Average Error: 62.0 → 52.0
Time: 2.3s
Precision: 64
\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)\]
↓
\[\sqrt[3]{{\left(\mathsf{fma}\left({x}^{4}, 9, -{y}^{4}\right)\right)}^{3}} - \left(y \cdot y\right) \cdot \left(-2\right)\]
9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)
↓
\sqrt[3]{{\left(\mathsf{fma}\left({x}^{4}, 9, -{y}^{4}\right)\right)}^{3}} - \left(y \cdot y\right) \cdot \left(-2\right)
double f(double x, double y) {
        double r64619 = 9.0;
        double r64620 = x;
        double r64621 = 4.0;
        double r64622 = pow(r64620, r64621);
        double r64623 = r64619 * r64622;
        double r64624 = y;
        double r64625 = r64624 * r64624;
        double r64626 = 2.0;
        double r64627 = r64625 - r64626;
        double r64628 = r64625 * r64627;
        double r64629 = r64623 - r64628;
        return r64629;
}
↓
double f(double x, double y) {
        double r64630 = x;
        double r64631 = 4.0;
        double r64632 = pow(r64630, r64631);
        double r64633 = 9.0;
        double r64634 = y;
        double r64635 = 4.0;
        double r64636 = pow(r64634, r64635);
        double r64637 = -r64636;
        double r64638 = fma(r64632, r64633, r64637);
        double r64639 = 3.0;
        double r64640 = pow(r64638, r64639);
        double r64641 = cbrt(r64640);
        double r64642 = r64634 * r64634;
        double r64643 = 2.0;
        double r64644 = -r64643;
        double r64645 = r64642 * r64644;
        double r64646 = r64641 - r64645;
        return r64646;
}

Error

Derivation

  1. Initial program 62.0

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

  3. Applied sub-neg62.0

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

  4. Applied distribute-lft-in62.0

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

  5. Applied associate--r+52.0

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

  6. Simplified52.0

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

  8. Applied add-cbrt-cube52.0

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

  9. Simplified52.0

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

  10. Final simplification52.0

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

Reproduce

herbie shell --seed 2020060 +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))))