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

To compute the square root of a number without using the built-in `sqrt()` function, you can use the **Newton-Raphson method** (also known as the **Heron's method**). This is an iterative method that converges to the square root of a number with high precision. ### Steps to Compute Square Root Using Newton-Raphson Method: 1. Start with an initial guess for the square root (e.g., `x0 = number / 2`). 2. Iteratively improve the guess using the formula: \[ x_{n+1} = \frac{x_n + \frac{number}{x_n}}{2} \] 3. Repeat the process until the difference between successive guesses is smaller than a very small tolerance value (e.g., `1e-10`). ### Python Implementation: Here’s a function `my_sqrt()` that implements this method:

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def my_sqrt(number, tolerance=1e-10):
    if number < 0:
        raise ValueError("Cannot compute square root of a negative number.")
    if number == 0:
        return 0
    
    # Initial guess
    x0 = number / 2.0
    
    while True:
        # Compute the next guess
        x1 = (x0 + number / x0) / 2.0
        
        # Check for convergence
        if abs(x1 - x0) < tolerance:
            return x1
        
        # Update the guess
        x0 = x1

# Example usage
print(my_sqrt(25))  # Output: 5.0
print(my_sqrt(2))   # Output: 1.414213562373095

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print(my_sqrt(25))  # Output: 5.0
print(my_sqrt(2))   # Output: 1.414213562373095
print(my_sqrt(0.25))# Output: 0.5

Extract Code

I extracted the following code from that output:

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def my_sqrt(number, tolerance=1e-10):
    if number < 0:
        raise ValueError("Cannot compute square root of a negative number.")
    if number == 0:
        return 0
    
    # Initial guess
    x0 = number / 2.0
    
    while True:
        # Compute the next guess
        x1 = (x0 + number / x0) / 2.0
        
        # Check for convergence
        if abs(x1 - x0) < tolerance:
            return x1
        
        # Update the guess
        x0 = x1

Check if all of the following conditions are true:

Run Code Interpreter

Running the following program:

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def my_sqrt(number, tolerance=1e-10):
    if number < 0:
        raise ValueError("Cannot compute square root of a negative number.")
    if number == 0:
        return 0
    
    # Initial guess
    x0 = number / 2.0
    
    while True:
        # Compute the next guess
        x1 = (x0 + number / x0) / 2.0
        
        # Check for convergence
        if abs(x1 - x0) < tolerance:
            return x1
        
        # Update the guess
        x0 = x1





answer = round(my_sqrt(12345), 1)
expected = 111.1
assert answer == expected, f'Wrong answer; got {answer} instead of {expected}'
print('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(number, tolerance=1e-10): if number < 0: raise ValueError("Cannot compute square root of a negative number.") if number == 0: return 0 # Initial guess x0 = number / 2.0 while True: # Compute the next guess x1 = (x0 + number / x0) / 2.0 # Check for convergence if abs(x1 - x0) < tolerance: return x1 # Update the guess x0 = x1

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 Code**: - The function `my_sqrt` takes two parameters: `number` and `tolerance`. - It first checks if the number is negative, raising a `ValueError` if so, which is a reasonable check since square roots of negative numbers are not real. - If the number is zero, it immediately returns zero, which is correct since the square root of zero is zero. - The function then initializes a guess `x0` as half of the input number. - It enters a `while` loop to iteratively improve the guess using the formula `x1 = (x0 + number / x0) / 2.0`. - This formula is known as the **Babylonian method** or **Heron's method** for computing square roots, which is a well-known iterative algorithm. - The loop continues until the difference between successive guesses is less than the specified `tolerance`, indicating convergence to the square root. 2. **Checking for Direct Calls to `sqrt`**: - The code does not contain any direct calls to a `sqrt` function from libraries like `math` or `numpy`. - The student has implemented an iterative method to approximate the square root, which is a valid approach to solving the problem from scratch. 3. **Conclusion**: - The student has implemented a classic algorithm for computing square roots without using a direct call to a library function like `sqrt`. - The use of the Babylonian method demonstrates an understanding of how to compute square roots iteratively. Based on this analysis, 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