Performance: 75% faster Project Euler 187 (#10503)

* Add comments and wikipedia link in calculate_prime_numbers

* Add improved calculate_prime_numbers

* Separate slow_solution and new_solution

* Use for loops in solution

* Separate while_solution and new solution

* Add performance benchmark

* Add doctest for calculate_prime_numbers

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

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* Removed white space

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Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>

Author: ManpreetXSingh

Diff for: project_euler/problem_187/sol1.py

@@ -14,29 +14,89 @@
 from math import isqrt
 
 
-def calculate_prime_numbers(max_number: int) -> list[int]:
+def slow_calculate_prime_numbers(max_number: int) -> list[int]:
     """
-    Returns prime numbers below max_number
+    Returns prime numbers below max_number.
+    See: https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
 
-    >>> calculate_prime_numbers(10)
+    >>> slow_calculate_prime_numbers(10)
     [2, 3, 5, 7]
+
+    >>> slow_calculate_prime_numbers(2)
+    []
     """
 
+    # List containing a bool value for every number below max_number/2
     is_prime = [True] * max_number
+
     for i in range(2, isqrt(max_number - 1) + 1):
         if is_prime[i]:
+            # Mark all multiple of i as not prime
             for j in range(i**2, max_number, i):
                 is_prime[j] = False
 
     return [i for i in range(2, max_number) if is_prime[i]]
 
 
-def solution(max_number: int = 10**8) -> int:
+def calculate_prime_numbers(max_number: int) -> list[int]:
+    """
+    Returns prime numbers below max_number.
+    See: https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
+
+    >>> calculate_prime_numbers(10)
+    [2, 3, 5, 7]
+
+    >>> calculate_prime_numbers(2)
+    []
+    """
+
+    if max_number <= 2:
+        return []
+
+    # List containing a bool value for every odd number below max_number/2
+    is_prime = [True] * (max_number // 2)
+
+    for i in range(3, isqrt(max_number - 1) + 1, 2):
+        if is_prime[i // 2]:
+            # Mark all multiple of i as not prime using list slicing
+            is_prime[i**2 // 2 :: i] = [False] * (
+                # Same as: (max_number - (i**2)) // (2 * i) + 1
+                # but faster than len(is_prime[i**2 // 2 :: i])
+                len(range(i**2 // 2, max_number // 2, i))
+            )
+
+    return [2] + [2 * i + 1 for i in range(1, max_number // 2) if is_prime[i]]
+
+
+def slow_solution(max_number: int = 10**8) -> int:
     """
     Returns the number of composite integers below max_number have precisely two,
-    not necessarily distinct, prime factors
+    not necessarily distinct, prime factors.
 
-    >>> solution(30)
+    >>> slow_solution(30)
+    10
+    """
+
+    prime_numbers = slow_calculate_prime_numbers(max_number // 2)
+
+    semiprimes_count = 0
+    left = 0
+    right = len(prime_numbers) - 1
+    while left <= right:
+        while prime_numbers[left] * prime_numbers[right] >= max_number:
+            right -= 1
+        semiprimes_count += right - left + 1
+        left += 1
+
+    return semiprimes_count
+
+
+def while_solution(max_number: int = 10**8) -> int:
+    """
+    Returns the number of composite integers below max_number have precisely two,
+    not necessarily distinct, prime factors.
+
+    >>> while_solution(30)
     10
     """
 
@@ -54,5 +114,49 @@ def solution(max_number: int = 10**8) -> int:
     return semiprimes_count
 
 
+def solution(max_number: int = 10**8) -> int:
+    """
+    Returns the number of composite integers below max_number have precisely two,
+    not necessarily distinct, prime factors.
+
+    >>> solution(30)
+    10
+    """
+
+    prime_numbers = calculate_prime_numbers(max_number // 2)
+
+    semiprimes_count = 0
+    right = len(prime_numbers) - 1
+    for left in range(len(prime_numbers)):
+        if left > right:
+            break
+        for r in range(right, left - 2, -1):
+            if prime_numbers[left] * prime_numbers[r] < max_number:
+                break
+        right = r
+        semiprimes_count += right - left + 1
+
+    return semiprimes_count
+
+
+def benchmark() -> None:
+    """
+    Benchmarks
+    """
+    # Running performance benchmarks...
+    # slow_solution : 108.50874730000032
+    # while_sol     : 28.09581200000048
+    # solution      : 25.063097400000515
+
+    from timeit import timeit
+
+    print("Running performance benchmarks...")
+
+    print(f"slow_solution : {timeit('slow_solution()', globals=globals(), number=10)}")
+    print(f"while_sol     : {timeit('while_solution()', globals=globals(), number=10)}")
+    print(f"solution      : {timeit('solution()', globals=globals(), number=10)}")
+
+
 if __name__ == "__main__":
-    print(f"{solution() = }")
+    print(f"Solution: {solution()}")
+    benchmark()