Average Error: 36.4 → 31.6
Time: 34.7s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 8.436549254635570642232365575110592726102 \cdot 10^{-159}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}}\\ \end{array}\]
\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 8.436549254635570642232365575110592726102 \cdot 10^{-159}:\\
\;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\

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

\end{array}
double f(double g, double h, double a) {
        double r91847 = 1.0;
        double r91848 = 2.0;
        double r91849 = a;
        double r91850 = r91848 * r91849;
        double r91851 = r91847 / r91850;
        double r91852 = g;
        double r91853 = -r91852;
        double r91854 = r91852 * r91852;
        double r91855 = h;
        double r91856 = r91855 * r91855;
        double r91857 = r91854 - r91856;
        double r91858 = sqrt(r91857);
        double r91859 = r91853 + r91858;
        double r91860 = r91851 * r91859;
        double r91861 = cbrt(r91860);
        double r91862 = r91853 - r91858;
        double r91863 = r91851 * r91862;
        double r91864 = cbrt(r91863);
        double r91865 = r91861 + r91864;
        return r91865;
}
↓
double f(double g, double h, double a) {
        double r91866 = g;
        double r91867 = 8.436549254635571e-159;
        bool r91868 = r91866 <= r91867;
        double r91869 = 1.0;
        double r91870 = 2.0;
        double r91871 = a;
        double r91872 = r91870 * r91871;
        double r91873 = r91869 / r91872;
        double r91874 = cbrt(r91873);
        double r91875 = -r91866;
        double r91876 = r91875 - r91866;
        double r91877 = cbrt(r91876);
        double r91878 = r91874 * r91877;
        double r91879 = r91866 * r91866;
        double r91880 = h;
        double r91881 = r91880 * r91880;
        double r91882 = r91879 - r91881;
        double r91883 = sqrt(r91882);
        double r91884 = r91875 - r91883;
        double r91885 = r91873 * r91884;
        double r91886 = cbrt(r91885);
        double r91887 = r91878 + r91886;
        double r91888 = r91883 - r91866;
        double r91889 = r91869 * r91888;
        double r91890 = cbrt(r91889);
        double r91891 = cbrt(r91872);
        double r91892 = r91890 / r91891;
        double r91893 = cbrt(r91883);
        double r91894 = r91893 * r91893;
        double r91895 = r91894 * r91893;
        double r91896 = r91875 - r91895;
        double r91897 = cbrt(r91896);
        double r91898 = r91874 * r91897;
        double r91899 = r91892 + r91898;
        double r91900 = r91868 ? r91887 : r91899;
        return r91900;
}

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 < 8.436549254635571e-159

    1. Initial program 37.5

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

    2. Simplified37.5

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

    4. Applied cbrt-prod33.4

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

    5. Taylor expanded around -inf 31.9

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

    6. Simplified31.9

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

    if 8.436549254635571e-159 < g

    1. Initial program 35.2

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

    2. Simplified35.2

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

    4. Applied associate-*l/35.2

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

    5. Applied cbrt-div35.2

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

    7. Applied cbrt-prod31.2

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

    9. Applied add-cube-cbrt31.2

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

  4. Final simplification31.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 8.436549254635570642232365575110592726102 \cdot 10^{-159}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right) \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019323 +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))))))))