03 Types, type conversions and floating point arithmetic

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Exercise 03.1

Compare the computed values of

d_0 = a \cdot b + a \cdot c

and

d_1 = a \cdot (b + c)

when $a = 100$ , $b = 0.1$ and $c = 0.2$ . Store $d_{0}$ in the variable d0 and $d_{1}$ in the variable d1.

Try checking for equality, e.g. print(d0 == d1).

a = ...
b = ...
c = ...

d0 = ...
print(d0)

d1 = ...
print(d1)

print(d0 == d1)

## tests ##
assert d0 == 30.0
assert d1 != 30.0
assert d0 != d1

练习 03.1

比较以下两个计算值：

d_0 = a \cdot b + a \cdot c

和

d_1 = a \cdot (b + c)

当 $a = 100$ , $b = 0.1$ 和 $c = 0.2$ 时。将 (d_{0}) 的值存储在变量 d0 中，将 (d_{1}) 的值存储在变量 d1 中。

尝试检查两者是否相等，例如：print(d0 == d1)。

Answer

当 $a = 100$ , $b = 0.1$ 和 $c = 0.2$ 时：

首先计算 $d_0$ :

$d_0 = a \cdot b + a \cdot c = 100 \times 0.1 + 100 \times 0.2$

然后计算 $d_1$ :

$d_1 = a \cdot (b + c) = 100 \times (0.1 + 0.2)$

接下来，我们可以将这两个计算值进行编程比较。

a = 100
b = 0.1
c = 0.2

# 计算d0和d1的值
d0 = a * b + a * c
d1 = a * (b + c)

# 输出d0和d1的值
print("d0 =", d0)
print("d1 =", d1)

# 检查d0和d1是否相等
print("d0 and d1 are equal:", d0 == d1)

运行上述代码，我们可以获得 $d_0$ 和 $d_1$ 的值，以及它们是否相等的结果。

Exercise 03.2

For the polynomial

\begin{align} f(x, y) &= (x + y)^{6} \\ &= x^6 + 6x^{5}y + 15x^{4}y^{2} + 20x^{3}y^{3} + 15x^{2}y^{4} + 6xy^{5} + y^{6} \end{align}

compute $f$ using: (i) the compact form $(x + y)^{6}$ ; and (ii) the expanded form for:

(a) $x = 10$ and $y = 10.1$

(b) $x = 10$ and $y = -10.1$

and compare the number of significant digits for which the answers are the same.
Store the answer for the compact version using the variable f0, and using the variable f1 for the expanded version.

For case (b), compare the computed and analytical solutions and consider the relative error.
Which approach would you recommend for computing this expression?

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