Octave 3.8, jcobi/2

Average Error: 23.5 → 11.1

Time: 37.0s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]

Math

\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 9.677751062610224 \cdot 10^{+196}:\\ \;\;\;\;\frac{1.0 + \sqrt[3]{\left(\frac{\beta + \alpha}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) + 2.0}\right) \cdot \left(\left(\frac{\beta + \alpha}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) + 2.0}\right) \cdot \left(\left(\frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) + 2.0} \cdot \frac{\beta + \alpha}{\sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}\right) \cdot \frac{1}{\sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r3269421 = alpha;
        double r3269422 = beta;
        double r3269423 = r3269421 + r3269422;
        double r3269424 = r3269422 - r3269421;
        double r3269425 = r3269423 * r3269424;
        double r3269426 = 2.0;
        double r3269427 = i;
        double r3269428 = r3269426 * r3269427;
        double r3269429 = r3269423 + r3269428;
        double r3269430 = r3269425 / r3269429;
        double r3269431 = 2.0;
        double r3269432 = r3269429 + r3269431;
        double r3269433 = r3269430 / r3269432;
        double r3269434 = 1.0;
        double r3269435 = r3269433 + r3269434;
        double r3269436 = r3269435 / r3269431;
        return r3269436;
}

↓

double f(double alpha, double beta, double i) {
        double r3269437 = alpha;
        double r3269438 = 9.677751062610224e+196;
        bool r3269439 = r3269437 <= r3269438;
        double r3269440 = 1.0;
        double r3269441 = beta;
        double r3269442 = r3269441 + r3269437;
        double r3269443 = i;
        double r3269444 = 2.0;
        double r3269445 = fma(r3269443, r3269444, r3269441);
        double r3269446 = r3269437 + r3269445;
        double r3269447 = r3269442 / r3269446;
        double r3269448 = r3269441 - r3269437;
        double r3269449 = 2.0;
        double r3269450 = r3269446 + r3269449;
        double r3269451 = r3269448 / r3269450;
        double r3269452 = r3269447 * r3269451;
        double r3269453 = cbrt(r3269446);
        double r3269454 = r3269442 / r3269453;
        double r3269455 = r3269451 * r3269454;
        double r3269456 = 1.0;
        double r3269457 = r3269453 * r3269453;
        double r3269458 = r3269456 / r3269457;
        double r3269459 = r3269455 * r3269458;
        double r3269460 = r3269452 * r3269459;
        double r3269461 = r3269452 * r3269460;
        double r3269462 = cbrt(r3269461);
        double r3269463 = r3269440 + r3269462;
        double r3269464 = r3269463 / r3269449;
        double r3269465 = r3269449 / r3269437;
        double r3269466 = 8.0;
        double r3269467 = r3269437 * r3269437;
        double r3269468 = r3269437 * r3269467;
        double r3269469 = r3269466 / r3269468;
        double r3269470 = 4.0;
        double r3269471 = r3269470 / r3269467;
        double r3269472 = r3269469 - r3269471;
        double r3269473 = r3269465 + r3269472;
        double r3269474 = r3269473 / r3269449;
        double r3269475 = r3269439 ? r3269464 : r3269474;
        return r3269475;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 9.677751062610224e+196

   1. Initial program 18.2

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   2. Simplified 18.2

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} + 1.0}{2.0}}\]
   3. Using strategy rm

   4. Applied add-cbrt-cube 25.8

      \[\leadsto \frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}}} + 1.0}{2.0}\]

   5. Applied add-cbrt-cube 28.1

      \[\leadsto \frac{\frac{\left(\beta + \alpha\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}}}{\sqrt[3]{\left(\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}} + 1.0}{2.0}\]

   6. Applied add-cbrt-cube 28.2

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)\right) \cdot \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}}{\sqrt[3]{\left(\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}} + 1.0}{2.0}\]

   7. Applied cbrt-unprod 30.6

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)\right) \cdot \left(\beta + \alpha\right)\right) \cdot \left(\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)\right)}}}{\sqrt[3]{\left(\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}} + 1.0}{2.0}\]

   8. Applied cbrt-undiv 30.6

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{\left(\left(\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)\right) \cdot \left(\beta + \alpha\right)\right) \cdot \left(\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}}} + 1.0}{2.0}\]

