### Problem Statement

A complex number is one of the form a+bi, where a and b are real numbers, and i represents the square root of -1. We call a the real part of the complex number, and b the imaginary part. When multiplying two complex numbers a+bi and c+di, the result is (a*c-b*d)+(a*d+b*c)i. When adding two complex numbers, a+bi and c+di, the result is (a+c)+(b+d)i. Finally, the magnitude of a complex number, a+bi, is sqrt(a*a+b*b).

The Mandelbrot set is a set of complex numbers. To determine whether or not a complex number, C = a+bi, is in the Mandelbrot set, we use the following recurrence:

Z0 = C

Zn+1 = Zn*Zn+C

If, for all n greater than or equal to 0, the magnitude of Zn is less than or equal to 2, then the complex number represented by C = a+bi is in the Mandelbrot set.

You will be given an int, max, representing the maximum value of n to check. For example, if max were 10, you would be required to check Zn for all 0 <= n <= 10. You will also be given two doubles, a and b, representing the real and imaginary parts of Z0, respectively. If the magnitude of Zn is always less than or equal to 2, for all 0 <= n <= max, you should return -1. Otherwise, you should return the smallest n such that the magnitude of Zn is greater than 2.

### Definition

 Class: Mandelbrot Method: iterations Parameters: int, double, double Returns: int Method signature: int iterations(int max, double a, double b) (be sure your method is public)

### Constraints

-max will be between 1 and 30, inclusive.
-a will be between -2 and 2, inclusive.
-b will be between -2 and 2, inclusive.
-To prevent rounding errors, the magnitude of Zn will not be within 1e-3 of 2, for any n between 0 and n', inclusive, where n' is the smallest number (or max) such that the magnitude of Zn' is greater than 2.

### Examples

0)

 `5` `1` `1`
`Returns: 1`
 Here, Z0 = 1 + 1i Thus, Z1 = 1 - 1 + 1i + 1i + 1 + 1i = 1 + 3i. The magnitude of 1+3i is sqrt(1*1 + 3*3) = sqrt(10) > 2. Therefore, the return is 1.
1)

 `2` `-1` `-1`
`Returns: 2`
 Z0 = -1 - 1i Z1 = -1 + 1i Z2 = -1 - 3i
2)

 `30` `.25` `.25`
`Returns: -1`
 In this example, the magnitude of Zn is never more than 2.
3)

 `30` `-1.9` `0`
`Returns: -1`
4)

 `30` `.375` `.3`
`Returns: 21`
5)

 `1` `2` `2`
`Returns: 0`

#### Problem url:

http://www.topcoder.com/stat?c=problem_statement&pm=1172

#### Problem stats url:

http://www.topcoder.com/tc?module=ProblemDetail&rd=4701&pm=1172

lars2520

#### Testers:

brett1479 , schveiguy

Math, Simulation