Two Big Points, and one small one.
After you evaluate f(x,y,z) itself, keeping too many significant figures, the big job is δf.
1. For each contribution to the uncertainty to f(x,y,z), analyze
δf(x,y,z) due to δx = ∂f/ ∂x times δx
δf(x,y,z) due to δy = ∂f /∂y times δy
δf(x,y,z) due to δz = ∂f /∂z times δz,
Using the partial derivatives, one-by-one.
2. Combine the contributions to the uncertainties in f(x,y,z) due to
uncorrelated δx, δy, δz by adding in quadrature, as synthesis.
δf = sqrt (δf due to δx 2 + δf due to δy 2 + δf due to δz 2 ), The
master equation of Taylor, page 75.
3. Then, round off δf appropriately
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One significant figure for δf, 76.3 80, unless
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The result starts with a one , 12.34 12.
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But it is ok if the answer ends with a 5 to keep it 0.650.65, or round it up 0.650.7.
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Then round off the answer for f(x,y,z) to be consistent with rounded δf
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Put a box around f +/- δf as the final result, or show it as f (δf).