Octave 3.8, jcobi/4

Average Error: 52.3 → 10.2

Time: 2.9m

Precision: 64

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

Math

\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;i \le 6.263709625600619 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left|\left(\alpha + \beta\right) + i \cdot 2\right| - \sqrt{1.0}} \cdot \frac{\frac{i}{\left(\alpha + \beta\right) + i \cdot 2} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt{1.0} + \left|\left(\alpha + \beta\right) + i \cdot 2\right|}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{i}{\left(\alpha + \beta\right) + i \cdot 2} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt{1.0} + \left|\left(\alpha + \beta\right) + i \cdot 2\right|} \cdot \frac{i \cdot \frac{1}{2} + \left(\alpha + \beta\right) \cdot \frac{1}{4}}{\left|\left(\alpha + \beta\right) + i \cdot 2\right| - \sqrt{1.0}}\right)}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r12267453 = i;
        double r12267454 = alpha;
        double r12267455 = beta;
        double r12267456 = r12267454 + r12267455;
        double r12267457 = r12267456 + r12267453;
        double r12267458 = r12267453 * r12267457;
        double r12267459 = r12267455 * r12267454;
        double r12267460 = r12267459 + r12267458;
        double r12267461 = r12267458 * r12267460;
        double r12267462 = 2.0;
        double r12267463 = r12267462 * r12267453;
        double r12267464 = r12267456 + r12267463;
        double r12267465 = r12267464 * r12267464;
        double r12267466 = r12267461 / r12267465;
        double r12267467 = 1.0;
        double r12267468 = r12267465 - r12267467;
        double r12267469 = r12267466 / r12267468;
        return r12267469;
}

↓

double f(double alpha, double beta, double i) {
        double r12267470 = i;
        double r12267471 = 6.263709625600619e+146;
        bool r12267472 = r12267470 <= r12267471;
        double r12267473 = alpha;
        double r12267474 = beta;
        double r12267475 = r12267473 * r12267474;
        double r12267476 = r12267473 + r12267474;
        double r12267477 = r12267476 + r12267470;
        double r12267478 = r12267470 * r12267477;
        double r12267479 = r12267475 + r12267478;
        double r12267480 = 2.0;
        double r12267481 = r12267470 * r12267480;
        double r12267482 = r12267476 + r12267481;
        double r12267483 = r12267479 / r12267482;
        double r12267484 = fabs(r12267482);
        double r12267485 = 1.0;
        double r12267486 = sqrt(r12267485);
        double r12267487 = r12267484 - r12267486;
        double r12267488 = r12267483 / r12267487;
        double r12267489 = r12267470 / r12267482;
        double r12267490 = r12267489 * r12267477;
        double r12267491 = r12267486 + r12267484;
        double r12267492 = r12267490 / r12267491;
        double r12267493 = r12267488 * r12267492;
        double r12267494 = 0.5;
        double r12267495 = r12267470 * r12267494;
        double r12267496 = 0.25;
        double r12267497 = r12267476 * r12267496;
        double r12267498 = r12267495 + r12267497;
        double r12267499 = r12267498 / r12267487;
        double r12267500 = r12267492 * r12267499;
        double r12267501 = log(r12267500);
        double r12267502 = exp(r12267501);
        double r12267503 = r12267472 ? r12267493 : r12267502;
        return r12267503;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 6.263709625600619e+146

   1. Initial program 41.9

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 41.9

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]

   4. Applied add-sqr-sqrt 41.9

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} - \sqrt{1.0} \cdot \sqrt{1.0}}\]

   5. Applied difference-of-squares 41.9

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}\right) \cdot \left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}\right)}}\]

   6. Applied *-un-lft-identity 41.9

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \color{blue}{\left(1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}}}{\left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}\right) \cdot \left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}\right)}\]

   7. Applied associate-*r* 41.9

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}\right) \cdot \left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}\right)}\]

   8. Applied *-un-lft-identity 41.9

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \color{blue}{\left(1 \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}\right) \cdot \left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}\right)}\]

   9. Applied associate-*r* 41.9

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot 1\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}\right) \cdot \left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}\right)}\]

   10. Applied times-frac 14.9

       \[\leadsto \frac{\color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot 1} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}\right) \cdot \left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}\right)}\]

   11. Applied times-frac 14.9

       \[\leadsto \color{blue}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot 1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}}}\]

   12. Simplified 14.9

       \[\leadsto \color{blue}{\frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}}\]

   13. Simplified 10.4

       \[\leadsto \frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}} \cdot \color{blue}{\frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}}}\]

   if 6.263709625600619e+146 < i

   1. Initial program 62.1

      \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 62.1

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]

   4. Applied add-sqr-sqrt 62.1

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} \cdot \sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} - \sqrt{1.0} \cdot \sqrt{1.0}}\]

   5. Applied difference-of-squares 62.1

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\color{blue}{\left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}\right) \cdot \left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}\right)}}\]

   6. Applied *-un-lft-identity 62.1

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \color{blue}{\left(1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right)}}}{\left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}\right) \cdot \left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}\right)}\]

   7. Applied associate-*r* 62.1

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\color{blue}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}\right) \cdot \left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}\right)}\]

   8. Applied *-un-lft-identity 62.1

      \[\leadsto \frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \color{blue}{\left(1 \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}\right) \cdot \left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}\right)}\]

