Average Error: 14.7 → 0.8
Time: 17.7s
Precision: 64
\[\sqrt[3]{\frac{g}{2 \cdot a}}\]
↓
\[\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{2}}\right) \cdot \sqrt[3]{\frac{1}{a}}\]
double f(double g, double a) {
        double r23037906 = g;
        double r23037907 = 2.0;
        double r23037908 = a;
        double r23037909 = r23037907 * r23037908;
        double r23037910 = r23037906 / r23037909;
        double r23037911 = cbrt(r23037910);
        return r23037911;
}
↓
double f(double g, double a) {
        double r23037912 = g;
        double r23037913 = cbrt(r23037912);
        double r23037914 = 0.5;
        double r23037915 = cbrt(r23037914);
        double r23037916 = r23037913 * r23037915;
        double r23037917 = 1.0;
        double r23037918 = a;
        double r23037919 = r23037917 / r23037918;
        double r23037920 = cbrt(r23037919);
        double r23037921 = r23037916 * r23037920;
        return r23037921;
}


\sqrt[3]{\frac{g}{2 \cdot a}}
↓
\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{2}}\right) \cdot \sqrt[3]{\frac{1}{a}}

Error

Bits error versus g

Bits error versus a

Derivation

  1. Initial program 14.7

    \[\sqrt[3]{\frac{g}{2 \cdot a}}\]
  2. Using strategy rm

  3. Applied div-inv14.7

    \[\leadsto \sqrt[3]{\color{blue}{g \cdot \frac{1}{2 \cdot a}}}\]

  4. Applied cbrt-prod0.9

    \[\leadsto \color{blue}{\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}}\]
  5. Using strategy rm

  6. Applied *-un-lft-identity0.9

    \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\frac{\color{blue}{1 \cdot 1}}{2 \cdot a}}\]

  7. Applied times-frac0.8

    \[\leadsto \sqrt[3]{g} \cdot \sqrt[3]{\color{blue}{\frac{1}{2} \cdot \frac{1}{a}}}\]

  8. Applied cbrt-prod0.8

    \[\leadsto \sqrt[3]{g} \cdot \color{blue}{\left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{\frac{1}{a}}\right)}\]

  9. Applied associate-*r*0.8

    \[\leadsto \color{blue}{\left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{2}}\right) \cdot \sqrt[3]{\frac{1}{a}}}\]

  10. Taylor expanded around inf 33.6

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{2}} \cdot {g}^{\frac{1}{3}}\right)} \cdot \sqrt[3]{\frac{1}{a}}\]

  11. Simplified0.8

    \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{1}{2}} \cdot \sqrt[3]{g}\right)} \cdot \sqrt[3]{\frac{1}{a}}\]

  12. Final simplification0.8

    \[\leadsto \left(\sqrt[3]{g} \cdot \sqrt[3]{\frac{1}{2}}\right) \cdot \sqrt[3]{\frac{1}{a}}\]

Reproduce

herbie shell --seed 2019101 +o rules:numerics
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  (cbrt (/ g (* 2 a))))