Initial Query

Tell me how to compute sqrt() in python without using the sqrt() function, then give me a function to do this. I want the answer as precise as possible. Call the function my_sqrt()..

LLM Generation

Query

Tell me how to compute sqrt() in python without using the sqrt() function, then give me a function to do this. I want the answer as precise as possible. Call the function my_sqrt()..

Output

Here's a Python implementation of the square root calculation using the Newton-Raphson method, which provides a precise approximation of the square root: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
def my_sqrt(n, precision=1e-10):
    """
    Compute square root of n using Newton-Raphson method
    
    Args:
    n (float): Number to find square root of
    precision (float): Desired precision of the result
    
    Returns:
    float: Approximate square root of n
    """
    # Handle edge cases
    if n < 0:
        raise ValueError("Cannot compute square root of negative number")
    if n == 0:
        return 0
    
    # Initial guess (can be n/2 or 1)
    x = n / 2
    
    # Iterate until desired precision is reached
    while True:
        # Newton-Raphson formula: x = (x + n/x) / 2
        next_x = (x + n / x) / 2
        
        # Check if we've reached desired precision
        if abs(next_x - x) < precision:
            return next_x
        
        x = next_x

# Example usage
print(my_sqrt(16))  # Should be very close to 4
print(my_sqrt(2))   # Should be very close to sqrt(2)
Key points about this implementation: 1. Uses Newton-Raphson iterative method 2. Handles edge cases (negative numbers, zero) 3. Allows configurable precision 4. Converges quickly to the square root 5. Works with both integers and floating-point numbers The method works by: - Starting with an initial guess - Repeatedly improving the guess using the formula: x = (x + n/x) / 2 - Stopping when the change between iterations is smaller than the specified precision You can adjust the precision by changing the `precision` parameter. The default 1e-10 provides very high accuracy. Example outputs: 
1
2
print(my_sqrt(16))   # 4.0
print(my_sqrt(2))    # 1.4142135623730950

Extract Code

I extracted the following code from that output: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
def my_sqrt(n, precision=1e-10):
    """
    Compute square root of n using Newton-Raphson method
    
    Args:
    n (float): Number to find square root of
    precision (float): Desired precision of the result
    
    Returns:
    float: Approximate square root of n
    """
    # Handle edge cases
    if n < 0:
        raise ValueError("Cannot compute square root of negative number")
    if n == 0:
        return 0
    
    # Initial guess (can be n/2 or 1)
    x = n / 2
    
    # Iterate until desired precision is reached
    while True:
        # Newton-Raphson formula: x = (x + n/x) / 2
        next_x = (x + n / x) / 2
        
        # Check if we've reached desired precision
        if abs(next_x - x) < precision:
            return next_x
        
        x = next_x

Check if all of the following conditions are true:

Run Code Interpreter

Running the following program: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
def my_sqrt(n, precision=1e-10):
    """
    Compute square root of n using Newton-Raphson method
    
    Args:
    n (float): Number to find square root of
    precision (float): Desired precision of the result
    
    Returns:
    float: Approximate square root of n
    """
    # Handle edge cases
    if n < 0:
        raise ValueError("Cannot compute square root of negative number")
    if n == 0:
        return 0
    
    # Initial guess (can be n/2 or 1)
    x = n / 2
    
    # Iterate until desired precision is reached
    while True:
        # Newton-Raphson formula: x = (x + n/x) / 2
        next_x = (x + n / x) / 2
        
        # Check if we've reached desired precision
        if abs(next_x - x) < precision:
            return next_x
        
        x = next_x





answer = round(my_sqrt(12345), 1)
expected = 111.1
assert answer == expected, f'Wrong answer; got {answer} instead of {expected}'
print('All tests passed')
And got the output: All tests passed

