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 r61807 = 9.0;
        double r61808 = x;
        double r61809 = 4.0;
        double r61810 = pow(r61808, r61809);
        double r61811 = r61807 * r61810;
        double r61812 = y;
        double r61813 = r61812 * r61812;
        double r61814 = 2.0;
        double r61815 = r61813 - r61814;
        double r61816 = r61813 * r61815;
        double r61817 = r61811 - r61816;
        return r61817;
}
↓
double f(double x, double y) {
        double r61818 = y;
        double r61819 = 2.0;
        double r61820 = pow(r61818, r61819);
        double r61821 = 2.0;
        double r61822 = 9.0;
        double r61823 = x;
        double r61824 = 4.0;
        double r61825 = pow(r61823, r61824);
        double r61826 = r61822 * r61825;
        double r61827 = pow(r61818, r61824);
        double r61828 = r61826 - r61827;
        double r61829 = 3.0;
        double r61830 = pow(r61828, r61829);
        double r61831 = cbrt(r61830);
        double r61832 = fma(r61820, r61821, r61831);
        return r61832;
}

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 2020081 +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))))