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

One common way to compute the square root of a number without using Python’s built‐in sqrt() is to use Newton’s method (also known as the Newton–Raphson method). The idea is to find a root of the function   f(y) = y² − x since the square root of x is the number y for which y² = x. Newton’s method provides an iterative formula:   yₙ₊₁ = yₙ − f(yₙ)/f′(yₙ) For our function, f′(y) = 2y and the update rule becomes:   yₙ₊₁ = yₙ − (yₙ² − x)/(2yₙ) = (yₙ + x/yₙ) / 2 You start with an initial guess (for example, x itself if x > 1 or 1 if x is less than 1) and continue iterating until the change between successive approximations is below a chosen tolerance. Below is the complete Python function my_sqrt() implementing this algorithm with a high precision tolerance: ----------------------------------------------------- def my_sqrt(x, tolerance=1e-10, max_iterations=1000): """ Compute the square root of a non-negative number x using Newton's method. Arguments: x -- the number to compute the square root of (must be non-negative). tolerance -- the precision tolerance (default is 1e-10). max_iterations -- the maximum number of iterations to prevent an infinite loop. Returns: The approximate square root of x. Raises: ValueError -- if x is negative. """ # Handle special cases: if x < 0: raise ValueError("Cannot compute the square root of a negative number.") if x == 0: return 0.0 # Choose an initial guess: guess = x if x >= 1 else 1.0 for _ in range(max_iterations): new_guess = 0.5 * (guess + x/guess) # If the change is within the tolerance, we consider the result good enough: if abs(new_guess - guess) < tolerance: return new_guess guess = new_guess # If we reach the maximum iterations, return the last approximation return guess # Example usage: if __name__ == "__main__": number = 25 print("The square root of", number, "is approximately:", my_sqrt(number)) ----------------------------------------------------- How it works: 1. If x is negative, a ValueError is raised. 2. For x = 0, the function immediately returns 0.0. 3. An initial guess is chosen based on the value of x. 4. The loop iteratively updates the guess using the formula and stops when the difference between successive guesses is less than the specified tolerance. 5. The function returns the final guess as the approximation of sqrt(x). This method is very efficient and converges quadratically, meaning that the number of correct digits roughly doubles with each iteration once it is close to the answer.

Extract Code

I extracted the following code from that output: 

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#!/usr/bin/env python3
def my_sqrt(x, tolerance=1e-10, max_iterations=1000):
    """
    Compute the square root of a non-negative number x using Newton's method.
    
    Arguments:
    x -- the number to compute the square root of (must be non-negative).
    tolerance -- the precision tolerance (default is 1e-10).
    max_iterations -- the maximum number of iterations to prevent an infinite loop.
    
    Returns:
    The approximate square root of x.
    
    Raises:
    ValueError -- if x is negative.
    """
    # Handle special cases:
    if x < 0:
        raise ValueError("Cannot compute the square root of a negative number.")
    if x == 0:
        return 0.0

    # Choose an initial guess:
    guess = x if x >= 1 else 1.0

    for _ in range(max_iterations):
        new_guess = 0.5 * (guess + x/guess)
        # If the change is within the tolerance, we consider the result good enough:
        if abs(new_guess - guess) < tolerance:
            return new_guess
        guess = new_guess

    # If we reach the maximum iterations, return the last approximation
    return guess

Check if all of the following conditions are true:

Run Code Interpreter

Running the following program: 

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#!/usr/bin/env python3
def my_sqrt(x, tolerance=1e-10, max_iterations=1000):
    """
    Compute the square root of a non-negative number x using Newton's method.
    
    Arguments:
    x -- the number to compute the square root of (must be non-negative).
    tolerance -- the precision tolerance (default is 1e-10).
    max_iterations -- the maximum number of iterations to prevent an infinite loop.
    
    Returns:
    The approximate square root of x.
    
