xprec 0.1.0

Fast emulated quadruple (double-double) precision arithmetic
Documentation 
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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.


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

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