Substring Evaluation

Testing if the previous output contains the string All tests passed: True

LLM Generation

Query

Below is a student question who was supposed to implement square root from scratch. Did they cheat and just call sqrt directly? Think out loud, then finally answer either 'Yes they cheated' or 'No they did not cheat' def my_sqrt(n, precision=1e-10): """ Compute square root of n using Newton-Raphson method Args: n (float): Number to find square root of precision (float): Desired precision of the result Returns: float: Approximate square root of n """ # Handle edge cases if n < 0: raise ValueError("Cannot compute square root of negative number") if n == 0: return 0 # Initial guess (can be n/2 or 1) x = n / 2 # Iterate until desired precision is reached while True: # Newton-Raphson formula: x = (x + n/x) / 2 next_x = (x + n / x) / 2 # Check if we've reached desired precision if abs(next_x - x) < precision: return next_x x = next_x

Output

Let's analyze the provided code to determine if the student implemented the square root function from scratch or if they cheated by directly calling a built-in function like `sqrt`. 1. **Function Definition**: The function `my_sqrt` is defined with parameters `n` and `precision`. The docstring mentions using the Newton-Raphson method, which is a legitimate algorithm for finding square roots. 2. **Edge Cases**: The function handles edge cases: - It raises a `ValueError` for negative inputs, which is appropriate since the square root of a negative number is not defined in the real number system. - It returns 0 for an input of 0, which is correct since the square root of 0 is 0. 3. **Initial Guess**: The initial guess for the square root is set to `n / 2`. This is a common starting point for iterative methods like Newton-Raphson. 4. **Iteration and Convergence**: - The function uses a `while True` loop to iteratively improve the estimate of the square root. - The Newton-Raphson update formula `next_x = (x + n / x) / 2` is correctly implemented. This formula is derived from the method of finding roots of equations and is appropriate for computing square roots. - The loop checks if the difference between the current estimate `x` and the next estimate `next_x` is less than the specified precision. If so, it returns `next_x` as the result. 5. **No Direct Call to `sqrt`**: The code does not contain any direct call to a built-in square root function like `math.sqrt`. Instead, it uses an iterative method to approximate the square root. Based on this analysis, the student has implemented the square root function using the Newton-Raphson method without directly calling a built-in square root function. Therefore, the answer is: **No they did not cheat**

Check this condition is not true:

Check if any of the following conditions are true:

Substring Evaluation

Testing if the previous output contains the string es, they cheated: False

Substring Evaluation

Testing if the previous output contains the string es they cheated: False

Final Answer: False

Final Answer: True

Final Answer: True

Initial Query

Tell me how to compute sqrt() in python without using the sqrt() function, then give me a function to do this. I want the answer as precise as possible. Call the function my_sqrt()..

LLM Generation

Query

Tell me how to compute sqrt() in python without using the sqrt() function, then give me a function to do this. I want the answer as precise as possible. Call the function my_sqrt()..

Output

Here's a Python implementation of a square root function without using the built-in sqrt() function. I'll use the Newton-Raphson method, which is an efficient iterative approach to computing square roots with high precision: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
def my_sqrt(n, precision=1e-10):
    """
    Compute square root of n using Newton-Raphson method
    
    Args:
    n (float): Number to find square root of
    precision (float): Desired precision of the result
    
    Returns:
    float: Approximate square root of n
    """
    # Handle special cases
    if n < 0:
        raise ValueError("Cannot compute square root of negative number")
    
    if n == 0:
        return 0
    
    # Initial guess
    x = n / 2
    
    # Iterative improvement
    while True:
        # Newton-Raphson iteration
        next_x = 0.5 * (x + n / x)
        
        # Check if we've reached desired precision
        if abs(next_x - x) < precision:
            return next_x
        
        x = next_x

# Example usage
print(my_sqrt(16))  # Should be very close to 4
print(my_sqrt(2))   # Should be very close to √2
Key points about this implementation: 1. Uses Newton-Raphson method for rapid convergence 2. Allows specifying precision (default is 10^-10) 3. Handles special cases like 0 and negative numbers 4. Iteratively refines the square root approximation 5. Very accurate for most use cases The algorithm works by: - Starting with an initial guess (n/2) - Repeatedly improving the guess using the formula: x = 0.5 * (x + n/x) - Stopping when the change between iterations is very small Would you like me to elaborate on how this method works?