    Raises:
    ValueError -- if x is negative.
    """
    # Handle special cases:
    if x < 0:
        raise ValueError("Cannot compute the square root of a negative number.")
    if x == 0:
        return 0.0

    # Choose an initial guess:
    guess = x if x >= 1 else 1.0

    for _ in range(max_iterations):
        new_guess = 0.5 * (guess + x/guess)
        # If the change is within the tolerance, we consider the result good enough:
        if abs(new_guess - guess) < tolerance:
            return new_guess
        guess = new_guess

    # If we reach the maximum iterations, return the last approximation
    return 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' #!/usr/bin/env python3 def my_sqrt(x, tolerance=1e-10, max_iterations=1000): """ Compute the square root of a non-negative number x using Newton's method. Arguments: x -- the number to compute the square root of (must be non-negative). tolerance -- the precision tolerance (default is 1e-10). max_iterations -- the maximum number of iterations to prevent an infinite loop. Returns: The approximate square root of x. Raises: ValueError -- if x is negative. """ # Handle special cases: if x < 0: raise ValueError("Cannot compute the square root of a negative number.") if x == 0: return 0.0 # Choose an initial guess: guess = x if x >= 1 else 1.0 for _ in range(max_iterations): new_guess = 0.5 * (guess + x/guess) # If the change is within the tolerance, we consider the result good enough: if abs(new_guess - guess) < tolerance: return new_guess guess = new_guess # If we reach the maximum iterations, return the last approximation return guess

Output

Let's analyze the provided code to determine if the student directly called the `sqrt` function or implemented their own version of the square root calculation. 1. **Function Definition**: The function `my_sqrt` is defined to compute the square root of a non-negative number `x` using Newton's 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 `x` is negative, which is appropriate since the square root of a negative number is not defined in the set of real numbers. - If `x` is zero, the function returns `0.0`, 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` if `x` is greater than or equal to 1, otherwise, it is set to 1.0. This is a reasonable starting point for the Newton's method iteration. 4. **Newton's Method Iteration**: - The loop iterates up to `max_iterations` times, updating the guess using the formula `new_guess = 0.5 * (guess + x/guess)`. This is the core of Newton's method for finding square roots. - The loop checks if the change between the new guess and the old guess is within the specified `tolerance`. If it is, the function returns the new guess as the approximate square root. 5. **Return Value**: - If the loop completes without finding a sufficiently accurate approximation (which is unlikely if `max_iterations` is large enough), the function returns the last computed guess. **Conclusion**: - The student has implemented the square root calculation using Newton's method, which is a legitimate and common approach to solving this problem. - There is no direct call to any built-in `sqrt` function in the code. The student has not cheated by using a direct call to `sqrt`. 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

One common way to compute a square root without using math.sqrt() is to use the Newton–Raphson (or Heron's) method. This iterative method approximates the square root by starting with a guess and refining it using the formula:   next_guess = (guess + x / guess) / 2 Here’s a Python function named my_sqrt() that implements this method with a tolerance to control precision: --------------------------------------------------------- def my_sqrt(x, tolerance=1e-10, max_iter=1000): """ Compute the square root of a non-negative number x using Newton's method. Parameters: x (float): The number to compute the square root of. Must be non-negative. tolerance (float): The precision tolerance for the result. max_iter (int): Maximum number of iterations to perform. Returns: float: An approximation of sqrt(x) within the specified tolerance. Raises: ValueError: If x is negative. """ if x < 0: raise ValueError("Cannot compute square root of a negative number") if x == 0: return 0.0 # The square root of 0 is 0. # Initial guess can be x/2.0, though other choices are possible. guess = x / 2.0 for _ in range(max_iter): new_guess = (guess + x / guess) / 2.0 # Check if the improvement is within tolerance. if abs(new_guess - guess) < tolerance: return new_guess guess = new_guess # If we reach here, max_iter iterations were reached without convergence. return guess # Example usage: if __name__ == "__main__": num = 25.0 result = my_sqrt(num) print("Square root of", num, "is approximately", result) --------------------------------------------------------- How It Works: 1. The function first handles edge cases: if x is negative (raising an error) or if x is zero. 2. An initial guess is made (x/2.0). This is then refined using the iterative formula. 3. The loop continues until the absolute difference between consecutive guesses is less than the specified tolerance. 4. If the loop finishes without converging within the max_iter limit, it returns the current guess. This method provides a very precise approximation of the square root for most practical purposes.

Extract Code

I extracted the following code from that output: 

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#!/usr/bin/env python3
import math

def my_sqrt(x, tolerance=1e-10, max_iter=1000):
    """
    Compute the square root of a non-negative number x using Newton's method.
    
    Parameters:
        x (float): The number to compute the square root of. Must be non-negative.
        tolerance (float): The precision tolerance for the result.
        max_iter (int): Maximum number of iterations to perform.
    
