prompt: You will be provided with a partial code base and an issue statement explaining a problem to resolve.
  <issue>
  maths/number_of_digits.py is naive!
  <code>[maths/number_of_digits.py](https://github.com/TheAlgorithms/Python/blob/master/maths/number_of_digits.py)</code> has a naive method. The suggested method has a time complexity of O(n). We can count number of digits of a number in O(1).
  
  </issue>
  <code>
  [start of maths/number_of_digits.py]
  1 def num_digits(n: int) -> int:
  2     """
  3     Find the number of digits in a number.
  4 
  5     >>> num_digits(12345)
  6     5
  7     >>> num_digits(123)
  8     3
  9     """
  10     digits = 0
  11     while n > 0:
  12         n = n // 10
  13         digits += 1
  14     return digits
  15 
  16 
  17 if __name__ == "__main__":
  18     print(num_digits(12345))  # ===> 5
  19
  [end of maths/number_of_digits.py]
  </code>
  
  I need you to solve this issue by generating a single patch file that I can apply directly to this repository using git apply. Please respond with a single patch file in the following format.
  <patch>
  diff --git a/file.py b/file.py
  --- a/file.py
  +++ b/file.py
  @@ -1,27 +1,35 @@
   def euclidean(a, b):
  -    while b:
  -        a, b = b, a % b
  -    return a
  +    if b == 0:
  +        return a
  +    return euclidean(b, a % b)
   
   
   def bresenham(x0, y0, x1, y1):
       points = []
       dx = abs(x1 - x0)
       dy = abs(y1 - y0)
  -    sx = 1 if x0 < x1 else -1
  -    sy = 1 if y0 < y1 else -1
  -    err = dx - dy
  +    x, y = x0, y0
  +    sx = -1 if x0 > x1 else 1
  +    sy = -1 if y0 > y1 else 1
   
  -    while True:
  -        points.append((x0, y0))
  -        if x0 == x1 and y0 == y1:
  -            break
  -        e2 = 2 * err
  -        if e2 > -dy:
  -            err -= dy
  -            x0 += sx
  -        if e2 < dx:
  -            err += dx
  -            y0 += sy
  +    if dx > dy:
  +        err = dx / 2.0
  +        while x != x1:
  +            points.append((x, y))
  +            err -= dy
  +            if err < 0:
  +                y += sy
  +                err += dx
  +            x += sx
  +    else:
  +        err = dy / 2.0
  +        while y != y1:
  +            points.append((x, y))
  +            err -= dx
  +            if err < 0:
  +                x += sx
  +                err += dy
  +            y += sy
  +
  +    points.append((x, y))
       return points
  
  </patch>
golden_diff: diff --git a/maths/number_of_digits.py b/maths/number_of_digits.py
  --- a/maths/number_of_digits.py
  +++ b/maths/number_of_digits.py
  @@ -1,3 +1,7 @@
  +import math
  +from timeit import timeit
  +
  +
   def num_digits(n: int) -> int:
       """
       Find the number of digits in a number.
  @@ -14,5 +18,82 @@
       return digits
   
   
  +def num_digits_fast(n: int) -> int:
  +    """
  +    Find the number of digits in a number.
  +    abs() is used as logarithm for negative numbers is not defined.
  +
  +    >>> num_digits_fast(12345)
  +    5
  +    >>> num_digits_fast(123)
  +    3
  +    """
  +    return (math.floor(math.log(abs(n), 10) + 1))
  +
  +
  +def num_digits_faster(n: int) -> int:
  +    """
  +    Find the number of digits in a number.
  +    abs() is used for negative numbers
  +
  +    >>> num_digits_faster(12345)
  +    5
  +    >>> num_digits_faster(123)
  +    3
  +    """
  +    return (len(str(abs(n))))
  +
  +
  +def benchmark() -> None:
  +    """
  +    Benchmark code for comparing 3 functions,
  +    with 3 different length int values.
  +    """
  +    print('\nFor small_num = ', small_num, ':')
  +    print("> num_digits()",
  +          '\t\tans =', num_digits(small_num),
  +          '\ttime =', timeit("z.num_digits(z.small_num)",
  +                             setup="import __main__ as z"), "seconds")
  +    print("> num_digits_fast()",
  +          '\tans =', num_digits_fast(small_num),
  +          '\ttime =', timeit("z.num_digits_fast(z.small_num)",
  +                             setup="import __main__ as z"), "seconds")
  +    print("> num_digits_faster()",
  +          '\tans =', num_digits_faster(small_num),
  +          '\ttime =', timeit("z.num_digits_faster(z.small_num)",
  +                             setup="import __main__ as z"), "seconds")
  +
  +    print('\nFor medium_num = ', medium_num, ':')
  +    print("> num_digits()",
  +          '\t\tans =', num_digits(medium_num),
  +          '\ttime =', timeit("z.num_digits(z.medium_num)",
  +                             setup="import __main__ as z"), "seconds")
  +    print("> num_digits_fast()",
  +          '\tans =', num_digits_fast(medium_num),
  +          '\ttime =', timeit("z.num_digits_fast(z.medium_num)",
  +                             setup="import __main__ as z"), "seconds")
  +    print("> num_digits_faster()",
  +          '\tans =', num_digits_faster(medium_num),
  +          '\ttime =', timeit("z.num_digits_faster(z.medium_num)",
  +                             setup="import __main__ as z"), "seconds")
  +
  +    print('\nFor large_num = ', large_num, ':')
  +    print("> num_digits()",
  +          '\t\tans =', num_digits(large_num),
  +          '\ttime =', timeit("z.num_digits(z.large_num)",
  +                             setup="import __main__ as z"), "seconds")
  +    print("> num_digits_fast()",
  +          '\tans =', num_digits_fast(large_num),
  +          '\ttime =', timeit("z.num_digits_fast(z.large_num)",
  +                             setup="import __main__ as z"), "seconds")
  +    print("> num_digits_faster()",
  +          '\tans =', num_digits_faster(large_num),
  +          '\ttime =', timeit("z.num_digits_faster(z.large_num)",
  +                             setup="import __main__ as z"), "seconds")
  +
  +
   if __name__ == "__main__":
  -    print(num_digits(12345))  # ===> 5
  +    small_num = 262144
  +    medium_num = 1125899906842624
  +    large_num = 1267650600228229401496703205376
  +    benchmark()
  
