Octave 3.8, jcobi/4

Average Error: 53.8 → 14.6

Time: 21.7s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 1.507581904402624758675010074327237763323 \cdot 10^{135}:\\ \;\;\;\;e^{\log \left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) - \left(\left(\left(\log \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) + \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) + \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(0.25, \frac{1}{{i}^{2}}, \log 0.0625\right)}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r135094 = i;
        double r135095 = alpha;
        double r135096 = beta;
        double r135097 = r135095 + r135096;
        double r135098 = r135097 + r135094;
        double r135099 = r135094 * r135098;
        double r135100 = r135096 * r135095;
        double r135101 = r135100 + r135099;
        double r135102 = r135099 * r135101;
        double r135103 = 2.0;
        double r135104 = r135103 * r135094;
        double r135105 = r135097 + r135104;
        double r135106 = r135105 * r135105;
        double r135107 = r135102 / r135106;
        double r135108 = 1.0;
        double r135109 = r135106 - r135108;
        double r135110 = r135107 / r135109;
        return r135110;
}

↓

double f(double alpha, double beta, double i) {
        double r135111 = i;
        double r135112 = 1.5075819044026248e+135;
        bool r135113 = r135111 <= r135112;
        double r135114 = alpha;
        double r135115 = beta;
        double r135116 = r135114 + r135115;
        double r135117 = r135116 + r135111;
        double r135118 = r135111 * r135117;
        double r135119 = log(r135118);
        double r135120 = 2.0;
        double r135121 = r135120 * r135111;
        double r135122 = r135116 + r135121;
        double r135123 = r135122 * r135122;
        double r135124 = 1.0;
        double r135125 = r135123 - r135124;
        double r135126 = log(r135125);
        double r135127 = fma(r135111, r135120, r135116);
        double r135128 = log(r135127);
        double r135129 = r135126 + r135128;
        double r135130 = r135129 + r135128;
        double r135131 = fma(r135115, r135114, r135118);
        double r135132 = log(r135131);
        double r135133 = r135130 - r135132;
        double r135134 = r135119 - r135133;
        double r135135 = exp(r135134);
        double r135136 = 0.25;
        double r135137 = 1.0;
        double r135138 = 2.0;
        double r135139 = pow(r135111, r135138);
        double r135140 = r135137 / r135139;
        double r135141 = 0.0625;
        double r135142 = log(r135141);
        double r135143 = fma(r135136, r135140, r135142);
        double r135144 = exp(r135143);
        double r135145 = r135113 ? r135135 : r135144;
        return r135145;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 1.5075819044026248e+135

   1. Initial program 40.6

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

   2. Simplified 38.2

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

   4. Applied add-exp-log 39.6

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

   5. Applied add-exp-log 39.5

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

   6. Applied add-exp-log 39.6

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

   7. Applied add-exp-log 39.8

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

   8. Applied prod-exp 40.0

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

   9. Applied prod-exp 39.8

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

   10. Applied div-exp 22.3

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

   11. Applied add-exp-log 22.1

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

   12. Applied add-exp-log 22.3

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

   13. Applied prod-exp 22.3

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

   14. Applied div-exp 20.0

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

   15. Simplified 38.2

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

   17. Applied add-exp-log 39.6

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

   18. Applied add-exp-log 39.5

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

   19. Applied add-exp-log 39.6

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

   20. Applied add-exp-log 39.8

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

   21. Applied prod-exp 40.0

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

   22. Applied prod-exp 39.8

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

   23. Applied div-exp 22.3

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

   24. Applied add-exp-log 22.3

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

   25. Applied div-exp 20.0

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

   26. Applied rem-log-exp 20.0

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

   if 1.5075819044026248e+135 < 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 63.8

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

   4. Applied add-exp-log 63.8

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

   5. Applied add-exp-log 63.8

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

   6. Applied add-exp-log 63.8

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

   7. Applied add-exp-log 63.8

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

   8. Applied prod-exp 63.8

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

   9. Applied prod-exp 63.8

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

   10. Applied div-exp 59.5

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

   11. Applied add-exp-log 59.5

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

   12. Applied add-exp-log 59.5

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

   13. Applied prod-exp 59.5

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

   14. Applied div-exp 59.3

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

   15. Simplified 63.8

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

   16. Taylor expanded around inf 10.5

       \[\leadsto e^{\color{blue}{0.25 \cdot \frac{1}{{i}^{2}} + \log 0.0625}}\]

   17. Simplified 10.5

       \[\leadsto e^{\color{blue}{\mathsf{fma}\left(0.25, \frac{1}{{i}^{2}}, \log 0.0625\right)}}\]
3. Recombined 2 regimes into one program.

4. Final simplification 14.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.507581904402624758675010074327237763323 \cdot 10^{135}:\\ \;\;\;\;e^{\log \left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) - \left(\left(\left(\log \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1\right) + \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) + \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\mathsf{fma}\left(0.25, \frac{1}{{i}^{2}}, \log 0.0625\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020002 +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)))