If I take your picture of a building and copy that's a copyright violation.
If I take a picture of a building from the exact same angle you did, that's not a copyright violation.
Similarly, if you copy the definitions of all functions/methods/classes/etc from one API, nobody will believe you independently came up with it.
So "add(a, b) returning the sum of the two" is probably not copyrightable by itself, but a whole namespace full of functions might be.
That's not how it works for copyright. Beyond that, how do you independently arrive at an API that you are trying to have an exact copy of? Photography is not programming. I cannot run a bunch of words through a program until it prints out some book I want and then say I arrived at it independently. You honestly need to read the copyright laws as you are missing some information.
[edit: the original premise of all this thread is IBM didn't try to copyright the BIOS API]
Your original post was about IBM not being able to copyright the PC Bios. Which is simply not true in any way shape or form. https://www.allaboutcircuits.com/news/how-compaqs-clone-comp...
Relevant portion "But one problem remained with their IBM-compatible machine. How would they recreate the IBM BIOS without breaching the copyright? Why did they need a compatible BIOS (basic input output system)? . . . Some companies, such as Eagle, copied the BIOS either by reading the IBM manual to understand which interrupt codes did what or by simply copying the hard-coded ROM chip that holds the BIOS. This breached the copyright that IBM had over the BIOS and so did not take IBM long to prevent such companies from selling more of those machines. However, there is a way to bypass these types of copyrights with the use of a “clean room”."
No, no, no, that is not what I said. I was talking about the API of the BIOS. The BIOS source code is, of course, copyrightable. That is why IBM was able to make it very hard for people to copy it until folks did a proper clean room. Why would I go into detail about the counter measure if it was not copyrightable? That makes no sense.
What "different way" are we talking about here? IBM asserted their copyright quite forcefully against Eagle and shut them down.
Eagle directly copied the source code of the BIOS, they were not sued for copying the API.
The API of a BIOS is not the whole thing! It just specifies the interface by which the BIOS can be called by the OS.
For a crude analogy, consider the interface specification that most modern cars adhere to: there will be a steering wheel with a certain shape, size, and position relative to the driver's seat; there will be two or three pedals, with the accelerator on the right, the brake pedal to its left, and the clutch pedal, if present, to the left of the brake; etc. etc. None of this tells you whether the car runs on gasoline, diesel, CNG, or electricity, or whether the motor is in the back or the front, how many cylinders it has, etc. — all these are implementation details that do not usually affect the interface much.
The implementation of the BIOS — the code that actually performs its various functions — is clearly copyrightable, and always has been. But the API is just the specification of what those functions are and how they can be invoked. Imagine if one car manufacturer had been able to copyright the steering wheel and pedal arrangement that we're all familiar with!