Octave 3.8, jcobi/2

Average Error: 23.8 → 11.4

Time: 26.1s

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 1.3944789768910781 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{e^{\log \left(\mathsf{fma}\left(\left(1.0 \cdot 1.0\right), 1.0, \left(\left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right) \cdot \left(\left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right) \cdot \left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(\left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right), \left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right), \left(1.0 \cdot \left(1.0 - \frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right)\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{80.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \frac{6.0}{\alpha}\right) - \frac{24.0}{\alpha \cdot \alpha}}{\mathsf{fma}\left(\left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right), \left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right), \left(1.0 \cdot \left(1.0 - \frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right)\right)\right)}}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r2558412 = alpha;
        double r2558413 = beta;
        double r2558414 = r2558412 + r2558413;
        double r2558415 = r2558413 - r2558412;
        double r2558416 = r2558414 * r2558415;
        double r2558417 = 2.0;
        double r2558418 = i;
        double r2558419 = r2558417 * r2558418;
        double r2558420 = r2558414 + r2558419;
        double r2558421 = r2558416 / r2558420;
        double r2558422 = 2.0;
        double r2558423 = r2558420 + r2558422;
        double r2558424 = r2558421 / r2558423;
        double r2558425 = 1.0;
        double r2558426 = r2558424 + r2558425;
        double r2558427 = r2558426 / r2558422;
        return r2558427;
}

↓

double f(double alpha, double beta, double i) {
        double r2558428 = alpha;
        double r2558429 = 1.3944789768910781e+137;
        bool r2558430 = r2558428 <= r2558429;
        double r2558431 = 1.0;
        double r2558432 = r2558431 * r2558431;
        double r2558433 = beta;
        double r2558434 = r2558433 + r2558428;
        double r2558435 = 2.0;
        double r2558436 = 2.0;
        double r2558437 = i;
        double r2558438 = fma(r2558436, r2558437, r2558434);
        double r2558439 = r2558435 + r2558438;
        double r2558440 = r2558434 / r2558439;
        double r2558441 = r2558433 - r2558428;
        double r2558442 = r2558441 / r2558438;
        double r2558443 = r2558440 * r2558442;
        double r2558444 = r2558443 * r2558443;
        double r2558445 = r2558443 * r2558444;
        double r2558446 = fma(r2558432, r2558431, r2558445);
        double r2558447 = log(r2558446);
        double r2558448 = exp(r2558447);
        double r2558449 = r2558431 - r2558443;
        double r2558450 = r2558431 * r2558449;
        double r2558451 = fma(r2558443, r2558443, r2558450);
        double r2558452 = r2558448 / r2558451;
        double r2558453 = r2558452 / r2558435;
        double r2558454 = 80.0;
        double r2558455 = r2558428 * r2558428;
        double r2558456 = r2558428 * r2558455;
        double r2558457 = r2558454 / r2558456;
        double r2558458 = 6.0;
        double r2558459 = r2558458 / r2558428;
        double r2558460 = r2558457 + r2558459;
        double r2558461 = 24.0;
        double r2558462 = r2558461 / r2558455;
        double r2558463 = r2558460 - r2558462;
        double r2558464 = r2558463 / r2558451;
        double r2558465 = r2558464 / r2558435;
        double r2558466 = r2558430 ? r2558453 : r2558465;
        return r2558466;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if alpha < 1.3944789768910781e+137

   1. Initial program 15.2

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

   2. Simplified 4.5

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

   4. Applied fma-udef 4.5

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

   6. Applied flip3-+ 4.5

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

   7. Simplified 4.5

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

   8. Simplified 4.5

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

   10. Applied add-exp-log 4.5

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

   if 1.3944789768910781e+137 < alpha

   1. Initial program 61.4

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

   2. Simplified 45.4

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

   4. Applied fma-udef 45.4

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

   6. Applied flip3-+ 45.4

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

   7. Simplified 45.4

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

   8. Simplified 45.4

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

   9. Taylor expanded around inf 41.7

      \[\leadsto \frac{\frac{\color{blue}{\left(6.0 \cdot \frac{1}{\alpha} + 80.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 24.0 \cdot \frac{1}{{\alpha}^{2}}}}{\mathsf{fma}\left(\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)} \cdot \frac{\alpha + \beta}{2.0 + \mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}\right), \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)} \cdot \frac{\alpha + \beta}{2.0 + \mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}\right), \left(1.0 \cdot \left(1.0 - \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)} \cdot \frac{\alpha + \beta}{2.0 + \mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}\right)\right)\right)}}{2.0}\]

   10. Simplified 41.7

       \[\leadsto \frac{\frac{\color{blue}{\left(\frac{6.0}{\alpha} + \frac{80.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)}\right) - \frac{24.0}{\alpha \cdot \alpha}}}{\mathsf{fma}\left(\left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)} \cdot \frac{\alpha + \beta}{2.0 + \mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}\right), \left(\frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)} \cdot \frac{\alpha + \beta}{2.0 + \mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}\right), \left(1.0 \cdot \left(1.0 - \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)} \cdot \frac{\alpha + \beta}{2.0 + \mathsf{fma}\left(2, i, \left(\alpha + \beta\right)\right)}\right)\right)\right)}}{2.0}\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.3944789768910781 \cdot 10^{+137}:\\ \;\;\;\;\frac{\frac{e^{\log \left(\mathsf{fma}\left(\left(1.0 \cdot 1.0\right), 1.0, \left(\left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right) \cdot \left(\left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right) \cdot \left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right)\right)\right)\right)\right)}}{\mathsf{fma}\left(\left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right), \left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right), \left(1.0 \cdot \left(1.0 - \frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right)\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\left(\frac{80.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \frac{6.0}{\alpha}\right) - \frac{24.0}{\alpha \cdot \alpha}}{\mathsf{fma}\left(\left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right), \left(\frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right), \left(1.0 \cdot \left(1.0 - \frac{\beta + \alpha}{2.0 + \mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)} \cdot \frac{\beta - \alpha}{\mathsf{fma}\left(2, i, \left(\beta + \alpha\right)\right)}\right)\right)\right)}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019130 +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))