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Stock compensation (Details) - USD ($) $ / shares in Units, $ in Millions | 3 Months Ended | 6 Months Ended
--- | --- | ---
Jun. 30, 2018 | Jun. 30, 2017 | Jun. 30, 2018 | Jun. 30, 2017
Stock compensation |  |  |  |
Stock compensation expense | $ 36.6 | $ 33.8 | $ 72.8 | $ 64.4
Weighted-average fair value assumptions |  |  |  |
Valuation method |  |  | Black-Scholes valuation model |
Dividend yield (as a percent) |  |  | 0.00% |
Research and Development Expense [Member] |  |  |  |
Stock compensation |  |  |  |
Stock compensation expense | 24.8 | 22.9 | $ 49.0 | 44.4
Selling, General and Administrative Expenses [Member] |  |  |  |
Stock compensation |  |  |  |
Stock compensation expense | $ 11.8 | $ 10.9 | $ 23.8 | $ 20.0
Stock Options |  |  |  |
Weighted-average fair value assumptions |  |  |  |
Average risk-free interest rates (as a percent) | 2.71% | 1.56% | 2.47% | 1.79%
Average expected life (in years) | 5 years 4 months 13 days | 5 years 4 months 28 days | 5 years 6 months | 5 years 4 months 10 days
Volatility (as a percent) | 46.00% | 50.00% | 45.00% | 49.00%
Weighted-average fair value (in dollars per share) | $ 28.65 | $ 60.44 | $ 39.47 | $ 52.87
Restricted Stock Units (RSUs) |  |  |  |
Weighted-average fair value assumptions |  |  |  |
Average risk-free interest rates (as a percent) | 2.52% | 1.38% | 2.50% | 1.31%
Average expected life (in years) | 6 months | 6 months | 6 months | 6 months
Volatility (as a percent) | 30.00% | 42.00% | 51.00% | 42.00%
Weighted-average fair value (in dollars per share) | $ 12.11 | $ 17.55 | $ 16.25 | $ 18.80

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- Definition Represents the expense recognized during the period arising from equity-based compensation arrangements (for example, shares of stock, unit, stock options or other equity instruments) with employees, directors and certain consultants qualifying for treatment as employees. + References Reference 1: http://www.xbrl.org/2003/role/presentationRef -Publisher FASB -Name Accounting Standards Codification -Topic 718 -SubTopic 10 -Section S99 -Paragraph 1 -Subparagraph (SAB Topic 14.F) -URI http://asc.fasb.org/extlink&oid=27013229&loc=d3e301413-122809 Reference 2: http://www.xbrl.org/2003/role/presentationRef -Publisher FASB -Name Accounting Standards Codification -Topic 718 -SubTopic 10 -Section 50 -Paragraph 1 -Subparagraph (b) -URI http://asc.fasb.org/extlink&oid=96867065&loc=d3e5047-113901 Reference 3: http://www.xbrl.org/2003/role/presentationRef -Publisher FASB -Name Accounting Standards Codification -Topic 718 -SubTopic 10 -Section 50 -Paragraph 2 -Subparagraph (h)(1)(i) -URI http://asc.fasb.org/extlink&oid=96867065&loc=d3e5070-113901 + Details
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- Definition The estimated dividend rate (a percentage of the share price) to be paid (expected dividends) to holders of the underlying shares over the option's term. + References Reference 1: http://www.xbrl.org/2003/role/presentationRef -Publisher FASB -Name Accounting Standards Codification -Topic 718 -SubTopic 10 -Section 50 -Paragraph 2 -Subparagraph (f)(2)(iii) -URI http://asc.fasb.org/extlink&oid=96867065&loc=d3e5070-113901 + Details
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- Definition The estimated measure of the percentage by which a share price is expected to fluctuate during a period. Volatility also may be defined as a probability-weighted measure of the dispersion of returns about the mean. The volatility of a share price is the standard deviation of the continuously compounded rates of return on the share over a specified period. That is the same as the standard deviation of the differences in the natural logarithms of the stock prices plus dividends, if any, over the period. + References Reference 1: http://www.xbrl.org/2003/role/presentationRef -Publisher FASB -Name Accounting Standards Codification -Topic 718 -SubTopic 10 -Section 50 -Paragraph 2 -Subparagraph (f)(2)(ii) -URI http://asc.fasb.org/extlink&oid=96867065&loc=d3e5070-113901 + Details
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- Definition For each plan, identification of the award pricing model or other valuation method used in calculating the weighted average fair values disclosed. The model is also used to calculate the compensation expense that is shown within the balance sheet, income statement, and cash flow. Examples of valuation techniques are lattice models (binomial model), closed-form models (Black-Scholes-Merton formula), and a Monte Carlo simulation technique. Fair value is the amount at which an asset or liability could be bought or incurred or sold or settled in a current transaction between willing parties, that is, other than in a forced or liquidation sale. May include disclosures about the assumptions underlying application of the method selected. + References Reference 1: http://www.xbrl.org/2003/role/presentationRef -Publisher FASB -Name Accounting Standards Codification -Topic 718 -SubTopic 10 -Section 50 -Paragraph 2 -Subparagraph (f)(1) -URI http://asc.fasb.org/extlink&oid=96867065&loc=d3e5070-113901 + Details
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- Definition The risk-free interest rate assumption that is used in valuing an option on its own shares. + References Reference 1: http://www.xbrl.org/2003/role/presentationRef -Publisher FASB -Name Accounting Standards Codification -Topic 718 -SubTopic 10 -Section 50 -Paragraph 2 -Subparagraph (f)(2)(iv) -URI http://asc.fasb.org/extlink&oid=96867065&loc=d3e5070-113901 + Details
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- Definition Line items represent financial concepts included in a table. These concepts are used to disclose reportable information associated with domain members defined in one or many axes to the table. + References No definition available. + Details
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- Definition The weighted average grant-date fair value of options granted during the reporting period as calculated by applying the disclosed option pricing methodology. + References Reference 1: http://www.xbrl.org/2003/role/presentationRef -Publisher FASB -Name Accounting Standards Codification -Topic 718 -SubTopic 10 -Section 50 -Paragraph 2 -Subparagraph (d)(1) -URI http://asc.fasb.org/extlink&oid=96867065&loc=d3e5070-113901 + Details
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- Definition Expected term of share-based compensation awards, in 'PnYnMnDTnHnMnS' format, for example, 'P1Y5M13D' represents the reported fact of one year, five months, and thirteen days. + References Reference 1: http://www.xbrl.org/2003/role/presentationRef -Publisher FASB -Name Accounting Standards Codification -Topic 718 -SubTopic 10 -Section 50 -Paragraph 2 -Subparagraph (f)(2)(i) -URI http://asc.fasb.org/extlink&oid=96867065&loc=d3e5070-113901 Reference 2: http://www.xbrl.org/2003/role/presentationRef -Publisher FASB -Name Accounting Standards Codification -Topic 718 -SubTopic 10 -Section S99 -Paragraph 1 -Subparagraph (SAB Topic 14.D.2) -URI http://asc.fasb.org/extlink&oid=27013229&loc=d3e301413-122809 + Details
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