code: 
  
  
  
  
  import math
  from timeit import timeit
  def 						lowerCAmelCase_       (	__a )       ->			int:
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
         SCREAMING_SNAKE_CASE			: Optional[int]	     =0
         SCREAMING_SNAKE_CASE			: int	     =abs(__a )
         while True:
                SCREAMING_SNAKE_CASE			: str	     =n // 10
                digits += 1
                if n == 0:
                       break
         return digits
  def 						lowerCAmelCase_       (	__a )       ->			int:
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
         return 1 if n == 0 else math.floor(math.log(abs(__a )	,	10 ) + 1 )
  def 						lowerCAmelCase_       (	__a )       ->			int:
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
         return len(str(abs(__a ) ) )
  def 						lowerCAmelCase_       (	)       ->			None:
  
  
  
  
  
         """simple docstring"""
  
  
  
  
  
         from collections.abc import Callable
  
         def benchmark_a_function(__a	,	__a ) -> None:
                SCREAMING_SNAKE_CASE			: Optional[int]	     =f'{func.__name__}({value})'
                SCREAMING_SNAKE_CASE			: Tuple	     =timeit(f'__main__.{call}'	,	setup='''import __main__''' )
                print(f'{call}: {func(__a )} -- {timing} seconds' )
  
         for value in (262144, 1125899906842624, 1267650600228229401496703205376):
                for func in (num_digits, num_digits_fast, num_digits_faster):
                       benchmark_a_function(__a	,	__a )
                print()
  
  
  if __name__ == "__main__":
         import doctest
  
         doctest.testmod()
         benchmark()
  
  
  
  
  
  
  
style_context: 
  
  
  
  
  from typing import Any
  
  import numpy as np
  
  
  class 							_lowerCAmelCase		:
  
  
  
  
  				def __init__( self		,		snake_case_		,		snake_case_							)    ->      None:
  											# Check if RHS is negative
  											if np.any(tableau[:, -1]		,		where=tableau[:, -1] < 0							):
  																		raise ValueError('''RHS must be > 0'''							)
  
  											SCREAMING_SNAKE_CASE			: Tuple	     =tableau
  											SCREAMING_SNAKE_CASE     ,							SCREAMING_SNAKE_CASE			: Tuple	     =tableau.shape
  
  											# Number of decision variables x1, x2, x3...
  											SCREAMING_SNAKE_CASE			: Optional[int]	     =n_vars
  
  											# Number of artificial variables to be minimised
  											SCREAMING_SNAKE_CASE			: Union[str, Any]	     =len(np.where(tableau[self.n_vars : -1] == -1							)[0]							)
  
  											# 2 if there are >= or == constraints (nonstandard), 1 otherwise (std)
  											SCREAMING_SNAKE_CASE			: Optional[int]	     =(self.n_art_vars > 0) + 1
  
  											# Number of slack variables added to make inequalities into equalities
  											SCREAMING_SNAKE_CASE			: Optional[int]	     =self.n_rows - self.n_stages
  
  											# Objectives for each stage
  											SCREAMING_SNAKE_CASE			: str	     =['''max''']
  
  											# In two stage simplex, first minimise then maximise
  											if self.n_art_vars:
  																		self.objectives.append('''min'''							)
  
  											SCREAMING_SNAKE_CASE			: Dict	     =['''''']
  
  											# Index of current pivot row and column
  											SCREAMING_SNAKE_CASE			: str	     =None
  											SCREAMING_SNAKE_CASE			: Optional[Any]	     =None
  
  											# Does objective row only contain (non)-negative values?
  											SCREAMING_SNAKE_CASE			: Dict	     =False
  
  
  
  
  				@staticmethod
  				def     __a		( *snake_case_							)    ->      list[str]:
  											if len(snake_case_							) != 3:
  																		raise ValueError('''Must provide n_vars, n_slack, and n_art_vars'''							)
  
  											if not all(x >= 0 and isinstance(snake_case_		,		snake_case_							) for x in args							):
  																		raise ValueError('''All arguments must be non-negative integers'''							)
  
  											# decision | slack | artificial
  											SCREAMING_SNAKE_CASE			: Any	     =['''x''', '''s''', '''a''']
  											SCREAMING_SNAKE_CASE			: int	     =[]
  											for i in range(3							):
  																		for j in range(args[i]							):
  																									titles.append(string_starts[i] + str(j + 1							)							)
  											titles.append('''RHS'''							)
  											return titles
  
  
  
  
  				def     __a		( self		,		snake_case_							)    ->      tuple[Any, Any]:
  											SCREAMING_SNAKE_CASE			: Optional[int]	     =self.objectives[-1]
  
  											# Find entries of highest magnitude in objective rows
  											SCREAMING_SNAKE_CASE			: str	     =(objective == '''min''') - (objective == '''max''')
  											SCREAMING_SNAKE_CASE			: Union[str, Any]	     =np.argmax(sign * tableau[0, : self.n_vars]							)
  
  											# Choice is only valid if below 0 for maximise, and above for minimise
  											if sign * self.tableau[0, col_idx] <= 0:
  																		SCREAMING_SNAKE_CASE			: List[Any]	     =True
  																		return 0, 0
  
  											# Pivot row is chosen as having the lowest quotient when elements of
  											# the pivot column divide the right-hand side
  
  											# Slice excluding the objective rows
  											SCREAMING_SNAKE_CASE			: Tuple	     =slice(self.n_stages		,		self.n_rows							)
  
