Real World Problems Caused by Rounding Error

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.
      http://www.esrin.esa.it/htdocs/tidc/Press/Press96/ariane5rep.html
      
   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.
   Sources
   1. Rounding error changes Parliament makeup
      Debora Weber-Wulff
      The Risks Digest
      Volume 13, Issue 37, 1992
      http://catless.ncl.ac.uk/Risks/

   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%.