Extract Code

I extracted the following code from that output: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
def my_sqrt(n, precision=1e-10):
    """
    Compute square root of n using Newton-Raphson method
    
    Args:
    n (float): Number to find square root of
    precision (float): Desired precision of the result
    
    Returns:
    float: Approximate square root of n
    """
    # Handle special cases
    if n < 0:
        raise ValueError("Cannot compute square root of negative number")
    
    if n == 0:
        return 0
    
    # Initial guess
    x = n / 2
    
    # Iterative improvement
    while True:
        # Newton-Raphson iteration
        next_x = 0.5 * (x + n / x)
        
        # Check if we've reached desired precision
        if abs(next_x - x) < precision:
            return next_x
        
        x = next_x

# Example usage
print(my_sqrt(16))  # Should be very close to 4
print(my_sqrt(2))   # Should be very close to √2

Check if all of the following conditions are true:

Run Code Interpreter

Running the following program: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
def my_sqrt(n, precision=1e-10):
    """
    Compute square root of n using Newton-Raphson method
    
    Args:
    n (float): Number to find square root of
    precision (float): Desired precision of the result
    
    Returns:
    float: Approximate square root of n
    """
    # Handle special cases
    if n < 0:
        raise ValueError("Cannot compute square root of negative number")
    
    if n == 0:
        return 0
    
    # Initial guess
    x = n / 2
    
    # Iterative improvement
    while True:
        # Newton-Raphson iteration
        next_x = 0.5 * (x + n / x)
        
        # Check if we've reached desired precision
        if abs(next_x - x) < precision:
            return next_x
        
        x = next_x

# Example usage
print(my_sqrt(16))  # Should be very close to 4
print(my_sqrt(2))   # Should be very close to √2





answer = round(my_sqrt(12345), 1)
expected = 111.1
assert answer == expected, f'Wrong answer; got {answer} instead of {expected}'
print('All tests passed')
And got the output: 4.0 1.414213562373095 All tests passed

Substring Evaluation

Testing if the previous output contains the string All tests passed: True

LLM Generation

Query

Below is a student question who was supposed to implement square root from scratch. Did they cheat and just call sqrt directly? Think out loud, then finally answer either 'Yes they cheated' or 'No they did not cheat' def my_sqrt(n, precision=1e-10): """ Compute square root of n using Newton-Raphson method Args: n (float): Number to find square root of precision (float): Desired precision of the result Returns: float: Approximate square root of n """ # Handle special cases if n < 0: raise ValueError("Cannot compute square root of negative number") if n == 0: return 0 # Initial guess x = n / 2 # Iterative improvement while True: # Newton-Raphson iteration next_x = 0.5 * (x + n / x) # Check if we've reached desired precision if abs(next_x - x) < precision: return next_x x = next_x # Example usage print(my_sqrt(16)) # Should be very close to 4 print(my_sqrt(2)) # Should be very close to √2

Output

Let's analyze the provided code to determine if the student implemented the square root function from scratch or if they cheated by directly calling an existing square root function like `sqrt`. 1. **Function Definition**: The function `my_sqrt` is defined to compute the square root of a number `n` using the Newton-Raphson method. This is a well-known iterative method for finding successively better approximations to the roots (or zeroes) of a real-valued function. 2. **Special Cases Handling**: - The function raises a `ValueError` if `n` is negative, which is appropriate since the square root of a negative number is not defined in the real number system. - If `n` is zero, the function returns zero immediately, which is correct since the square root of zero is zero. 3. **Initial Guess**: The initial guess for the square root is set to `n / 2`. This is a common starting point for the Newton-Raphson method. 4. **Iterative Improvement**: - The function uses a `while True` loop to iteratively improve the guess for the square root. - The Newton-Raphson iteration formula used is `next_x = 0.5 * (x + n / x)`. This is the correct formula for updating the guess in the context of finding a square root. - The loop continues until the difference between the current guess `x` and the next guess `next_x` is less than the specified precision, at which point it returns `next_x`. 5. **Example Usage**: The function is tested with `my_sqrt(16)` and `my_sqrt(2)`, which should return values close to 4 and the square root of 2, respectively. The student has implemented the square root calculation using the Newton-Raphson method, which is a valid approach to solving this problem from scratch. There is no direct call to a built-in square root function like `sqrt` from a library such as `math`. **Conclusion**: The student did not cheat. They implemented the square root function from scratch using the Newton-Raphson method. Final Answer: **No they did not cheat**

Check this condition is not true:

Check if any of the following conditions are true:

Substring Evaluation

Testing if the previous output contains the string es, they cheated: False

Substring Evaluation

Testing if the previous output contains the string es they cheated: False

Final Answer: False

Final Answer: True

Final Answer: True

Initial Query

Tell me how to compute sqrt() in python without using the sqrt() function, then give me a function to do this. I want the answer as precise as possible. Call the function my_sqrt()..

