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Share-Based Compensation - Narrative (Details) - USD ($)
$ / shares in Units, shares in Millions, $ in Millions
12 Months Ended
Sep. 30, 2024
Sep. 30, 2023
Sep. 30, 2022
Share-Based Payment Arrangement [Abstract]      
Share-based compensation arrangement by share-based payment award, number of shares authorized (in shares) 3.5    
Stock options authorized (in shares) 0.4    
Allocated share-based compensation expense $ 37.2 $ 28.0 $ 17.8
Weighted average grant date fair value, granted (in dollar per share) $ 26.77 $ 22.14 $ 15.21
Employee service share-based compensation, non vested awards, total compensation cost not yet recognized, stock options $ 43.1    
Employee service share-based compensation, non vested awards, total compensation cost not yet recognized, period for recognition 3 years 3 days    
Share-based compensation arrangement by share-based payment award, options, exercises in period, total intrinsic value $ 9.8 $ 4.2 $ 39.0
Employee service share-based compensation, nonvested awards, total compensation cost not yet recognized $ 43.9    
Employee service share-based compensation, nonvested awards, total compensation cost not yet recognized, period for recognition, restricted stock 1 year 1 month 17 days