   9. Applied associate-*r* 62.1

      \[\leadsto \frac{\frac{\color{blue}{\left(\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot 1\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot 1\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}\right) \cdot \left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}\right)}\]

   10. Applied times-frac 60.2

       \[\leadsto \frac{\color{blue}{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot 1} \cdot \frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}\right) \cdot \left(\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}\right)}\]

   11. Applied times-frac 60.2

       \[\leadsto \color{blue}{\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot 1}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot 1}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} + \sqrt{1.0}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}}}\]

   12. Simplified 60.2

       \[\leadsto \color{blue}{\frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}}} \cdot \frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)} - \sqrt{1.0}}\]

   13. Simplified 60.1

       \[\leadsto \frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}} \cdot \color{blue}{\frac{\frac{\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}}}\]

   14. Taylor expanded around 0 10.1

       \[\leadsto \frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}} \cdot \frac{\color{blue}{\frac{1}{2} \cdot i + \left(\frac{1}{4} \cdot \beta + \frac{1}{4} \cdot \alpha\right)}}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}}\]

   15. Simplified 10.1

       \[\leadsto \frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}} \cdot \frac{\color{blue}{\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)}}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}}\]
   16. Using strategy rm

   17. Applied add-exp-log 15.5

       \[\leadsto \frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}} \cdot \frac{\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)}{\color{blue}{e^{\log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}\right)}}}\]

   18. Applied add-exp-log 14.4

       \[\leadsto \frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}} \cdot \frac{\color{blue}{e^{\log \left(\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)\right)}}}{e^{\log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}\right)}}\]

   19. Applied div-exp 14.4

       \[\leadsto \frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}} \cdot \color{blue}{e^{\log \left(\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)\right) - \log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}\right)}}\]

   20. Applied add-exp-log 15.6

       \[\leadsto \frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\color{blue}{e^{\log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}\right)}}} \cdot e^{\log \left(\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)\right) - \log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}\right)}\]

   21. Applied add-exp-log 16.2

       \[\leadsto \frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \color{blue}{e^{\log \left(\left(\alpha + \beta\right) + i\right)}}}{e^{\log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}\right)}} \cdot e^{\log \left(\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)\right) - \log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}\right)}\]

   22. Applied add-exp-log 16.6

       \[\leadsto \frac{\frac{i}{\color{blue}{e^{\log \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}} \cdot e^{\log \left(\left(\alpha + \beta\right) + i\right)}}{e^{\log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}\right)}} \cdot e^{\log \left(\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)\right) - \log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}\right)}\]

   23. Applied add-exp-log 16.8

       \[\leadsto \frac{\frac{\color{blue}{e^{\log i}}}{e^{\log \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \cdot e^{\log \left(\left(\alpha + \beta\right) + i\right)}}{e^{\log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}\right)}} \cdot e^{\log \left(\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)\right) - \log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}\right)}\]

   24. Applied div-exp 16.8

       \[\leadsto \frac{\color{blue}{e^{\log i - \log \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}} \cdot e^{\log \left(\left(\alpha + \beta\right) + i\right)}}{e^{\log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}\right)}} \cdot e^{\log \left(\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)\right) - \log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}\right)}\]

   25. Applied prod-exp 16.8

       \[\leadsto \frac{\color{blue}{e^{\left(\log i - \log \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) + \log \left(\left(\alpha + \beta\right) + i\right)}}}{e^{\log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}\right)}} \cdot e^{\log \left(\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)\right) - \log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}\right)}\]

   26. Applied div-exp 16.8

       \[\leadsto \color{blue}{e^{\left(\left(\log i - \log \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) + \log \left(\left(\alpha + \beta\right) + i\right)\right) - \log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}\right)}} \cdot e^{\log \left(\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)\right) - \log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}\right)}\]

   27. Applied prod-exp 16.8

       \[\leadsto \color{blue}{e^{\left(\left(\left(\log i - \log \left(\left(\alpha + \beta\right) + 2 \cdot i\right)\right) + \log \left(\left(\alpha + \beta\right) + i\right)\right) - \log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}\right)\right) + \left(\log \left(\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)\right) - \log \left(\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}\right)\right)}}\]

   28. Simplified 10.1

       \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{i}{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| + \sqrt{1.0}} \cdot \frac{\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\beta + \alpha\right)}{\left|\left(\alpha + \beta\right) + 2 \cdot i\right| - \sqrt{1.0}}\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 10.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 6.263709625600619 \cdot 10^{+146}:\\ \;\;\;\;\frac{\frac{\alpha \cdot \beta + i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\left|\left(\alpha + \beta\right) + i \cdot 2\right| - \sqrt{1.0}} \cdot \frac{\frac{i}{\left(\alpha + \beta\right) + i \cdot 2} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt{1.0} + \left|\left(\alpha + \beta\right) + i \cdot 2\right|}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{\frac{i}{\left(\alpha + \beta\right) + i \cdot 2} \cdot \left(\left(\alpha + \beta\right) + i\right)}{\sqrt{1.0} + \left|\left(\alpha + \beta\right) + i \cdot 2\right|} \cdot \frac{i \cdot \frac{1}{2} + \left(\alpha + \beta\right) \cdot \frac{1}{4}}{\left|\left(\alpha + \beta\right) + i \cdot 2\right| - \sqrt{1.0}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019158 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))