Average Error: 35.9 → 33.7
Time: 35.6s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\sqrt[3]{\frac{\sqrt{1}}{\sqrt[3]{\frac{2 \cdot a}{\sqrt{g \cdot g - h \cdot h} - g}} \cdot \sqrt[3]{\frac{2 \cdot a}{\sqrt{g \cdot g - h \cdot h} - g}}}} \cdot \sqrt[3]{\frac{\sqrt{1}}{\sqrt[3]{\frac{2 \cdot a}{\sqrt{g \cdot g - h \cdot h} - g}}}} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}\]
\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{\sqrt{1}}{\sqrt[3]{\frac{2 \cdot a}{\sqrt{g \cdot g - h \cdot h} - g}} \cdot \sqrt[3]{\frac{2 \cdot a}{\sqrt{g \cdot g - h \cdot h} - g}}}} \cdot \sqrt[3]{\frac{\sqrt{1}}{\sqrt[3]{\frac{2 \cdot a}{\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 r109552 = 1.0;
        double r109553 = 2.0;
        double r109554 = a;
        double r109555 = r109553 * r109554;
        double r109556 = r109552 / r109555;
        double r109557 = g;
        double r109558 = -r109557;
        double r109559 = r109557 * r109557;
        double r109560 = h;
        double r109561 = r109560 * r109560;
        double r109562 = r109559 - r109561;
        double r109563 = sqrt(r109562);
        double r109564 = r109558 + r109563;
        double r109565 = r109556 * r109564;
        double r109566 = cbrt(r109565);
        double r109567 = r109558 - r109563;
        double r109568 = r109556 * r109567;
        double r109569 = cbrt(r109568);
        double r109570 = r109566 + r109569;
        return r109570;
}
↓
double f(double g, double h, double a) {
        double r109571 = 1.0;
        double r109572 = sqrt(r109571);
        double r109573 = 2.0;
        double r109574 = a;
        double r109575 = r109573 * r109574;
        double r109576 = g;
        double r109577 = r109576 * r109576;
        double r109578 = h;
        double r109579 = r109578 * r109578;
        double r109580 = r109577 - r109579;
        double r109581 = sqrt(r109580);
        double r109582 = r109581 - r109576;
        double r109583 = r109575 / r109582;
        double r109584 = cbrt(r109583);
        double r109585 = r109584 * r109584;
        double r109586 = r109572 / r109585;
        double r109587 = cbrt(r109586);
        double r109588 = r109572 / r109584;
        double r109589 = cbrt(r109588);
        double r109590 = r109587 * r109589;
        double r109591 = -r109576;
        double r109592 = r109591 - r109581;
        double r109593 = r109571 * r109592;
        double r109594 = cbrt(r109593);
        double r109595 = cbrt(r109575);
        double r109596 = r109594 / r109595;
        double r109597 = r109590 + r109596;
        return r109597;
}

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 35.9

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

  3. Applied associate-*l/35.9

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

  4. Applied cbrt-div33.8

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

  6. Applied add-cbrt-cube33.8

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

  7. Simplified33.9

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

  9. Applied add-cube-cbrt33.9

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

  10. Applied add-sqr-sqrt33.9

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

  11. Applied times-frac33.9

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

  12. Applied cbrt-prod33.7

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

  13. Final simplification33.7

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

Reproduce

herbie shell --seed 2019306 +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))))))))