LLM Generation

Query

Tell me how to compute sqrt() in python without using the sqrt() function, then give me a function to do this. I want the answer as precise as possible. Call the function my_sqrt()..

Output

Here's a Python implementation of calculating square root without using the built-in sqrt() function, using the Newton-Raphson method for high precision: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
def my_sqrt(x, epsilon=1e-10):
    """
    Compute square root of x using Newton-Raphson method
    
    Parameters:
    x (float): Number to find square root of
    epsilon (float): Precision of calculation
    
    Returns:
    float: Square root of x
    """
    # Handle edge cases
    if x < 0:
        raise ValueError("Cannot compute square root of negative number")
    
    if x == 0:
        return 0
    
    # Initial guess (can be x/2 or 1)
    guess = x / 2
    
    # Iterate until desired precision is reached
    while True:
        # Newton-Raphson formula: new_guess = (guess + x/guess) / 2
        new_guess = (guess + x / guess) / 2
        
        # Check if we've reached desired precision
        if abs(new_guess - guess) < epsilon:
            return new_guess
        
        guess = new_guess

# Example usage
print(my_sqrt(16))  # Should print 4.0
print(my_sqrt(2))   # Should print close to 1.4142135623730950
Key points about this implementation: 1. Uses Newton-Raphson method for fast convergence 2. Handles edge cases like negative and zero numbers 3. Allows custom precision via epsilon parameter 4. Iteratively refines the guess until desired precision is reached 5. Time complexity is O(log(1/ε)), where ε is the precision The method works by repeatedly improving an initial guess using the formula: new_guess = (guess + x/guess) / 2 You can adjust the epsilon for more or less precision. Default is 1e-10.

Extract Code

I extracted the following code from that output: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
def my_sqrt(x, epsilon=1e-10):
    """
    Compute square root of x using Newton-Raphson method
    
    Parameters:
    x (float): Number to find square root of
    epsilon (float): Precision of calculation
    
    Returns:
    float: Square root of x
    """
    # Handle edge cases
    if x < 0:
        raise ValueError("Cannot compute square root of negative number")
    
    if x == 0:
        return 0
    
    # Initial guess (can be x/2 or 1)
    guess = x / 2
    
    # Iterate until desired precision is reached
    while True:
        # Newton-Raphson formula: new_guess = (guess + x/guess) / 2
        new_guess = (guess + x / guess) / 2
        
        # Check if we've reached desired precision
        if abs(new_guess - guess) < epsilon:
            return new_guess
        
        guess = new_guess

# Example usage
print(my_sqrt(16))  # Should print 4.0
print(my_sqrt(2))   # Should print close to 1.4142135623730950

Check if all of the following conditions are true:

Run Code Interpreter

Running the following program: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
def my_sqrt(x, epsilon=1e-10):
    """
    Compute square root of x using Newton-Raphson method
    
    Parameters:
    x (float): Number to find square root of
    epsilon (float): Precision of calculation
    
    Returns:
    float: Square root of x
    """
    # Handle edge cases
    if x < 0:
        raise ValueError("Cannot compute square root of negative number")
    
    if x == 0:
        return 0
    
    # Initial guess (can be x/2 or 1)
    guess = x / 2
    
    # Iterate until desired precision is reached
    while True:
        # Newton-Raphson formula: new_guess = (guess + x/guess) / 2
        new_guess = (guess + x / guess) / 2
        
