1. The Patriot and the Scud.

Sources
1. General Accounting Office Report GAO/IMTEC-92-26.
2. Robert Skeel, "Roundoff Error Cripples Patriot Missile," SIAM News, July 1992.

On February 25, 1991, during the Gulf War, a Patriot missile defense system let a Scud get through. It hit a barracks, killing 28 people. The problem was in the differencing of floating point numbers obtained by converting and scaling an integer timing register. The GAO report has less than the full story. For that see Skeel's excellent article.

2. The short flight of the Ariane 5.

Source
1. Report by the Inquiry Board.

On June 4, 1996, the first Ariane 5 was launched. All went well for 36 seconds. Then the Ariane veered off course and self-destructed. The problem was in the Inertial Reference System, which produced an operation exception trying to convert a 64-bit floating-point number to a 12-bit integer. It sent a diagnostic word to the On-Board Computer, which interpreted it as flight data. Finis.

Ironically, the computation was done by legacy software from the Ariane 4, and its results were not needed after lift-off.

3. The Vancouver Stock Exchange.

Sources
1. The Wall Street Journal November 8, 1983, p.37.
2. The Toronto Star, November 19, 1983.
3. B.D. McCullough and H.D. Vinod Journal of Economic Literature Vol XXXVII (June 1999), pp. 633-665. (References communicated by Valerie Fraysse)

In 1982 (I figure) the Vancouver Stock Exchange instituted a new index initialized to a value of 1000.000. The index was updated after each transaction. Twenty two months later it had fallen to 520. The cause was that the updated value was truncated rather than rounded. The rounded calculation gave a value of 1098.892.

4. Parliamentary elections in Schleswig-Holstein.

Source
1. Rounding error changes Parliament makeup, Debora Weber-Wulff, The Risks Digest, Volume 13, Issue 37, 1992.

In German parliamentary elections, a party with less than 5.0% of the vote cannot be seated. The Greens appeared to have a cliff-hanging 5.0%, until it was discovered (after the results had been announced) that they really had only 4.97%. The printout was to two figures, and the actual percentage was rounded to 5.0%.

5. A few more examples

Try searching the RISK Digest with the key word "rounding" (David Goldberg's idea).