Average Error: 35.5 → 31.2
Time: 9.2s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -2.10518339907436567 \cdot 10^{-158}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) + \left|\sqrt[3]{g \cdot g - h \cdot h}\right| \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]
\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)}
↓
\begin{array}{l}
\mathbf{if}\;g \le -2.10518339907436567 \cdot 10^{-158}:\\
\;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) + \left|\sqrt[3]{g \cdot g - h \cdot h}\right| \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\

\mathbf{else}:\\
\;\;\;\;\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]{\left(-g\right) - g}\\

\end{array}
double f(double g, double h, double a) {
        double r152531 = 1.0;
        double r152532 = 2.0;
        double r152533 = a;
        double r152534 = r152532 * r152533;
        double r152535 = r152531 / r152534;
        double r152536 = g;
        double r152537 = -r152536;
        double r152538 = r152536 * r152536;
        double r152539 = h;
        double r152540 = r152539 * r152539;
        double r152541 = r152538 - r152540;
        double r152542 = sqrt(r152541);
        double r152543 = r152537 + r152542;
        double r152544 = r152535 * r152543;
        double r152545 = cbrt(r152544);
        double r152546 = r152537 - r152542;
        double r152547 = r152535 * r152546;
        double r152548 = cbrt(r152547);
        double r152549 = r152545 + r152548;
        return r152549;
}
↓
double f(double g, double h, double a) {
        double r152550 = g;
        double r152551 = -2.1051833990743657e-158;
        bool r152552 = r152550 <= r152551;
        double r152553 = 1.0;
        double r152554 = 2.0;
        double r152555 = a;
        double r152556 = r152554 * r152555;
        double r152557 = r152553 / r152556;
        double r152558 = cbrt(r152557);
        double r152559 = -r152550;
        double r152560 = r152550 * r152550;
        double r152561 = h;
        double r152562 = r152561 * r152561;
        double r152563 = r152560 - r152562;
        double r152564 = cbrt(r152563);
        double r152565 = fabs(r152564);
        double r152566 = sqrt(r152564);
        double r152567 = r152565 * r152566;
        double r152568 = r152559 + r152567;
        double r152569 = cbrt(r152568);
        double r152570 = r152558 * r152569;
        double r152571 = sqrt(r152563);
        double r152572 = r152559 - r152571;
        double r152573 = cbrt(r152572);
        double r152574 = r152558 * r152573;
        double r152575 = r152570 + r152574;
        double r152576 = r152559 + r152571;
        double r152577 = r152557 * r152576;
        double r152578 = cbrt(r152577);
        double r152579 = r152559 - r152550;
        double r152580 = cbrt(r152579);
        double r152581 = r152558 * r152580;
        double r152582 = r152578 + r152581;
        double r152583 = r152552 ? r152575 : r152582;
        return r152583;
}

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. Split input into 2 regimes

  2. if g < -2.1051833990743657e-158

    1. Initial program 33.7

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

    3. Applied cbrt-prod33.7

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

    5. Applied cbrt-prod30.1

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

    7. Applied add-cube-cbrt30.1

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

    8. Applied sqrt-prod30.1

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

    9. Simplified30.1

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

    if -2.1051833990743657e-158 < g

    1. Initial program 37.1

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

    3. Applied cbrt-prod33.6

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

    4. Taylor expanded around inf 32.2

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \color{blue}{g}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification31.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -2.10518339907436567 \cdot 10^{-158}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) + \left|\sqrt[3]{g \cdot g - h \cdot h}\right| \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]

Reproduce

herbie shell --seed 2020060 +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))))))))