Average Error: 36.0 → 31.6
Time: 12.5s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -1.2934885954313043 \cdot 10^{-169}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \left(-g\right)\right)}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]
\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 -1.2934885954313043 \cdot 10^{-169}:\\
\;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \left(-g\right)\right)}}{\sqrt[3]{2 \cdot a}}\\

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

\end{array}
double f(double g, double h, double a) {
        double r245896 = 1.0;
        double r245897 = 2.0;
        double r245898 = a;
        double r245899 = r245897 * r245898;
        double r245900 = r245896 / r245899;
        double r245901 = g;
        double r245902 = -r245901;
        double r245903 = r245901 * r245901;
        double r245904 = h;
        double r245905 = r245904 * r245904;
        double r245906 = r245903 - r245905;
        double r245907 = sqrt(r245906);
        double r245908 = r245902 + r245907;
        double r245909 = r245900 * r245908;
        double r245910 = cbrt(r245909);
        double r245911 = r245902 - r245907;
        double r245912 = r245900 * r245911;
        double r245913 = cbrt(r245912);
        double r245914 = r245910 + r245913;
        return r245914;
}
↓
double f(double g, double h, double a) {
        double r245915 = g;
        double r245916 = -1.2934885954313043e-169;
        bool r245917 = r245915 <= r245916;
        double r245918 = 1.0;
        double r245919 = r245915 * r245915;
        double r245920 = h;
        double r245921 = r245920 * r245920;
        double r245922 = r245919 - r245921;
        double r245923 = sqrt(r245922);
        double r245924 = r245923 - r245915;
        double r245925 = r245918 * r245924;
        double r245926 = cbrt(r245925);
        double r245927 = 2.0;
        double r245928 = a;
        double r245929 = r245927 * r245928;
        double r245930 = cbrt(r245929);
        double r245931 = r245926 / r245930;
        double r245932 = -r245915;
        double r245933 = r245932 - r245932;
        double r245934 = r245918 * r245933;
        double r245935 = cbrt(r245934);
        double r245936 = r245935 / r245930;
        double r245937 = r245931 + r245936;
        double r245938 = r245918 / r245929;
        double r245939 = r245938 * r245924;
        double r245940 = cbrt(r245939);
        double r245941 = r245932 - r245915;
        double r245942 = r245918 * r245941;
        double r245943 = cbrt(r245942);
        double r245944 = r245943 / r245930;
        double r245945 = r245940 + r245944;
        double r245946 = r245917 ? r245937 : r245945;
        return r245946;
}

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 < -1.2934885954313043e-169

    1. Initial program 34.8

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

    2. Simplified34.8

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

    4. Applied associate-*l/34.8

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

    5. Applied cbrt-div34.7

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

    7. Applied associate-*l/34.7

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

    8. Applied cbrt-div30.9

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

    9. Taylor expanded around -inf 31.2

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

    10. Simplified31.2

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

    if -1.2934885954313043e-169 < g

    1. Initial program 37.2

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

    2. Simplified37.2

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

    4. Applied associate-*l/37.2

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

    5. Applied cbrt-div33.2

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

    6. Taylor expanded around inf 31.9

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

  4. Final simplification31.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -1.2934885954313043 \cdot 10^{-169}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \left(-g\right)\right)}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(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))))))))