2-ancestry mixing, zero discriminant

Average Error: 15.8 → 0.9

Time: 19.3s

Precision: 64

Math

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

↓

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

C

double f(double g, double a) {
        double r5268541 = g;
        double r5268542 = 2.0;
        double r5268543 = a;
        double r5268544 = r5268542 * r5268543;
        double r5268545 = r5268541 / r5268544;
        double r5268546 = cbrt(r5268545);
        return r5268546;
}

↓

double f(double g, double a) {
        double r5268547 = 0.5;
        double r5268548 = cbrt(r5268547);
        double r5268549 = cbrt(r5268548);
        double r5268550 = a;
        double r5268551 = r5268549 / r5268550;
        double r5268552 = cbrt(r5268551);
        double r5268553 = r5268548 * r5268548;
        double r5268554 = cbrt(r5268553);
        double r5268555 = cbrt(r5268554);
        double r5268556 = r5268552 * r5268555;
        double r5268557 = g;
        double r5268558 = cbrt(r5268557);
        double r5268559 = r5268554 * r5268558;
        double r5268560 = r5268556 * r5268559;
        return r5268560;
}

Error

Bits error versus g

Bits error versus a

Derivation

1. Initial program 15.8

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

3. Applied div-inv 15.8

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

4. Applied cbrt-prod 0.9

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

5. Simplified 0.8

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

7. Applied *-un-lft-identity 0.8

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

8. Applied *-un-lft-identity 0.8

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

9. Applied times-frac 0.8

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

10. Applied cbrt-prod 0.8

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

11. Applied associate-*r* 0.8

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

12. Simplified 0.8

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

14. Applied *-un-lft-identity 0.8

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

15. Applied add-cube-cbrt 0.8

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

16. Applied times-frac 0.8

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

17. Applied cbrt-prod 0.9

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

18. Applied associate-*r* 0.9

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

19. Simplified 0.9

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

21. Applied *-un-lft-identity 0.9

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

22. Applied add-cube-cbrt 0.9

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

23. Applied cbrt-prod 0.9

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

24. Applied times-frac 0.9

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

25. Applied cbrt-prod 0.9

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

26. Simplified 0.9

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

27. Final simplification 0.9

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

Reproduce

herbie shell --seed 2019142 +o rules:numerics
(FPCore (g a)
  :name "2-ancestry mixing, zero discriminant"
  (cbrt (/ g (* 2 a))))