Fixes TheAlgorithms#9943 Added Doctests to binary_exponentiation_3.py (TheAlgorithms#10121)

* Python mirror_formulae.py is added to the repository

* Changes done after reading readme.md

* Changes for running doctest on all platforms

* [pre-commit.ci] auto fixes from pre-commit.com hooks

for more information, see https://pre-commit.ci

* Change 2 for Doctests

* Changes for doctest 2

* updating DIRECTORY.md

* Doctest whitespace error rectification to mirror_formulae.py

* updating DIRECTORY.md

* Adding Thermodynamic Work Done Formulae

* Work done on/by body in a thermodynamic setting

* updating DIRECTORY.md

* updating DIRECTORY.md

* Doctest adiition to binary_exponentiation_3.py

* Change 1

* updating DIRECTORY.md

* Rename binary_exponentiation_3.py to binary_exponentiation_2.py

* updating DIRECTORY.md

* updating DIRECTORY.md

* Formatting

---------

Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>
Co-authored-by: Christian Clauss <cclauss@me.com>
Co-authored-by: Tianyi Zheng <tianyizheng02@gmail.com>

Author: 5 people

DIRECTORY.md

@@ -54,13 +54,12 @@
   * [Largest Pow Of Two Le Num](bit_manipulation/largest_pow_of_two_le_num.py)
   * [Missing Number](bit_manipulation/missing_number.py)
   * [Numbers Different Signs](bit_manipulation/numbers_different_signs.py)
+  * [Power Of 4](bit_manipulation/power_of_4.py)
   * [Reverse Bits](bit_manipulation/reverse_bits.py)
   * [Single Bit Manipulation Operations](bit_manipulation/single_bit_manipulation_operations.py)
 
 ## Blockchain
-  * [Chinese Remainder Theorem](blockchain/chinese_remainder_theorem.py)
   * [Diophantine Equation](blockchain/diophantine_equation.py)
-  * [Modular Division](blockchain/modular_division.py)
 
 ## Boolean Algebra
   * [And Gate](boolean_algebra/and_gate.py)
@@ -101,11 +100,13 @@
   * [Diffie Hellman](ciphers/diffie_hellman.py)
   * [Elgamal Key Generator](ciphers/elgamal_key_generator.py)
   * [Enigma Machine2](ciphers/enigma_machine2.py)
+  * [Fractionated Morse Cipher](ciphers/fractionated_morse_cipher.py)
   * [Hill Cipher](ciphers/hill_cipher.py)
   * [Mixed Keyword Cypher](ciphers/mixed_keyword_cypher.py)
   * [Mono Alphabetic Ciphers](ciphers/mono_alphabetic_ciphers.py)
   * [Morse Code](ciphers/morse_code.py)
   * [Onepad Cipher](ciphers/onepad_cipher.py)
+  * [Permutation Cipher](ciphers/permutation_cipher.py)
   * [Playfair Cipher](ciphers/playfair_cipher.py)
   * [Polybius](ciphers/polybius.py)
   * [Porta Cipher](ciphers/porta_cipher.py)
@@ -172,6 +173,7 @@
 
 ## Data Structures
   * Arrays
+    * [Equilibrium Index In Array](data_structures/arrays/equilibrium_index_in_array.py)
     * [Median Two Array](data_structures/arrays/median_two_array.py)
     * [Permutations](data_structures/arrays/permutations.py)
     * [Prefix Sum](data_structures/arrays/prefix_sum.py)
@@ -352,6 +354,7 @@
   * [Smith Waterman](dynamic_programming/smith_waterman.py)
   * [Subset Generation](dynamic_programming/subset_generation.py)
   * [Sum Of Subset](dynamic_programming/sum_of_subset.py)
+  * [Trapped Water](dynamic_programming/trapped_water.py)
   * [Tribonacci](dynamic_programming/tribonacci.py)
   * [Viterbi](dynamic_programming/viterbi.py)
   * [Word Break](dynamic_programming/word_break.py)
@@ -360,6 +363,7 @@
   * [Apparent Power](electronics/apparent_power.py)
   * [Builtin Voltage](electronics/builtin_voltage.py)
   * [Carrier Concentration](electronics/carrier_concentration.py)
+  * [Charging Capacitor](electronics/charging_capacitor.py)
   * [Circular Convolution](electronics/circular_convolution.py)
   * [Coulombs Law](electronics/coulombs_law.py)
   * [Electric Conductivity](electronics/electric_conductivity.py)
@@ -466,6 +470,8 @@
     * [Test Min Spanning Tree Prim](graphs/tests/test_min_spanning_tree_prim.py)
 