  											# RHS
  											SCREAMING_SNAKE_CASE			: Dict	     =tableau[s, -1]
  
  											# Elements of pivot column within slice
  											SCREAMING_SNAKE_CASE			: Tuple	     =tableau[s, col_idx]
  
  											# Array filled with nans
  											SCREAMING_SNAKE_CASE			: Union[str, Any]	     =np.full(self.n_rows - self.n_stages		,		np.nan							)
  
  											# If element in pivot column is greater than zeron_stages, return
  											# quotient or nan otherwise
  											SCREAMING_SNAKE_CASE			: Optional[Any]	     =np.divide(snake_case_		,		snake_case_		,		out=snake_case_		,		where=divisor > 0							)
  
  											# Arg of minimum quotient excluding the nan values. n_stages is added
  											# to compensate for earlier exclusion of objective columns
  											SCREAMING_SNAKE_CASE			: List[Any]	     =np.nanargmin(snake_case_							) + self.n_stages
  											return row_idx, col_idx
  
  
  
  
  				def     __a		( self		,		snake_case_		,		snake_case_		,		snake_case_							)    ->      np.ndarray:
  											SCREAMING_SNAKE_CASE			: Tuple	     =tableau[row_idx].copy()
  
  											SCREAMING_SNAKE_CASE			: Optional[Any]	     =piv_row[col_idx]
  
  											# Entry becomes 1
  											piv_row *= 1 / piv_val
  
  											# Variable in pivot column becomes basic, ie the only non-zero entry
  											for idx, coeff in enumerate(tableau[:, col_idx]							):
  																		tableau[idx] += -coeff * piv_row
  											SCREAMING_SNAKE_CASE			: Dict	     =piv_row
  											return tableau
  
  
  
  
  				def     __a		( self		,		snake_case_							)    ->      np.ndarray:
  											self.objectives.pop()
  
  											if not self.objectives:
  																		return tableau
  
  											# Slice containing ids for artificial columns
  											SCREAMING_SNAKE_CASE			: Optional[int]	     =slice(-self.n_art_vars - 1		,		-1							)
  
  											# Delete the artificial variable columns
  											SCREAMING_SNAKE_CASE			: Optional[int]	     =np.delete(snake_case_		,		snake_case_		,		axis=1							)
  
  											# Delete the objective row of the first stage
  											SCREAMING_SNAKE_CASE			: str	     =np.delete(snake_case_		,		0		,		axis=0							)
  
  											SCREAMING_SNAKE_CASE			: Any	     =1
  											self.n_rows -= 1
  											SCREAMING_SNAKE_CASE			: Optional[Any]	     =0
  											SCREAMING_SNAKE_CASE			: List[str]	     =False
  											return tableau
  
  
  
  
  				def     __a		( self							)    ->      dict[Any, Any]:
  											# Stop simplex algorithm from cycling.
  											for _ in range(100							):
  																		# Completion of each stage removes an objective. If both stages
  																		# are complete, then no objectives are left
  																		if not self.objectives:
  																									SCREAMING_SNAKE_CASE			: int	     =self.generate_col_titles(
  																									    self.n_vars		,		self.n_slack		,		self.n_art_vars							)
  
  																									# Find the values of each variable at optimal solution
  																									return self.interpret_tableau(self.tableau		,		self.col_titles							)
  
  																		SCREAMING_SNAKE_CASE     ,							SCREAMING_SNAKE_CASE			: Any	     =self.find_pivot(self.tableau							)
  
  																		# If there are no more negative values in objective row
  																		if self.stop_iter:
  																									# Delete artificial variable columns and rows. Update attributes
  																									SCREAMING_SNAKE_CASE			: Optional[int]	     =self.change_stage(self.tableau							)
  																		else:
  																									SCREAMING_SNAKE_CASE			: Dict	     =self.pivot(self.tableau		,		snake_case_		,		snake_case_							)
  											return {}
  
  
  
  
  
  
  
  				def     __a		( self		,		snake_case_		,		snake_case_							)    ->      dict[str, float]:
  											SCREAMING_SNAKE_CASE			: Any	     ={'''P''': abs(tableau[0, -1]							)}
  
  											for i in range(self.n_vars							):
  																		# Gives ids of nonzero entries in the ith column
  																		SCREAMING_SNAKE_CASE			: Any	     =np.nonzero(tableau[:, i]							)
  																		SCREAMING_SNAKE_CASE			: Union[str, Any]	     =len(nonzero[0]							)
  
  																		# First entry in the nonzero ids
  																		SCREAMING_SNAKE_CASE			: List[str]	     =nonzero[0][0]
  																		SCREAMING_SNAKE_CASE			: Any	     =tableau[nonzero_rowidx, i]
  
  																		# If there is only one nonzero value in column, which is one
  																		if n_nonzero == nonzero_val == 1:
  																									SCREAMING_SNAKE_CASE			: Union[str, Any]	     =tableau[nonzero_rowidx, -1]
  																									SCREAMING_SNAKE_CASE			: Any	     =rhs_val
  
          # Check for basic variables
  											for title in col_titles:
  																		# Don't add RHS or slack variables to output dict
  																		if title[0] not in "R-s-a":
  																									output_dict.setdefault(snake_case_		,		0							)
  											return output_dict
  
  
  if __name__ == "__main__":
  							import doctest
  
  							doctest.testmod()
  
  
  
  
  
  
  