Fast emulated quadruple precision in Rust

Emulates quadruple precision with a pair of doubles. This roughly doubles the mantissa bits (and thus squares the precision of double). The range is almost the same as double, with a larger area of denormalized numbers. This is also called double-double arithmetic, compensated arithmetic, or Dekker arithmetic.

The rough cost in floating point operations (fl) and relative error as multiples of u² = 1.32e-32 (round-off error or half the machine epsilon) is as follows:

The error bounds are mostly tight analytical bounds (except for divisions).[^1] An asterisk indicates the need for one or two double divisions, which are about an order of magnitude more expensive than regular flops on a modern CPU.

The table can be distilled into two rules of thumb: double-double arithmetic roughly doubles the number of significant digits at the cost of a roughly 15x slowdown compared to double arithmetic.

[^1]: M. Joldes, et al., ACM Trans. Math. Softw. 44, 1-27 (2018) and J.-M. Muller and L. Rideau, ACM Trans. Math. Softw. 48, 1, 9 (2022). The flop count has been reduced by 3 for divisons/reciprocals. In the case of double-double division, the bound is 10u² but largest observed error is 6u². In double by double division, we expect u². We report the largest observed error.

License and Copying

Copyright (C) 2023-2025 Markus Wallerberger and others.

Released under the MIT license (see LICENSE for details).