Average Error: 62.0 → 52.0
Time: 11.2s
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 r100323 = 9.0;
        double r100324 = x;
        double r100325 = 4.0;
        double r100326 = pow(r100324, r100325);
        double r100327 = r100323 * r100326;
        double r100328 = y;
        double r100329 = r100328 * r100328;
        double r100330 = 2.0;
        double r100331 = r100329 - r100330;
        double r100332 = r100329 * r100331;
        double r100333 = r100327 - r100332;
        return r100333;
}
↓
double f(double x, double y) {
        double r100334 = x;
        double r100335 = 4.0;
        double r100336 = pow(r100334, r100335);
        double r100337 = 9.0;
        double r100338 = y;
        double r100339 = 4.0;
        double r100340 = pow(r100338, r100339);
        double r100341 = -r100340;
        double r100342 = fma(r100336, r100337, r100341);
        double r100343 = 3.0;
        double r100344 = pow(r100342, r100343);
        double r100345 = cbrt(r100344);
        double r100346 = r100338 * r100338;
        double r100347 = 2.0;
        double r100348 = -r100347;
        double r100349 = r100346 * r100348;
        double r100350 = r100345 - r100349;
        return r100350;
}

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 2020043 +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))))