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In this section, we will see how the errors of two numbers, x and y, are propagated with the four arithmetic operations. Error propagation means that the errors in the input numbers of a process or an operation cause the errors in the output numbers.

Let their absolute errors be ε_x and ε_y, respectively. Then the magnitudes of the absolute/relative errors in the sum and difference are

(1.2 13)

(1.2 14)

From this, we can see why the relative error is magnified to cause the ‘loss of significance’ in the case of subtraction when the two numbers X and Y are almost equal so that |X − Y| ≈ 0.

The magnitudes of the absolute and relative errors in the multiplication/division are

(1.2 15)

(1.2 16)

(1.2 17)

(1.2 18)

This implies that, in the worst‐case, the relative error in multiplication/division may be as large as the sum of the relative errors of the two numbers.