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 r77726 = 9.0;
        double r77727 = x;
        double r77728 = 4.0;
        double r77729 = pow(r77727, r77728);
        double r77730 = r77726 * r77729;
        double r77731 = y;
        double r77732 = pow(r77731, r77728);
        double r77733 = r77730 - r77732;
        double r77734 = 2.0;
        double r77735 = r77731 * r77731;
        double r77736 = r77734 * r77735;
        double r77737 = r77733 + r77736;
        return r77737;
}
↓
double f(double x, double y) {
        double r77738 = 2.0;
        double r77739 = y;
        double r77740 = r77738 * r77739;
        double r77741 = 9.0;
        double r77742 = x;
        double r77743 = 4.0;
        double r77744 = pow(r77742, r77743);
        double r77745 = r77741 * r77744;
        double r77746 = pow(r77739, r77743);
        double r77747 = r77745 - r77746;
        double r77748 = 3.0;
        double r77749 = pow(r77747, r77748);
        double r77750 = cbrt(r77749);
        double r77751 = fma(r77740, r77739, r77750);
        return r77751;
}

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 2020025 +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))))