Average Error: 34.9 → 30.5
Time: 31.8s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 4.0263109172966606 \cdot 10^{-175}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} + g\right) \cdot \frac{\frac{-1}{2}}{a}} + \frac{\sqrt[3]{\left(\left(-g\right) - g\right) \cdot \frac{1}{2}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{a} \cdot \frac{1}{2}} + \sqrt[3]{\sqrt{g \cdot g - h \cdot h} + g} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{a}}\\ \end{array}\]
\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 4.0263109172966606 \cdot 10^{-175}:\\
\;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} + g\right) \cdot \frac{\frac{-1}{2}}{a}} + \frac{\sqrt[3]{\left(\left(-g\right) - g\right) \cdot \frac{1}{2}}}{\sqrt[3]{a}}\\

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

\end{array}
double f(double g, double h, double a) {
        double r5032441 = 1.0;
        double r5032442 = 2.0;
        double r5032443 = a;
        double r5032444 = r5032442 * r5032443;
        double r5032445 = r5032441 / r5032444;
        double r5032446 = g;
        double r5032447 = -r5032446;
        double r5032448 = r5032446 * r5032446;
        double r5032449 = h;
        double r5032450 = r5032449 * r5032449;
        double r5032451 = r5032448 - r5032450;
        double r5032452 = sqrt(r5032451);
        double r5032453 = r5032447 + r5032452;
        double r5032454 = r5032445 * r5032453;
        double r5032455 = cbrt(r5032454);
        double r5032456 = r5032447 - r5032452;
        double r5032457 = r5032445 * r5032456;
        double r5032458 = cbrt(r5032457);
        double r5032459 = r5032455 + r5032458;
        return r5032459;
}
↓
double f(double g, double h, double a) {
        double r5032460 = g;
        double r5032461 = 4.0263109172966606e-175;
        bool r5032462 = r5032460 <= r5032461;
        double r5032463 = r5032460 * r5032460;
        double r5032464 = h;
        double r5032465 = r5032464 * r5032464;
        double r5032466 = r5032463 - r5032465;
        double r5032467 = sqrt(r5032466);
        double r5032468 = r5032467 + r5032460;
        double r5032469 = -0.5;
        double r5032470 = a;
        double r5032471 = r5032469 / r5032470;
        double r5032472 = r5032468 * r5032471;
        double r5032473 = cbrt(r5032472);
        double r5032474 = -r5032460;
        double r5032475 = r5032474 - r5032460;
        double r5032476 = 0.5;
        double r5032477 = r5032475 * r5032476;
        double r5032478 = cbrt(r5032477);
        double r5032479 = cbrt(r5032470);
        double r5032480 = r5032478 / r5032479;
        double r5032481 = r5032473 + r5032480;
        double r5032482 = r5032467 - r5032460;
        double r5032483 = r5032482 / r5032470;
        double r5032484 = r5032483 * r5032476;
        double r5032485 = cbrt(r5032484);
        double r5032486 = cbrt(r5032468);
        double r5032487 = cbrt(r5032471);
        double r5032488 = r5032486 * r5032487;
        double r5032489 = r5032485 + r5032488;
        double r5032490 = r5032462 ? r5032481 : r5032489;
        return r5032490;
}

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 < 4.0263109172966606e-175

    1. Initial program 35.9

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

    2. Simplified35.9

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

    4. Applied associate-*l/35.9

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

    5. Applied cbrt-div31.8

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

    6. Taylor expanded around -inf 30.9

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

    7. Simplified30.9

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

    if 4.0263109172966606e-175 < g

    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. Simplified33.7

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

    4. Applied cbrt-prod30.1

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

  4. Final simplification30.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 4.0263109172966606 \cdot 10^{-175}:\\ \;\;\;\;\sqrt[3]{\left(\sqrt{g \cdot g - h \cdot h} + g\right) \cdot \frac{\frac{-1}{2}}{a}} + \frac{\sqrt[3]{\left(\left(-g\right) - g\right) \cdot \frac{1}{2}}}{\sqrt[3]{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{\sqrt{g \cdot g - h \cdot h} - g}{a} \cdot \frac{1}{2}} + \sqrt[3]{\sqrt{g \cdot g - h \cdot h} + g} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019143 +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))))))))