Octave 3.8, jcobi/4

Average Error: 54.1 → 12.0

Time: 32.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 1.647212354283295028652665553828540921944 \cdot 10^{79}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\left(i \cdot 0.5\right) \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\alpha + \left(\sqrt{1} + \left(\beta + i \cdot 2\right)\right)}}{\frac{\left(\beta + i \cdot 2\right) + \left(\alpha - \sqrt{1}\right)}{i}}\right)}\\ \mathbf{elif}\;\beta \le 4.444149846130252102755247128183742063862 \cdot 10^{156}:\\ \;\;\;\;\frac{\frac{\sqrt{i \cdot \left(\left(i + \alpha\right) + \beta\right) + \beta \cdot \alpha}}{\left(\beta + i \cdot 2\right) + \alpha}}{\frac{\left(\beta + i \cdot 2\right) + \left(\alpha - \sqrt{1}\right)}{i}} \cdot \frac{\sqrt{i \cdot \left(\left(i + \alpha\right) + \beta\right) + \beta \cdot \alpha}}{\left(\left(\beta + i \cdot 2\right) + \alpha\right) \cdot \frac{\left(\alpha + \sqrt{1}\right) + \left(\beta + i \cdot 2\right)}{\left(i + \alpha\right) + \beta}}\\ \mathbf{elif}\;\beta \le 3.185312918840883354690651673329007970501 \cdot 10^{198}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\left(i \cdot 0.5\right) \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\alpha + \left(\sqrt{1} + \left(\beta + i \cdot 2\right)\right)}}{\frac{\left(\beta + i \cdot 2\right) + \left(\alpha - \sqrt{1}\right)}{i}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\left(\alpha + \sqrt{1}\right) + \left(\beta + i \cdot 2\right)} \cdot \frac{\left(i + \alpha\right) + \beta}{\left(\beta + i \cdot 2\right) + \alpha}}{\frac{\left(\left(\beta + i \cdot 2\right) + \alpha\right) - \sqrt{1}}{i}}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r135895 = i;
        double r135896 = alpha;
        double r135897 = beta;
        double r135898 = r135896 + r135897;
        double r135899 = r135898 + r135895;
        double r135900 = r135895 * r135899;
        double r135901 = r135897 * r135896;
        double r135902 = r135901 + r135900;
        double r135903 = r135900 * r135902;
        double r135904 = 2.0;
        double r135905 = r135904 * r135895;
        double r135906 = r135898 + r135905;
        double r135907 = r135906 * r135906;
        double r135908 = r135903 / r135907;
        double r135909 = 1.0;
        double r135910 = r135907 - r135909;
        double r135911 = r135908 / r135910;
        return r135911;
}

↓

double f(double alpha, double beta, double i) {
        double r135912 = beta;
        double r135913 = 1.647212354283295e+79;
        bool r135914 = r135912 <= r135913;
        double r135915 = i;
        double r135916 = 0.5;
        double r135917 = r135915 * r135916;
        double r135918 = alpha;
        double r135919 = r135912 + r135918;
        double r135920 = r135915 + r135919;
        double r135921 = 2.0;
        double r135922 = r135915 * r135921;
        double r135923 = r135919 + r135922;
        double r135924 = r135920 / r135923;
        double r135925 = r135917 * r135924;
        double r135926 = 1.0;
        double r135927 = sqrt(r135926);
        double r135928 = r135912 + r135922;
        double r135929 = r135927 + r135928;
        double r135930 = r135918 + r135929;
        double r135931 = r135925 / r135930;
        double r135932 = r135918 - r135927;
        double r135933 = r135928 + r135932;
        double r135934 = r135933 / r135915;
        double r135935 = r135931 / r135934;
        double r135936 = log(r135935);
        double r135937 = exp(r135936);
        double r135938 = 4.444149846130252e+156;
        bool r135939 = r135912 <= r135938;
        double r135940 = r135915 + r135918;
        double r135941 = r135940 + r135912;
        double r135942 = r135915 * r135941;
        double r135943 = r135912 * r135918;
        double r135944 = r135942 + r135943;
        double r135945 = sqrt(r135944);
        double r135946 = r135928 + r135918;
        double r135947 = r135945 / r135946;
        double r135948 = r135947 / r135934;
        double r135949 = r135918 + r135927;
        double r135950 = r135949 + r135928;
        double r135951 = r135950 / r135941;
        double r135952 = r135946 * r135951;
        double r135953 = r135945 / r135952;
        double r135954 = r135948 * r135953;
        double r135955 = 3.1853129188408834e+198;
        bool r135956 = r135912 <= r135955;
        double r135957 = r135940 / r135950;
        double r135958 = r135941 / r135946;
        double r135959 = r135957 * r135958;
        double r135960 = r135946 - r135927;
        double r135961 = r135960 / r135915;
        double r135962 = r135959 / r135961;
        double r135963 = r135956 ? r135937 : r135962;
        double r135964 = r135939 ? r135954 : r135963;
        double r135965 = r135914 ? r135937 : r135964;
        return r135965;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if beta < 1.647212354283295e+79 or 4.444149846130252e+156 < beta < 3.1853129188408834e+198

