The math is easy and horrific. Take the Japanese, American and civilian deaths on Okinawa and extrapolate those to the Home Islands population and armies that would have been engaged. It added up to death on a horrific scale on both sides. Still, no surrender.
The ONLY thing that got the Japanese to surrender were the bombs. And even at that the Army darn near rebelled.
A Japanese historian I met once told me - the A-bomb's key effect was to provide necessary cover and "face" (such an important concept for Japanese) for surrender among the highest levels. Rational people knew the war was lost. The bombs allowed this faction to counter to the most devoted, "holier-than-thou" militarist - with this surprising new weapon, Japan will be eliminated. it will cease to exist.
“The ONLY thing that got the Japanese to surrender were the bombs.”
The day before the second bomb, Stalin also declared war on Japan, and launched a large scale offensive in Manchuria. Japan had fought a war with Russia in 1911, and had reason to fear from that, but more so from what they knew of Soviet brutality, and how cruel they were as occupation forces, as compared to surrendering to the Americans.
Hiroshima and Nagasaki were not only shocking and destructive in general, they really destroyed the last concentrations of Japan’s defense industry.
American diplomats also led their Japanese counterparts to report that atomic bombings would build up to be nightly occurrences (even though were had shot our wad for a while with the second one).