⚡️ Speed up function compute_and_sort by 719%

The optimized code achieves a **718% speedup** through two key algorithmic improvements:

**1. Mathematical Optimization of `calculate_pairwise_products`:**
- **Original:** O(n²) nested loops explicitly computing all pairs `arr[i] * arr[j]` where `i ≠ j`
- **Optimized:** O(n) mathematical formula using the identity: sum of all pairwise products = `total² - total_sq` where `total = sum(arr)` and `total_sq = sum(x² for x in arr)`
- **Impact:** Reduces ~28.2 seconds to ~7ms for large arrays (99.97% reduction in this function's runtime)

**2. Bubble Sort Early Termination:**
- **Original:** Always performs full passes through the entire array
- **Optimized:** 
  - Reduces inner loop range by `i` each iteration since last `i` elements are already sorted
  - Adds `already_sorted` flag to exit early when no swaps occur
- **Impact:** Reduces sorting time from ~16.2 seconds to ~6.6 seconds (59% improvement)

**Performance by Test Case Type:**
- **Large uniform arrays** (1000 elements, same values): ~77,000% faster due to mathematical optimization
- **Large sequential/random arrays**: ~300-800% faster from combined optimizations
- **Small arrays** (<10 elements): Minimal improvement (0-20%) as overhead dominates
- **Edge cases** (zeros, negatives): Consistent improvements across all sizes

The mathematical optimization dominates performance gains for large arrays since it eliminates the quadratic bottleneck in pairwise product calculation.

Author: codeflash-ai[bot]

code_to_optimize/bubble_sort.py

@@ -1,10 +1,14 @@
 def sorter(arr):
     print("codeflash stdout: Sorting list")
     for i in range(len(arr)):
-        for j in range(len(arr) - 1):
+        already_sorted = True
+        for j in range(len(arr) - i - 1):
             if arr[j] > arr[j + 1]:
                 temp = arr[j]
                 arr[j] = arr[j + 1]
                 arr[j + 1] = temp
+                already_sorted = False
+        if already_sorted:
+            break
     print(f"result: {arr}")
     return arr

code_to_optimize/process_and_bubble_sort_codeflash_trace.py

@@ -1,21 +1,17 @@
 from code_to_optimize.bubble_sort import sorter
 from codeflash.benchmarking.codeflash_trace import codeflash_trace
 
-def calculate_pairwise_products(arr):
-    """
-    Calculate the average of all pairwise products in the array.
-    """
-    sum_of_products = 0
-    count = 0
 
-    for i in range(len(arr)):
-        for j in range(len(arr)):
-            if i != j:
-                sum_of_products += arr[i] * arr[j]
-                count += 1
+def calculate_pairwise_products(arr):
+    """Calculate the average of all pairwise products in the array."""
+    if len(arr) < 2:
+        return 0
+    total = sum(arr)
+    total_sq = sum(x * x for x in arr)
+    sum_of_products = total * total - total_sq
+    count = len(arr) * (len(arr) - 1)
+    return sum_of_products / count
 
-    # The average of all pairwise products
-    return sum_of_products / count if count > 0 else 0
 
 @codeflash_trace
 def compute_and_sort(arr):