Average Error: 16.6 → 6.1
Time: 14.0s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 15848726308903979000:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2} \cdot \left(\left(\alpha + \beta\right) - 2\right) - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 15848726308903979000:\\
\;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2} \cdot \left(\left(\alpha + \beta\right) - 2\right) - 1\right)}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\

\end{array}
double f(double alpha, double beta) {
        double r66297 = beta;
        double r66298 = alpha;
        double r66299 = r66297 - r66298;
        double r66300 = r66298 + r66297;
        double r66301 = 2.0;
        double r66302 = r66300 + r66301;
        double r66303 = r66299 / r66302;
        double r66304 = 1.0;
        double r66305 = r66303 + r66304;
        double r66306 = r66305 / r66301;
        return r66306;
}
↓
double f(double alpha, double beta) {
        double r66307 = alpha;
        double r66308 = 1.584872630890398e+19;
        bool r66309 = r66307 <= r66308;
        double r66310 = beta;
        double r66311 = r66307 + r66310;
        double r66312 = 2.0;
        double r66313 = r66311 + r66312;
        double r66314 = r66310 / r66313;
        double r66315 = r66311 * r66311;
        double r66316 = r66312 * r66312;
        double r66317 = r66315 - r66316;
        double r66318 = r66307 / r66317;
        double r66319 = r66311 - r66312;
        double r66320 = r66318 * r66319;
        double r66321 = 1.0;
        double r66322 = r66320 - r66321;
        double r66323 = r66314 - r66322;
        double r66324 = r66323 / r66312;
        double r66325 = cbrt(r66314);
        double r66326 = r66325 * r66325;
        double r66327 = r66326 * r66325;
        double r66328 = 4.0;
        double r66329 = 2.0;
        double r66330 = pow(r66307, r66329);
        double r66331 = r66328 / r66330;
        double r66332 = r66312 / r66307;
        double r66333 = r66331 - r66332;
        double r66334 = 8.0;
        double r66335 = 3.0;
        double r66336 = pow(r66307, r66335);
        double r66337 = r66334 / r66336;
        double r66338 = r66333 - r66337;
        double r66339 = r66327 - r66338;
        double r66340 = r66339 / r66312;
        double r66341 = r66309 ? r66324 : r66340;
        return r66341;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 1.584872630890398e+19

    1. Initial program 0.6

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied div-sub0.6

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]

    4. Applied associate-+l-0.6

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
    5. Using strategy rm

    6. Applied flip-+0.6

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\color{blue}{\frac{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2}{\left(\alpha + \beta\right) - 2}}} - 1\right)}{2}\]

    7. Applied associate-/r/0.6

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\color{blue}{\frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2} \cdot \left(\left(\alpha + \beta\right) - 2\right)} - 1\right)}{2}\]

    if 1.584872630890398e+19 < alpha

    1. Initial program 50.6

      \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
    2. Using strategy rm

    3. Applied div-sub50.6

      \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]

    4. Applied associate-+l-49.1

      \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt49.1

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

    7. Taylor expanded around inf 17.9

      \[\leadsto \frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]

    8. Simplified17.9

      \[\leadsto \frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \color{blue}{\left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 15848726308903979000:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) \cdot \left(\alpha + \beta\right) - 2 \cdot 2} \cdot \left(\left(\alpha + \beta\right) - 2\right) - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))