xprec - fast emulated quadruple (double-double) 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 (flops) and relative error as multiples of u² = 1.32e-32 (round-off error or half the machine epsilon) is as follows:
| (op) | d (op) d | error | dd (op) d | error | dd (op) dd | error |
|---|---|---|---|---|---|---|
| add_small | 3 flops | 0u² | 7 flops | 2u² | 17 flops | 3u² |
| + - | 6 flops | 0u² | 10 flops | 2u² | 20 flops | 3u² |
| * | 2 flops | 0u² | 6 flops | 2u² | 9 flops | 4u² |
| / | 3* flops | 1u² | 7* flops | 3u² | 28* flops | 6u² |
| reciprocal | 3* flops | 1u² | 19* flops | 2.3u² |
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.
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Copyright (C) 2023-2025 Markus Wallerberger and others.
Released under the MIT license (see LICENSE for details).