-level supercharger stage, both fighters had almost exactly the same speed both low and high blower from 5000 feet altitude up to service ceiling! ...
"The reason the Corsair was faster in the main stage blower was that its engine was provided with ram air coming in directly from the forward facing wing duct into the carburetor, whereas the Hellcat had the carburetor air coming in from the accessory compartment of the fuselage just behind the engine, with no ram air effect... Our engineering department defended their position because taking the warmer air for the main stage blower would prevent inadvertent carbureter icing engine failures.
"After noting the 20 knots indicated airspeed difference that had caused all the 'lower performance' ruckus for our Hellcat, we eagerly decided to change the airspeed system so that it would read even with the Corsair when they were in formation... We liked our simple and less complicated airspeed system, with the static and dynamic orifices on the same boom, but we decided to go whole hog and put the static orifice on the fuselage like the Corsair to tailor the system to read 20 knots higher."
But they made a mistake by using only one orifice. Pax River discovered a side slip in landing configuration could drop the indicated airspeed to zero! As the senior engineering test pilot, Meyer took the blame for missing that. Grumman's solution was a dual orifice system like the Corsair. "That was the last we heard of the Hellcat's performance gap with the Corsair."
He admits Grumman could not match the "delightful low forces and high rolling performance the Corsair had so ably demonstrated." Their engineers tried everything, even copies of the Corsair ailerons, but the Hellcat still couldn't match the Corsair roll. The problem was the higher lateral stability inherent in its wing design. And a new wing was out of the question at the height of the war. Later, when spring tabs were introduced in the -5 Hellcat and retrofitted to the -3, they made up much of the deficit.
Back to the airspeed, I view the Meyer account with some skepticism. Wasn't the Navy aware that you can't compare speeds without correcting for position error? (In those days I think it was often called "installation error.") On the other hand, the airspeed correction tables in the Hellcat pilot handbook are consistent with his story of Grumman's airspeed tweak. At all speeds the -3 indicates lower than CAS, and the -5 higher than CAS. At an efficient cruise speed of 140 knots, the -3 indicates 13 knots lower than CAS and the -5 indicates 7 knots higher. At 200 knots (as high as the table goes) the numbers are 14 and 17, respectively.
References:
Corwin Meyer and Steve Ginter, "Grumman F6F Hellcat," 2012.
"Pilots Handbook of Flight Operating Instructions for Navy Models F6F-3, F6F-3N, F6F-5, F5F-5N Airplanes" (year not shown)