For the Desktop version you can switch to Forms Mode and open the worksheet to see it. Click Forms in the upper right (upper left for Mac) and look through the list and open the Qualified Dividends and Capital Gain Tax Worksheet.
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The online calculator that you used says it provides "an estimate of federal income tax owed" (in the paragraph at the top of the screen). It is not an exact calculation of your tax. The figure it gave you, $31,809, is a pretty good estimate of the actual tax of $31,487.
In your attachment you said that you used the "tax rate table." I assume you mean the Tax Computation Worksheet, since your taxable income is over $100,000. The Tax Computation Worksheet is only one of seven different methods that can be used to calculate the tax, depending on what is in your return. Your return must require one of the other methods. You said you had no long-term capital gains, but you might have had qualified dividends or something else that requires using a different method to calculate the tax. Qualified dividends is one of the most common reasons that you cannot use the Tax Computation Worksheet.
In the 2013 IRS instructions for Form 1040, look at the instructions for line 44, starting where it says "Use one of the following methods to figure your tax" at the bottom of the middle column on page 40. The seven methods are explained there.
You can see which method TurboTax used by looking at Form 1040 in forms mode. In the Tax Smart Worksheet between lines 43 and 44 of Form 1040 there are seven checkboxes corresponding to the seven methods for calculating the tax. The box that is checked shows you which method TurboTax used.
Also, as Critter#2 said, if you had really underpaid your tax by $322 you would have heard from the IRS long before now.