code: 
  
  
  
  
  
  
  from __future__ import annotations
  
  
  
  
  
  
  
  def 			lowercase__  (				A_:	List[Any]     ,			A_:	Optional[int]		)  ->							Dict:
  
  
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
         __UpperCAmelCase    							=0
         __UpperCAmelCase    							=len(UpperCamelCase__		) - 1
         while left <= right:
                __UpperCAmelCase    							=(left + right) // 2
                if left == right:
                       if sorted_list[middle] < item:
                              __UpperCAmelCase    							=middle + 1
                       break
                elif sorted_list[middle] < item:
                       __UpperCAmelCase    							=middle + 1
                else:
                       __UpperCAmelCase    							=middle - 1
         sorted_list.insert(UpperCamelCase__     ,			UpperCamelCase__		)
         return sorted_list
  
  
  
  
  
  
  
  def 			lowercase__  (				A_:	Tuple     ,			A_:	List[str]		)  ->							List[str]:
  
  
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
         __UpperCAmelCase    							=[]
         while left and right:
                if left[0][0] < right[0][0]:
                       result.append(left.pop(0		)		)
                else:
                       result.append(right.pop(0		)		)
         return result + left + right
  
  
  
  
  
  
  
  def 			lowercase__  (				A_:	Dict		)  ->							Union[str, Any]:
  
  
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
         __UpperCAmelCase    							=len(UpperCamelCase__		)
         if length <= 1:
                return list_ad
         __UpperCAmelCase    							=length // 2
         return merge(sortlist_ad(list_ad[:middle]		)     ,			sortlist_ad(list_ad[middle:]		)		)
  
  
  
  
  
  
  
  def 			lowercase__  (				A_:	list[int]		)  ->							int:
  
  
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
         if len(UpperCamelCase__		) <= 1:
                return collection
         __UpperCAmelCase    							=[]
         __UpperCAmelCase    							=False
         for i in range(0     ,			len(UpperCamelCase__		)     ,			2		):
                if i == len(UpperCamelCase__		) - 1:
                       __UpperCAmelCase    							=True
                else:
                       if collection[i] < collection[i + 1]:
                              two_paired_list.append([collection[i], collection[i + 1]]		)
                       else:
                              two_paired_list.append([collection[i + 1], collection[i]]		)
         __UpperCAmelCase    							=sortlist_ad(UpperCamelCase__		)
         __UpperCAmelCase    							=[i[0] for i in sorted_list_ad]
         result.append(sorted_list_ad[-1][1]		)
         if has_last_odd_item:
                __UpperCAmelCase    							=collection[-1]
                __UpperCAmelCase    							=binary_search_insertion(UpperCamelCase__     ,			UpperCamelCase__		)
         __UpperCAmelCase    							=False
         for i in range(len(UpperCamelCase__		) - 1		):
                if result[i] == collection[-1] and has_last_odd_item:
                       __UpperCAmelCase    							=True
                __UpperCAmelCase    							=sorted_list_ad[i][1]
                # If last_odd_item is inserted before the item's index,
                # you should forward index one more.
                if is_last_odd_item_inserted_before_this_index:
                       __UpperCAmelCase    							=result[: i + 2] + binary_search_insertion(result[i + 2 :]     ,			UpperCamelCase__		)
                else:
                       __UpperCAmelCase    							=result[: i + 1] + binary_search_insertion(result[i + 1 :]     ,			UpperCamelCase__		)
  
         return result
  
  
  if __name__ == "__main__":
     __A  							=  input("Enter numbers separated by a comma:\n").strip()
     __A  							=  [int(item) for item in user_input.split(",")]
     print(merge_insertion_sort(unsorted))
  
  
style_context: 
  
  
  
  
  
  
  import math as m
  
  
  
  
  
  
  
  def 			lowercase__  (				A_:	Tuple     ,			A_:	str     ,			A_:	Union[str, Any]=0.0_0_1		)  ->							List[str]:
  
  
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
         return (f(a + h		) - f(a - h		)) / (2 * h)
  
  
  
  
  
  
  
  def 			lowercase__  (				A_:	str     ,			A_:	Union[str, Any]=0     ,			A_:	List[Any]=100     ,			A_:	Optional[Any]=0.0_0_0_1     ,			A_:	Tuple=1e-6     ,			A_:	Union[str, Any]=False		)  ->							Union[str, Any]:
  
  
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
  
         __UpperCAmelCase    							=xa  # set the initial guess
         __UpperCAmelCase    							=[a]
         __UpperCAmelCase    							=abs(f(A_		)		)
         __UpperCAmelCase    							=lambda A_		: calc_derivative(A_     ,			A_     ,			h=A_		)  # noqa: E731  Derivative of f(x)
         for _ in range(A_		):
                if fa(A_		) == 0:
                       raise ValueError("""No converging solution found"""		)
                __UpperCAmelCase    							=a - f(A_		) / fa(A_		)  # Calculate the next estimate
                if logsteps:
                       steps.append(A_		)
                if error < maxerror:
                       break
         else:
                raise ValueError("""Iteration limit reached, no converging solution found"""		)
         if logsteps:
                # If logstep is true, then log intermediate steps
                return a, error, steps
         return a, error
  
  
  if __name__ == "__main__":
     from matplotlib import pyplot as plt
  
     __A  							=  lambda x: m.tanh(x) ** 2 - m.exp(3 * x)  # noqa: E731
     __A							,    __A							,    __A  							=  newton_raphson(
         f, xa=10, maxiter=10_00, step=1E-6, logsteps=True
     )
     plt.plot([abs(f(x)) for x in steps])
     plt.xlabel("step")
     plt.ylabel("error")
     plt.show()
     print(F"""solution = {{{solution:f}}}, error = {{{error:f}}}""")
  
  
  
  
  