[xoranth]

What GCC is doing at -O3 is encoding even (resp. odd) numbers in a bitmap.

43690 = 0xAAAA = 0b1010101010101010

21845 = 0x5555 = 0b0101010101010101

Then

    sal     rax, cl
    test    eax, 43690
    jne     .L3

    // .L3
    mov     edi, OFFSET FLAT:.LC1 // This is a pointer to the string "odd"
    call    puts
    jmp     .L2

is equivalent to C:

    if ((1 << number) & 0xAAAA) {
        puts("odd"); // x64 is little endian
    }

So execution is constant time (it only needs to check against two constants), there's no loop.

It should be advantageous compared to a jump table because it doesn't have to do two indirect jumps (though you now have branches that might mispredict).

Edit: code formatting.