$cd ~ # SICP Exercise 2.13 Let$C_a$$be the center of an interval and$$T_a be the tolerance of ‘a’. Then the interval ‘a’ spans:

$a = [C_a \times (1 - 0.5 T_a), C_a \times (1 + 0.5 T_a)]$

Similarly for ‘b’:

$b = [C_b \times (1 - 0.5 T_b), C_b \times (1 + 0.5 T_b)]$

The interval of ‘a*b’:

\begin {align} a * b = [&C_a C_b (1 - 0.5 T_a)(1 - 0.5 T_b), \\ &C_a C_b (1 + 0.5 T_a)(1 + 0.5 T_b)] \\ a * b = [&C_a C_b (1 - 0.5 T_b - 0.5 T_a + 0.25 T_a T_b), \\ &C_a C_b (1 + 0.5 T_b + 0.5 T_a + 0.25 T_a T_b)] \\ \end{align}

The term $$T_a T_b$$ can be ignored when $$T_a$$ and $$T_b$$ are sufficiently low.

Therefore:

\begin{align} a * b = [&C_a C_b (1 - 0.5 (T_a + T_b)), \\ &C_a C_b (1 + 0.5 (T_a + T_b))] \end{align}

Thus, for sufficiently small tolerances, the resultant interval of the product of two intervals has a tolerance of the sum of each tolerance.