Average Error: 34.6 → 30.4
Time: 31.8s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 4.099263178664206 \cdot 10^{-157}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\left(-g\right) + \left(-g\right)} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} + \left(-g\right)}\\ \end{array}\]
\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 4.099263178664206 \cdot 10^{-157}:\\
\;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\left(-g\right) + \left(-g\right)} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}\\

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

\end{array}
double f(double g, double h, double a) {
        double r5545508 = 1.0;
        double r5545509 = 2.0;
        double r5545510 = a;
        double r5545511 = r5545509 * r5545510;
        double r5545512 = r5545508 / r5545511;
        double r5545513 = g;
        double r5545514 = -r5545513;
        double r5545515 = r5545513 * r5545513;
        double r5545516 = h;
        double r5545517 = r5545516 * r5545516;
        double r5545518 = r5545515 - r5545517;
        double r5545519 = sqrt(r5545518);
        double r5545520 = r5545514 + r5545519;
        double r5545521 = r5545512 * r5545520;
        double r5545522 = cbrt(r5545521);
        double r5545523 = r5545514 - r5545519;
        double r5545524 = r5545512 * r5545523;
        double r5545525 = cbrt(r5545524);
        double r5545526 = r5545522 + r5545525;
        return r5545526;
}
↓
double f(double g, double h, double a) {
        double r5545527 = g;
        double r5545528 = 4.099263178664206e-157;
        bool r5545529 = r5545527 <= r5545528;
        double r5545530 = 1.0;
        double r5545531 = 2.0;
        double r5545532 = a;
        double r5545533 = r5545531 * r5545532;
        double r5545534 = r5545530 / r5545533;
        double r5545535 = -r5545527;
        double r5545536 = r5545527 * r5545527;
        double r5545537 = h;
        double r5545538 = r5545537 * r5545537;
        double r5545539 = r5545536 - r5545538;
        double r5545540 = sqrt(r5545539);
        double r5545541 = r5545535 - r5545540;
        double r5545542 = r5545534 * r5545541;
        double r5545543 = cbrt(r5545542);
        double r5545544 = r5545535 + r5545535;
        double r5545545 = cbrt(r5545544);
        double r5545546 = cbrt(r5545534);
        double r5545547 = r5545545 * r5545546;
        double r5545548 = r5545543 + r5545547;
        double r5545549 = cbrt(r5545540);
        double r5545550 = r5545549 * r5545549;
        double r5545551 = r5545549 * r5545550;
        double r5545552 = r5545535 - r5545551;
        double r5545553 = cbrt(r5545552);
        double r5545554 = r5545546 * r5545553;
        double r5545555 = r5545540 + r5545535;
        double r5545556 = cbrt(r5545555);
        double r5545557 = r5545546 * r5545556;
        double r5545558 = r5545554 + r5545557;
        double r5545559 = r5545529 ? r5545548 : r5545558;
        return r5545559;
}

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 < 4.099263178664206e-157

    1. Initial program 35.6

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

    3. Applied cbrt-prod31.8

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

    4. Taylor expanded around -inf 31.0

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

    5. Simplified31.0

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

    if 4.099263178664206e-157 < g

    1. Initial program 33.4

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

    3. Applied cbrt-prod33.4

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

    5. Applied cbrt-prod29.7

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

    7. Applied add-cube-cbrt29.7

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

  4. Final simplification30.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 4.099263178664206 \cdot 10^{-157}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\left(-g\right) + \left(-g\right)} \cdot \sqrt[3]{\frac{1}{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \left(\sqrt[3]{\sqrt{g \cdot g - h \cdot h}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h}}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} + \left(-g\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 +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))))))))