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Share-Based Compensation - Narrative (Details) - USD ($)
$ / shares in Units, shares in Millions, $ in Millions
12 Months Ended
Sep. 30, 2023
Sep. 30, 2022
Sep. 30, 2021
Share-Based Payment Arrangement [Abstract]      
Share-based compensation arrangement by share-based payment award, number of shares authorized (in shares) 2.4    
Stock options authorized (in shares) 1.7    
Allocated share-based compensation expense $ 28.0 $ 17.8 $ 13.9
Weighted average grant date fair value, granted (in dollar per share) $ 33.21 $ 22.82 $ 19.83
Employee service share-based compensation, non vested awards, total compensation cost not yet recognized, stock options $ 9.2    
Employee service share-based compensation, non vested awards, total compensation cost not yet recognized, period for recognition (in years) 3 years 11 months 12 days    
Share-based compensation arrangement by share-based payment award, options, exercises in period, total intrinsic value $ 4.2 $ 39.0 $ 11.7
Employee service share-based compensation, nonvested awards, total compensation cost not yet recognized $ 40.5    
Employee service share-based compensation, nonvested awards, total compensation cost not yet recognized, period for recognition, restricted stock 1 year 2 months 26 days