Average Error: 35.3 → 31.4
Time: 28.2s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\frac{\sqrt[3]{\frac{-1}{2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)}}{\sqrt[3]{a}} + \left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}\right) \cdot \sqrt[3]{\frac{1}{2}}\]
\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)}
↓
\frac{\sqrt[3]{\frac{-1}{2} \cdot \left(\sqrt{g \cdot g - h \cdot h} + g\right)}}{\sqrt[3]{a}} + \left(\sqrt[3]{\frac{1}{a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}\right) \cdot \sqrt[3]{\frac{1}{2}}
double f(double g, double h, double a) {
        double r3575810 = 1.0;
        double r3575811 = 2.0;
        double r3575812 = a;
        double r3575813 = r3575811 * r3575812;
        double r3575814 = r3575810 / r3575813;
        double r3575815 = g;
        double r3575816 = -r3575815;
        double r3575817 = r3575815 * r3575815;
        double r3575818 = h;
        double r3575819 = r3575818 * r3575818;
        double r3575820 = r3575817 - r3575819;
        double r3575821 = sqrt(r3575820);
        double r3575822 = r3575816 + r3575821;
        double r3575823 = r3575814 * r3575822;
        double r3575824 = cbrt(r3575823);
        double r3575825 = r3575816 - r3575821;
        double r3575826 = r3575814 * r3575825;
        double r3575827 = cbrt(r3575826);
        double r3575828 = r3575824 + r3575827;
        return r3575828;
}
↓
double f(double g, double h, double a) {
        double r3575829 = -0.5;
        double r3575830 = g;
        double r3575831 = r3575830 * r3575830;
        double r3575832 = h;
        double r3575833 = r3575832 * r3575832;
        double r3575834 = r3575831 - r3575833;
        double r3575835 = sqrt(r3575834);
        double r3575836 = r3575835 + r3575830;
        double r3575837 = r3575829 * r3575836;
        double r3575838 = cbrt(r3575837);
        double r3575839 = a;
        double r3575840 = cbrt(r3575839);
        double r3575841 = r3575838 / r3575840;
        double r3575842 = 1.0;
        double r3575843 = r3575842 / r3575839;
        double r3575844 = cbrt(r3575843);
        double r3575845 = r3575835 - r3575830;
        double r3575846 = cbrt(r3575845);
        double r3575847 = r3575844 * r3575846;
        double r3575848 = 0.5;
        double r3575849 = cbrt(r3575848);
        double r3575850 = r3575847 * r3575849;
        double r3575851 = r3575841 + r3575850;
        return r3575851;
}

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 35.3

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

  2. Simplified35.2

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

  4. Applied associate-*l/35.2

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

  5. Applied cbrt-div33.2

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

  7. Applied cbrt-prod33.2

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

  9. Applied div-inv33.2

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

  10. Applied cbrt-prod31.4

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

  11. Final simplification31.4

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

Reproduce

herbie shell --seed 2019152 +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))))))))