**Exercise 5.8.10.** On a computer using a decimal floating-point system with *ϵ*_{mach} = 10^{−8} and an exponent range of [−16, 15], what is the result of the following computations?

1. 10^{4} + 10^{−5}

2. 1 − 10^{−4}

3. 10^{10}/10^{9}

4. 10^{−10}/10^{9}

5. 10^{15} × 10^{−16}

6. 10^{15} × 10^{16}

**Exercise 5.8.11.** Which of the following two formulas to compute the midpoint of an interval [*a*, *b*] is preferable in floating-point arithmetic?

$m\text{}=\text{}\frac{a\text{}+\text{}b}{2},\hspace{1em}m\text{}=\text{}a\text{}+\text{}\frac{b\text{}-\text{}a}{2}$

Is it possible to compute *m* that lies outside the interval with either one of these formulas?

**Exercise 5.8.12.** The standard deviation of a set of real numbers can be computed in two mathematically equivalent ways

$\sigma \text{}=\text{}{\left[\frac{1}{n\text{}-\text{}1}{\displaystyle \sum _{i\text{}=\text{}1}^{n}{\left({x}_{i}\text{}-\text{}\overline{x}\right)}^{2}}\right]}^{1/2},\hspace{1em}\sigma \text{}=\text{}{\left[\frac{1}{n\text{}-\text{}1}{\displaystyle \sum _{i\text{}=\text{}1}^{n}{x}_{i}^{2}\text{}-\text{}{\overline{x}}^{2}}\right]}^{1/2}$

where $\overline{x}\text{}=\text{}{\displaystyle {\sum}_{i\text{}=\text{}1}^{n}{x}_{i}}$ is the mean. Which one ...

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