ive had my body fat measured twice at two different gyms. one used the measuring tape and inputed information into the computer to get the number, while my other gym now uses a small hand held machine where the trainer just inputs height and weight and then you squeeze the machine and you get the number that way.

which way is the most accurate?

Those methods are both extremely inaccurate. Skin fold caliper measurement is much more accurate, if done properly by someone who knows what they are doing, but even that is not perfect. I believe the only truely accurate method is the water immersion method, but that usually is very expensive, if you can even find a place that does it.