Consolidate binary exponentiation files (TheAlgorithms#10742)

* Consolidate binary exponentiation files

* updating DIRECTORY.md

* Fix typos in doctests

* Add suggestions from code review

* Fix timeit benchmarks

---------

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

Author: 2 people

Diff for: DIRECTORY.md

@@ -578,9 +578,7 @@
   * [Bailey Borwein Plouffe](maths/bailey_borwein_plouffe.py)
   * [Base Neg2 Conversion](maths/base_neg2_conversion.py)
   * [Basic Maths](maths/basic_maths.py)
-  * [Binary Exp Mod](maths/binary_exp_mod.py)
   * [Binary Exponentiation](maths/binary_exponentiation.py)
-  * [Binary Exponentiation 2](maths/binary_exponentiation_2.py)
   * [Binary Multiplication](maths/binary_multiplication.py)
   * [Binomial Coefficient](maths/binomial_coefficient.py)
   * [Binomial Distribution](maths/binomial_distribution.py)

Diff for: maths/binary_exp_mod.py

@@ -1,48 +1,196 @@
-"""Binary Exponentiation."""
+"""
+Binary Exponentiation
 
-# Author : Junth Basnet
-# Time Complexity : O(logn)
+This is a method to find a^b in O(log b) time complexity and is one of the most commonly
+used methods of exponentiation. The method is also useful for modular exponentiation,
+when the solution to (a^b) % c is required.
 
+To calculate a^b:
+- If b is even, then a^b = (a * a)^(b / 2)
+- If b is odd, then a^b = a * a^(b - 1)
+Repeat until b = 1 or b = 0
 
-def binary_exponentiation(a: int, n: int) -> int:
+For modular exponentiation, we use the fact that (a * b) % c = ((a % c) * (b % c)) % c
+"""
+
+
+def binary_exp_recursive(base: float, exponent: int) -> float:
     """
-    Compute a number raised by some quantity
-    >>> binary_exponentiation(-1, 3)
+    Computes a^b recursively, where a is the base and b is the exponent
+
+    >>> binary_exp_recursive(3, 5)
+    243
+    >>> binary_exp_recursive(11, 13)
+    34522712143931
+    >>> binary_exp_recursive(-1, 3)
     -1
-    >>> binary_exponentiation(-1, 4)
+    >>> binary_exp_recursive(0, 5)
+    0
+    >>> binary_exp_recursive(3, 1)
+    3
+    >>> binary_exp_recursive(3, 0)
     1
-    >>> binary_exponentiation(2, 2)
-    4
-    >>> binary_exponentiation(3, 5)
+    >>> binary_exp_recursive(1.5, 4)
+    5.0625
+    >>> binary_exp_recursive(3, -1)
+    Traceback (most recent call last):
+        ...
+    ValueError: Exponent must be a non-negative integer
+    """
+    if exponent < 0:
+        raise ValueError("Exponent must be a non-negative integer")
+
+    if exponent == 0:
+        return 1
+
+    if exponent % 2 == 1:
+        return binary_exp_recursive(base, exponent - 1) * base
+
+    b = binary_exp_recursive(base, exponent // 2)
+    return b * b
+
+
+def binary_exp_iterative(base: float, exponent: int) -> float:
+    """
+    Computes a^b iteratively, where a is the base and b is the exponent
+
+    >>> binary_exp_iterative(3, 5)
     243
-    >>> binary_exponentiation(10, 3)
-    1000
-    >>> binary_exponentiation(5e3, 1)
-    5000.0
-    >>> binary_exponentiation(-5e3, 1)
-    -5000.0
-    """
-    if n == 0:
+    >>> binary_exp_iterative(11, 13)
+    34522712143931
+    >>> binary_exp_iterative(-1, 3)
+    -1
+    >>> binary_exp_iterative(0, 5)
+    0
+    >>> binary_exp_iterative(3, 1)
+    3
+    >>> binary_exp_iterative(3, 0)
+    1
+    >>> binary_exp_iterative(1.5, 4)
+    5.0625
+    >>> binary_exp_iterative(3, -1)
+    Traceback (most recent call last):
+        ...
+    ValueError: Exponent must be a non-negative integer
+    """
+    if exponent < 0:
+        raise ValueError("Exponent must be a non-negative integer")
+
+    res: int | float = 1
+    while exponent > 0:
+        if exponent & 1:
+            res *= base
+
+        base *= base
+        exponent >>= 1
+
+    return res
+
+
+def binary_exp_mod_recursive(base: float, exponent: int, modulus: int) -> float:
+    """
+    Computes a^b % c recursively, where a is the base, b is the exponent, and c is the
+    modulus
+
+    >>> binary_exp_mod_recursive(3, 4, 5)
+    1
+    >>> binary_exp_mod_recursive(11, 13, 7)
+    4
+    >>> binary_exp_mod_recursive(1.5, 4, 3)
+    2.0625
+    >>> binary_exp_mod_recursive(7, -1, 10)
+    Traceback (most recent call last):
+        ...
+    ValueError: Exponent must be a non-negative integer
+    >>> binary_exp_mod_recursive(7, 13, 0)
+    Traceback (most recent call last):
+        ...
+    ValueError: Modulus must be a positive integer
+    """
+    if exponent < 0:
+        raise ValueError("Exponent must be a non-negative integer")
+    if modulus <= 0:
+        raise ValueError("Modulus must be a positive integer")
+
+    if exponent == 0:
         return 1
 
