When it idles for a while and then you step on the gas blue smoke comes out of the tail pipe 2004 bmw 545i

fuel injectors. they are most likely pouring fuel into the cylinders instead of spraying a fine mist.

what kind of smoke by that I mean what colour is it and when does is happen under hard acceleration, at idle, when first started, under light acceleration, idling a long time? blue smoke is burning oil, white smoke is burning coolant sometimes accompanied with a sweet smell, black smoke its running rich. those are the basics.

Is it black? black smoke means you are burning too much gasoline could be a bad choke or even a dirty air filter.
Blue smoke means you are burning oil out of your engine
.White means water getting into crank case or pistons.

Are you saying you have smoke coming out of the tail pipe ? Or is something dripping onto the exhaust pipe and burning ?
If the smoke is coming out of the tail pipe, is it blue, white, or black ?

Replaced the Idle Air Control?

Smoke from tail pipe or smoke from under hood?
A general rule of thumb if it is coming from tail pipe the color of smoke will tell you a lot.

White and thick means head gasket failure, Blue means oil is being burned.

I would recommend you check the seal of everything and make sure it is cleaned that had to do with vacuum or fuel distribution. EGR valve (gasket fails sometimes) IAC valve, MAP and MAF sensors, Fuel injectors, FPR (this will need replaced if leaking),

Also check you coolant and oil look for signs of either one mixing with one another, smell you coolant and make sure it don't smell like gas either, look around where the head meets the block and make sure you don't have any coolant leaking there. The smoke if from the tail pipe makes me think head gasket and these are simple checks. They are not 100% diagnosis but can lead you to the problem.

Also one obvious thing would be make sure no one put anything in your gas tank.

I would most likely look at the injectors first then the head gasket next.

you have the n62 engine in this x5 valve guide seals are leaking causing the blue smoke but i hate to tell you people havnt had much luck with repair in fact i am replacing the cylinder heads on a 750 with the same engine right now after an attempt to replace the valve guide seals failed to resolve the problem

It sounds like piston rings are worn you can try some engine restore thats what its called when you change the oil next time put that in in place of about a 1/2 of quart of oil run for 500-1000 miles and see if its better if not need to rebuild motor.

If the smoke comming from the tail pipe is black your running rich. If thats the case the entire fuel system has to be ckecked out. Injectors could be leaking, a bad pressure regulator,stuck return line ,a coolant temp sensor could be telling the ECU that the engines cold give her more gas. I don't recommend doing fuel service unless your qualified to do so for the obvious reasons.
If the smoke coming out of the tail is grayish you have a bad head gasket. If the smoke coming out of the tail is blue you have engine oil getting into the combustion chamber. keep me posted

RINGS AROUN THE PISTON, OIL LEAKING INTO THE COMBUSTION CHAMBER