 ## Greedy Methods
+  * [Best Time To Buy And Sell Stock](greedy_methods/best_time_to_buy_and_sell_stock.py)
+  * [Fractional Cover Problem](greedy_methods/fractional_cover_problem.py)
   * [Fractional Knapsack](greedy_methods/fractional_knapsack.py)
   * [Fractional Knapsack 2](greedy_methods/fractional_knapsack_2.py)
   * [Gas Station](greedy_methods/gas_station.py)
@@ -524,6 +530,10 @@
   * Local Weighted Learning
     * [Local Weighted Learning](machine_learning/local_weighted_learning/local_weighted_learning.py)
   * [Logistic Regression](machine_learning/logistic_regression.py)
+  * Loss Functions
+    * [Binary Cross Entropy](machine_learning/loss_functions/binary_cross_entropy.py)
+    * [Huber Loss](machine_learning/loss_functions/huber_loss.py)
+    * [Mean Squared Error](machine_learning/loss_functions/mean_squared_error.py)
   * [Mfcc](machine_learning/mfcc.py)
   * [Multilayer Perceptron Classifier](machine_learning/multilayer_perceptron_classifier.py)
   * [Polynomial Regression](machine_learning/polynomial_regression.py)
@@ -564,7 +574,9 @@
   * [Carmichael Number](maths/carmichael_number.py)
   * [Catalan Number](maths/catalan_number.py)
   * [Ceil](maths/ceil.py)
+  * [Chebyshev Distance](maths/chebyshev_distance.py)
   * [Check Polygon](maths/check_polygon.py)
+  * [Chinese Remainder Theorem](maths/chinese_remainder_theorem.py)
   * [Chudnovsky Algorithm](maths/chudnovsky_algorithm.py)
   * [Collatz Sequence](maths/collatz_sequence.py)
   * [Combinations](maths/combinations.py)
@@ -591,6 +603,7 @@
   * [Gaussian](maths/gaussian.py)
   * [Gaussian Error Linear Unit](maths/gaussian_error_linear_unit.py)
   * [Gcd Of N Numbers](maths/gcd_of_n_numbers.py)
+  * [Germain Primes](maths/germain_primes.py)
   * [Greatest Common Divisor](maths/greatest_common_divisor.py)
   * [Greedy Coin Change](maths/greedy_coin_change.py)
   * [Hamming Numbers](maths/hamming_numbers.py)
@@ -618,7 +631,9 @@
   * [Matrix Exponentiation](maths/matrix_exponentiation.py)
   * [Max Sum Sliding Window](maths/max_sum_sliding_window.py)
   * [Median Of Two Arrays](maths/median_of_two_arrays.py)
+  * [Minkowski Distance](maths/minkowski_distance.py)
   * [Mobius Function](maths/mobius_function.py)
+  * [Modular Division](maths/modular_division.py)
   * [Modular Exponential](maths/modular_exponential.py)
   * [Monte Carlo](maths/monte_carlo.py)
   * [Monte Carlo Dice](maths/monte_carlo_dice.py)
@@ -720,12 +735,16 @@
 ## Neural Network
   * [2 Hidden Layers Neural Network](neural_network/2_hidden_layers_neural_network.py)
   * Activation Functions
+    * [Binary Step](neural_network/activation_functions/binary_step.py)
     * [Exponential Linear Unit](neural_network/activation_functions/exponential_linear_unit.py)
     * [Leaky Rectified Linear Unit](neural_network/activation_functions/leaky_rectified_linear_unit.py)
     * [Mish](neural_network/activation_functions/mish.py)
     * [Rectified Linear Unit](neural_network/activation_functions/rectified_linear_unit.py)
     * [Scaled Exponential Linear Unit](neural_network/activation_functions/scaled_exponential_linear_unit.py)
     * [Sigmoid Linear Unit](neural_network/activation_functions/sigmoid_linear_unit.py)
+    * [Soboleva Modified Hyperbolic Tangent](neural_network/activation_functions/soboleva_modified_hyperbolic_tangent.py)
+    * [Softplus](neural_network/activation_functions/softplus.py)
+    * [Squareplus](neural_network/activation_functions/squareplus.py)
   * [Back Propagation Neural Network](neural_network/back_propagation_neural_network.py)
   * [Convolution Neural Network](neural_network/convolution_neural_network.py)
   * [Perceptron](neural_network/perceptron.py)
@@ -779,6 +798,7 @@
   * [Newtons Second Law Of Motion](physics/newtons_second_law_of_motion.py)
   * [Photoelectric Effect](physics/photoelectric_effect.py)
   * [Potential Energy](physics/potential_energy.py)
+  * [Reynolds Number](physics/reynolds_number.py)
   * [Rms Speed Of Molecule](physics/rms_speed_of_molecule.py)
   * [Shear Stress](physics/shear_stress.py)
   * [Speed Of Sound](physics/speed_of_sound.py)
@@ -1101,6 +1121,7 @@
   * [Interpolation Search](searches/interpolation_search.py)
   * [Jump Search](searches/jump_search.py)
   * [Linear Search](searches/linear_search.py)
+  * [Median Of Medians](searches/median_of_medians.py)
   * [Quick Select](searches/quick_select.py)
   * [Sentinel Linear Search](searches/sentinel_linear_search.py)
   * [Simple Binary Search](searches/simple_binary_search.py)
@@ -1201,6 +1222,7 @@
   * [Snake Case To Camel Pascal Case](strings/snake_case_to_camel_pascal_case.py)
   * [Split](strings/split.py)
   * [String Switch Case](strings/string_switch_case.py)
+  * [Strip](strings/strip.py)
   * [Text Justification](strings/text_justification.py)
   * [Top K Frequent Words](strings/top_k_frequent_words.py)
   * [Upper](strings/upper.py)

