Average Error: 35.3 → 31.6
Time: 28.9s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\frac{\sqrt[3]{\frac{-1}{2} \cdot \left(g + \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right)}}{\sqrt[3]{a}} + \left(\sqrt[3]{\frac{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}{\sqrt[3]{a}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}\right) \cdot \sqrt[3]{\frac{1}{2}}\]
\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(g + \sqrt{\left(h + g\right) \cdot \left(g - h\right)}\right)}}{\sqrt[3]{a}} + \left(\sqrt[3]{\frac{\sqrt{\left(h + g\right) \cdot \left(g - h\right)} - g}{\sqrt[3]{a}}} \cdot \sqrt[3]{\frac{1}{\sqrt[3]{a} \cdot \sqrt[3]{a}}}\right) \cdot \sqrt[3]{\frac{1}{2}}
double f(double g, double h, double a) {
        double r5302532 = 1.0;
        double r5302533 = 2.0;
        double r5302534 = a;
        double r5302535 = r5302533 * r5302534;
        double r5302536 = r5302532 / r5302535;
        double r5302537 = g;
        double r5302538 = -r5302537;
        double r5302539 = r5302537 * r5302537;
        double r5302540 = h;
        double r5302541 = r5302540 * r5302540;
        double r5302542 = r5302539 - r5302541;
        double r5302543 = sqrt(r5302542);
        double r5302544 = r5302538 + r5302543;
        double r5302545 = r5302536 * r5302544;
        double r5302546 = cbrt(r5302545);
        double r5302547 = r5302538 - r5302543;
        double r5302548 = r5302536 * r5302547;
        double r5302549 = cbrt(r5302548);
        double r5302550 = r5302546 + r5302549;
        return r5302550;
}
↓
double f(double g, double h, double a) {
        double r5302551 = -0.5;
        double r5302552 = g;
        double r5302553 = h;
        double r5302554 = r5302553 + r5302552;
        double r5302555 = r5302552 - r5302553;
        double r5302556 = r5302554 * r5302555;
        double r5302557 = sqrt(r5302556);
        double r5302558 = r5302552 + r5302557;
        double r5302559 = r5302551 * r5302558;
        double r5302560 = cbrt(r5302559);
        double r5302561 = a;
        double r5302562 = cbrt(r5302561);
        double r5302563 = r5302560 / r5302562;
        double r5302564 = r5302557 - r5302552;
        double r5302565 = r5302564 / r5302562;
        double r5302566 = cbrt(r5302565);
        double r5302567 = 1.0;
        double r5302568 = r5302562 * r5302562;
        double r5302569 = r5302567 / r5302568;
        double r5302570 = cbrt(r5302569);
        double r5302571 = r5302566 * r5302570;
        double r5302572 = 0.5;
        double r5302573 = cbrt(r5302572);
        double r5302574 = r5302571 * r5302573;
        double r5302575 = r5302563 + r5302574;
        return r5302575;
}

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. Simplified35.3

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

  4. Applied associate-*l/35.3

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

  5. Applied cbrt-div33.4

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

  7. Applied cbrt-prod33.4

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

  9. Applied add-cube-cbrt33.5

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

  10. Applied *-un-lft-identity33.5

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

  11. Applied times-frac33.5

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

  12. Applied cbrt-prod31.6

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

  13. Final simplification31.6

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

Reproduce

herbie shell --seed 2019163 +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))))))))