2-ancestry mixing, positive discriminant

Average Error: 35.3 → 30.6

Time: 32.5s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;g \le 9.191731187329127127438182160139194041057 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{\sqrt[3]{a} \cdot \sqrt{2}}} \cdot \sqrt[3]{\frac{1}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt{2}}} + \sqrt[3]{\left(\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right) \cdot \frac{1}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt{2}}} \cdot \frac{\sqrt[3]{1 \cdot \frac{h \cdot \left(-h\right)}{g + \sqrt{g \cdot g - h \cdot h}}}}{\sqrt[3]{\sqrt[3]{a} \cdot \sqrt{2}}} + \sqrt[3]{\frac{\frac{1}{a}}{2}} \cdot \sqrt[3]{-\left(g + \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right)}\\ \end{array}\]

C

double f(double g, double h, double a) {
        double r148561 = 1.0;
        double r148562 = 2.0;
        double r148563 = a;
        double r148564 = r148562 * r148563;
        double r148565 = r148561 / r148564;
        double r148566 = g;
        double r148567 = -r148566;
        double r148568 = r148566 * r148566;
        double r148569 = h;
        double r148570 = r148569 * r148569;
        double r148571 = r148568 - r148570;
        double r148572 = sqrt(r148571);
        double r148573 = r148567 + r148572;
        double r148574 = r148565 * r148573;
        double r148575 = cbrt(r148574);
        double r148576 = r148567 - r148572;
        double r148577 = r148565 * r148576;
        double r148578 = cbrt(r148577);
        double r148579 = r148575 + r148578;
        return r148579;
}

↓

double f(double g, double h, double a) {
        double r148580 = g;
        double r148581 = 9.191731187329127e-156;
        bool r148582 = r148580 <= r148581;
        double r148583 = 1.0;
        double r148584 = -r148580;
        double r148585 = r148584 - r148580;
        double r148586 = r148583 * r148585;
        double r148587 = cbrt(r148586);
        double r148588 = a;
        double r148589 = cbrt(r148588);
        double r148590 = 2.0;
        double r148591 = sqrt(r148590);
        double r148592 = r148589 * r148591;
        double r148593 = cbrt(r148592);
        double r148594 = r148587 / r148593;
        double r148595 = 1.0;
        double r148596 = r148589 * r148589;
        double r148597 = r148596 * r148591;
        double r148598 = r148595 / r148597;
        double r148599 = cbrt(r148598);
        double r148600 = r148594 * r148599;
        double r148601 = h;
        double r148602 = r148601 + r148580;
        double r148603 = r148580 - r148601;
        double r148604 = r148602 * r148603;
        double r148605 = sqrt(r148604);
        double r148606 = r148584 - r148605;
        double r148607 = r148588 * r148590;
        double r148608 = r148583 / r148607;
        double r148609 = r148606 * r148608;
        double r148610 = cbrt(r148609);
        double r148611 = r148600 + r148610;
        double r148612 = -r148601;
        double r148613 = r148601 * r148612;
        double r148614 = r148580 * r148580;
        double r148615 = r148601 * r148601;
        double r148616 = r148614 - r148615;
        double r148617 = sqrt(r148616);
        double r148618 = r148580 + r148617;
        double r148619 = r148613 / r148618;
        double r148620 = r148583 * r148619;
        double r148621 = cbrt(r148620);
        double r148622 = r148621 / r148593;
        double r148623 = r148599 * r148622;
        double r148624 = r148583 / r148588;
        double r148625 = r148624 / r148590;
        double r148626 = cbrt(r148625);
        double r148627 = r148580 + r148605;
        double r148628 = -r148627;
        double r148629 = cbrt(r148628);
        double r148630 = r148626 * r148629;
        double r148631 = r148623 + r148630;
        double r148632 = r148582 ? r148611 : r148631;
        return r148632;
}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Derivation

