Average Error: 16.1 → 6.4
Time: 1.0m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.30567877700455932 \cdot 10^{40}:\\ \;\;\;\;\frac{\log \left(e^{\sqrt[3]{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.30567877700455932 \cdot 10^{40}:\\
\;\;\;\;\frac{\log \left(e^{\sqrt[3]{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r146354 = beta;
        double r146355 = alpha;
        double r146356 = r146354 - r146355;
        double r146357 = r146355 + r146354;
        double r146358 = 2.0;
        double r146359 = r146357 + r146358;
        double r146360 = r146356 / r146359;
        double r146361 = 1.0;
        double r146362 = r146360 + r146361;
        double r146363 = r146362 / r146358;
        return r146363;
}
↓
double f(double alpha, double beta) {
        double r146364 = alpha;
        double r146365 = 1.3056787770045593e+40;
        bool r146366 = r146364 <= r146365;
        double r146367 = beta;
        double r146368 = r146364 + r146367;
        double r146369 = 2.0;
        double r146370 = r146368 + r146369;
        double r146371 = r146367 / r146370;
        double r146372 = cbrt(r146371);
        double r146373 = 6.0;
        double r146374 = pow(r146372, r146373);
        double r146375 = cbrt(r146374);
        double r146376 = exp(r146375);
        double r146377 = log(r146376);
        double r146378 = r146377 * r146372;
        double r146379 = r146364 / r146370;
        double r146380 = 1.0;
        double r146381 = r146379 - r146380;
        double r146382 = r146378 - r146381;
        double r146383 = r146382 / r146369;
        double r146384 = 4.0;
        double r146385 = r146364 * r146364;
        double r146386 = r146384 / r146385;
        double r146387 = r146369 / r146364;
        double r146388 = r146386 - r146387;
        double r146389 = 8.0;
        double r146390 = 3.0;
        double r146391 = pow(r146364, r146390);
        double r146392 = r146389 / r146391;
        double r146393 = r146388 - r146392;
        double r146394 = r146371 - r146393;
        double r146395 = r146394 / r146369;
        double r146396 = r146366 ? r146383 : r146395;
        return r146396;
}

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 < 1.3056787770045593e+40

    1. Initial program 1.9

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

    3. Applied div-sub1.9

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

    4. Applied associate-+l-1.9

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

    6. Applied add-cube-cbrt1.9

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

    8. Applied pow1/321.7

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

    9. Applied pow1/321.7

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

    10. Applied pow-prod-down1.9

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

    11. Simplified1.9

      \[\leadsto \frac{{\color{blue}{\left({\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}\right)}}^{\frac{1}{3}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]
    12. Using strategy rm

    13. Applied add-log-exp1.9

      \[\leadsto \frac{\color{blue}{\log \left(e^{{\left({\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}\right)}^{\frac{1}{3}}}\right)} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]

    14. Simplified1.9

      \[\leadsto \frac{\log \color{blue}{\left(e^{\sqrt[3]{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}}\right)} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]

    if 1.3056787770045593e+40 < alpha

    1. Initial program 51.1

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

    3. Applied div-sub51.1

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

    4. Applied associate-+l-49.5

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

    5. Taylor expanded around inf 17.5

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

    6. Simplified17.5

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.30567877700455932 \cdot 10^{40}:\\ \;\;\;\;\frac{\log \left(e^{\sqrt[3]{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))