Average Error: 35.0 → 30.6
Time: 37.1s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -1.213814236807820980958444544158686748103 \cdot 10^{-179}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]
\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 -1.213814236807820980958444544158686748103 \cdot 10^{-179}:\\
\;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\

\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 r111556 = 1.0;
        double r111557 = 2.0;
        double r111558 = a;
        double r111559 = r111557 * r111558;
        double r111560 = r111556 / r111559;
        double r111561 = g;
        double r111562 = -r111561;
        double r111563 = r111561 * r111561;
        double r111564 = h;
        double r111565 = r111564 * r111564;
        double r111566 = r111563 - r111565;
        double r111567 = sqrt(r111566);
        double r111568 = r111562 + r111567;
        double r111569 = r111560 * r111568;
        double r111570 = cbrt(r111569);
        double r111571 = r111562 - r111567;
        double r111572 = r111560 * r111571;
        double r111573 = cbrt(r111572);
        double r111574 = r111570 + r111573;
        return r111574;
}
↓
double f(double g, double h, double a) {
        double r111575 = g;
        double r111576 = -1.213814236807821e-179;
        bool r111577 = r111575 <= r111576;
        double r111578 = 1.0;
        double r111579 = 2.0;
        double r111580 = a;
        double r111581 = r111579 * r111580;
        double r111582 = r111578 / r111581;
        double r111583 = cbrt(r111582);
        double r111584 = r111575 * r111575;
        double r111585 = h;
        double r111586 = r111585 * r111585;
        double r111587 = r111584 - r111586;
        double r111588 = sqrt(r111587);
        double r111589 = r111588 - r111575;
        double r111590 = cbrt(r111589);
        double r111591 = r111583 * r111590;
        double r111592 = -r111575;
        double r111593 = r111592 - r111588;
        double r111594 = r111582 * r111593;
        double r111595 = cbrt(r111594);
        double r111596 = r111591 + r111595;
        double r111597 = r111592 + r111588;
        double r111598 = r111582 * r111597;
        double r111599 = cbrt(r111598);
        double r111600 = r111592 - r111575;
        double r111601 = cbrt(r111600);
        double r111602 = r111583 * r111601;
        double r111603 = r111599 + r111602;
        double r111604 = r111577 ? r111596 : r111603;
        return r111604;
}

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 < -1.213814236807821e-179

    1. Initial program 34.5

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

    3. Applied cbrt-prod30.8

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

    4. Simplified30.8

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

    if -1.213814236807821e-179 < g

    1. Initial program 35.4

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

    3. Applied cbrt-prod31.4

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

    4. Taylor expanded around inf 30.5

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

  4. Final simplification30.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -1.213814236807820980958444544158686748103 \cdot 10^{-179}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - g}\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))))))))