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:
| (op) | f64 f64 | error | Df64 f64 | error | Df64 Df64 | error |
|---|---|---|---|---|---|---|
| add_fast | 3 fl | 0u² | 7 fl | 2u² | 17 fl | 3u² |
| + - | 6 fl | 0u² | 10 fl | 2u² | 20 fl | 3u² |
| * | 2 fl | 0u² | 6 fl | 2u² | 9 fl | 4u² |
| / | 3* fl | 1u² | 7* fl | 3u² | 28* fl | 6u² |
| reciprocal | 3* fl | 1u² | 19* fl | 2.3u² | ||
| sqrt | 4* fl | 2u² | 8* fl | 4u² |
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.
Copyright (C) 2023-2025 Markus Wallerberger and others.
Released under the MIT license (see LICENSE for details).
Footnotes
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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. ↩