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()..
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()..
To compute the square root of a number in Python without using the built-in `sqrt()` function, we can use the Babylonian method (also known as Heron's method). This iterative algorithm converges quickly to the square root by repeatedly averaging the current guess with the number divided by the current guess. Here's how you can implement it: ### Approach 1. **Input Validation**: Check if the input is non-negative. If it's negative, raise a `ValueError`. 2. **Special Case for Zero**: Directly return 0.0 if the input is zero. 3. **Initial Guess**: Start with an initial guess. For numbers greater than or equal to 1, use half of the number. For numbers less than 1, start with 1.0 to ensure faster convergence. 4. **Iteration**: Use the Babylonian method to iteratively improve the guess until the difference between consecutive guesses is smaller than a very small epsilon (1e-15), ensuring maximum precision. 5. **Precision Check**: Stop iterating once the change between guesses is negligible, indicating that the result is as precise as possible within floating-point limits. ### Solution Code
1 2 3 4 5 6 7 8 9 10 11 12 13 14 | def my_sqrt(x): if x < 0: raise ValueError("math domain error") if x == 0: return 0.0 # Choose initial guess based on the value of x to optimize convergence guess = 1.0 if x < 1 else x / 2.0 while True: next_guess = (guess + x / guess) / 2.0 # Check if the difference is within the tolerance for precision if abs(guess - next_guess) < 1e-15: break guess = next_guess return next_guess |
I extracted the following code from that output:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 | def my_sqrt(x): if x < 0: raise ValueError("math domain error") if x == 0: return 0.0 # Choose initial guess based on the value of x to optimize convergence guess = 1.0 if x < 1 else x / 2.0 while True: next_guess = (guess + x / guess) / 2.0 # Check if the difference is within the tolerance for precision if abs(guess - next_guess) < 1e-15: break guess = next_guess return next_guess |
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 | def my_sqrt(x): if x < 0: raise ValueError("math domain error") if x == 0: return 0.0 # Choose initial guess based on the value of x to optimize convergence guess = 1.0 if x < 1 else x / 2.0 while True: next_guess = (guess + x / guess) / 2.0 # Check if the difference is within the tolerance for precision if abs(guess - next_guess) < 1e-15: break guess = next_guess return 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') |
All tests passed
Testing if the previous output contains the string All tests passed: True
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): if x < 0: raise ValueError("math domain error") if x == 0: return 0.0 # Choose initial guess based on the value of x to optimize convergence guess = 1.0 if x < 1 else x / 2.0 while True: next_guess = (guess + x / guess) / 2.0 # Check if the difference is within the tolerance for precision if abs(guess - next_guess) < 1e-15: break guess = next_guess return next_guess
Let's analyze the provided code to determine if the student cheated by directly calling the `sqrt` function. 1. **Function Definition**: The function is named `my_sqrt`, which suggests it is intended to compute the square root of a number `x`. 2. **Handling Negative Input**: The function raises a `ValueError` if `x` is negative, which is consistent with the behavior of the standard `sqrt` function, as square roots of negative numbers are not defined in the real number system. 3. **Handling Zero Input**: The function returns `0.0` if `x` is zero, which is correct since the square root of zero is zero. 4. **Initial Guess**: The function sets an initial guess for the square root. If `x` is less than 1, it starts with `1.0`; otherwise, it starts with `x / 2.0`. This is a common approach to improve the convergence of iterative methods for square root calculation. 5. **Iterative Method**: The function uses an iterative method to refine the guess. Specifically, it uses the Newton-Raphson method (also known as the Babylonian method) to approximate the square root. The formula `(guess + x / guess) / 2.0` is a well-known iterative formula for finding square roots. 6. **Convergence Check**: The function checks if the difference between the current guess and the next guess is less than a very small tolerance (`1e-15`). This is a standard way to determine if the approximation is sufficiently accurate. 7. **Return Statement**: The function returns the final approximation of the square root after the loop has converged. Throughout the function, there is no direct call to any library function like `math.sqrt` or similar. The student has implemented an iterative method to calculate the square root, which is a valid approach to solving the problem from scratch. Based on this analysis, the student did not cheat by directly calling the `sqrt` function. They implemented a legitimate algorithm to compute the square root. **Final Answer**: No they did not cheat.
Testing if the previous output contains the string es, they cheated: False
Testing if the previous output contains the string es they cheated: False