Average Error: 35.6 → 31.7
Time: 7.6s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 1.7494555242604437 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + -1 \cdot g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \end{array}\]
\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 1.7494555242604437 \cdot 10^{-220}:\\
\;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + -1 \cdot 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)}\\

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

\end{array}
double f(double g, double h, double a) {
        double r124813 = 1.0;
        double r124814 = 2.0;
        double r124815 = a;
        double r124816 = r124814 * r124815;
        double r124817 = r124813 / r124816;
        double r124818 = g;
        double r124819 = -r124818;
        double r124820 = r124818 * r124818;
        double r124821 = h;
        double r124822 = r124821 * r124821;
        double r124823 = r124820 - r124822;
        double r124824 = sqrt(r124823);
        double r124825 = r124819 + r124824;
        double r124826 = r124817 * r124825;
        double r124827 = cbrt(r124826);
        double r124828 = r124819 - r124824;
        double r124829 = r124817 * r124828;
        double r124830 = cbrt(r124829);
        double r124831 = r124827 + r124830;
        return r124831;
}
↓
double f(double g, double h, double a) {
        double r124832 = g;
        double r124833 = 1.7494555242604437e-220;
        bool r124834 = r124832 <= r124833;
        double r124835 = 1.0;
        double r124836 = -r124832;
        double r124837 = -1.0;
        double r124838 = r124837 * r124832;
        double r124839 = r124836 + r124838;
        double r124840 = r124835 * r124839;
        double r124841 = cbrt(r124840);
        double r124842 = 2.0;
        double r124843 = a;
        double r124844 = r124842 * r124843;
        double r124845 = cbrt(r124844);
        double r124846 = r124841 / r124845;
        double r124847 = r124835 / r124844;
        double r124848 = r124832 * r124832;
        double r124849 = h;
        double r124850 = r124849 * r124849;
        double r124851 = r124848 - r124850;
        double r124852 = sqrt(r124851);
        double r124853 = r124836 - r124852;
        double r124854 = r124847 * r124853;
        double r124855 = cbrt(r124854);
        double r124856 = r124846 + r124855;
        double r124857 = r124836 + r124832;
        double r124858 = r124835 * r124857;
        double r124859 = cbrt(r124858);
        double r124860 = r124859 / r124845;
        double r124861 = cbrt(r124847);
        double r124862 = cbrt(r124853);
        double r124863 = r124861 * r124862;
        double r124864 = r124860 + r124863;
        double r124865 = r124834 ? r124856 : r124864;
        return r124865;
}

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 < 1.7494555242604437e-220

    1. Initial program 36.5

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

    3. Applied associate-*l/36.5

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

    4. Applied cbrt-div33.1

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

    5. Taylor expanded around -inf 32.3

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

    if 1.7494555242604437e-220 < 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

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

    4. Applied cbrt-div34.5

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

    6. Applied cbrt-prod30.9

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

    7. Taylor expanded around inf 31.1

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

  4. Final simplification31.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 1.7494555242604437 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + -1 \cdot g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) + g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 +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))))))))