-    elif n % 2 == 1:
-        return binary_exponentiation(a, n - 1) * a
+    if exponent % 2 == 1:
+        return (binary_exp_mod_recursive(base, exponent - 1, modulus) * base) % modulus
 
-    else:
-        b = binary_exponentiation(a, n // 2)
-        return b * b
+    r = binary_exp_mod_recursive(base, exponent // 2, modulus)
+    return (r * r) % modulus
 
 
-if __name__ == "__main__":
-    import doctest
+def binary_exp_mod_iterative(base: float, exponent: int, modulus: int) -> float:
+    """
+    Computes a^b % c iteratively, where a is the base, b is the exponent, and c is the
+    modulus
 
-    doctest.testmod()
+    >>> binary_exp_mod_iterative(3, 4, 5)
+    1
+    >>> binary_exp_mod_iterative(11, 13, 7)
+    4
+    >>> binary_exp_mod_iterative(1.5, 4, 3)
+    2.0625
+    >>> binary_exp_mod_iterative(7, -1, 10)
+    Traceback (most recent call last):
+        ...
+    ValueError: Exponent must be a non-negative integer
+    >>> binary_exp_mod_iterative(7, 13, 0)
+    Traceback (most recent call last):
+        ...
+    ValueError: Modulus must be a positive integer
+    """
+    if exponent < 0:
+        raise ValueError("Exponent must be a non-negative integer")
+    if modulus <= 0:
+        raise ValueError("Modulus must be a positive integer")
+
+    res: int | float = 1
+    while exponent > 0:
+        if exponent & 1:
+            res = ((res % modulus) * (base % modulus)) % modulus
+
+        base *= base
+        exponent >>= 1
+
+    return res
+
+
+if __name__ == "__main__":
+    from timeit import timeit
 
-    try:
-        BASE = int(float(input("Enter Base : ").strip()))
-        POWER = int(input("Enter Power : ").strip())
-    except ValueError:
-        print("Invalid literal for integer")
+    a = 1269380576
+    b = 374
+    c = 34
 
-    RESULT = binary_exponentiation(BASE, POWER)
-    print(f"{BASE}^({POWER}) : {RESULT}")
+    runs = 100_000
+    print(
+        timeit(
+            f"binary_exp_recursive({a}, {b})",
+            setup="from __main__ import binary_exp_recursive",
+            number=runs,
+        )
+    )
+    print(
+        timeit(
+            f"binary_exp_iterative({a}, {b})",
+            setup="from __main__ import binary_exp_iterative",
+            number=runs,
+        )
+    )
+    print(
+        timeit(
+            f"binary_exp_mod_recursive({a}, {b}, {c})",
+            setup="from __main__ import binary_exp_mod_recursive",
+            number=runs,
+        )
+    )
+    print(
+        timeit(
+            f"binary_exp_mod_iterative({a}, {b}, {c})",
+            setup="from __main__ import binary_exp_mod_iterative",
+            number=runs,
+        )
+    )