Average Error: 35.2 → 31.2
Time: 31.3s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 4.732713041008813 \cdot 10^{-185}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{-\left(g + g\right)} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right) \cdot \frac{1}{2 \cdot a}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]
\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 4.732713041008813 \cdot 10^{-185}:\\
\;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{-\left(g + g\right)} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}\\

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

\end{array}
double f(double g, double h, double a) {
        double r4999616 = 1.0;
        double r4999617 = 2.0;
        double r4999618 = a;
        double r4999619 = r4999617 * r4999618;
        double r4999620 = r4999616 / r4999619;
        double r4999621 = g;
        double r4999622 = -r4999621;
        double r4999623 = r4999621 * r4999621;
        double r4999624 = h;
        double r4999625 = r4999624 * r4999624;
        double r4999626 = r4999623 - r4999625;
        double r4999627 = sqrt(r4999626);
        double r4999628 = r4999622 + r4999627;
        double r4999629 = r4999620 * r4999628;
        double r4999630 = cbrt(r4999629);
        double r4999631 = r4999622 - r4999627;
        double r4999632 = r4999620 * r4999631;
        double r4999633 = cbrt(r4999632);
        double r4999634 = r4999630 + r4999633;
        return r4999634;
}
↓
double f(double g, double h, double a) {
        double r4999635 = g;
        double r4999636 = 4.732713041008813e-185;
        bool r4999637 = r4999635 <= r4999636;
        double r4999638 = 1.0;
        double r4999639 = 2.0;
        double r4999640 = a;
        double r4999641 = r4999639 * r4999640;
        double r4999642 = r4999638 / r4999641;
        double r4999643 = -r4999635;
        double r4999644 = r4999635 * r4999635;
        double r4999645 = h;
        double r4999646 = r4999645 * r4999645;
        double r4999647 = r4999644 - r4999646;
        double r4999648 = sqrt(r4999647);
        double r4999649 = r4999643 - r4999648;
        double r4999650 = r4999642 * r4999649;
        double r4999651 = cbrt(r4999650);
        double r4999652 = r4999635 + r4999635;
        double r4999653 = -r4999652;
        double r4999654 = cbrt(r4999653);
        double r4999655 = cbrt(r4999642);
        double r4999656 = r4999654 * r4999655;
        double r4999657 = r4999651 + r4999656;
        double r4999658 = r4999648 + r4999643;
        double r4999659 = r4999658 * r4999642;
        double r4999660 = cbrt(r4999659);
        double r4999661 = cbrt(r4999648);
        double r4999662 = r4999661 * r4999661;
        double r4999663 = r4999661 * r4999662;
        double r4999664 = r4999643 - r4999663;
        double r4999665 = cbrt(r4999664);
        double r4999666 = cbrt(r4999641);
        double r4999667 = r4999665 / r4999666;
        double r4999668 = r4999660 + r4999667;
        double r4999669 = r4999637 ? r4999657 : r4999668;
        return r4999669;
}

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 < 4.732713041008813e-185

    1. Initial program 35.9

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

    3. Applied cbrt-prod32.2

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

    4. Taylor expanded around -inf 31.7

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

    5. Simplified31.7

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

    if 4.732713041008813e-185 < g

    1. Initial program 34.6

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

    3. Applied associate-*l/34.6

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

    4. Applied cbrt-div30.7

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

    5. Simplified30.7

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

    7. Applied add-cube-cbrt30.7

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

  4. Final simplification31.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 4.732713041008813 \cdot 10^{-185}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{-\left(g + g\right)} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} + \left(-g\right)\right) \cdot \frac{1}{2 \cdot a}} + \frac{\sqrt[3]{\left(-g\right) - \sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019129 +o rules:numerics
(FPCore (g h a)
  :name "2-ancestry mixing, positive discriminant"
  (+ (cbrt (* (/ 1 (* 2 a)) (+ (- g) (sqrt (- (* g g) (* h h)))))) (cbrt (* (/ 1 (* 2 a)) (- (- g) (sqrt (- (* g g) (* h h))))))))