Octave 3.8, jcobi/4

Average Error: 54.4 → 11.2

Time: 17.9s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 1.24769679993065537 \cdot 10^{131}:\\ \;\;\;\;\frac{\sqrt{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\frac{\sqrt{i} \cdot \sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\frac{\sqrt{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.5, i, 0.25 \cdot \beta\right)\right)}}\right)\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r121955 = i;
        double r121956 = alpha;
        double r121957 = beta;
        double r121958 = r121956 + r121957;
        double r121959 = r121958 + r121955;
        double r121960 = r121955 * r121959;
        double r121961 = r121957 * r121956;
        double r121962 = r121961 + r121960;
        double r121963 = r121960 * r121962;
        double r121964 = 2.0;
        double r121965 = r121964 * r121955;
        double r121966 = r121958 + r121965;
        double r121967 = r121966 * r121966;
        double r121968 = r121963 / r121967;
        double r121969 = 1.0;
        double r121970 = r121967 - r121969;
        double r121971 = r121968 / r121970;
        return r121971;
}

↓

double f(double alpha, double beta, double i) {
        double r121972 = i;
        double r121973 = 1.2476967999306554e+131;
        bool r121974 = r121972 <= r121973;
        double r121975 = sqrt(r121972);
        double r121976 = alpha;
        double r121977 = beta;
        double r121978 = r121976 + r121977;
        double r121979 = 2.0;
        double r121980 = r121979 * r121972;
        double r121981 = r121978 + r121980;
        double r121982 = 1.0;
        double r121983 = sqrt(r121982);
        double r121984 = r121981 + r121983;
        double r121985 = r121975 / r121984;
        double r121986 = r121978 + r121972;
        double r121987 = r121972 * r121986;
        double r121988 = fma(r121977, r121976, r121987);
        double r121989 = sqrt(r121988);
        double r121990 = r121975 * r121989;
        double r121991 = r121981 - r121983;
        double r121992 = r121990 / r121991;
        double r121993 = fma(r121972, r121979, r121978);
        double r121994 = r121989 / r121993;
        double r121995 = r121993 / r121994;
        double r121996 = r121986 / r121995;
        double r121997 = r121992 * r121996;
        double r121998 = r121985 * r121997;
        double r121999 = r121975 / r121991;
        double r122000 = 0.25;
        double r122001 = 0.5;
        double r122002 = r122000 * r121977;
        double r122003 = fma(r122001, r121972, r122002);
        double r122004 = fma(r122000, r121976, r122003);
        double r122005 = r121993 / r122004;
        double r122006 = r121986 / r122005;
        double r122007 = r121999 * r122006;
        double r122008 = r121985 * r122007;
        double r122009 = r121974 ? r121998 : r122008;
        return r122009;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 2 regimes

2. if i < 1.2476967999306554e+131

   1. Initial program 40.9

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

   2. Simplified 37.5

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

   4. Applied associate-/l* 28.5

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

   6. Applied *-un-lft-identity 28.5

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

   7. Applied *-un-lft-identity 28.5

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

   8. Applied times-frac 28.5

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

   9. Applied times-frac 18.1

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

   10. Applied times-frac 15.2

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

   11. Simplified 15.2

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

   13. Applied add-sqr-sqrt 15.2

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

   14. Applied difference-of-squares 15.2

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

   15. Applied add-sqr-sqrt 15.4

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

   16. Applied times-frac 12.1

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

   17. Applied associate-*l* 10.5

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

   19. Applied *-un-lft-identity 10.5

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

   20. Applied add-sqr-sqrt 10.6

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

   21. Applied times-frac 10.5

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

   22. Applied *-un-lft-identity 10.5

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

   23. Applied times-frac 10.6

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

   24. Applied *-un-lft-identity 10.6

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

   25. Applied times-frac 10.6

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

   26. Applied associate-*r* 10.6

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

   27. Simplified 10.5

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

   if 1.2476967999306554e+131 < 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 associate-/l* 63.8

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

   6. Applied *-un-lft-identity 63.8

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

   7. Applied *-un-lft-identity 63.8

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

   8. Applied times-frac 63.8

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

   9. Applied times-frac 58.5

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

   10. Applied times-frac 58.4

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

   11. Simplified 58.4

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

   13. Applied add-sqr-sqrt 58.4

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

   14. Applied difference-of-squares 58.4

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

   15. Applied add-sqr-sqrt 58.5

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

   16. Applied times-frac 58.3

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

   17. Applied associate-*l* 58.3

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

   18. Taylor expanded around 0 11.7

       \[\leadsto \frac{\sqrt{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\frac{\sqrt{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\color{blue}{0.25 \cdot \alpha + \left(0.5 \cdot i + 0.25 \cdot \beta\right)}}}\right)\]

   19. Simplified 11.7

       \[\leadsto \frac{\sqrt{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\frac{\sqrt{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\color{blue}{\mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.5, i, 0.25 \cdot \beta\right)\right)}}}\right)\]
3. Recombined 2 regimes into one program.

4. Final simplification 11.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.24769679993065537 \cdot 10^{131}:\\ \;\;\;\;\frac{\sqrt{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\frac{\sqrt{i} \cdot \sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\frac{\sqrt{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}} \cdot \left(\frac{\sqrt{i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}} \cdot \frac{\left(\alpha + \beta\right) + i}{\frac{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}{\mathsf{fma}\left(0.25, \alpha, \mathsf{fma}\left(0.5, i, 0.25 \cdot \beta\right)\right)}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020089 +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)))