Octave 3.8, jcobi/2

Average Error: 23.5 → 11.9

Time: 22.1s

Precision: 64

\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.841782349532883 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt[3]{\left(1.0 + \left(\beta + \alpha\right) \cdot \left(\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right)\right) \cdot \left(\left(1.0 + \left(\beta + \alpha\right) \cdot \left(\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right)\right) \cdot \left(1.0 + \left(\beta + \alpha\right) \cdot \left(\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right)\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.8551287992761453 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} + \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 8.621431291668078 \cdot 10^{+174}:\\ \;\;\;\;\frac{1.0 + \left(\frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0} \cdot \left(\sqrt[3]{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0} \cdot \sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right)}\right) \cdot \left(\beta + \alpha\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} + \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{4.0}{\alpha \cdot \alpha}}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r1851888 = alpha;
        double r1851889 = beta;
        double r1851890 = r1851888 + r1851889;
        double r1851891 = r1851889 - r1851888;
        double r1851892 = r1851890 * r1851891;
        double r1851893 = 2.0;
        double r1851894 = i;
        double r1851895 = r1851893 * r1851894;
        double r1851896 = r1851890 + r1851895;
        double r1851897 = r1851892 / r1851896;
        double r1851898 = 2.0;
        double r1851899 = r1851896 + r1851898;
        double r1851900 = r1851897 / r1851899;
        double r1851901 = 1.0;
        double r1851902 = r1851900 + r1851901;
        double r1851903 = r1851902 / r1851898;
        return r1851903;
}

↓

double f(double alpha, double beta, double i) {
        double r1851904 = alpha;
        double r1851905 = 5.841782349532883e+51;
        bool r1851906 = r1851904 <= r1851905;
        double r1851907 = 1.0;
        double r1851908 = beta;
        double r1851909 = r1851908 + r1851904;
        double r1851910 = r1851908 - r1851904;
        double r1851911 = cbrt(r1851910);
        double r1851912 = i;
        double r1851913 = 2.0;
        double r1851914 = r1851912 * r1851913;
        double r1851915 = r1851909 + r1851914;
        double r1851916 = 2.0;
        double r1851917 = r1851915 + r1851916;
        double r1851918 = cbrt(r1851917);
        double r1851919 = r1851911 / r1851918;
        double r1851920 = r1851919 * r1851919;
        double r1851921 = r1851911 / r1851915;
        double r1851922 = r1851921 / r1851918;
        double r1851923 = r1851920 * r1851922;
        double r1851924 = r1851909 * r1851923;
        double r1851925 = r1851907 + r1851924;
        double r1851926 = r1851925 * r1851925;
        double r1851927 = r1851925 * r1851926;
        double r1851928 = cbrt(r1851927);
        double r1851929 = r1851928 / r1851916;
        double r1851930 = 1.8551287992761453e+84;
        bool r1851931 = r1851904 <= r1851930;
        double r1851932 = r1851916 / r1851904;
        double r1851933 = 8.0;
        double r1851934 = r1851904 * r1851904;
        double r1851935 = r1851934 * r1851904;
        double r1851936 = r1851933 / r1851935;
        double r1851937 = r1851932 + r1851936;
        double r1851938 = 4.0;
        double r1851939 = r1851938 / r1851934;
        double r1851940 = r1851937 - r1851939;
        double r1851941 = r1851940 / r1851916;
        double r1851942 = 8.621431291668078e+174;
        bool r1851943 = r1851904 <= r1851942;
        double r1851944 = r1851911 * r1851911;
        double r1851945 = r1851918 * r1851918;
        double r1851946 = cbrt(r1851945);
        double r1851947 = cbrt(r1851918);
        double r1851948 = r1851946 * r1851947;
        double r1851949 = r1851918 * r1851948;
        double r1851950 = r1851944 / r1851949;
        double r1851951 = r1851922 * r1851950;
        double r1851952 = r1851951 * r1851909;
        double r1851953 = r1851907 + r1851952;
        double r1851954 = r1851953 / r1851916;
        double r1851955 = r1851943 ? r1851954 : r1851941;
        double r1851956 = r1851931 ? r1851941 : r1851955;
        double r1851957 = r1851906 ? r1851929 : r1851956;
        return r1851957;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 5.841782349532883e+51

   1. Initial program 11.6

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

   3. Applied *-un-lft-identity 11.6

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

   4. Applied *-un-lft-identity 11.6

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   5. Applied times-frac 1.2

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   6. Applied times-frac 1.2

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   7. Simplified 1.2

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

   9. Applied add-cube-cbrt 1.4

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

   10. Applied *-un-lft-identity 1.4

       \[\leadsto \frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   11. Applied add-cube-cbrt 1.3

       \[\leadsto \frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \sqrt[3]{\beta - \alpha}}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   12. Applied times-frac 1.3

       \[\leadsto \frac{\left(\alpha + \beta\right) \cdot \frac{\color{blue}{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   13. Applied times-frac 1.3

       \[\leadsto \frac{\left(\alpha + \beta\right) \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)} + 1.0}{2.0}\]
   14. Using strategy rm

