That claim can only be made with respect to using a Kulisch accumulator (what he calls the quire) for accumulation.
This can represent the exact sum of any number of floating point values (or products of floating point values) with the intermediate operations preserved at infinite precision, up to a single rounding at the end.
e.g., for computing an inner product of vectors \sum_i x_i y_i with x_i and y_i in floating point, a fused multiply-add operation (as on today's computers) would perform something like:
where what is performed within the rounding function r() is done to infinite precision and a single rounding at the end. This is also only true if you are accumulating into the same value, and not splitting this operation across multiple values, otherwise there would be additional rounding steps performed.
Using a Kulisch accumulator the result would be:
r(\sum_i x_i y_i)
similarly also only preserving precision if everything is done in a Kulisch accumulator.
You can add a Kulisch accumulator to IEEE floating point or any other floating point as well. In fact the second? ever computer with floating point (the Zuse Z3) performed FP addition using a similar accumulator, but sums-of-products probably didn't preserve the multiplied value though.
This can represent the exact sum of any number of floating point values (or products of floating point values) with the intermediate operations preserved at infinite precision, up to a single rounding at the end.
e.g., for computing an inner product of vectors \sum_i x_i y_i with x_i and y_i in floating point, a fused multiply-add operation (as on today's computers) would perform something like:
r(x_n y_n + r(... + r(x_2 y_2 + r(x_1 y_1 + 0))...))
where what is performed within the rounding function r() is done to infinite precision and a single rounding at the end. This is also only true if you are accumulating into the same value, and not splitting this operation across multiple values, otherwise there would be additional rounding steps performed.
Using a Kulisch accumulator the result would be:
r(\sum_i x_i y_i)
similarly also only preserving precision if everything is done in a Kulisch accumulator.
You can add a Kulisch accumulator to IEEE floating point or any other floating point as well. In fact the second? ever computer with floating point (the Zuse Z3) performed FP addition using a similar accumulator, but sums-of-products probably didn't preserve the multiplied value though.