2-ancestry mixing, zero discriminant

Average Error: 15.4 → 0.8

Time: 13.2s

Precision: 64

Math

\[\sqrt[3]{\frac{g}{2 \cdot a}}\]

↓

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

C

double f(double g, double a) {
        double r123009 = g;
        double r123010 = 2.0;
        double r123011 = a;
        double r123012 = r123010 * r123011;
        double r123013 = r123009 / r123012;
        double r123014 = cbrt(r123013);
        return r123014;
}

↓

double f(double g, double a) {
        double r123015 = 1.0;
        double r123016 = 2.0;
        double r123017 = a;
        double r123018 = r123016 * r123017;
        double r123019 = cbrt(r123018);
        double r123020 = g;
        double r123021 = cbrt(r123020);
        double r123022 = r123019 / r123021;
        double r123023 = r123015 / r123022;
        return r123023;
}

Error

Bits error versus g

Bits error versus a

Derivation

1. Initial program 15.4

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

3. Applied cbrt-div 0.8

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

4. Simplified 0.8

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

6. Applied *-un-lft-identity 0.8

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

7. Applied cbrt-prod 0.8

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

8. Applied associate-/l* 0.8

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

9. Simplified 0.8

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

11. Applied add-cube-cbrt 0.8

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

12. Applied associate-/l* 0.8

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

13. Simplified 0.8

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

14. Final simplification 0.8

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

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  (cbrt (/ g (* 2.0 a))))