Octave 3.8, jcobi/2

Average Error: 24.1 → 12.9

Time: 34.2s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.442037061945920645369018327902562528872 \cdot 10^{124}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \frac{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot i}}\right)\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \le 1.157236759942197475904943781835715492446 \cdot 10^{199}:\\ \;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} - \frac{4}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot i}}\right)}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}\right)}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r153382 = alpha;
        double r153383 = beta;
        double r153384 = r153382 + r153383;
        double r153385 = r153383 - r153382;
        double r153386 = r153384 * r153385;
        double r153387 = 2.0;
        double r153388 = i;
        double r153389 = r153387 * r153388;
        double r153390 = r153384 + r153389;
        double r153391 = r153386 / r153390;
        double r153392 = r153390 + r153387;
        double r153393 = r153391 / r153392;
        double r153394 = 1.0;
        double r153395 = r153393 + r153394;
        double r153396 = r153395 / r153387;
        return r153396;
}

↓

double f(double alpha, double beta, double i) {
        double r153397 = alpha;
        double r153398 = 1.4420370619459206e+124;
        bool r153399 = r153397 <= r153398;
        double r153400 = 1.0;
        double r153401 = beta;
        double r153402 = r153397 + r153401;
        double r153403 = r153401 - r153397;
        double r153404 = 2.0;
        double r153405 = i;
        double r153406 = r153404 * r153405;
        double r153407 = r153402 + r153406;
        double r153408 = r153403 / r153407;
        double r153409 = cbrt(r153408);
        double r153410 = cbrt(r153403);
        double r153411 = 1.0;
        double r153412 = r153411 / r153407;
        double r153413 = cbrt(r153412);
        double r153414 = r153410 * r153413;
        double r153415 = r153409 * r153414;
        double r153416 = r153415 * r153409;
        double r153417 = r153407 + r153404;
        double r153418 = r153416 / r153417;
        double r153419 = r153402 * r153418;
        double r153420 = r153400 + r153419;
        double r153421 = 3.0;
        double r153422 = pow(r153420, r153421);
        double r153423 = cbrt(r153422);
        double r153424 = r153423 / r153404;
        double r153425 = 1.1572367599421975e+199;
        bool r153426 = r153397 <= r153425;
        double r153427 = 8.0;
        double r153428 = pow(r153397, r153421);
        double r153429 = r153427 / r153428;
        double r153430 = 4.0;
        double r153431 = 2.0;
        double r153432 = pow(r153397, r153431);
        double r153433 = r153430 / r153432;
        double r153434 = r153429 - r153433;
        double r153435 = r153404 / r153397;
        double r153436 = r153434 + r153435;
        double r153437 = r153436 / r153404;
        double r153438 = sqrt(r153417);
        double r153439 = r153415 / r153438;
        double r153440 = r153439 * r153409;
        double r153441 = r153402 * r153440;
        double r153442 = r153441 / r153438;
        double r153443 = r153442 + r153400;
        double r153444 = r153443 / r153404;
        double r153445 = r153426 ? r153437 : r153444;
        double r153446 = r153399 ? r153424 : r153445;
        return r153446;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 1.4420370619459206e+124

   1. Initial program 14.7

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

   3. Applied *-un-lft-identity 14.7

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

   4. Applied *-un-lft-identity 14.7

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

   5. Applied times-frac 4.3

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

   6. Applied times-frac 4.3

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

   7. Simplified 4.3

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

   9. Applied add-sqr-sqrt 4.3

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

   10. Applied add-cube-cbrt 4.3

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

   11. Applied times-frac 4.3

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

   13. Applied div-inv 4.3

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

   14. Applied cbrt-prod 4.4

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

   16. Applied add-cbrt-cube 4.3

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

   17. Simplified 4.3

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

   if 1.4420370619459206e+124 < alpha < 1.1572367599421975e+199

   1. Initial program 57.1

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

   3. Applied *-un-lft-identity 57.1

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

   4. Applied *-un-lft-identity 57.1

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

   5. Applied times-frac 38.3

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

   6. Applied times-frac 38.3

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

   7. Simplified 38.3

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

   9. Applied add-sqr-sqrt 38.4

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

   10. Applied add-cube-cbrt 38.4

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

   11. Applied times-frac 38.4

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

   12. Taylor expanded around inf 40.6

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

   13. Simplified 40.6

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

   if 1.1572367599421975e+199 < alpha

   1. Initial program 64.0

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

   3. Applied *-un-lft-identity 64.0

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

   4. Applied *-un-lft-identity 64.0

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

   5. Applied times-frac 50.7

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

   6. Applied times-frac 50.8

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

   7. Simplified 50.8

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

   9. Applied add-sqr-sqrt 51.2

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

   10. Applied add-cube-cbrt 51.2

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

   11. Applied times-frac 51.4

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

   13. Applied div-inv 51.4

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

   14. Applied cbrt-prod 51.4

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

   16. Applied associate-*r/ 51.4

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

   17. Applied associate-*r/ 51.4

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

4. Final simplification 12.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.442037061945920645369018327902562528872 \cdot 10^{124}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \frac{\left(\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot i}}\right)\right) \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \le 1.157236759942197475904943781835715492446 \cdot 10^{199}:\\ \;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} - \frac{4}{{\alpha}^{2}}\right) + \frac{2}{\alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\frac{\sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}} \cdot \left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\frac{1}{\left(\alpha + \beta\right) + 2 \cdot i}}\right)}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \sqrt[3]{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}\right)}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} + 1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))