verification_info: {"golden_diff": "diff --git a/maths/number_of_digits.py b/maths/number_of_digits.py\n--- a/maths/number_of_digits.py\n+++ b/maths/number_of_digits.py\n@@ -1,3 +1,7 @@\n+import math\n+from timeit import timeit\n+\n+\n def num_digits(n: int) -> int:\n     \"\"\"\n     Find the number of digits in a number.\n@@ -14,5 +18,82 @@\n     return digits\n \n \n+def num_digits_fast(n: int) -> int:\n+    \"\"\"\n+    Find the number of digits in a number.\n+    abs() is used as logarithm for negative numbers is not defined.\n+\n+    >>> num_digits_fast(12345)\n+    5\n+    >>> num_digits_fast(123)\n+    3\n+    \"\"\"\n+    return (math.floor(math.log(abs(n), 10) + 1))\n+\n+\n+def num_digits_faster(n: int) -> int:\n+    \"\"\"\n+    Find the number of digits in a number.\n+    abs() is used for negative numbers\n+\n+    >>> num_digits_faster(12345)\n+    5\n+    >>> num_digits_faster(123)\n+    3\n+    \"\"\"\n+    return (len(str(abs(n))))\n+\n+\n+def benchmark() -> None:\n+    \"\"\"\n+    Benchmark code for comparing 3 functions,\n+    with 3 different length int values.\n+    \"\"\"\n+    print('\\nFor small_num = ', small_num, ':')\n+    print(\"> num_digits()\",\n+          '\\t\\tans =', num_digits(small_num),\n+          '\\ttime =', timeit(\"z.num_digits(z.small_num)\",\n+                             setup=\"import __main__ as z\"), \"seconds\")\n+    print(\"> num_digits_fast()\",\n+          '\\tans =', num_digits_fast(small_num),\n+          '\\ttime =', timeit(\"z.num_digits_fast(z.small_num)\",\n+                             setup=\"import __main__ as z\"), \"seconds\")\n+    print(\"> num_digits_faster()\",\n+          '\\tans =', num_digits_faster(small_num),\n+          '\\ttime =', timeit(\"z.num_digits_faster(z.small_num)\",\n+                             setup=\"import __main__ as z\"), \"seconds\")\n+\n+    print('\\nFor medium_num = ', medium_num, ':')\n+    print(\"> num_digits()\",\n+          '\\t\\tans =', num_digits(medium_num),\n+          '\\ttime =', timeit(\"z.num_digits(z.medium_num)\",\n+                             setup=\"import __main__ as z\"), \"seconds\")\n+    print(\"> num_digits_fast()\",\n+          '\\tans =', num_digits_fast(medium_num),\n+          '\\ttime =', timeit(\"z.num_digits_fast(z.medium_num)\",\n+                             setup=\"import __main__ as z\"), \"seconds\")\n+    print(\"> num_digits_faster()\",\n+          '\\tans =', num_digits_faster(medium_num),\n+          '\\ttime =', timeit(\"z.num_digits_faster(z.medium_num)\",\n+                             setup=\"import __main__ as z\"), \"seconds\")\n+\n+    print('\\nFor large_num = ', large_num, ':')\n+    print(\"> num_digits()\",\n+          '\\t\\tans =', num_digits(large_num),\n+          '\\ttime =', timeit(\"z.num_digits(z.large_num)\",\n+                             setup=\"import __main__ as z\"), \"seconds\")\n+    print(\"> num_digits_fast()\",\n+          '\\tans =', num_digits_fast(large_num),\n+          '\\ttime =', timeit(\"z.num_digits_fast(z.large_num)\",\n+                             setup=\"import __main__ as z\"), \"seconds\")\n+    print(\"> num_digits_faster()\",\n+          '\\tans =', num_digits_faster(large_num),\n+          '\\ttime =', timeit(\"z.num_digits_faster(z.large_num)\",\n+                             setup=\"import __main__ as z\"), \"seconds\")\n+\n+\n if __name__ == \"__main__\":\n-    print(num_digits(12345))  # ===> 5\n+    small_num = 262144\n+    medium_num = 1125899906842624\n+    large_num = 1267650600228229401496703205376\n+    benchmark()\n", "issue": "maths/number_of_digits.py is naive!\n<code>[maths/number_of_digits.py](https://github.com/TheAlgorithms/Python/blob/master/maths/number_of_digits.py)</code> has a naive method. The suggested method has a time complexity of O(n). We can count number of digits of a number in O(1).\n", "before_files": [{"content": "def num_digits(n: int) -> int:\n    \"\"\"\n    Find the number of digits in a number.\n\n    >>> num_digits(12345)\n    5\n    >>> num_digits(123)\n    3\n    \"\"\"\n    digits = 0\n    while n > 0:\n        n = n // 10\n        digits += 1\n    return digits\n\n\nif __name__ == \"__main__\":\n    print(num_digits(12345))  # ===> 5\n", "path": "maths/number_of_digits.py"}]}