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Compasses cost virtually nothing in terms of power, the chips are cheap their signal is easy to read. Usually they're not even read constantly, an accelerometer is read to constantly adjust the compass which avoids even asking the fractionally slower chip. The accelerometer is essentially free to call. The compass is only queried periodically to ensure that the accelerometer hasn't caused any drift in value.
All-in-all... compasses cost nothing.
Having the screen powered... costs almost everything.
Velocio
edscoble
Does it drain the battery a bit using the compass?