Octave 3.8, jcobi/4

Average Error: 54.4 → 10.4

Time: 42.7s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 7.507322636276203348538938900712812441086 \cdot 10^{112}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{1}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \frac{\frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\left(\alpha + \beta\right) + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{1}}}{\frac{2}{i}} \cdot \frac{\frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\left(\alpha + \beta\right) + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r266439 = i;
        double r266440 = alpha;
        double r266441 = beta;
        double r266442 = r266440 + r266441;
        double r266443 = r266442 + r266439;
        double r266444 = r266439 * r266443;
        double r266445 = r266441 * r266440;
        double r266446 = r266445 + r266444;
        double r266447 = r266444 * r266446;
        double r266448 = 2.0;
        double r266449 = r266448 * r266439;
        double r266450 = r266442 + r266449;
        double r266451 = r266450 * r266450;
        double r266452 = r266447 / r266451;
        double r266453 = 1.0;
        double r266454 = r266451 - r266453;
        double r266455 = r266452 / r266454;
        return r266455;
}

↓

double f(double alpha, double beta, double i) {
        double r266456 = i;
        double r266457 = 7.507322636276203e+112;
        bool r266458 = r266456 <= r266457;
        double r266459 = 1.0;
        double r266460 = alpha;
        double r266461 = beta;
        double r266462 = r266460 + r266461;
        double r266463 = 2.0;
        double r266464 = r266463 * r266456;
        double r266465 = r266462 + r266464;
        double r266466 = 1.0;
        double r266467 = sqrt(r266466);
        double r266468 = r266465 + r266467;
        double r266469 = r266468 / r266459;
        double r266470 = r266459 / r266469;
        double r266471 = fma(r266456, r266463, r266462);
        double r266472 = r266462 + r266456;
        double r266473 = r266456 * r266472;
        double r266474 = fma(r266461, r266460, r266473);
        double r266475 = r266471 / r266474;
        double r266476 = r266470 / r266475;
        double r266477 = r266465 - r266467;
        double r266478 = r266477 / r266472;
        double r266479 = r266456 / r266478;
        double r266480 = r266479 / r266471;
        double r266481 = r266476 * r266480;
        double r266482 = r266463 / r266456;
        double r266483 = r266470 / r266482;
        double r266484 = r266483 * r266480;
        double r266485 = r266458 ? r266481 : r266484;
        return r266485;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 7.507322636276203e+112

   1. Initial program 37.4

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

   2. Simplified 34.0

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

   4. Applied associate-/l* 23.6

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

   6. Applied *-un-lft-identity 23.6

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

   7. Applied *-un-lft-identity 23.6

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

   8. Applied times-frac 23.6

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

   9. Applied times-frac 17.8

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

   10. Applied associate-/r* 14.4

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

   11. Simplified 14.3

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

   13. Applied associate-/r/ 14.5

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

   14. Applied *-un-lft-identity 14.5

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

   15. Applied add-sqr-sqrt 14.5

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

   16. Applied difference-of-squares 14.5

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

   17. Applied times-frac 10.0

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

   18. Applied *-un-lft-identity 10.0

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

   19. Applied times-frac 10.0

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

   20. Applied times-frac 10.0

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

   if 7.507322636276203e+112 < i

   1. Initial program 64.0

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

   2. Simplified 63.7

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

   4. Applied associate-/l* 63.7

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

   6. Applied *-un-lft-identity 63.7

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

   7. Applied *-un-lft-identity 63.7

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

   8. Applied times-frac 63.7

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

   9. Applied times-frac 54.8

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

   10. Applied associate-/r* 54.6

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

   11. Simplified 54.6

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

   13. Applied associate-/r/ 54.6

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

   14. Applied *-un-lft-identity 54.6

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

   15. Applied add-sqr-sqrt 54.6

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

   16. Applied difference-of-squares 54.6

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

   17. Applied times-frac 54.0

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

   18. Applied *-un-lft-identity 54.0

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

   19. Applied times-frac 53.9

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

   20. Applied times-frac 53.9

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

   21. Taylor expanded around inf 10.6

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

4. Final simplification 10.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 7.507322636276203348538938900712812441086 \cdot 10^{112}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{1}}}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}} \cdot \frac{\frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\left(\alpha + \beta\right) + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{1}}}{\frac{2}{i}} \cdot \frac{\frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}{\left(\alpha + \beta\right) + i}}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019354 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :precision binary64
  :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)))