Octave 3.8, jcobi/4

Average Error: 53.4 → 12.2

Time: 14.6s

Precision: 64

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

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;i \le 1.49387319812956663 \cdot 10^{43}:\\ \;\;\;\;\frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{elif}\;i \le 1.68420550280345399 \cdot 10^{81}:\\ \;\;\;\;\frac{i}{4} \cdot \left(\left(\sqrt[3]{\left(\alpha + \beta\right) + i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + i}\right) \cdot \frac{\frac{\sqrt[3]{\left(\alpha + \beta\right) + i}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\\ \mathbf{elif}\;i \le 1.49881899548770838 \cdot 10^{146}:\\ \;\;\;\;\frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r119914 = i;
        double r119915 = alpha;
        double r119916 = beta;
        double r119917 = r119915 + r119916;
        double r119918 = r119917 + r119914;
        double r119919 = r119914 * r119918;
        double r119920 = r119916 * r119915;
        double r119921 = r119920 + r119919;
        double r119922 = r119919 * r119921;
        double r119923 = 2.0;
        double r119924 = r119923 * r119914;
        double r119925 = r119917 + r119924;
        double r119926 = r119925 * r119925;
        double r119927 = r119922 / r119926;
        double r119928 = 1.0;
        double r119929 = r119926 - r119928;
        double r119930 = r119927 / r119929;
        return r119930;
}

↓

double f(double alpha, double beta, double i) {
        double r119931 = i;
        double r119932 = 1.4938731981295666e+43;
        bool r119933 = r119931 <= r119932;
        double r119934 = alpha;
        double r119935 = beta;
        double r119936 = r119934 + r119935;
        double r119937 = 2.0;
        double r119938 = r119937 * r119931;
        double r119939 = r119936 + r119938;
        double r119940 = 1.0;
        double r119941 = sqrt(r119940);
        double r119942 = r119939 + r119941;
        double r119943 = r119936 + r119931;
        double r119944 = r119931 * r119943;
        double r119945 = fma(r119935, r119934, r119944);
        double r119946 = r119939 - r119941;
        double r119947 = r119945 / r119946;
        double r119948 = r119942 / r119947;
        double r119949 = r119931 / r119948;
        double r119950 = fma(r119931, r119937, r119936);
        double r119951 = r119943 / r119950;
        double r119952 = r119951 / r119950;
        double r119953 = r119949 * r119952;
        double r119954 = 1.684205502803454e+81;
        bool r119955 = r119931 <= r119954;
        double r119956 = 4.0;
        double r119957 = r119931 / r119956;
        double r119958 = cbrt(r119943);
        double r119959 = r119958 * r119958;
        double r119960 = r119958 / r119950;
        double r119961 = r119960 / r119950;
        double r119962 = r119959 * r119961;
        double r119963 = r119957 * r119962;
        double r119964 = 1.4988189954877084e+146;
        bool r119965 = r119931 <= r119964;
        double r119966 = r119957 * r119952;
        double r119967 = log(r119966);
        double r119968 = exp(r119967);
        double r119969 = r119965 ? r119953 : r119968;
        double r119970 = r119955 ? r119963 : r119969;
        double r119971 = r119933 ? r119953 : r119970;
        return r119971;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if i < 1.4938731981295666e+43 or 1.684205502803454e+81 < i < 1.4988189954877084e+146

   1. Initial program 45.8

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

   2. Simplified 43.2

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

   4. Applied associate-/l* 34.3

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

   6. Applied div-inv 34.3

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

   7. Applied times-frac 17.8

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

   8. Applied times-frac 15.1

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

   9. Simplified 15.0

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

   11. Applied add-sqr-sqrt 15.0

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

   12. Applied difference-of-squares 15.0

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

   13. Applied associate-/l* 11.6

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

   if 1.4938731981295666e+43 < i < 1.684205502803454e+81

   1. Initial program 30.3

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

   2. Simplified 28.8

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

   4. Applied associate-/l* 21.8

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

   6. Applied div-inv 21.9

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

   7. Applied times-frac 20.2

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

   8. Applied times-frac 16.5

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

   9. Simplified 16.4

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

   10. Taylor expanded around 0 28.1

       \[\leadsto \frac{i}{\color{blue}{4}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\]
   11. Using strategy rm

   12. Applied *-un-lft-identity 28.1

       \[\leadsto \frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\color{blue}{1 \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}\]