   15. Applied add-cbrt-cube 1.3

       \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(\left(\alpha + \beta\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + 1.0\right) \cdot \left(\left(\alpha + \beta\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + 1.0\right)\right) \cdot \left(\left(\alpha + \beta\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + 1.0\right)}}}{2.0}\]

   16. Simplified 1.3

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

   if 5.841782349532883e+51 < alpha < 1.8551287992761453e+84 or 8.621431291668078e+174 < alpha

   1. Initial program 57.7

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

   2. Taylor expanded around inf 41.3

      \[\leadsto \frac{\color{blue}{\left(2.0 \cdot \frac{1}{\alpha} + 8.0 \cdot \frac{1}{{\alpha}^{3}}\right) - 4.0 \cdot \frac{1}{{\alpha}^{2}}}}{2.0}\]

   3. Simplified 41.3

      \[\leadsto \frac{\color{blue}{\left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} + \frac{2.0}{\alpha}\right) - \frac{4.0}{\alpha \cdot \alpha}}}{2.0}\]

   if 1.8551287992761453e+84 < alpha < 8.621431291668078e+174

   1. Initial program 48.0

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

   3. Applied *-un-lft-identity 48.0

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\color{blue}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)}} + 1.0}{2.0}\]

   4. Applied *-un-lft-identity 48.0

      \[\leadsto \frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   5. Applied times-frac 35.2

      \[\leadsto \frac{\frac{\color{blue}{\frac{\alpha + \beta}{1} \cdot \frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}}{1 \cdot \left(\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0\right)} + 1.0}{2.0}\]

   6. Applied times-frac 35.1

      \[\leadsto \frac{\color{blue}{\frac{\frac{\alpha + \beta}{1}}{1} \cdot \frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   7. Simplified 35.1

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

   9. Applied add-cube-cbrt 35.2

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

   10. Applied *-un-lft-identity 35.2

       \[\leadsto \frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\beta - \alpha}{\color{blue}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   11. Applied add-cube-cbrt 35.3

       \[\leadsto \frac{\left(\alpha + \beta\right) \cdot \frac{\frac{\color{blue}{\left(\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}\right) \cdot \sqrt[3]{\beta - \alpha}}}{1 \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   12. Applied times-frac 35.2

       \[\leadsto \frac{\left(\alpha + \beta\right) \cdot \frac{\color{blue}{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} + 1.0}{2.0}\]

   13. Applied times-frac 35.2

       \[\leadsto \frac{\left(\alpha + \beta\right) \cdot \color{blue}{\left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)} + 1.0}{2.0}\]
   14. Using strategy rm

   15. Applied add-cube-cbrt 35.2

       \[\leadsto \frac{\left(\alpha + \beta\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\color{blue}{\left(\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}\right) \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + 1.0}{2.0}\]

   16. Applied cbrt-prod 35.1

       \[\leadsto \frac{\left(\alpha + \beta\right) \cdot \left(\frac{\frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{1}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \color{blue}{\left(\sqrt[3]{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} \cdot \sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right)}} \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0}}\right) + 1.0}{2.0}\]
3. Recombined 3 regimes into one program.

4. Final simplification 11.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.841782349532883 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt[3]{\left(1.0 + \left(\beta + \alpha\right) \cdot \left(\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right)\right) \cdot \left(\left(1.0 + \left(\beta + \alpha\right) \cdot \left(\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right)\right) \cdot \left(1.0 + \left(\beta + \alpha\right) \cdot \left(\left(\frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right) \cdot \frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right)\right)\right)}}{2.0}\\ \mathbf{elif}\;\alpha \le 1.8551287992761453 \cdot 10^{+84}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} + \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{4.0}{\alpha \cdot \alpha}}{2.0}\\ \mathbf{elif}\;\alpha \le 8.621431291668078 \cdot 10^{+174}:\\ \;\;\;\;\frac{1.0 + \left(\frac{\frac{\sqrt[3]{\beta - \alpha}}{\left(\beta + \alpha\right) + i \cdot 2}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}} \cdot \frac{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0} \cdot \left(\sqrt[3]{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0} \cdot \sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}} \cdot \sqrt[3]{\sqrt[3]{\left(\left(\beta + \alpha\right) + i \cdot 2\right) + 2.0}}\right)}\right) \cdot \left(\beta + \alpha\right)}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{2.0}{\alpha} + \frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha}\right) - \frac{4.0}{\alpha \cdot \alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1) (> beta -1) (> i 0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2.0)) 1.0) 2.0))