   1. Initial program 52.3

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

   2. Simplified 37.5

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

   4. Applied add-sqr-sqrt 37.5

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

   5. Applied difference-of-squares 37.5

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

   6. Applied times-frac 36.1

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

   7. Applied associate-/r* 35.4

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

   8. Simplified 35.4

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

   9. Taylor expanded around inf 9.1

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

   10. Simplified 9.1

       \[\leadsto \frac{\frac{\color{blue}{i \cdot 0.5}}{\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)} \cdot \frac{\left(i + \alpha\right) + \beta}{\alpha + \left(\beta + i \cdot 2\right)}}{\frac{\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}}{i}}\]
   11. Using strategy rm

   12. Applied add-exp-log 13.7

       \[\leadsto \frac{\frac{i \cdot 0.5}{\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)} \cdot \frac{\left(i + \alpha\right) + \beta}{\alpha + \left(\beta + i \cdot 2\right)}}{\frac{\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}}{\color{blue}{e^{\log i}}}}\]

   13. Applied add-exp-log 13.5

       \[\leadsto \frac{\frac{i \cdot 0.5}{\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)} \cdot \frac{\left(i + \alpha\right) + \beta}{\alpha + \left(\beta + i \cdot 2\right)}}{\frac{\color{blue}{e^{\log \left(\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}\right)}}}{e^{\log i}}}\]

   14. Applied div-exp 13.4

       \[\leadsto \frac{\frac{i \cdot 0.5}{\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)} \cdot \frac{\left(i + \alpha\right) + \beta}{\alpha + \left(\beta + i \cdot 2\right)}}{\color{blue}{e^{\log \left(\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}\right) - \log i}}}\]

   15. Applied add-exp-log 14.3

       \[\leadsto \frac{\frac{i \cdot 0.5}{\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)} \cdot \frac{\left(i + \alpha\right) + \beta}{\color{blue}{e^{\log \left(\alpha + \left(\beta + i \cdot 2\right)\right)}}}}{e^{\log \left(\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}\right) - \log i}}\]

   16. Applied add-exp-log 14.2

       \[\leadsto \frac{\frac{i \cdot 0.5}{\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)} \cdot \frac{\color{blue}{e^{\log \left(\left(i + \alpha\right) + \beta\right)}}}{e^{\log \left(\alpha + \left(\beta + i \cdot 2\right)\right)}}}{e^{\log \left(\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}\right) - \log i}}\]

   17. Applied div-exp 14.2

       \[\leadsto \frac{\frac{i \cdot 0.5}{\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)} \cdot \color{blue}{e^{\log \left(\left(i + \alpha\right) + \beta\right) - \log \left(\alpha + \left(\beta + i \cdot 2\right)\right)}}}{e^{\log \left(\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}\right) - \log i}}\]

   18. Applied add-exp-log 14.6

       \[\leadsto \frac{\frac{i \cdot 0.5}{\color{blue}{e^{\log \left(\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)\right)}}} \cdot e^{\log \left(\left(i + \alpha\right) + \beta\right) - \log \left(\alpha + \left(\beta + i \cdot 2\right)\right)}}{e^{\log \left(\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}\right) - \log i}}\]

   19. Applied add-exp-log 14.6

       \[\leadsto \frac{\frac{i \cdot \color{blue}{e^{\log 0.5}}}{e^{\log \left(\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)\right)}} \cdot e^{\log \left(\left(i + \alpha\right) + \beta\right) - \log \left(\alpha + \left(\beta + i \cdot 2\right)\right)}}{e^{\log \left(\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}\right) - \log i}}\]

   20. Applied add-exp-log 14.6

       \[\leadsto \frac{\frac{\color{blue}{e^{\log i}} \cdot e^{\log 0.5}}{e^{\log \left(\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)\right)}} \cdot e^{\log \left(\left(i + \alpha\right) + \beta\right) - \log \left(\alpha + \left(\beta + i \cdot 2\right)\right)}}{e^{\log \left(\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}\right) - \log i}}\]

   21. Applied prod-exp 14.8

       \[\leadsto \frac{\frac{\color{blue}{e^{\log i + \log 0.5}}}{e^{\log \left(\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)\right)}} \cdot e^{\log \left(\left(i + \alpha\right) + \beta\right) - \log \left(\alpha + \left(\beta + i \cdot 2\right)\right)}}{e^{\log \left(\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}\right) - \log i}}\]

   22. Applied div-exp 14.8

       \[\leadsto \frac{\color{blue}{e^{\left(\log i + \log 0.5\right) - \log \left(\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)\right)}} \cdot e^{\log \left(\left(i + \alpha\right) + \beta\right) - \log \left(\alpha + \left(\beta + i \cdot 2\right)\right)}}{e^{\log \left(\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}\right) - \log i}}\]

