Average Error: 35.8 → 31.7
Time: 20.4s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le 2.28726507085781537 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]
\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 2.28726507085781537 \cdot 10^{-210}:\\
\;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}} + \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(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}\\

\end{array}
double f(double g, double h, double a) {
        double r126327 = 1.0;
        double r126328 = 2.0;
        double r126329 = a;
        double r126330 = r126328 * r126329;
        double r126331 = r126327 / r126330;
        double r126332 = g;
        double r126333 = -r126332;
        double r126334 = r126332 * r126332;
        double r126335 = h;
        double r126336 = r126335 * r126335;
        double r126337 = r126334 - r126336;
        double r126338 = sqrt(r126337);
        double r126339 = r126333 + r126338;
        double r126340 = r126331 * r126339;
        double r126341 = cbrt(r126340);
        double r126342 = r126333 - r126338;
        double r126343 = r126331 * r126342;
        double r126344 = cbrt(r126343);
        double r126345 = r126341 + r126344;
        return r126345;
}
↓
double f(double g, double h, double a) {
        double r126346 = g;
        double r126347 = 2.2872650708578154e-210;
        bool r126348 = r126346 <= r126347;
        double r126349 = 1.0;
        double r126350 = -r126346;
        double r126351 = r126350 - r126346;
        double r126352 = r126349 * r126351;
        double r126353 = cbrt(r126352);
        double r126354 = 2.0;
        double r126355 = a;
        double r126356 = r126354 * r126355;
        double r126357 = cbrt(r126356);
        double r126358 = r126353 / r126357;
        double r126359 = r126349 / r126356;
        double r126360 = r126346 * r126346;
        double r126361 = h;
        double r126362 = r126361 * r126361;
        double r126363 = r126360 - r126362;
        double r126364 = sqrt(r126363);
        double r126365 = r126350 - r126364;
        double r126366 = r126359 * r126365;
        double r126367 = cbrt(r126366);
        double r126368 = r126358 + r126367;
        double r126369 = r126364 - r126346;
        double r126370 = r126359 * r126369;
        double r126371 = cbrt(r126370);
        double r126372 = r126349 * r126365;
        double r126373 = cbrt(r126372);
        double r126374 = r126373 / r126357;
        double r126375 = r126371 + r126374;
        double r126376 = r126348 ? r126368 : r126375;
        return r126376;
}

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 < 2.2872650708578154e-210

    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

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

    4. Applied associate-*l/36.5

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

    5. Applied cbrt-div32.8

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

    6. Taylor expanded around -inf 32.1

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

    7. Simplified32.1

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

    if 2.2872650708578154e-210 < g

    1. Initial program 35.2

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

    2. Simplified35.2

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

    4. Applied associate-*l/35.2

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

    5. Applied cbrt-div31.4

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

  4. Final simplification31.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le 2.28726507085781537 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - g\right)}}{\sqrt[3]{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\sqrt{g \cdot g - h \cdot h} - g\right)} + \frac{\sqrt[3]{1 \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020043 
(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))))))))