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Problem 2.13
Friday, August 23, 2019; ago; Download .md

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.