Average Error: 35.7 → 31.4
Time: 39.4s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 4.046018607306917 \cdot 10^{-169}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} + g\right) \cdot \frac{\frac{-1}{2}}{a}} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{-1}{2}}{a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} + g} + \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}}\\ \end{array}\]
\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 4.046018607306917 \cdot 10^{-169}:\\
\;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} + g\right) \cdot \frac{\frac{-1}{2}}{a}} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}}\\

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

\end{array}
double f(double g, double h, double a) {
        double r24851505 = 1.0;
        double r24851506 = 2.0;
        double r24851507 = a;
        double r24851508 = r24851506 * r24851507;
        double r24851509 = r24851505 / r24851508;
        double r24851510 = g;
        double r24851511 = -r24851510;
        double r24851512 = r24851510 * r24851510;
        double r24851513 = h;
        double r24851514 = r24851513 * r24851513;
        double r24851515 = r24851512 - r24851514;
        double r24851516 = sqrt(r24851515);
        double r24851517 = r24851511 + r24851516;
        double r24851518 = r24851509 * r24851517;
        double r24851519 = cbrt(r24851518);
        double r24851520 = r24851511 - r24851516;
        double r24851521 = r24851509 * r24851520;
        double r24851522 = cbrt(r24851521);
        double r24851523 = r24851519 + r24851522;
        return r24851523;
}
↓
double f(double g, double h, double a) {
        double r24851524 = g;
        double r24851525 = 4.046018607306917e-169;
        bool r24851526 = r24851524 <= r24851525;
        double r24851527 = r24851524 * r24851524;
        double r24851528 = h;
        double r24851529 = r24851528 * r24851528;
        double r24851530 = r24851527 - r24851529;
        double r24851531 = sqrt(r24851530);
        double r24851532 = r24851531 + r24851524;
        double r24851533 = -0.5;
        double r24851534 = a;
        double r24851535 = r24851533 / r24851534;
        double r24851536 = r24851532 * r24851535;
        double r24851537 = cbrt(r24851536);
        double r24851538 = -r24851524;
        double r24851539 = r24851538 - r24851524;
        double r24851540 = cbrt(r24851539);
        double r24851541 = 2.0;
        double r24851542 = r24851541 * r24851534;
        double r24851543 = cbrt(r24851542);
        double r24851544 = r24851540 / r24851543;
        double r24851545 = r24851537 + r24851544;
        double r24851546 = cbrt(r24851535);
        double r24851547 = cbrt(r24851532);
        double r24851548 = r24851546 * r24851547;
        double r24851549 = r24851531 - r24851524;
        double r24851550 = r24851549 / r24851542;
        double r24851551 = cbrt(r24851550);
        double r24851552 = r24851548 + r24851551;
        double r24851553 = r24851526 ? r24851545 : r24851552;
        return r24851553;
}

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 < 4.046018607306917e-169

    1. Initial program 36.5

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

    2. Simplified36.5

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

    4. Applied cbrt-div32.5

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

    5. Taylor expanded around -inf 31.6

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

    6. Simplified31.6

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

    if 4.046018607306917e-169 < g

    1. Initial program 34.9

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

    2. Simplified34.8

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

    4. Applied cbrt-prod31.2

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

  4. Final simplification31.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 4.046018607306917 \cdot 10^{-169}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} + g\right) \cdot \frac{\frac{-1}{2}}{a}} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{-1}{2}}{a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} + g} + \sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019120 +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))))))))