Average Error: 52.0 → 52.0
Time: 2.6s
Precision: 64
\[x = 10864 \land y = 18817\]
\[\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)\]
↓
\[\mathsf{fma}\left(2 \cdot y, y, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)\]
\left(9 \cdot {x}^{4} - {y}^{4}\right) + 2 \cdot \left(y \cdot y\right)
↓
\mathsf{fma}\left(2 \cdot y, y, \sqrt[3]{{\left(9 \cdot {x}^{4} - {y}^{4}\right)}^{3}}\right)
double f(double x, double y) {
        double r72850 = 9.0;
        double r72851 = x;
        double r72852 = 4.0;
        double r72853 = pow(r72851, r72852);
        double r72854 = r72850 * r72853;
        double r72855 = y;
        double r72856 = pow(r72855, r72852);
        double r72857 = r72854 - r72856;
        double r72858 = 2.0;
        double r72859 = r72855 * r72855;
        double r72860 = r72858 * r72859;
        double r72861 = r72857 + r72860;
        return r72861;
}
↓
double f(double x, double y) {
        double r72862 = 2.0;
        double r72863 = y;
        double r72864 = r72862 * r72863;
        double r72865 = 9.0;
        double r72866 = x;
        double r72867 = 4.0;
        double r72868 = pow(r72866, r72867);
        double r72869 = r72865 * r72868;
        double r72870 = pow(r72863, r72867);
        double r72871 = r72869 - r72870;
        double r72872 = 3.0;
        double r72873 = pow(r72871, r72872);
        double r72874 = cbrt(r72873);
        double r72875 = fma(r72864, r72863, r72874);
        return r72875;
}

Error

Derivation

  1. Initial program 52.0

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

  2. Simplified52.0

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

  4. Applied add-cbrt-cube52.0

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

  5. Simplified52.0

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

  6. Final simplification52.0

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

Reproduce

herbie shell --seed 2020083 +o rules:numerics
(FPCore (x y)
  :name "From Rump in a 1983 paper"
  :precision binary64
  :pre (and (== x 10864) (== y 18817))
  (+ (- (* 9 (pow x 4)) (pow y 4)) (* 2 (* y y))))