Problem E
Different Distances
Some people say ‘The shortest distance between two points is a straight line.’ However, this depends on the distance metric employed. Between points $(x_1,y_1)$ and $(x_2,y_2)$, the Euclidean (aka straight-line) distance is
\[ \sqrt {(x_1 - x_2)^2 + (y_1 - y_2)^2} \]However, other distance metrics are often useful. For instance, in a city full of buildings, it is often impossible to travel in a straight line between two points, since buildings are in the way. In this case, the so-called Manhattan (or city-block) distance is the most useful:
\[ |x_1 - x_2| + |y_1 - y_2| \]Both Euclidean and city-block distance are specific instances of what is more generally called the family of $p$-norms. The distance according to norm $p$ is given by
\[ \left( |x_1 - x_2| ^ p + |y_1 - y_2| ^ p \right)^{1/p} \]If we look at Euclidean and Manhattan distances, these are both just specific instances of $p = 2$ and $p=1$, respectively.
For $p < 1$ this distance measure is not actually a metric, but it may still be interesting sometimes. For this problem, write a program to compute the $p$-norm distance between pairs of points, for a given value of $p$.
Input
The input file contains up to $1\, 000$ test cases, each of which contains five real numbers, $x_1~ y_1~ x_2~ y_2~ p$, each of which have at most $10$ digits past the decimal point. All coordinates are in the range $(0, 100]$ and $p$ is in the range $[0.1, 10]$. The last test case is followed by a line containing a single zero.
Output
For each test case output the $p$-norm distance between the two points $(x_1,y_1)$ and $(x_2,y_2)$. Your answer may have absolute or relative error of at most $0.0001$.
| Sample Input 1 | Sample Output 1 |
|---|---|
1.0 1.0 2.0 2.0 2.0 1.0 1.0 2.0 2.0 1.0 1.0 1.0 20.0 20.0 10.0 0 |
1.4142135624 2.0000000000 20.3636957882 |
