Average Error: 35.3 → 29.4
Time: 28.4s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -6.3921017940024375 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{\frac{-1}{2}}{a}} \cdot \sqrt[3]{\frac{h \cdot h}{g - \sqrt{\left(g - h\right) \cdot \left(g + h\right)}}}\\ \mathbf{elif}\;g \le 1.1669440241605785 \cdot 10^{-166}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g\right)} + \sqrt[3]{g + g} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-h \cdot h}{g + \sqrt{\left(g - h\right) \cdot \left(g + h\right)}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}} + \sqrt[3]{g + \sqrt{\left(g - h\right) \cdot \left(g + h\right)}} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{a}}\\ \end{array}\]
\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 -6.3921017940024375 \cdot 10^{-161}:\\
\;\;\;\;\frac{\sqrt[3]{\frac{1}{2} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{\frac{-1}{2}}{a}} \cdot \sqrt[3]{\frac{h \cdot h}{g - \sqrt{\left(g - h\right) \cdot \left(g + h\right)}}}\\

\mathbf{elif}\;g \le 1.1669440241605785 \cdot 10^{-166}:\\
\;\;\;\;\sqrt[3]{\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g\right)} + \sqrt[3]{g + g} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{a}}\\

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

\end{array}
double f(double g, double h, double a) {
        double r4927328 = 1.0;
        double r4927329 = 2.0;
        double r4927330 = a;
        double r4927331 = r4927329 * r4927330;
        double r4927332 = r4927328 / r4927331;
        double r4927333 = g;
        double r4927334 = -r4927333;
        double r4927335 = r4927333 * r4927333;
        double r4927336 = h;
        double r4927337 = r4927336 * r4927336;
        double r4927338 = r4927335 - r4927337;
        double r4927339 = sqrt(r4927338);
        double r4927340 = r4927334 + r4927339;
        double r4927341 = r4927332 * r4927340;
        double r4927342 = cbrt(r4927341);
        double r4927343 = r4927334 - r4927339;
        double r4927344 = r4927332 * r4927343;
        double r4927345 = cbrt(r4927344);
        double r4927346 = r4927342 + r4927345;
        return r4927346;
}
↓
double f(double g, double h, double a) {
        double r4927347 = g;
        double r4927348 = -6.3921017940024375e-161;
        bool r4927349 = r4927347 <= r4927348;
        double r4927350 = 0.5;
        double r4927351 = h;
        double r4927352 = r4927347 - r4927351;
        double r4927353 = r4927347 + r4927351;
        double r4927354 = r4927352 * r4927353;
        double r4927355 = sqrt(r4927354);
        double r4927356 = r4927355 - r4927347;
        double r4927357 = r4927350 * r4927356;
        double r4927358 = cbrt(r4927357);
        double r4927359 = a;
        double r4927360 = cbrt(r4927359);
        double r4927361 = r4927358 / r4927360;
        double r4927362 = -0.5;
        double r4927363 = r4927362 / r4927359;
        double r4927364 = cbrt(r4927363);
        double r4927365 = r4927351 * r4927351;
        double r4927366 = r4927347 - r4927355;
        double r4927367 = r4927365 / r4927366;
        double r4927368 = cbrt(r4927367);
        double r4927369 = r4927364 * r4927368;
        double r4927370 = r4927361 + r4927369;
        double r4927371 = 1.1669440241605785e-166;
        bool r4927372 = r4927347 <= r4927371;
        double r4927373 = r4927350 / r4927359;
        double r4927374 = r4927373 * r4927356;
        double r4927375 = cbrt(r4927374);
        double r4927376 = r4927347 + r4927347;
        double r4927377 = cbrt(r4927376);
        double r4927378 = r4927377 * r4927364;
        double r4927379 = r4927375 + r4927378;
        double r4927380 = -r4927365;
        double r4927381 = r4927347 + r4927355;
        double r4927382 = r4927380 / r4927381;
        double r4927383 = cbrt(r4927382);
        double r4927384 = cbrt(r4927373);
        double r4927385 = r4927383 * r4927384;
        double r4927386 = cbrt(r4927381);
        double r4927387 = r4927386 * r4927364;
        double r4927388 = r4927385 + r4927387;
        double r4927389 = r4927372 ? r4927379 : r4927388;
        double r4927390 = r4927349 ? r4927370 : r4927389;
        return r4927390;
}

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 3 regimes

  2. if g < -6.3921017940024375e-161

    1. Initial program 34.4

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

    2. Simplified34.4

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

    4. Applied cbrt-prod34.3

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

    6. Applied associate-*l/34.3

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

    7. Applied cbrt-div30.4

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

    9. Applied flip-+30.2

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

    10. Simplified29.1

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

    if -6.3921017940024375e-161 < g < 1.1669440241605785e-166

    1. Initial program 55.0

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

    2. Simplified55.0

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

    4. Applied cbrt-prod51.1

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

    5. Taylor expanded around inf 36.0

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

    if 1.1669440241605785e-166 < g

    1. Initial program 33.8

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

    2. Simplified33.8

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

    4. Applied cbrt-prod30.1

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

    6. Applied cbrt-prod30.1

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

    8. Applied flip--29.9

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

    9. Simplified28.9

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

  4. Final simplification29.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -6.3921017940024375 \cdot 10^{-161}:\\ \;\;\;\;\frac{\sqrt[3]{\frac{1}{2} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g\right)}}{\sqrt[3]{a}} + \sqrt[3]{\frac{\frac{-1}{2}}{a}} \cdot \sqrt[3]{\frac{h \cdot h}{g - \sqrt{\left(g - h\right) \cdot \left(g + h\right)}}}\\ \mathbf{elif}\;g \le 1.1669440241605785 \cdot 10^{-166}:\\ \;\;\;\;\sqrt[3]{\frac{\frac{1}{2}}{a} \cdot \left(\sqrt{\left(g - h\right) \cdot \left(g + h\right)} - g\right)} + \sqrt[3]{g + g} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{a}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\frac{-h \cdot h}{g + \sqrt{\left(g - h\right) \cdot \left(g + h\right)}}} \cdot \sqrt[3]{\frac{\frac{1}{2}}{a}} + \sqrt[3]{g + \sqrt{\left(g - h\right) \cdot \left(g + h\right)}} \cdot \sqrt[3]{\frac{\frac{-1}{2}}{a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(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))))))))