Trigonometric Optimization

Time limit2sMemory limit128 MB

Summary

Choose positive integers x, y, z summing to S to optimize sin/cos(x)+sin/cos(y)+sin/cos(z), rounded to 10 decimals.

Level: Medium4 of 10
Topics: Math, Brute force, Greedy
Solved: No attempts yet

Problem

Many problems that arise in practice can be stated as optimization problems: we must maximize or minimize a so-called criterion function subject to some constraints.

Consider the following trigonometric optimization problem. Maximize or minimize the criterion function

$F_1(x) + F_2(y) + F_3(z)$

subject to the constraint $x + y + z = S$ , where $x$ , $y$ , $z$ are the variables and $S$ is a given parameter. All of $x$ , $y$ , $z$ , $S$ are natural numbers (positive integers, so $x, y, z \ge 1$ ). Each of the functions $F_1$ , $F_2$ , $F_3$ is a trigonometric function — either $\sin$ or $\cos$ — and its argument is measured in radians.

Write a program that solves this trigonometric optimization problem.

Input

The input consists of 5 lines.

Line 1 describes the function $F_1$ and contains either sin or cos.
Line 2 describes the function $F_2$ in the same format.
Line 3 describes the function $F_3$ in the same format.
Line 4 contains either min or max. If it is min, the criterion function must be minimized; if it is max, it must be maximized.
Line 5 contains the parameter $S$ ( $3 \le S \le 10^6$ ).

Output

Print a single line containing the optimal value of the criterion function: its maximum when line 4 is max, or its minimum when line 4 is min, taken over all natural numbers $x$ , $y$ , $z$ with $x + y + z = S$ .

Print the value rounded to exactly 10 digits after the decimal point (format %.10f).

Examples3

Example 1
Input
```
sin
cos
sin
max
10
```
Expected output
```
2.7787651403
```
Example 2
Input
```
sin
sin
sin
max
3
```
Expected output
```
2.5244129544
```
Example 3
Input
```
cos
cos
cos
min
3
```
Expected output
```
1.6209069176
```