Average Error: 36.0 → 32.1
Time: 26.1s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\frac{\sqrt[3]{1 \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)}}{\sqrt[3]{2 \cdot a}} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}\]
\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)}
↓
\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}}
double f(double g, double h, double a) {
        double r107853 = 1.0;
        double r107854 = 2.0;
        double r107855 = a;
        double r107856 = r107854 * r107855;
        double r107857 = r107853 / r107856;
        double r107858 = g;
        double r107859 = -r107858;
        double r107860 = r107858 * r107858;
        double r107861 = h;
        double r107862 = r107861 * r107861;
        double r107863 = r107860 - r107862;
        double r107864 = sqrt(r107863);
        double r107865 = r107859 + r107864;
        double r107866 = r107857 * r107865;
        double r107867 = cbrt(r107866);
        double r107868 = r107859 - r107864;
        double r107869 = r107857 * r107868;
        double r107870 = cbrt(r107869);
        double r107871 = r107867 + r107870;
        return r107871;
}
↓
double f(double g, double h, double a) {
        double r107872 = 1.0;
        double r107873 = g;
        double r107874 = r107873 * r107873;
        double r107875 = h;
        double r107876 = r107875 * r107875;
        double r107877 = r107874 - r107876;
        double r107878 = sqrt(r107877);
        double r107879 = r107878 - r107873;
        double r107880 = r107872 * r107879;
        double r107881 = cbrt(r107880);
        double r107882 = 2.0;
        double r107883 = a;
        double r107884 = r107882 * r107883;
        double r107885 = cbrt(r107884);
        double r107886 = r107881 / r107885;
        double r107887 = -r107873;
        double r107888 = r107887 - r107878;
        double r107889 = r107872 * r107888;
        double r107890 = cbrt(r107889);
        double r107891 = r107890 / r107885;
        double r107892 = r107886 + r107891;
        return r107892;
}

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. Initial program 36.0

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

  2. Simplified36.0

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

  4. Applied associate-*l/36.0

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

  5. Applied cbrt-div34.1

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

  7. Applied associate-*l/34.1

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

  8. Applied cbrt-div32.1

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

  9. Final simplification32.1

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

Reproduce

herbie shell --seed 2019212 +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))))))))