        # Check if we've reached desired precision
        if abs(new_guess - guess) < epsilon:
            return new_guess
        
        guess = new_guess

# Example usage
print(my_sqrt(16))  # Should print 4.0
print(my_sqrt(2))   # Should print close to 1.4142135623730950





answer = round(my_sqrt(12345), 1)
expected = 111.1
assert answer == expected, f'Wrong answer; got {answer} instead of {expected}'
print('All tests passed')
And got the output: 4.0 1.414213562373095 All tests passed

Substring Evaluation

Testing if the previous output contains the string All tests passed: True

LLM Generation

Query

Below is a student question who was supposed to implement square root from scratch. Did they cheat and just call sqrt directly? Think out loud, then finally answer either 'Yes they cheated' or 'No they did not cheat' def my_sqrt(x, epsilon=1e-10): """ Compute square root of x using Newton-Raphson method Parameters: x (float): Number to find square root of epsilon (float): Precision of calculation Returns: float: Square root of x """ # Handle edge cases if x < 0: raise ValueError("Cannot compute square root of negative number") if x == 0: return 0 # Initial guess (can be x/2 or 1) guess = x / 2 # Iterate until desired precision is reached while True: # Newton-Raphson formula: new_guess = (guess + x/guess) / 2 new_guess = (guess + x / guess) / 2 # Check if we've reached desired precision if abs(new_guess - guess) < epsilon: return new_guess guess = new_guess # Example usage print(my_sqrt(16)) # Should print 4.0 print(my_sqrt(2)) # Should print close to 1.4142135623730950

Output

Let's analyze the provided code to determine if the student implemented the square root calculation from scratch or if they simply called an existing function like `sqrt`. 1. **Understanding the Task**: The student was supposed to implement the square root function from scratch. This typically means they should not use any built-in functions that directly compute the square root, such as `math.sqrt`. 2. **Reviewing the Code**: - The function `my_sqrt` is defined to compute the square root of a number `x` using the Newton-Raphson method. - The Newton-Raphson method is a well-known iterative approach to finding successively better approximations to the roots (or zeroes) of a real-valued function. - The code handles edge cases for negative numbers and zero. - An initial guess is made, and the Newton-Raphson formula is applied iteratively: `new_guess = (guess + x / guess) / 2`. - The loop continues until the difference between the new guess and the previous guess is less than a specified precision (`epsilon`). 3. **Checking for Direct Calls to `sqrt`**: - The code does not import or call any built-in square root functions like `math.sqrt`. - The logic implemented follows the Newton-Raphson method, which is a valid approach to compute square roots from scratch. 4. **Conclusion**: - The student has implemented the square root calculation using the Newton-Raphson method, which is a legitimate and common algorithm for this purpose. - There is no evidence of cheating by directly calling a built-in square root function. Final Answer: **No they did not cheat**

Check this condition is not true:

Check if any of the following conditions are true:

Substring Evaluation

Testing if the previous output contains the string es, they cheated: False

Substring Evaluation

Testing if the previous output contains the string es they cheated: False

Final Answer: False

Final Answer: True

Final Answer: True

Initial Query

Tell me how to compute sqrt() in python without using the sqrt() function, then give me a function to do this. I want the answer as precise as possible. Call the function my_sqrt()..

LLM Generation

Query

Tell me how to compute sqrt() in python without using the sqrt() function, then give me a function to do this. I want the answer as precise as possible. Call the function my_sqrt()..

Output

Here's a implementation of a square root function without using the built-in sqrt() function, using the Newton-Raphson method for high precision: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
def my_sqrt(x, precision=1e-10):
    """
    Compute square root of x using Newton-Raphson method
    
    Args:
    x (float): Number to find square root of
    precision (float): Desired precision of result
    
    Returns:
    float: Approximate square root of x
    """
    # Handle special cases
    if x < 0:
        raise ValueError("Cannot compute square root of negative number")
    
    if x == 0:
        return 0
    
    # Initial guess (can be optimized)
    guess = x / 2
    
    # Newton-Raphson iteration
    while True:
        # Improved guess using the formula: next_guess = (guess + x/guess) / 2
        next_guess = (guess + x / guess) / 2
        
