Average Error: 36.2 → 31.6
Time: 36.6s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -3.421984391530689600469067551996753796002 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{h \cdot h}{\frac{2 \cdot a}{1}}}}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}} + \frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]
\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 -3.421984391530689600469067551996753796002 \cdot 10^{-165}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{h \cdot h}{\frac{2 \cdot a}{1}}}}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}} + \frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}}\\

\mathbf{else}:\\
\;\;\;\;\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 \sqrt[3]{\left(-g\right) - g}\\

\end{array}
double f(double g, double h, double a) {
        double r85597 = 1.0;
        double r85598 = 2.0;
        double r85599 = a;
        double r85600 = r85598 * r85599;
        double r85601 = r85597 / r85600;
        double r85602 = g;
        double r85603 = -r85602;
        double r85604 = r85602 * r85602;
        double r85605 = h;
        double r85606 = r85605 * r85605;
        double r85607 = r85604 - r85606;
        double r85608 = sqrt(r85607);
        double r85609 = r85603 + r85608;
        double r85610 = r85601 * r85609;
        double r85611 = cbrt(r85610);
        double r85612 = r85603 - r85608;
        double r85613 = r85601 * r85612;
        double r85614 = cbrt(r85613);
        double r85615 = r85611 + r85614;
        return r85615;
}
↓
double f(double g, double h, double a) {
        double r85616 = g;
        double r85617 = -3.4219843915306896e-165;
        bool r85618 = r85616 <= r85617;
        double r85619 = h;
        double r85620 = r85619 * r85619;
        double r85621 = 2.0;
        double r85622 = a;
        double r85623 = r85621 * r85622;
        double r85624 = 1.0;
        double r85625 = r85623 / r85624;
        double r85626 = r85620 / r85625;
        double r85627 = cbrt(r85626);
        double r85628 = r85616 * r85616;
        double r85629 = r85628 - r85620;
        double r85630 = sqrt(r85629);
        double r85631 = r85630 - r85616;
        double r85632 = cbrt(r85631);
        double r85633 = r85627 / r85632;
        double r85634 = r85624 * r85631;
        double r85635 = cbrt(r85634);
        double r85636 = cbrt(r85623);
        double r85637 = r85635 / r85636;
        double r85638 = r85633 + r85637;
        double r85639 = r85624 / r85623;
        double r85640 = -r85616;
        double r85641 = r85640 + r85630;
        double r85642 = r85639 * r85641;
        double r85643 = cbrt(r85642);
        double r85644 = cbrt(r85639);
        double r85645 = r85640 - r85616;
        double r85646 = cbrt(r85645);
        double r85647 = r85644 * r85646;
        double r85648 = r85643 + r85647;
        double r85649 = r85618 ? r85638 : r85648;
        return r85649;
}

Error

Bits error versus g

Bits error versus h

Bits error versus a

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if g < -3.4219843915306896e-165

    1. Initial program 35.3

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

    3. Applied associate-*l/35.3

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

    4. Applied cbrt-div31.4

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

    5. Simplified31.4

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

    7. Applied flip--31.3

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

    8. Applied associate-*r/31.4

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

    9. Applied cbrt-div31.4

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

    10. Simplified31.5

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

    11. Simplified31.5

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

    if -3.4219843915306896e-165 < g

    1. Initial program 36.9

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

    3. Applied cbrt-prod33.0

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

    4. Taylor expanded around inf 31.7

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

  4. Final simplification31.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -3.421984391530689600469067551996753796002 \cdot 10^{-165}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{h \cdot h}{\frac{2 \cdot a}{1}}}}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}} + \frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  :precision binary64
  (+ (cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))