How do I slide player along a wall smoothly?

I'm trying to create sliding along a wall mechanic for my game and can't seem to work out the vector math needed. Basically I'd like smooth sliding across walls like that in games like: Quake, Half-Life 1, counter-strike etc. But my player slides weirdly and even starts bouncing in some cases. That happens because the distance from the wall becomes higher than threshold. The problem seems to be with sliding direction I set. This is 3D game by the way.

Here's the code snippet, that makes the...

How do I slide player along a wall smoothly?