   13. Applied *-un-lft-identity 28.1

       \[\leadsto \frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\color{blue}{1 \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{1 \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\]

   14. Applied add-cube-cbrt 28.6

       \[\leadsto \frac{i}{4} \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\left(\alpha + \beta\right) + i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + i}\right) \cdot \sqrt[3]{\left(\alpha + \beta\right) + i}}}{1 \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{1 \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\]

   15. Applied times-frac 28.6

       \[\leadsto \frac{i}{4} \cdot \frac{\color{blue}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + i}}{1} \cdot \frac{\sqrt[3]{\left(\alpha + \beta\right) + i}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}}{1 \cdot \mathsf{fma}\left(i, 2, \alpha + \beta\right)}\]

   16. Applied times-frac 19.0

       \[\leadsto \frac{i}{4} \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{\left(\alpha + \beta\right) + i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + i}}{1}}{1} \cdot \frac{\frac{\sqrt[3]{\left(\alpha + \beta\right) + i}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}\]

   17. Simplified 19.0

       \[\leadsto \frac{i}{4} \cdot \left(\color{blue}{\left(\sqrt[3]{\left(\alpha + \beta\right) + i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + i}\right)} \cdot \frac{\frac{\sqrt[3]{\left(\alpha + \beta\right) + i}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\]

   if 1.4988189954877084e+146 < 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 63.9

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

   4. Applied associate-/l* 63.9

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

   6. Applied div-inv 63.9

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

   7. Applied times-frac 62.1

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

   8. Applied times-frac 62.0

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

   9. Simplified 62.0

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

   10. Taylor expanded around 0 11.0

       \[\leadsto \frac{i}{\color{blue}{4}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\]
   11. Using strategy rm

   12. Applied add-exp-log 16.3

       \[\leadsto \frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\color{blue}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}}\]

   13. Applied add-exp-log 17.1

       \[\leadsto \frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\color{blue}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}\]

   14. Applied add-exp-log 17.2

       \[\leadsto \frac{i}{4} \cdot \frac{\frac{\color{blue}{e^{\log \left(\left(\alpha + \beta\right) + i\right)}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}\]

   15. Applied div-exp 17.2

       \[\leadsto \frac{i}{4} \cdot \frac{\color{blue}{e^{\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}}{e^{\log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}\]

   16. Applied div-exp 17.2

       \[\leadsto \frac{i}{4} \cdot \color{blue}{e^{\left(\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}}\]

   17. Applied add-exp-log 17.2

       \[\leadsto \frac{i}{\color{blue}{e^{\log 4}}} \cdot e^{\left(\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}\]

   18. Applied add-exp-log 17.1

       \[\leadsto \frac{\color{blue}{e^{\log i}}}{e^{\log 4}} \cdot e^{\left(\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}\]

   19. Applied div-exp 17.4

       \[\leadsto \color{blue}{e^{\log i - \log 4}} \cdot e^{\left(\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)}\]

   20. Applied prod-exp 17.4

       \[\leadsto \color{blue}{e^{\left(\log i - \log 4\right) + \left(\left(\log \left(\left(\alpha + \beta\right) + i\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right) - \log \left(\mathsf{fma}\left(i, 2, \alpha + \beta\right)\right)\right)}}\]

   21. Simplified 10.9

       \[\leadsto e^{\color{blue}{\log \left(\frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}}\]
3. Recombined 3 regimes into one program.

4. Final simplification 12.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.49387319812956663 \cdot 10^{43}:\\ \;\;\;\;\frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{elif}\;i \le 1.68420550280345399 \cdot 10^{81}:\\ \;\;\;\;\frac{i}{4} \cdot \left(\left(\sqrt[3]{\left(\alpha + \beta\right) + i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + i}\right) \cdot \frac{\frac{\sqrt[3]{\left(\alpha + \beta\right) + i}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)\\ \mathbf{elif}\;i \le 1.49881899548770838 \cdot 10^{146}:\\ \;\;\;\;\frac{i}{\frac{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1}}{\frac{\mathsf{fma}\left(\beta, \alpha, i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1}}}} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\frac{i}{4} \cdot \frac{\frac{\left(\alpha + \beta\right) + i}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}}{\mathsf{fma}\left(i, 2, \alpha + \beta\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020083 +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)))