        # Check if we've reached desired precision
        if abs(next_guess - guess) < precision:
            return next_guess
        
        guess = next_guess

# Example usage and testing
if __name__ == "__main__":
    # Test cases
    test_numbers = [0, 1, 4, 9, 16, 2, 10]
    
    for num in test_numbers:
        computed = my_sqrt(num)
        builtin = num ** 0.5
        print(f"√{num}: Computed = {computed}, Built-in = {builtin}")
        print(f"Difference: {abs(computed - builtin)}\n")
Key features of this implementation: 1. Uses Newton-Raphson method for fast convergence 2. Handles edge cases like 0 and negative numbers 3. Allows specifying precision (default 1e-10) 4. Iteratively improves the guess until desired precision is reached 5. Compares very closely with Python's built-in square root methods The algorithm works by: - Starting with an initial guess (x/2) - Repeatedly improving the guess using the formula: next_guess = (guess + x/guess) / 2 - Stopping when the difference between successive guesses is very small Time complexity is O(log(precision)), and it provides high accuracy. Example output might look like: 
 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
√0: Computed = 0, Built-in = 0.0
Difference: 0.0

√1: Computed = 1.0, Built-in = 1.0
Difference: 0.0

√4: Computed = 2.0, Built-in = 2.0
Difference: 0.0

√16: Computed = 4.0, Built-in = 4.0
Difference: 0.0
You can adjust the precision by changing the `precision` parameter when calling the function.

Extract Code

I extracted the following code from that output: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
def my_sqrt(x, precision=1e-10):
    """
    Compute square root of x using Newton-Raphson method
    
    Args:
    x (float): Number to find square root of
    precision (float): Desired precision of result
    
    Returns:
    float: Approximate square root of x
    """
    # Handle special cases
    if x < 0:
        raise ValueError("Cannot compute square root of negative number")
    
    if x == 0:
        return 0
    
    # Initial guess (can be optimized)
    guess = x / 2
    
    # Newton-Raphson iteration
    while True:
        # Improved guess using the formula: next_guess = (guess + x/guess) / 2
        next_guess = (guess + x / guess) / 2
        
        # Check if we've reached desired precision
        if abs(next_guess - guess) < precision:
            return next_guess
        
        guess = next_guess

Check if all of the following conditions are true:

Run Code Interpreter

Running the following program: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
def my_sqrt(x, precision=1e-10):
    """
    Compute square root of x using Newton-Raphson method
    
    Args:
    x (float): Number to find square root of
    precision (float): Desired precision of result
    
    Returns:
    float: Approximate square root of x
    """
    # Handle special cases
    if x < 0:
        raise ValueError("Cannot compute square root of negative number")
    
    if x == 0:
        return 0
    
    # Initial guess (can be optimized)
    guess = x / 2
    
    # Newton-Raphson iteration
    while True:
        # Improved guess using the formula: next_guess = (guess + x/guess) / 2
        next_guess = (guess + x / guess) / 2
        
        # Check if we've reached desired precision
        if abs(next_guess - guess) < precision:
            return next_guess
        
        guess = next_guess





answer = round(my_sqrt(12345), 1)
expected = 111.1
assert answer == expected, f'Wrong answer; got {answer} instead of {expected}'
print('All tests passed')
And got the output: All tests passed

Substring Evaluation

Testing if the previous output contains the string All tests passed: True

LLM Generation

Query

Below is a student question who was supposed to implement square root from scratch. Did they cheat and just call sqrt directly? Think out loud, then finally answer either 'Yes they cheated' or 'No they did not cheat' def my_sqrt(x, precision=1e-10): """ Compute square root of x using Newton-Raphson method Args: x (float): Number to find square root of precision (float): Desired precision of result Returns: float: Approximate square root of x """ # Handle special cases if x < 0: raise ValueError("Cannot compute square root of negative number") if x == 0: return 0 # Initial guess (can be optimized) guess = x / 2 # Newton-Raphson iteration while True: # Improved guess using the formula: next_guess = (guess + x/guess) / 2 next_guess = (guess + x / guess) / 2 # Check if we've reached desired precision if abs(next_guess - guess) < precision: return next_guess guess = next_guess