1. Split input into 2 regimes

2. if g < 9.191731187329127e-156

   1. Initial program 36.3

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

   2. Simplified 36.3

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

   4. Applied add-cube-cbrt 36.4

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

   5. Applied *-un-lft-identity 36.4

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

   6. Applied times-frac 36.4

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

   7. Applied add-sqr-sqrt 36.4

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

   8. Applied times-frac 36.4

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

   9. Applied *-un-lft-identity 36.4

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

   10. Applied times-frac 36.4

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

   11. Applied cbrt-prod 32.6

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

   12. Simplified 32.6

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

   13. Simplified 32.6

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

   15. Applied frac-times 32.6

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

   16. Applied cbrt-div 32.6

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

   17. Simplified 32.6

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

   18. Taylor expanded around -inf 31.4

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

   19. Simplified 31.4

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

   if 9.191731187329127e-156 < g

   1. Initial program 34.3

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

   2. Simplified 34.3

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

   4. Applied add-cube-cbrt 34.3

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

   5. Applied *-un-lft-identity 34.3

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

   6. Applied times-frac 34.3

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

   7. Applied add-sqr-sqrt 34.3

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

   8. Applied times-frac 34.3

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

   9. Applied *-un-lft-identity 34.3

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

   10. Applied times-frac 34.3

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

   11. Applied cbrt-prod 34.3

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

   12. Simplified 34.3

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

   13. Simplified 34.3

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

   15. Applied frac-times 34.3

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

   16. Applied cbrt-div 34.3

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

   17. Simplified 34.3

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

   19. Applied cbrt-prod 30.4

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

   20. Simplified 30.4

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

   21. Simplified 30.4

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

   23. Applied flip-- 30.3

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

   24. Simplified 29.8

       \[\leadsto \sqrt[3]{\frac{1}{\sqrt{2} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)}} \cdot \frac{\sqrt[3]{\frac{\color{blue}{\left(-h \cdot h\right) + 0}}{\sqrt{\left(g - h\right) \cdot \left(h + g\right)} + g} \cdot 1}}{\sqrt[3]{\sqrt{2} \cdot \sqrt[3]{a}}} + \sqrt[3]{\frac{\frac{1}{a}}{2}} \cdot \sqrt[3]{-\left(g + \sqrt{\left(g - h\right) \cdot \left(h + g\right)}\right)}\]

   25. Simplified 29.8

       \[\leadsto \sqrt[3]{\frac{1}{\sqrt{2} \cdot \left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right)}} \cdot \frac{\sqrt[3]{\frac{\left(-h \cdot h\right) + 0}{\color{blue}{\sqrt{g \cdot g - h \cdot h} + g}} \cdot 1}}{\sqrt[3]{\sqrt{2} \cdot \sqrt[3]{a}}} + \sqrt[3]{\frac{\frac{1}{a}}{2}} \cdot \sqrt[3]{-\left(g + \sqrt{\left(g - h\right) \cdot \left(h + g\right)}\right)}\]
3. Recombined 2 regimes into one program.

4. Final simplification 30.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 9.191731187329127127438182160139194041057 \cdot 10^{-156}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{\sqrt[3]{a} \cdot \sqrt{2}}} \cdot \sqrt[3]{\frac{1}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt{2}}} + \sqrt[3]{\left(\left(-g\right) - \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right) \cdot \frac{1}{a \cdot 2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{\left(\sqrt[3]{a} \cdot \sqrt[3]{a}\right) \cdot \sqrt{2}}} \cdot \frac{\sqrt[3]{1 \cdot \frac{h \cdot \left(-h\right)}{g + \sqrt{g \cdot g - h \cdot h}}}}{\sqrt[3]{\sqrt[3]{a} \cdot \sqrt{2}}} + \sqrt[3]{\frac{\frac{1}{a}}{2}} \cdot \sqrt[3]{-\left(g + \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019179 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  (+ (cbrt (* (/ 1.0 (* 2.0 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1.0 (* 2.0 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))