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Stock-Based Compensation - Schedule of Stock-Based Compensation Expense (Income) (Details) - USD ($) $ in Millions | 12 Months Ended
--- | ---
Sep. 30, 2021 | Sep. 30, 2020 | Sep. 30, 2019
Equity-based compensation plans |  |  |
Stock-based compensation expense | $ 29.8 | $ 30.2 | $ 30.0
Income tax benefit recognized for stock-based compensation | (4.4) | (3.6) | (4.9)
Stock-based compensation cost, net of tax | 25.4 | 26.6 | 25.1
Stock options |  |  |
Equity-based compensation plans |  |  |
Stock-based compensation expense | 1.8 | 6.8 | 6.9
Nonvested Stock Awards [Member] |  |  |
Equity-based compensation plans |  |  |
Stock-based compensation expense | 19.3 | 15.8 | 15.0
Income tax benefit recognized for stock-based compensation | (2.0) | (3.1) | (2.1)
Performance Shares [Member] |  |  |
Equity-based compensation plans |  |  |
Stock-based compensation expense | 6.1 | 6.7 | 7.1
Stock Appreciation Rights (SARs) |  |  |
Equity-based compensation plans |  |  |
Stock-based compensation expense | 1.1 | 0.2 | 0.4
Restricted Stock Units (RSUs) |  |  |
Equity-based compensation plans |  |  |
Stock-based compensation expense | $ 1.5 | $ 0.7 | $ 0.6

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