Average Error: 36.1 → 31.9
Time: 35.3s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -8.359356862874829159472492534077743049279 \cdot 10^{-185}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}} - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]
\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 -8.359356862874829159472492534077743049279 \cdot 10^{-185}:\\
\;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}} - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\

\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 r81787 = 1.0;
        double r81788 = 2.0;
        double r81789 = a;
        double r81790 = r81788 * r81789;
        double r81791 = r81787 / r81790;
        double r81792 = g;
        double r81793 = -r81792;
        double r81794 = r81792 * r81792;
        double r81795 = h;
        double r81796 = r81795 * r81795;
        double r81797 = r81794 - r81796;
        double r81798 = sqrt(r81797);
        double r81799 = r81793 + r81798;
        double r81800 = r81791 * r81799;
        double r81801 = cbrt(r81800);
        double r81802 = r81793 - r81798;
        double r81803 = r81791 * r81802;
        double r81804 = cbrt(r81803);
        double r81805 = r81801 + r81804;
        return r81805;
}
↓
double f(double g, double h, double a) {
        double r81806 = g;
        double r81807 = -8.359356862874829e-185;
        bool r81808 = r81806 <= r81807;
        double r81809 = 1.0;
        double r81810 = r81806 * r81806;
        double r81811 = h;
        double r81812 = r81811 * r81811;
        double r81813 = r81810 - r81812;
        double r81814 = sqrt(r81813);
        double r81815 = sqrt(r81814);
        double r81816 = r81815 * r81815;
        double r81817 = r81816 - r81806;
        double r81818 = r81809 * r81817;
        double r81819 = cbrt(r81818);
        double r81820 = 2.0;
        double r81821 = a;
        double r81822 = r81820 * r81821;
        double r81823 = cbrt(r81822);
        double r81824 = r81819 / r81823;
        double r81825 = r81809 / r81822;
        double r81826 = cbrt(r81825);
        double r81827 = -r81806;
        double r81828 = r81827 - r81814;
        double r81829 = cbrt(r81828);
        double r81830 = r81826 * r81829;
        double r81831 = r81824 + r81830;
        double r81832 = r81827 + r81814;
        double r81833 = r81825 * r81832;
        double r81834 = cbrt(r81833);
        double r81835 = r81827 - r81806;
        double r81836 = cbrt(r81835);
        double r81837 = r81826 * r81836;
        double r81838 = r81834 + r81837;
        double r81839 = r81808 ? r81831 : r81838;
        return r81839;
}

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 < -8.359356862874829e-185

    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 cbrt-prod35.2

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

    5. Applied associate-*l/35.2

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

    6. Applied cbrt-div31.3

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

    7. Simplified31.3

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

    9. Applied add-sqr-sqrt31.3

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

    10. Applied sqrt-prod31.3

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

    if -8.359356862874829e-185 < g

    1. Initial program 36.8

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

    3. Applied cbrt-prod33.4

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

    4. Taylor expanded around inf 32.5

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

  4. Final simplification31.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -8.359356862874829159472492534077743049279 \cdot 10^{-185}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt{\sqrt{g \cdot g - h \cdot h}} - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]

Reproduce

herbie shell --seed 2019326 +o rules:numerics
(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))))))))