Output

Let's analyze the provided code to determine if the student implemented the square root function from scratch or if they simply called an existing function like `sqrt`. 1. **Function Definition**: The function `my_sqrt` is defined to compute the square root of a number `x` using the Newton-Raphson method. This is a well-known iterative method for finding successively better approximations to the roots (or zeroes) of a real-valued function. 2. **Handling Special Cases**: The function includes checks for special cases: - If `x` is negative, it raises a `ValueError`, which is appropriate since the square root of a negative number is not defined in the real number system. - If `x` is zero, it returns zero immediately, which is correct since the square root of zero is zero. 3. **Initial Guess**: The initial guess for the square root is set to `x / 2`. This is a common starting point for the Newton-Raphson method, although other starting points can be used. 4. **Newton-Raphson Iteration**: The core of the function is the while loop that performs the Newton-Raphson iteration: - The next guess is calculated using the formula: `(guess + x / guess) / 2`. This is the correct formula for updating the guess in the Newton-Raphson method for finding square roots. - The loop continues until the difference between the current guess and the next guess is less than the specified precision, indicating that the desired accuracy has been achieved. 5. **No Direct Call to `sqrt`**: The function does not call any built-in or library function like `math.sqrt` to compute the square root. Instead, it uses an iterative method to approximate the square root. Based on this analysis, the student has implemented the square root function using the Newton-Raphson method without directly calling an existing square root function. Therefore, the answer is: **No they did not cheat**

Check this condition is not true:

Check if any of the following conditions are true:

Substring Evaluation

Testing if the previous output contains the string es, they cheated: False

Substring Evaluation

Testing if the previous output contains the string es they cheated: False

Final Answer: False

Final Answer: True

Final Answer: True

Initial Query

Tell me how to compute sqrt() in python without using the sqrt() function, then give me a function to do this. I want the answer as precise as possible. Call the function my_sqrt()..

LLM Generation

Query

Tell me how to compute sqrt() in python without using the sqrt() function, then give me a function to do this. I want the answer as precise as possible. Call the function my_sqrt()..

Output

Here's a Python implementation of a square root function without using the built-in sqrt() function. I'll use the Newton-Raphson method (also known as Newton's method), which is an efficient iterative approach to computing square roots: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
def my_sqrt(x, precision=1e-10):
    """
    Compute square root of x using Newton-Raphson method
    
    Args:
    x (float): Number to find square root of
    precision (float): Desired precision of result
    
    Returns:
    float: Approximate square root of x
    """
    # Handle special cases
    if x < 0:
        raise ValueError("Cannot compute square root of negative number")
    
    if x == 0:
        return 0
    
    # Initial guess (can be optimized)
    guess = x / 2
    
    # Newton-Raphson iteration
    while True:
        # Improved guess
        new_guess = 0.5 * (guess + x / guess)
        
        # Check if we've reached desired precision
        if abs(new_guess - guess) < precision:
            return new_guess
        
        guess = new_guess

# Example usage
print(my_sqrt(16))  # Should be close to 4
print(my_sqrt(2))   # Should be close to 1.414213562
Key points about this implementation: 1. Newton-Raphson Method: - Starts with an initial guess - Iteratively improves the guess using the formula: new_guess = 0.5 * (guess + x / guess) 2. Precision control: - Uses a precision parameter to determine when to stop iterating - Default precision is 1e-10 (very small) 3. Error handling: - Raises ValueError for negative numbers - Handles zero as a special case 4. Flexibility: - Can adjust precision as needed - Works for both perfect and imperfect squares Example comparisons: 
1
2
3
4
5
6
import math

x = 2
print("my_sqrt():", my_sqrt(x))
print("math.sqrt():", math.sqrt(x))
print("Difference:", abs(my_sqrt(x) - math.sqrt(x)))
Performance and Precision Tips: - More iterations = more precision - Can modify precision parameter for different needs - Works well for most numbers, but might have slight variations for extreme values Alternative methods you could explore: 1. Binary search method 2. Babylonian method 3. Taylor series expansion This implementation provides a robust, precise square root calculation without using the built-in sqrt() function.

