Average Error: 62.0 → 52.0
Time: 2.3s
Precision: 64
\[x = 10864 \land y = 18817\]
\[9 \cdot {x}^{4} - \left(y \cdot y\right) \cdot \left(y \cdot y - 2\right)\]
↓
\[\sqrt[3]{{\left(\mathsf{fma}\left({x}^{4}, 9, -{y}^{4}\right)\right)}^{3}} - \left(-2\right) \cdot \left(y \cdot y\right)\]
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(-2\right) \cdot \left(y \cdot y\right)
double f(double x, double y) {
        double r79198 = 9.0;
        double r79199 = x;
        double r79200 = 4.0;
        double r79201 = pow(r79199, r79200);
        double r79202 = r79198 * r79201;
        double r79203 = y;
        double r79204 = r79203 * r79203;
        double r79205 = 2.0;
        double r79206 = r79204 - r79205;
        double r79207 = r79204 * r79206;
        double r79208 = r79202 - r79207;
        return r79208;
}
↓
double f(double x, double y) {
        double r79209 = x;
        double r79210 = 4.0;
        double r79211 = pow(r79209, r79210);
        double r79212 = 9.0;
        double r79213 = y;
        double r79214 = 4.0;
        double r79215 = pow(r79213, r79214);
        double r79216 = -r79215;
        double r79217 = fma(r79211, r79212, r79216);
        double r79218 = 3.0;
        double r79219 = pow(r79217, r79218);
        double r79220 = cbrt(r79219);
        double r79221 = 2.0;
        double r79222 = -r79221;
        double r79223 = r79213 * r79213;
        double r79224 = r79222 * r79223;
        double r79225 = r79220 - r79224;
        return r79225;
}

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-rgt-in62.0

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

  5. Applied associate--r+52.0

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

  6. Simplified52.0

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

  8. Applied add-cbrt-cube52.0

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

  9. Simplified52.0

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

  10. Final simplification52.0

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

Reproduce

herbie shell --seed 2019354 +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))))