Average Error: 34.8 → 30.5
Time: 22.0s
Precision: 64
\[\sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right)} + \sqrt[3]{\frac{1}{2 \cdot a} \cdot \left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;g \le -1.2129274808481617 \cdot 10^{-158}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left|\sqrt[3]{g \cdot g - h \cdot h}\right| \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}} + \left(-g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{2 \cdot a}} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]
\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 -1.2129274808481617 \cdot 10^{-158}:\\
\;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left|\sqrt[3]{g \cdot g - h \cdot h}\right| \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}} + \left(-g\right)}\\

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

\end{array}
double f(double g, double h, double a) {
        double r2098270 = 1.0;
        double r2098271 = 2.0;
        double r2098272 = a;
        double r2098273 = r2098271 * r2098272;
        double r2098274 = r2098270 / r2098273;
        double r2098275 = g;
        double r2098276 = -r2098275;
        double r2098277 = r2098275 * r2098275;
        double r2098278 = h;
        double r2098279 = r2098278 * r2098278;
        double r2098280 = r2098277 - r2098279;
        double r2098281 = sqrt(r2098280);
        double r2098282 = r2098276 + r2098281;
        double r2098283 = r2098274 * r2098282;
        double r2098284 = cbrt(r2098283);
        double r2098285 = r2098276 - r2098281;
        double r2098286 = r2098274 * r2098285;
        double r2098287 = cbrt(r2098286);
        double r2098288 = r2098284 + r2098287;
        return r2098288;
}
↓
double f(double g, double h, double a) {
        double r2098289 = g;
        double r2098290 = -1.2129274808481617e-158;
        bool r2098291 = r2098289 <= r2098290;
        double r2098292 = -r2098289;
        double r2098293 = r2098289 * r2098289;
        double r2098294 = h;
        double r2098295 = r2098294 * r2098294;
        double r2098296 = r2098293 - r2098295;
        double r2098297 = sqrt(r2098296);
        double r2098298 = r2098292 - r2098297;
        double r2098299 = 1.0;
        double r2098300 = 2.0;
        double r2098301 = a;
        double r2098302 = r2098300 * r2098301;
        double r2098303 = r2098299 / r2098302;
        double r2098304 = r2098298 * r2098303;
        double r2098305 = cbrt(r2098304);
        double r2098306 = cbrt(r2098303);
        double r2098307 = cbrt(r2098296);
        double r2098308 = fabs(r2098307);
        double r2098309 = sqrt(r2098307);
        double r2098310 = r2098308 * r2098309;
        double r2098311 = r2098310 + r2098292;
        double r2098312 = cbrt(r2098311);
        double r2098313 = r2098306 * r2098312;
        double r2098314 = r2098305 + r2098313;
        double r2098315 = r2098292 + r2098297;
        double r2098316 = r2098315 * r2098303;
        double r2098317 = cbrt(r2098316);
        double r2098318 = r2098292 - r2098289;
        double r2098319 = cbrt(r2098318);
        double r2098320 = cbrt(r2098302);
        double r2098321 = r2098319 / r2098320;
        double r2098322 = r2098317 + r2098321;
        double r2098323 = r2098291 ? r2098314 : r2098322;
        return r2098323;
}

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 < -1.2129274808481617e-158

    1. Initial program 34.1

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

    3. Applied cbrt-prod30.4

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

    5. Applied add-cube-cbrt30.4

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

    6. Applied sqrt-prod30.4

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

    7. Simplified30.4

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

    if -1.2129274808481617e-158 < 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 associate-*l/35.4

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

    4. Applied cbrt-div31.6

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

    5. Taylor expanded around inf 30.6

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

  4. Final simplification30.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;g \le -1.2129274808481617 \cdot 10^{-158}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) - \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{2 \cdot a}} + \sqrt[3]{\frac{1}{2 \cdot a}} \cdot \sqrt[3]{\left|\sqrt[3]{g \cdot g - h \cdot h}\right| \cdot \sqrt{\sqrt[3]{g \cdot g - h \cdot h}} + \left(-g\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{\left(\left(-g\right) + \sqrt{g \cdot g - h \cdot h}\right) \cdot \frac{1}{2 \cdot a}} + \frac{\sqrt[3]{\left(-g\right) - g}}{\sqrt[3]{2 \cdot a}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 +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))))))))