Average Error: 16.3 → 5.7
Time: 5.1s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 20359646.5331954099:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\frac{\left(\alpha + \beta\right) + 2}{\left(\sqrt[3]{\sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\beta}}}\right)}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 20359646.5331954099:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\frac{\left(\alpha + \beta\right) + 2}{\left(\sqrt[3]{\sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\beta}}}\right)}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r98307 = beta;
        double r98308 = alpha;
        double r98309 = r98307 - r98308;
        double r98310 = r98308 + r98307;
        double r98311 = 2.0;
        double r98312 = r98310 + r98311;
        double r98313 = r98309 / r98312;
        double r98314 = 1.0;
        double r98315 = r98313 + r98314;
        double r98316 = r98315 / r98311;
        return r98316;
}
↓
double f(double alpha, double beta) {
        double r98317 = alpha;
        double r98318 = 20359646.53319541;
        bool r98319 = r98317 <= r98318;
        double r98320 = beta;
        double r98321 = cbrt(r98320);
        double r98322 = r98321 * r98321;
        double r98323 = r98317 + r98320;
        double r98324 = 2.0;
        double r98325 = r98323 + r98324;
        double r98326 = cbrt(r98321);
        double r98327 = r98326 * r98326;
        double r98328 = cbrt(r98322);
        double r98329 = cbrt(r98328);
        double r98330 = cbrt(r98326);
        double r98331 = r98329 * r98330;
        double r98332 = r98327 * r98331;
        double r98333 = r98325 / r98332;
        double r98334 = r98322 / r98333;
        double r98335 = r98317 / r98325;
        double r98336 = 1.0;
        double r98337 = r98335 - r98336;
        double r98338 = r98334 - r98337;
        double r98339 = r98338 / r98324;
        double r98340 = r98320 / r98325;
        double r98341 = 4.0;
        double r98342 = r98341 / r98317;
        double r98343 = r98342 / r98317;
        double r98344 = r98324 / r98317;
        double r98345 = 8.0;
        double r98346 = -r98345;
        double r98347 = 3.0;
        double r98348 = pow(r98317, r98347);
        double r98349 = r98346 / r98348;
        double r98350 = r98344 - r98349;
        double r98351 = r98343 - r98350;
        double r98352 = r98340 - r98351;
        double r98353 = r98352 / r98324;
        double r98354 = r98319 ? r98339 : r98353;
        return r98354;
}

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 < 20359646.53319541

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied add-cube-cbrt0.3

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

    7. Applied associate-/l*0.3

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

    9. Applied add-cube-cbrt0.3

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

    11. Applied add-cube-cbrt0.3

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

    12. Applied cbrt-prod0.3

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

    13. Applied cbrt-prod0.3

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

    if 20359646.53319541 < alpha

    1. Initial program 50.1

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

    3. Applied div-sub50.0

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

    4. Applied associate-+l-48.7

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

    5. Taylor expanded around inf 16.8

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

    6. Simplified16.8

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

  4. Final simplification5.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 20359646.5331954099:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\frac{\left(\alpha + \beta\right) + 2}{\left(\sqrt[3]{\sqrt[3]{\beta}} \cdot \sqrt[3]{\sqrt[3]{\beta}}\right) \cdot \left(\sqrt[3]{\sqrt[3]{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}} \cdot \sqrt[3]{\sqrt[3]{\sqrt[3]{\beta}}}\right)}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\frac{4}{\alpha}}{\alpha} - \left(\frac{2}{\alpha} - \frac{-8}{{\alpha}^{3}}\right)\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020024 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))