Average Error: 35.8 → 32.2
Time: 16.5s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}\]
\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)}
↓
\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \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 r172714 = 1.0;
        double r172715 = 2.0;
        double r172716 = a;
        double r172717 = r172715 * r172716;
        double r172718 = r172714 / r172717;
        double r172719 = g;
        double r172720 = -r172719;
        double r172721 = r172719 * r172719;
        double r172722 = h;
        double r172723 = r172722 * r172722;
        double r172724 = r172721 - r172723;
        double r172725 = sqrt(r172724);
        double r172726 = r172720 + r172725;
        double r172727 = r172718 * r172726;
        double r172728 = cbrt(r172727);
        double r172729 = r172720 - r172725;
        double r172730 = r172718 * r172729;
        double r172731 = cbrt(r172730);
        double r172732 = r172728 + r172731;
        return r172732;
}
↓
double f(double g, double h, double a) {
        double r172733 = 1.0;
        double r172734 = 2.0;
        double r172735 = a;
        double r172736 = r172734 * r172735;
        double r172737 = r172733 / r172736;
        double r172738 = cbrt(r172737);
        double r172739 = g;
        double r172740 = r172739 * r172739;
        double r172741 = h;
        double r172742 = r172741 * r172741;
        double r172743 = r172740 - r172742;
        double r172744 = sqrt(r172743);
        double r172745 = r172744 - r172739;
        double r172746 = cbrt(r172745);
        double r172747 = r172738 * r172746;
        double r172748 = -r172739;
        double r172749 = r172748 - r172744;
        double r172750 = r172733 * r172749;
        double r172751 = cbrt(r172750);
        double r172752 = cbrt(r172736);
        double r172753 = r172751 / r172752;
        double r172754 = r172747 + r172753;
        return r172754;
}

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 35.8

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

  2. Simplified35.8

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

  4. Applied associate-*l/35.8

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

  5. Applied cbrt-div34.0

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

  7. Applied cbrt-prod32.2

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

  8. Final simplification32.2

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

Reproduce

herbie shell --seed 2020042 +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))))))))