Significance of ios_base::sync_with_stdio(false); cin.tie(NULL);

The two calls have different meanings that have nothing to do with performance; the fact that it speeds up the execution time is (or might be) just a side effect. You should understand what each of them does and not blindly include them in every program because they look like an optimization.

ios_base::sync_with_stdio(false);

This disables the synchronization between the C and C++ standard streams. By default, all standard streams are synchronized, which in practice allows you to mix C- and C++-style I/O and get sensible and expected results. If you disable the synchronization, then C++ streams are allowed to have their own independent buffers, which makes mixing C- and C++-style I/O an adventure.

Also keep in mind that synchronized C++ streams are thread-safe (output from different threads may interleave, but you get no data races).

cin.tie(NULL);

This unties cin from cout. Tied streams ensure that one stream is flushed automatically before each I/O operation on the other stream.

By default cin is tied to cout to ensure a sensible user interaction. For example:

std::cout << "Enter name:";
std::cin >> name;

If cin and cout are tied, you can expect the output to be flushed (i.e., visible on the console) before the program prompts input from the user. If you untie the streams, the program might block waiting for the user to enter their name but the "Enter name" message is not yet visible (because cout is buffered by default, output is flushed/displayed on the console only on demand or when the buffer is full).

So if you untie cin from cout, you must make sure to flush cout manually every time you want to display something before expecting input on cin.

In conclusion, know what each of them does, understand the consequences, and then decide if you really want or need the possible side effect of speed improvement.

This is to synchronize IOs from C and C++ world. If you synchronize, then you have a guaranty that the orders of all IOs is exactly what you expect. In general, the problem is the buffering of IOs that causes the problem, synchronizing let both worlds to share the same buffers. For example cout << "Hello"; printf("World"); cout << "Ciao";; without synchronization you'll never know if you'll get HelloCiaoWorld or HelloWorldCiao or WorldHelloCiao...

tie lets you have the guaranty that IOs channels in C++ world are tied one to each other, which means for example that every output have been flushed before inputs occurs (think about cout << "What's your name ?"; cin >> name;).

You can always mix C or C++ IOs, but if you want some reasonable behavior you must synchronize both worlds. Beware that in general it is not recommended to mix them, if you program in C use C stdio, and if you program in C++ use streams. But you may want to mix existing C libraries into C++ code, and in such a case it is needed to synchronize both.