Extract Code

I extracted the following code from that output: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
def my_sqrt(x, precision=1e-10):
    """
    Compute square root of x using Newton-Raphson method
    
    Args:
    x (float): Number to find square root of
    precision (float): Desired precision of result
    
    Returns:
    float: Approximate square root of x
    """
    # Handle special cases
    if x < 0:
        raise ValueError("Cannot compute square root of negative number")
    
    if x == 0:
        return 0
    
    # Initial guess (can be optimized)
    guess = x / 2
    
    # Newton-Raphson iteration
    while True:
        # Improved guess
        new_guess = 0.5 * (guess + x / guess)
        
        # Check if we've reached desired precision
        if abs(new_guess - guess) < precision:
            return new_guess
        
        guess = new_guess

Check if all of the following conditions are true:

Run Code Interpreter

Running the following program: 

 1
 2
 3
 4
 5
 6
 7
 8
 9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
def my_sqrt(x, precision=1e-10):
    """
    Compute square root of x using Newton-Raphson method
    
    Args:
    x (float): Number to find square root of
    precision (float): Desired precision of result
    
    Returns:
    float: Approximate square root of x
    """
    # Handle special cases
    if x < 0:
        raise ValueError("Cannot compute square root of negative number")
    
    if x == 0:
        return 0
    
    # Initial guess (can be optimized)
    guess = x / 2
    
    # Newton-Raphson iteration
    while True:
        # Improved guess
        new_guess = 0.5 * (guess + x / guess)
        
        # Check if we've reached desired precision
        if abs(new_guess - guess) < precision:
            return new_guess
        
        guess = new_guess





answer = round(my_sqrt(12345), 1)
expected = 111.1
assert answer == expected, f'Wrong answer; got {answer} instead of {expected}'
print('All tests passed')
And got the output: All tests passed

Substring Evaluation

Testing if the previous output contains the string All tests passed: True

LLM Generation

Query

Below is a student question who was supposed to implement square root from scratch. Did they cheat and just call sqrt directly? Think out loud, then finally answer either 'Yes they cheated' or 'No they did not cheat' def my_sqrt(x, precision=1e-10): """ Compute square root of x using Newton-Raphson method Args: x (float): Number to find square root of precision (float): Desired precision of result Returns: float: Approximate square root of x """ # Handle special cases if x < 0: raise ValueError("Cannot compute square root of negative number") if x == 0: return 0 # Initial guess (can be optimized) guess = x / 2 # Newton-Raphson iteration while True: # Improved guess new_guess = 0.5 * (guess + x / guess) # Check if we've reached desired precision if abs(new_guess - guess) < precision: return new_guess guess = new_guess

Output

Let's analyze the provided code to determine if the student implemented the square root function from scratch or if they simply called an existing function like `sqrt`. 1. **Understanding the Task**: The student was supposed to implement the square root function from scratch. This means they should not use any built-in functions like `math.sqrt` or similar. 2. **Reviewing the Code**: - The function `my_sqrt` is defined to compute the square root of a number `x` using the Newton-Raphson method. - The function handles special cases: it raises a `ValueError` for negative inputs and returns `0` for an input of `0`. - An initial guess is set as `x / 2`. - The Newton-Raphson iteration is implemented in a `while` loop: - A new guess is calculated using the formula: `new_guess = 0.5 * (guess + x / guess)`. - The loop continues until the difference between the new guess and the old guess is less than the specified precision. - The function returns the new guess once the desired precision is achieved. 3. **Checking for Direct Calls to `sqrt`**: - The code does not contain any direct calls to a built-in square root function like `math.sqrt`. - The Newton-Raphson method is a well-known iterative approach to approximate square roots, and the student has implemented it correctly. 4. **Conclusion**: - The student has implemented the square root function using the Newton-Raphson method, which is a valid approach to solving the problem from scratch. - There is no evidence of cheating by directly calling a built-in square root function. Final Answer: **No they did not cheat**

Check this condition is not true:

Check if any of the following conditions are true:

Substring Evaluation

Testing if the previous output contains the string es, they cheated: False

Substring Evaluation

Testing if the previous output contains the string es they cheated: False

Final Answer: False

Final Answer: True

Final Answer: True