Problem p00639 - Generation 2

Orig Description

Problem C: Accelerated Railgun
She catched the thrown coin that draws parabolic curve with her sparkling fingers. She is an ESPer. Yes, she is an electro-master who has the third strongest power among more than one million ESPers in the city. Being flicked by her thumb, the coin is accelerated by electromagnetic force and is shot as Fleming's right-hand rule. Even if she holds back the initial velocity of the coin exceeds three times of the speed of sound. The coin that is shot in such velocity is heated because of air friction and adiabatic compression. As a result coin melts and shines in orange. This is her special ability, called railgun. The strength of railgun can make a hole of two meters in diameter on a concrete wall.
She had defeated criminals such as post office robberies and bank robberies in the city with her ability. And today, she decided to destroy a laboratory that is suspected to making some inhumane experiments on human body. Only her railgun can shoot it.
The railgun with a coin cannot destroy the laboratory because of lack of power. Since she have found a powered-suit nearby, so she decided to use it as a projectile. However, in this case it is difficult to take sight properly because the suit is much bigger and heavier than coins. Therefore she only can shoot the suit with certain velocity vector from her current position. Depending on the position of the laboratory, her railgun may not hit it and become a firework.
Therefore she asked cooperation to the strongest ESPer in the city. He can change direction of a moving object as one of uncountable application of his ESP. Let's consider a 2-dimensional plane where the laboratory is on the origin (0, 0). She shoots a projectile from P = (px, py) with velocity vector V = (vx, vy). His ESP makes a virtual wall of radius R (= 1.0) centered on the origin. When projectile collides with the wall, it is reflected so that incident angle will be equal to reflection angle.
Range of railgun is limited to D, because air friction decreases velocity of projectile and heat may melts projectile completely. Under these conditions, please write a program that judges if the railgun hits the laboratory. Size of the laboratory and the suit is ignorablly small. After the railgun is shot, it is allowed to pass through P again.
Input
Input consists of several datasets.
The first line of each dataset contains a real number D.
Next line contains 4 real numbers, which means px, py, vx, vy, respectively.
Input terminates when D = 0.
Output
For each dataset, if the railgun hits, output the distance it moved until hits. Otherwise output 'impossible' (without quotes).
You can print any number of digits and answer with an error less than 1.0e-6 will be accepted. 
Constraints
Judge data never include dataset where the answer is (D - 1.0e-3) or bigger. 
0.0001 ≤ |V| ≤ 0.9999
0.0001 ≤ |P| ≤ 0.9999
D ≤ 50
Sample Input
10.0
0.5 0.0 -0.2 0.0
1.0
0.1 0.0 0.2 0.2
0
Output for the Sample Input
0.50000000
impossible

Extracted Specification

An input consists of several datasets.
- The first line of each dataset contains a real number D.
- The next line contains 4 real numbers, which means px, py, vx, vy.
- Input terminates when D = 0.

### Example Input:

```
10.0
0.5 0.0 -0.2 0.0
1.0
0.1 0.0 0.2 0.2
0
```

### Function Signature:
Write a function f(inputs) that takes in a list of tuples, each containing (D, (px, py, vx, vy)).
def f(inputs: List[Tuple[float, Tuple[float, float, float, float]]]): 
    ''' 
    inputs: a list of tuples, each containing:
        - D: a real number
        - (px, py, vx, vy): a tuple of four real numbers
    '''

Test Case Generator

Sure, here is a test case generator for the given problem statement "Problem C: Accelerated Railgun":

```python
import random

def generate_real(min_val, max_val, precision=6):
    """Generate a random real number between min_val and max_val with specified precision."""
    return round(random.uniform(min_val, max_val), precision)

def tcgen():
    test_cases = []
    while True:
        # Randomly decide whether to terminate the input
        if random.random() < 0.1:  # 10% chance to terminate
            test_cases.append("0")
            break
        
        D = generate_real(0.0001, 50.0)
        
        px = generate_real(0.0001, 0.9999)
        py = generate_real(0.0001, 0.9999)
        
        vx = generate_real(0.0001, 0.9999)
        vy = generate_real(0.0001, 0.9999)
        
        test_cases.append(f"{D}")
        test_cases.append(f"{px} {py} {vx} {vy}")
    
    return "\n".join(test_cases)

# Example usage:
print(tcgen())
```

### Explanation:
1. **`generate_real` Function**: This function generates a random real number within a specified range and precision.
2. **`tcgen` Function**: 
   - It generates a series of test cases until a termination condition is met (10% chance to stop).
   - Each test case consists of a real number `D` and four real numbers for the coordinates and velocity vectors.
   - The test cases are accumulated in a list and joined into a single string with newlines separating each case.

### Note:
- The generator creates a random number `D` between 0.0001 and 50.
- It generates the coordinates `(px, py)` and velocity vectors `(vx, vy)` between 0.0001 and 0.9999 to meet the problem constraints.
- The loop randomly decides when to add the termination condition `0` to the test cases.

This script will generate realistic and varied test cases for the problem, ensuring good coverage for different scenarios.

Extract Arguments

def extract_arguments(fh):
    datasets = []
    while True:
        D = float(fh.readline().strip())
        if D == 0:
            break
        px, py, vx, vy = map(float, fh.readline().strip().split())
        datasets.append((D, (px, py, vx, vy)))
    return datasets