maths/binary_exponentiation_2.py

@@ -1,17 +1,33 @@
 """
-* Binary Exponentiation for Powers
-* This is a method to find a^b in a time complexity of O(log b)
-* This is one of the most commonly used methods of finding powers.
-* Also useful in cases where solution to (a^b)%c is required,
-* where a,b,c can be numbers over the computers calculation limits.
-* Done using iteration, can also be done using recursion
-
-* @author chinmoy159
-* @version 1.0 dated 10/08/2017
+Binary Exponentiation
+This is a method to find a^b in O(log b) time complexity
+This is one of the most commonly used methods of exponentiation
+It's also useful when the solution to (a^b) % c is required because a, b, c may be
+over the computer's calculation limits
+
+Let's say you need to calculate a ^ b
+- RULE 1 : a ^ b = (a*a) ^ (b/2) ---- example : 4 ^ 4 = (4*4) ^ (4/2) = 16 ^ 2
+- RULE 2 : IF b is odd, then a ^ b = a * (a ^ (b - 1)), where b - 1 is even
+Once b is even, repeat the process until b = 1 or b = 0, because a^1 = a and a^0 = 1
+
+For modular exponentiation, we use the fact that (a*b) % c = ((a%c) * (b%c)) % c
+Now apply RULE 1 or 2 as required
+
+@author chinmoy159
 """
 
 
 def b_expo(a: int, b: int) -> int:
+    """
+    >>> b_expo(2, 10)
+    1024
+    >>> b_expo(9, 0)
+    1
+    >>> b_expo(0, 12)
+    0
+    >>> b_expo(4, 12)
+    16777216
+    """
     res = 1
     while b > 0:
         if b & 1:
@@ -24,6 +40,16 @@ def b_expo(a: int, b: int) -> int:
 
 
 def b_expo_mod(a: int, b: int, c: int) -> int:
+    """
+    >>> b_expo_mod(2, 10, 1000000007)
+    1024
+    >>> b_expo_mod(11, 13, 19)
+    11
+    >>> b_expo_mod(0, 19, 20)
+    0
+    >>> b_expo_mod(15, 5, 4)
+    3
+    """
     res = 1
     while b > 0:
         if b & 1:
@@ -33,18 +59,3 @@ def b_expo_mod(a: int, b: int, c: int) -> int:
         b >>= 1
 
     return res
-
-
-"""
-* Wondering how this method works !
-* It's pretty simple.
-* Let's say you need to calculate a ^ b
-* RULE 1 : a ^ b = (a*a) ^ (b/2) ---- example : 4 ^ 4 = (4*4) ^ (4/2) = 16 ^ 2
-* RULE 2 : IF b is ODD, then ---- a ^ b = a * (a ^ (b - 1)) :: where (b - 1) is even.
-* Once b is even, repeat the process to get a ^ b
-* Repeat the process till b = 1 OR b = 0, because a^1 = a AND a^0 = 1
-*
-* As far as the modulo is concerned,
-* the fact : (a*b) % c = ((a%c) * (b%c)) % c
-* Now apply RULE 1 OR 2 whichever is required.
-"""