   9. Simplified 6.9

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(\left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right) \cdot \left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)\right) \cdot \left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)}} + 1.0}{2.0}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 6.9

       \[\leadsto \frac{\sqrt[3]{\left(\left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right) \cdot \left(\frac{\alpha + \beta}{\color{blue}{\left(\sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right) \cdot \sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)\right) \cdot \left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)} + 1.0}{2.0}\]

   12. Applied *-un-lft-identity 6.9

       \[\leadsto \frac{\sqrt[3]{\left(\left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right) \cdot \left(\frac{\color{blue}{1 \cdot \left(\alpha + \beta\right)}}{\left(\sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}\right) \cdot \sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)\right) \cdot \left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)} + 1.0}{2.0}\]

   13. Applied times-frac 6.9

       \[\leadsto \frac{\sqrt[3]{\left(\left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right) \cdot \left(\color{blue}{\left(\frac{1}{\sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}} \cdot \frac{\alpha + \beta}{\sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)\right) \cdot \left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)} + 1.0}{2.0}\]

   14. Applied associate-*l* 6.9

       \[\leadsto \frac{\sqrt[3]{\left(\left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right) \cdot \color{blue}{\left(\frac{1}{\sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}} \cdot \left(\frac{\alpha + \beta}{\sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)\right)}\right) \cdot \left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)} + 1.0}{2.0}\]

   if 9.677751062610224e+196 < alpha

   1. Initial program 63.2

      \[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]

   2. Simplified 62.5

      \[\leadsto \color{blue}{\frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)} + 1.0}{2.0}}\]
   3. Using strategy rm

   4. Applied add-cbrt-cube 62.5

      \[\leadsto \frac{\frac{\left(\beta + \alpha\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}}} + 1.0}{2.0}\]

   5. Applied add-cbrt-cube 62.5

      \[\leadsto \frac{\frac{\left(\beta + \alpha\right) \cdot \color{blue}{\sqrt[3]{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}}}{\sqrt[3]{\left(\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}} + 1.0}{2.0}\]

   6. Applied add-cbrt-cube 62.5

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)\right) \cdot \left(\beta + \alpha\right)}} \cdot \sqrt[3]{\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)}}{\sqrt[3]{\left(\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}} + 1.0}{2.0}\]

   7. Applied cbrt-unprod 62.5

      \[\leadsto \frac{\frac{\color{blue}{\sqrt[3]{\left(\left(\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)\right) \cdot \left(\beta + \alpha\right)\right) \cdot \left(\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)\right)}}}{\sqrt[3]{\left(\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}} + 1.0}{2.0}\]

   8. Applied cbrt-undiv 62.5

      \[\leadsto \frac{\color{blue}{\sqrt[3]{\frac{\left(\left(\left(\beta + \alpha\right) \cdot \left(\beta + \alpha\right)\right) \cdot \left(\beta + \alpha\right)\right) \cdot \left(\left(\left(\beta - \alpha\right) \cdot \left(\beta - \alpha\right)\right) \cdot \left(\beta - \alpha\right)\right)}{\left(\mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)\right) \cdot \mathsf{fma}\left(2.0, \mathsf{fma}\left(2, i, \beta + \alpha\right), \mathsf{fma}\left(2, i, \beta + \alpha\right) \cdot \mathsf{fma}\left(2, i, \beta + \alpha\right)\right)}}} + 1.0}{2.0}\]

   9. Simplified 50.0

      \[\leadsto \frac{\sqrt[3]{\color{blue}{\left(\left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right) \cdot \left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)\right) \cdot \left(\frac{\alpha + \beta}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{2.0 + \left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right)}\right)}} + 1.0}{2.0}\]

   10. Taylor expanded around inf 42.6

       \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

   11. Simplified 42.6

       \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}}{2.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 9.677751062610224 \cdot 10^{+196}:\\ \;\;\;\;\frac{1.0 + \sqrt[3]{\left(\frac{\beta + \alpha}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) + 2.0}\right) \cdot \left(\left(\frac{\beta + \alpha}{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) + 2.0}\right) \cdot \left(\left(\frac{\beta - \alpha}{\left(\alpha + \mathsf{fma}\left(i, 2, \beta\right)\right) + 2.0} \cdot \frac{\beta + \alpha}{\sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}\right) \cdot \frac{1}{\sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)} \cdot \sqrt[3]{\alpha + \mathsf{fma}\left(i, 2, \beta\right)}}\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019149 +o rules:numerics
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))