"The ratio of the unrecovered taxed contributions to the accumulated earnings (U/A) establishes the fraction of the earnings due to the taxed contributions."
This is correct only if only nondeductible contributions were made. Regardless, it's a meaningless number with regard to determining the taxable and nontaxable amounts.
"Multiplying that ratio (U/A) times the amount of the distribution (D) determines the non-taxable portion of the distribution."
No, that's wrong.
In your example with $5,000 of basis in nondeductible traditional IRA contributions, a $32,117 distribution during the year and a $900,000 balance at the end of the year, the fraction that is nontaxable is:
$5,000 / ($900,000 + $32,117) = 0.005364
You instead incorrectly calculated that fraction as:
$5,000/ ($900,000 + $32,117 - 5000) = 0.005393
The only reason that these calculations are close is that you used a very large year-end balance. If the year-end balance was instead $10,000, the correct calculation would produce a nontaxable fraction of:
$5,000 / ($10,000 + $32,117) = 0.1187
while your incorrectly calculated fraction would be:
$5,000/ ($10,000 + $32,117 - 5000) = 0.1347
