Octave 3.8, jcobi/1

Average Error: 15.9 → 5.9

Time: 25.7s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1845090834.4376380443572998046875:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta \cdot \left(1 \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right) + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2}\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{{1}^{6} + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)}^{3}}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2} \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) \cdot \left(1 \cdot 1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

C

double f(double alpha, double beta) {
        double r117362 = beta;
        double r117363 = alpha;
        double r117364 = r117362 - r117363;
        double r117365 = r117363 + r117362;
        double r117366 = 2.0;
        double r117367 = r117365 + r117366;
        double r117368 = r117364 / r117367;
        double r117369 = 1.0;
        double r117370 = r117368 + r117369;
        double r117371 = r117370 / r117366;
        return r117371;
}

↓

double f(double alpha, double beta) {
        double r117372 = alpha;
        double r117373 = 1845090834.437638;
        bool r117374 = r117372 <= r117373;
        double r117375 = beta;
        double r117376 = 1.0;
        double r117377 = r117372 + r117375;
        double r117378 = 2.0;
        double r117379 = r117377 + r117378;
        double r117380 = r117372 / r117379;
        double r117381 = r117380 + r117376;
        double r117382 = r117376 * r117381;
        double r117383 = 2.0;
        double r117384 = pow(r117380, r117383);
        double r117385 = r117382 + r117384;
        double r117386 = r117375 * r117385;
        double r117387 = 3.0;
        double r117388 = pow(r117380, r117387);
        double r117389 = pow(r117376, r117387);
        double r117390 = r117388 - r117389;
        double r117391 = r117379 * r117390;
        double r117392 = r117386 - r117391;
        double r117393 = 6.0;
        double r117394 = pow(r117376, r117393);
        double r117395 = r117380 * r117381;
        double r117396 = pow(r117395, r117387);
        double r117397 = r117394 + r117396;
        double r117398 = r117392 / r117397;
        double r117399 = r117379 * r117378;
        double r117400 = r117398 / r117399;
        double r117401 = r117395 * r117395;
        double r117402 = r117376 * r117376;
        double r117403 = r117402 * r117402;
        double r117404 = r117395 * r117402;
        double r117405 = r117403 - r117404;
        double r117406 = r117401 + r117405;
        double r117407 = r117400 * r117406;
        double r117408 = log(r117407);
        double r117409 = exp(r117408);
        double r117410 = r117375 / r117379;
        double r117411 = 4.0;
        double r117412 = r117372 * r117372;
        double r117413 = r117411 / r117412;
        double r117414 = r117378 / r117372;
        double r117415 = r117413 - r117414;
        double r117416 = 8.0;
        double r117417 = pow(r117372, r117387);
        double r117418 = r117416 / r117417;
        double r117419 = r117415 - r117418;
        double r117420 = r117410 - r117419;
        double r117421 = r117420 / r117378;
        double r117422 = r117374 ? r117409 : r117421;
        return r117422;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Split input into 2 regimes

2. if alpha < 1845090834.437638

   1. Initial program 0.1

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

   3. Applied div-sub 0.1

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

   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 flip3-- 0.1

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

   7. Applied frac-sub 0.1

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

   8. Applied associate-/l/ 0.1

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

   9. Simplified 0.1

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

   11. Applied add-exp-log 0.1

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

   12. Applied add-exp-log 2.1

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

   13. Applied prod-exp 2.1

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

   14. Applied add-exp-log 2.1

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

   15. Applied prod-exp 2.1

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

   16. Applied add-exp-log 0.7

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

   17. Applied div-exp 0.7

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

   18. Simplified 0.1

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

   20. Applied flip3-+ 0.1

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

   21. Applied associate-*l/ 0.1

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

   22. Applied associate-/r/ 0.1

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

   23. Simplified 0.1

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

   if 1845090834.437638 < alpha

   1. Initial program 49.2

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

   3. Applied div-sub 49.1

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

   4. Applied associate-+l- 47.7

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

   5. Taylor expanded around inf 18.1

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

   6. Simplified 18.1

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

4. Final simplification 5.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1845090834.4376380443572998046875:\\ \;\;\;\;e^{\log \left(\frac{\frac{\beta \cdot \left(1 \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right) + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{2}\right) - \left(\left(\alpha + \beta\right) + 2\right) \cdot \left({\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)}^{3} - {1}^{3}\right)}{{1}^{6} + {\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right)}^{3}}}{\left(\left(\alpha + \beta\right) + 2\right) \cdot 2} \cdot \left(\left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) + \left(\left(1 \cdot 1\right) \cdot \left(1 \cdot 1\right) - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} \cdot \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} + 1\right)\right) \cdot \left(1 \cdot 1\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019304 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))