    Returns:
        float: An approximation of sqrt(x) within the specified tolerance.
    
    Raises:
        ValueError: If x is negative.
    """
    if x < 0:
        raise ValueError("Cannot compute square root of a negative number")
    if x == 0:
        return 0.0  # The square root of 0 is 0.
    
    # Initial guess can be x/2.0, though other choices are possible.
    guess = x / 2.0
    for _ in range(max_iter):
        new_guess = (guess + x / guess) / 2.0
        # Check if the improvement is within tolerance.
        if abs(new_guess - guess) < tolerance:
            return new_guess
        guess = new_guess
    
    # If we reach here, max_iter iterations were reached without convergence.
    return guess

Check if all of the following conditions are true:

Run Code Interpreter

Running the following program: 

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#!/usr/bin/env python3
import math

def my_sqrt(x, tolerance=1e-10, max_iter=1000):
    """
    Compute the square root of a non-negative number x using Newton's method.
    
    Parameters:
        x (float): The number to compute the square root of. Must be non-negative.
        tolerance (float): The precision tolerance for the result.
        max_iter (int): Maximum number of iterations to perform.
    
    Returns:
        float: An approximation of sqrt(x) within the specified tolerance.
    
    Raises:
        ValueError: If x is negative.
    """
    if x < 0:
        raise ValueError("Cannot compute square root of a negative number")
    if x == 0:
        return 0.0  # The square root of 0 is 0.
    
    # Initial guess can be x/2.0, though other choices are possible.
    guess = x / 2.0
    for _ in range(max_iter):
        new_guess = (guess + x / guess) / 2.0
        # Check if the improvement is within tolerance.
        if abs(new_guess - guess) < tolerance:
            return new_guess
        guess = new_guess
    
    # If we reach here, max_iter iterations were reached without convergence.
    return 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' #!/usr/bin/env python3 import math def my_sqrt(x, tolerance=1e-10, max_iter=1000): """ Compute the square root of a non-negative number x using Newton's method. Parameters: x (float): The number to compute the square root of. Must be non-negative. tolerance (float): The precision tolerance for the result. max_iter (int): Maximum number of iterations to perform. Returns: float: An approximation of sqrt(x) within the specified tolerance. Raises: ValueError: If x is negative. """ if x < 0: raise ValueError("Cannot compute square root of a negative number") if x == 0: return 0.0 # The square root of 0 is 0. # Initial guess can be x/2.0, though other choices are possible. guess = x / 2.0 for _ in range(max_iter): new_guess = (guess + x / guess) / 2.0 # Check if the improvement is within tolerance. if abs(new_guess - guess) < tolerance: return new_guess guess = new_guess # If we reach here, max_iter iterations were reached without convergence. return guess

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 `math.sqrt`. 1. **Function Purpose**: The function `my_sqrt` is designed to compute the square root of a non-negative number `x` using Newton's method. This is a well-known iterative method for finding successively better approximations to the roots (or zeroes) of a real-valued function. 2. **Parameters and Return**: The function takes three parameters: `x` (the number to find the square root of), `tolerance` (the precision tolerance for the result), and `max_iter` (the maximum number of iterations). It returns a float that approximates the square root of `x`. 3. **Error Handling**: The function raises a `ValueError` if `x` is negative, which is appropriate since the square root of a negative number is not defined in the real number system. 4. **Special Case for Zero**: The function correctly handles the case where `x` is zero by returning `0.0`. 5. **Initial Guess**: The initial guess for the square root is set to `x / 2.0`. This is a common starting point for Newton's method, though other initial guesses could also be used. 6. **Iterative Process**: The function uses a loop to iteratively improve the guess for the square root. In each iteration, it calculates a new guess using the formula `(guess + x / guess) / 2.0`, which is the core of Newton's method for square roots. 7. **Convergence Check**: The loop checks if the absolute difference between the new guess and the old guess is less than the specified tolerance. If so, it returns the new guess as the result. 8. **Max Iterations**: If the loop completes without finding a sufficiently accurate result, the function returns the last computed guess. 9. **No Direct Call to `math.sqrt`**: The function does not call `math.sqrt` or any other built-in square root function. It implements the square root calculation using an iterative method from scratch. Based on this analysis, the student did not cheat. They implemented the square root function using Newton's method, which is a valid approach for this task. **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