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The Formula that the IRS uses is from 

Internal Revenue Manual 25.18.5

Injured Spouse's Separate Tax Liability÷Total of Spouses' Separate Tax Liabilities×Joint Tax Liability Shown on Return=Injured Spouse's Share of Tax Liability

 

Separate Tax Liabilities are calculated using Married Filing Separate Tax Tables for each spouse as mentioned earlier. The form 8379 has instructions for  breaking out the amounts for each spouse to be used in this calculation.