IEEE 754, floating-point multiply/add, and accurate summation

["Nelson H. F. Beebe" <beebe@xxxxxxxxxxxxx>]

The new paper listed below arrived in a new journal issue yesterday,
and appears to be a significant advance in the problem of accurate
floating-point summation, exploiting the floating-point multiply/add
instruction (or its library emulation, such as C99's fma() function),
plus double-compensated summation (Priest 1992).  There may well be
fodder here for floating-point hardware designers.

@String{j-TOMS                  = "ACM Transactions on Mathematical Software"}

@Article{Nievergelt:2003:SFM,
  author =       "Yves Nievergelt",
  title =        "Scalar fused multiply-add instructions produce
                 floating-point matrix arithmetic provably accurate to
                 the penultimate digit",
  journal =      j-TOMS,
  volume =       "29",
  number =       "1",
  pages =        "27--48",
  month =        mar,
  year =         "2003",
  CODEN =        "ACMSCU",
  ISSN =         "0098-3500",
  bibdate =      "Fri Mar 28 08:17:55 MST 2003",
  bibsource =    "http://www.acm.org/pubs/contents/journals/toms/";,
  URL =          "http://doi.acm.org/10.1145/641876.641878";,
  abstract =     "Combined with doubly compensated summation, scalar
                 fused multiply-add instructions redefine the concept of
                 floating-point arithmetic, because they allow for the
                 computation of sums of real or complex matrix products
                 accurate to the penultimate digit. Particular cases
                 include complex arithmetic, dot products, cross
                 products, residuals of linear systems, determinants of
                 small matrices, discriminants of quadratic, cubic, or
                 quartic equations, and polynomials.",
  acknowledgement = ack-nhfb,
  keywords =     "algorithms; design; languages; standardization;
                 theory; doubly compensated summation; floating-point
                 arithmetic; fused multiply-add instruction; matrix
                 arithmetic; provable accuracy; rounding error",
}

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