Average Error: 36.5 → 32.6
Time: 38.4s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 1.571183301273906648720737884084174539254 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(h \cdot h\right)}}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \end{array}\]
\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 1.571183301273906648720737884084174539254 \cdot 10^{-162}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(h \cdot h\right)}}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}\\

\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) - \sqrt{g \cdot g - h \cdot h}}\\

\end{array}
double f(double g, double h, double a) {
        double r122630 = 1.0;
        double r122631 = 2.0;
        double r122632 = a;
        double r122633 = r122631 * r122632;
        double r122634 = r122630 / r122633;
        double r122635 = g;
        double r122636 = -r122635;
        double r122637 = r122635 * r122635;
        double r122638 = h;
        double r122639 = r122638 * r122638;
        double r122640 = r122637 - r122639;
        double r122641 = sqrt(r122640);
        double r122642 = r122636 + r122641;
        double r122643 = r122634 * r122642;
        double r122644 = cbrt(r122643);
        double r122645 = r122636 - r122641;
        double r122646 = r122634 * r122645;
        double r122647 = cbrt(r122646);
        double r122648 = r122644 + r122647;
        return r122648;
}
↓
double f(double g, double h, double a) {
        double r122649 = g;
        double r122650 = 1.5711833012739066e-162;
        bool r122651 = r122649 <= r122650;
        double r122652 = 1.0;
        double r122653 = 2.0;
        double r122654 = a;
        double r122655 = r122653 * r122654;
        double r122656 = r122652 / r122655;
        double r122657 = h;
        double r122658 = r122657 * r122657;
        double r122659 = r122656 * r122658;
        double r122660 = cbrt(r122659);
        double r122661 = r122649 * r122649;
        double r122662 = r122661 - r122658;
        double r122663 = sqrt(r122662);
        double r122664 = r122663 - r122649;
        double r122665 = cbrt(r122664);
        double r122666 = r122660 / r122665;
        double r122667 = cbrt(r122656);
        double r122668 = r122667 * r122665;
        double r122669 = r122666 + r122668;
        double r122670 = -r122649;
        double r122671 = r122670 + r122663;
        double r122672 = r122656 * r122671;
        double r122673 = cbrt(r122672);
        double r122674 = r122670 - r122663;
        double r122675 = cbrt(r122674);
        double r122676 = r122667 * r122675;
        double r122677 = r122673 + r122676;
        double r122678 = r122651 ? r122669 : r122677;
        return r122678;
}

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 < 1.5711833012739066e-162

    1. Initial program 37.6

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

    3. Applied cbrt-prod33.6

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

    4. Simplified33.6

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

    6. Applied flip--33.4

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

    7. Applied associate-*r/33.5

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

    8. Applied cbrt-div33.5

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

    9. Simplified33.6

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \frac{\color{blue}{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(h \cdot h + 0\right)}}}{\sqrt[3]{\left(-g\right) + \sqrt{g \cdot g - h \cdot h}}}\]

    10. Simplified33.6

      \[\leadsto \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g} + \frac{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(h \cdot h + 0\right)}}{\color{blue}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}}}\]

    if 1.5711833012739066e-162 < 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-prod31.5

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

  4. Final simplification32.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 1.571183301273906648720737884084174539254 \cdot 10^{-162}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(h \cdot h\right)}}{\sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\sqrt{g \cdot g - h \cdot h} - g}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left(-g\right) - \sqrt{g \cdot g - h \cdot h}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019322 +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))))))))