How can I improve the speed of my slow quicksort algorithm?

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There are a few ways to improve the speed of a slow quicksort algorithm:

1. Randomized pivot selection: Choose a random element from the array as the pivot instead of always choosing the first or last element. This can help avoid worst-case scenarios where the pivot is always the smallest or largest element in the array.

2. Median-of-three pivot selection: Choose the median of the first, middle, and last elements as the pivot. This can also help avoid worst-case scenarios and improve overall performance.

3. Tail recursion elimination: Convert the quicksort function to use a loop instead of recursion. This can eliminate the overhead of recursive function calls and improve performance.

4. Insertion sort for small arrays: For arrays of a certain size (usually around 10-20 elements), switch to insertion sort instead of continuing with quicksort. Insertion sort has lower overhead and can be faster for small arrays.

5. Use a different sorting algorithm for small arrays: Instead of insertion sort, consider using a different sorting algorithm (such as selection sort or bubble sort) for small arrays that may perform better than quicksort.