   23. Applied prod-exp 14.8

       \[\leadsto \frac{\color{blue}{e^{\left(\left(\log i + \log 0.5\right) - \log \left(\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)\right)\right) + \left(\log \left(\left(i + \alpha\right) + \beta\right) - \log \left(\alpha + \left(\beta + i \cdot 2\right)\right)\right)}}}{e^{\log \left(\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}\right) - \log i}}\]

   24. Applied div-exp 14.8

       \[\leadsto \color{blue}{e^{\left(\left(\left(\log i + \log 0.5\right) - \log \left(\left(\beta + i \cdot 2\right) + \left(\alpha + \sqrt{1}\right)\right)\right) + \left(\log \left(\left(i + \alpha\right) + \beta\right) - \log \left(\alpha + \left(\beta + i \cdot 2\right)\right)\right)\right) - \left(\log \left(\left(\left(2 \cdot i + \beta\right) + \alpha\right) - \sqrt{1}\right) - \log i\right)}}\]

   25. Simplified 9.1

       \[\leadsto e^{\color{blue}{\log \left(\frac{\frac{\left(i \cdot 0.5\right) \cdot \frac{i + \left(\alpha + \beta\right)}{2 \cdot i + \left(\beta + \alpha\right)}}{\left(\left(\beta + 2 \cdot i\right) + \sqrt{1}\right) + \alpha}}{\frac{\left(\beta + i \cdot 2\right) + \left(\alpha - \sqrt{1}\right)}{i}}\right)}}\]

   if 1.647212354283295e+79 < beta < 4.444149846130252e+156

   1. Initial program 57.4

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

   2. Simplified 38.3

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

   4. Applied add-sqr-sqrt 38.3

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

   5. Applied difference-of-squares 38.3

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

   6. Applied times-frac 36.6

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

   7. Applied *-un-lft-identity 36.6

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

   8. Applied add-sqr-sqrt 36.7

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

   9. Applied times-frac 36.7

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

   10. Applied times-frac 36.2

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

   11. Simplified 36.2

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

   12. Simplified 36.2

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

   if 3.1853129188408834e+198 < beta

   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 57.5

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

   4. Applied add-sqr-sqrt 57.5

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

   5. Applied difference-of-squares 57.5

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

   6. Applied times-frac 57.4

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

   7. Applied associate-/r* 55.6

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

   8. Simplified 55.6

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

   9. Taylor expanded around 0 14.1

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

   10. Simplified 14.1

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

4. Final simplification 12.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 1.647212354283295028652665553828540921944 \cdot 10^{79}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\left(i \cdot 0.5\right) \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\alpha + \left(\sqrt{1} + \left(\beta + i \cdot 2\right)\right)}}{\frac{\left(\beta + i \cdot 2\right) + \left(\alpha - \sqrt{1}\right)}{i}}\right)}\\ \mathbf{elif}\;\beta \le 4.444149846130252102755247128183742063862 \cdot 10^{156}:\\ \;\;\;\;\frac{\frac{\sqrt{i \cdot \left(\left(i + \alpha\right) + \beta\right) + \beta \cdot \alpha}}{\left(\beta + i \cdot 2\right) + \alpha}}{\frac{\left(\beta + i \cdot 2\right) + \left(\alpha - \sqrt{1}\right)}{i}} \cdot \frac{\sqrt{i \cdot \left(\left(i + \alpha\right) + \beta\right) + \beta \cdot \alpha}}{\left(\left(\beta + i \cdot 2\right) + \alpha\right) \cdot \frac{\left(\alpha + \sqrt{1}\right) + \left(\beta + i \cdot 2\right)}{\left(i + \alpha\right) + \beta}}\\ \mathbf{elif}\;\beta \le 3.185312918840883354690651673329007970501 \cdot 10^{198}:\\ \;\;\;\;e^{\log \left(\frac{\frac{\left(i \cdot 0.5\right) \cdot \frac{i + \left(\beta + \alpha\right)}{\left(\beta + \alpha\right) + i \cdot 2}}{\alpha + \left(\sqrt{1} + \left(\beta + i \cdot 2\right)\right)}}{\frac{\left(\beta + i \cdot 2\right) + \left(\alpha - \sqrt{1}\right)}{i}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{i + \alpha}{\left(\alpha + \sqrt{1}\right) + \left(\beta + i \cdot 2\right)} \cdot \frac{\left(i + \alpha\right) + \beta}{\left(\beta + i \cdot 2\right) + \alpha}}{\frac{\left(\left(\beta + i \cdot 2\right) + \alpha\right) - \sqrt{1}}{i}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 1.0))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i)))) (- (* (+ (+ alpha beta) (* 2.0 i)) (+ (+ alpha beta) (* 2.0 i))) 1.0)))