Average Error: 35.2 → 31.2
Time: 23.8s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 1.33522342398327 \cdot 10^{-206}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{\frac{-1}{2}}{a}} + \sqrt[3]{\frac{\left(-g\right) - g}{2}} \cdot \sqrt[3]{\frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{g - g}{2}} + \sqrt[3]{g + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{a}}\\ \end{array}\]
\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 1.33522342398327 \cdot 10^{-206}:\\
\;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{\frac{-1}{2}}{a}} + \sqrt[3]{\frac{\left(-g\right) - g}{2}} \cdot \sqrt[3]{\frac{1}{a}}\\

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

\end{array}
double f(double g, double h, double a) {
        double r2408814 = 1.0;
        double r2408815 = 2.0;
        double r2408816 = a;
        double r2408817 = r2408815 * r2408816;
        double r2408818 = r2408814 / r2408817;
        double r2408819 = g;
        double r2408820 = -r2408819;
        double r2408821 = r2408819 * r2408819;
        double r2408822 = h;
        double r2408823 = r2408822 * r2408822;
        double r2408824 = r2408821 - r2408823;
        double r2408825 = sqrt(r2408824);
        double r2408826 = r2408820 + r2408825;
        double r2408827 = r2408818 * r2408826;
        double r2408828 = cbrt(r2408827);
        double r2408829 = r2408820 - r2408825;
        double r2408830 = r2408818 * r2408829;
        double r2408831 = cbrt(r2408830);
        double r2408832 = r2408828 + r2408831;
        return r2408832;
}
↓
double f(double g, double h, double a) {
        double r2408833 = g;
        double r2408834 = 1.33522342398327e-206;
        bool r2408835 = r2408833 <= r2408834;
        double r2408836 = r2408833 * r2408833;
        double r2408837 = h;
        double r2408838 = r2408837 * r2408837;
        double r2408839 = r2408836 - r2408838;
        double r2408840 = sqrt(r2408839);
        double r2408841 = r2408833 + r2408840;
        double r2408842 = -0.5;
        double r2408843 = a;
        double r2408844 = r2408842 / r2408843;
        double r2408845 = r2408841 * r2408844;
        double r2408846 = cbrt(r2408845);
        double r2408847 = -r2408833;
        double r2408848 = r2408847 - r2408833;
        double r2408849 = 2.0;
        double r2408850 = r2408848 / r2408849;
        double r2408851 = cbrt(r2408850);
        double r2408852 = 1.0;
        double r2408853 = r2408852 / r2408843;
        double r2408854 = cbrt(r2408853);
        double r2408855 = r2408851 * r2408854;
        double r2408856 = r2408846 + r2408855;
        double r2408857 = r2408833 - r2408833;
        double r2408858 = r2408857 / r2408849;
        double r2408859 = cbrt(r2408858);
        double r2408860 = r2408854 * r2408859;
        double r2408861 = cbrt(r2408841);
        double r2408862 = cbrt(r2408844);
        double r2408863 = r2408861 * r2408862;
        double r2408864 = r2408860 + r2408863;
        double r2408865 = r2408835 ? r2408856 : r2408864;
        return r2408865;
}

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 < 1.33522342398327e-206

    1. Initial program 35.7

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

    2. Simplified35.7

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

    4. Applied *-un-lft-identity35.7

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

    5. Applied *-un-lft-identity35.7

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

    6. Applied distribute-lft-out--35.7

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

    7. Applied times-frac35.7

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

    8. Applied cbrt-prod32.1

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

    9. Taylor expanded around -inf 31.5

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

    10. Simplified31.5

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

    if 1.33522342398327e-206 < g

    1. Initial program 34.7

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

    2. Simplified34.6

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

    4. Applied *-un-lft-identity34.6

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

    5. Applied *-un-lft-identity34.6

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

    6. Applied distribute-lft-out--34.6

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

    7. Applied times-frac34.6

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

    8. Applied cbrt-prod34.5

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

    10. Applied cbrt-prod31.1

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

    11. Taylor expanded around inf 31.0

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

  4. Final simplification31.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 1.33522342398327 \cdot 10^{-206}:\\ \;\;\;\;\sqrt[3]{\left(g + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{\frac{-1}{2}}{a}} + \sqrt[3]{\frac{\left(-g\right) - g}{2}} \cdot \sqrt[3]{\frac{1}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\frac{g - g}{2}} + \sqrt[3]{g + \sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019128 +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))))))))