Exact Arithmetic

The most fundamental operation in numerical computing — the dot product — is also the primary source of precision loss and non-reproducibility in floating-point arithmetic.

The Kulisch super-accumulator eliminates this problem at its root: a fixed-point register wide enough to hold the exact sum of any number of floating-point products, with only a single rounding operation at the very end.

The Universal Numbers Library implements this concept as the generalized quire — a C++ template that provides exact dot products for any number system: IEEE-style floats, posits, fixed-point, logarithmic, and double-base.

In This Section

• The Kulisch Super-Accumulator — History, theory, and practice of exact dot products, from Kulisch’s original work through the Universal library’s generalized quire. Includes worked examples, code, and a comprehensive reference list.

• Quire API Reference — Template parameters, operations, and usage patterns for the quire type.

Quick Start

#include <universal/number/cfloat/cfloat.hpp>
#include <universal/number/cfloat/fdp.hpp>

using Real = sw::universal::cfloat<32, 8, uint32_t, true, false, false>;

std::vector<Real> x = { 1e8f, -1e8f, 0.5f, 0.25f };
std::vector<Real> y = { 1.0f,  1.0f, 1.0f, 1.0f  };

// Naive accumulation: 0.0 (WRONG — residual lost to cancellation)
// Fused dot product:  0.75 (CORRECT — exact via quire)
Real result = sw::universal::fdp(x, y);

Supported Number Systems

Type	Header	FDP Available
float	<universal/native/fdp.hpp>	Yes
double	<universal/native/fdp.hpp>	Yes
cfloat	<universal/number/cfloat/fdp.hpp>	Yes
posit	<universal/number/posit/fdp.hpp>	Yes
fixpnt	<universal/number/fixpnt/fdp.hpp>	Yes
lns	<universal/number/lns/fdp.hpp>	Yes
dbns	<universal/number/dbns/fdp.hpp>	Yes
integer	<universal/number/integer/integer_impl.hpp>	Yes (via normalize)