Average Error: 62.0 → 52.0
Time: 1.7s
Precision: 64
\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)\]
↓
\[\mathsf{fma}\left({y}^{2}, 2, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)\]
9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
↓
\mathsf{fma}\left({y}^{2}, 2, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)
double f(double x, double y) {
        double r81582 = 9.0;
        double r81583 = x;
        double r81584 = 4.0;
        double r81585 = pow(r81583, r81584);
        double r81586 = r81582 * r81585;
        double r81587 = y;
        double r81588 = r81587 * r81587;
        double r81589 = 2.0;
        double r81590 = r81588 - r81589;
        double r81591 = r81588 * r81590;
        double r81592 = r81586 - r81591;
        return r81592;
}
↓
double f(double x, double y) {
        double r81593 = y;
        double r81594 = 2.0;
        double r81595 = pow(r81593, r81594);
        double r81596 = 2.0;
        double r81597 = 9.0;
        double r81598 = x;
        double r81599 = 4.0;
        double r81600 = pow(r81598, r81599);
        double r81601 = r81597 * r81600;
        double r81602 = pow(r81593, r81599);
        double r81603 = r81601 - r81602;
        double r81604 = 3.0;
        double r81605 = pow(r81603, r81604);
        double r81606 = cbrt(r81605);
        double r81607 = fma(r81595, r81596, r81606);
        return r81607;
}

Error

Derivation

  1. Initial program 62.0

    \[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)\]

  2. Taylor expanded around 0 62.0

    \[\leadsto \color{blue}{\left(2 \cdot {y}^{2} + 9 \cdot {x}^{4}\right) - {y}^{4}}\]

  3. Simplified52.0

    \[\leadsto \color{blue}{\mathsf{fma}\left({y}^{2}, 2, 9 \cdot {x}^{4} - {y}^{4}\right)}\]
  4. Using strategy rm

  5. Applied add-cbrt-cube52.0

    \[\leadsto \mathsf{fma}\left({y}^{2}, 2, \color{blue}{\sqrt[3]{\left(\left(9 \cdot {x}^{4} - {y}^{4}\right) \cdot \left(9 \cdot {x}^{4} - {y}^{4}\right)\right) \cdot \left(9 \cdot {x}^{4} - {y}^{4}\right)}}\right)\]

  6. Simplified52.0

    \[\leadsto \mathsf{fma}\left({y}^{2}, 2, \sqrt[3]{\color{blue}{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}}\right)\]

  7. Final simplification52.0

    \[\leadsto \mathsf{fma}\left({y}^{2}, 2, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)\]

Reproduce

herbie shell --seed 2019362 +o rules:numerics
(FPCore (x y)
  :name "From Rump in a 1983 paper, rewritten"
  :precision binary64
  :pre (and (== x 10864) (== y 18817))
  (- (* 9 (pow x 4)) (* (* y y) (- (* y y) 2))))