Octave 3.8, jcobi/2

Average Error: 23.8 → 12.5

Time: 26.4s

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 2.0760474421781283 \cdot 10^{+39}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\left(\beta + \alpha\right) - 2 \cdot i}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}} \cdot \left(\frac{\beta + \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) - 2 \cdot i}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 3.6345013768301078 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 1.480588010249348 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{{1.0}^{3} + {\left(\left(\left(\frac{\beta + \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\beta + \alpha\right) - 2 \cdot i}\right) \cdot \frac{\left(\beta + \alpha\right) - 2 \cdot i}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}\right)}^{3}}{\left(\left(\left(\frac{\beta + \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\beta + \alpha\right) - 2 \cdot i}\right) \cdot \frac{\left(\beta + \alpha\right) - 2 \cdot i}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}\right) \cdot \left(\left(\left(\frac{\beta + \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\beta + \alpha\right) - 2 \cdot i}\right) \cdot \frac{\left(\beta + \alpha\right) - 2 \cdot i}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}\right) + \left(1.0 \cdot 1.0 - 1.0 \cdot \left(\left(\left(\frac{\beta + \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\beta + \alpha\right) - 2 \cdot i}\right) \cdot \frac{\left(\beta + \alpha\right) - 2 \cdot i}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

C

double f(double alpha, double beta, double i) {
        double r2055739 = alpha;
        double r2055740 = beta;
        double r2055741 = r2055739 + r2055740;
        double r2055742 = r2055740 - r2055739;
        double r2055743 = r2055741 * r2055742;
        double r2055744 = 2.0;
        double r2055745 = i;
        double r2055746 = r2055744 * r2055745;
        double r2055747 = r2055741 + r2055746;
        double r2055748 = r2055743 / r2055747;
        double r2055749 = 2.0;
        double r2055750 = r2055747 + r2055749;
        double r2055751 = r2055748 / r2055750;
        double r2055752 = 1.0;
        double r2055753 = r2055751 + r2055752;
        double r2055754 = r2055753 / r2055749;
        return r2055754;
}

↓

double f(double alpha, double beta, double i) {
        double r2055755 = alpha;
        double r2055756 = 2.0760474421781283e+39;
        bool r2055757 = r2055755 <= r2055756;
        double r2055758 = 1.0;
        double r2055759 = beta;
        double r2055760 = r2055759 + r2055755;
        double r2055761 = 2.0;
        double r2055762 = i;
        double r2055763 = r2055761 * r2055762;
        double r2055764 = r2055760 - r2055763;
        double r2055765 = r2055763 + r2055760;
        double r2055766 = 2.0;
        double r2055767 = r2055765 + r2055766;
        double r2055768 = sqrt(r2055767);
        double r2055769 = r2055764 / r2055768;
        double r2055770 = r2055769 / r2055768;
        double r2055771 = r2055760 / r2055765;
        double r2055772 = r2055759 - r2055755;
        double r2055773 = r2055772 / r2055764;
        double r2055774 = r2055771 * r2055773;
        double r2055775 = r2055770 * r2055774;
        double r2055776 = r2055758 + r2055775;
        double r2055777 = r2055776 / r2055766;
        double r2055778 = 3.6345013768301078e+118;
        bool r2055779 = r2055755 <= r2055778;
        double r2055780 = r2055766 / r2055755;
        double r2055781 = 8.0;
        double r2055782 = r2055755 * r2055755;
        double r2055783 = r2055782 * r2055755;
        double r2055784 = r2055781 / r2055783;
        double r2055785 = 4.0;
        double r2055786 = r2055785 / r2055782;
        double r2055787 = r2055784 - r2055786;
        double r2055788 = r2055780 + r2055787;
        double r2055789 = r2055788 / r2055766;
        double r2055790 = 1.480588010249348e+179;
        bool r2055791 = r2055755 <= r2055790;
        double r2055792 = 3.0;
        double r2055793 = pow(r2055758, r2055792);
        double r2055794 = r2055771 * r2055772;
        double r2055795 = 1.0;
        double r2055796 = r2055795 / r2055764;
        double r2055797 = r2055794 * r2055796;
        double r2055798 = r2055764 / r2055767;
        double r2055799 = r2055797 * r2055798;
        double r2055800 = pow(r2055799, r2055792);
        double r2055801 = r2055793 + r2055800;
        double r2055802 = r2055799 * r2055799;
        double r2055803 = r2055758 * r2055758;
        double r2055804 = r2055758 * r2055799;
        double r2055805 = r2055803 - r2055804;
        double r2055806 = r2055802 + r2055805;
        double r2055807 = r2055801 / r2055806;
        double r2055808 = r2055807 / r2055766;
        double r2055809 = r2055791 ? r2055808 : r2055789;
        double r2055810 = r2055779 ? r2055789 : r2055809;
        double r2055811 = r2055757 ? r2055777 : r2055810;
        return r2055811;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Derivation

1. Split input into 3 regimes

2. if alpha < 2.0760474421781283e+39

   1. Initial program 11.3

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

   3. Applied *-un-lft-identity 11.3

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

   4. Applied flip-+ 13.5

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

   5. Applied associate-/r/ 13.5

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

   6. Applied times-frac 13.5

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

   7. Simplified 1.0

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

   9. Applied add-sqr-sqrt 1.1

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

   10. Applied associate-/r* 1.1

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

   if 2.0760474421781283e+39 < alpha < 3.6345013768301078e+118 or 1.480588010249348e+179 < alpha

   1. Initial program 54.5

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

   3. Applied *-un-lft-identity 54.5

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

   4. Applied flip-+ 54.1

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

   5. Applied associate-/r/ 54.1

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

   6. Applied times-frac 54.1

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

   7. Simplified 41.7

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

   8. Taylor expanded around inf 40.8

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

   9. Simplified 40.8

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

   if 3.6345013768301078e+118 < alpha < 1.480588010249348e+179

   1. Initial program 51.7

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

   3. Applied *-un-lft-identity 51.7

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

   4. Applied flip-+ 51.4

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

   5. Applied associate-/r/ 51.4

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

   6. Applied times-frac 51.4

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

   7. Simplified 37.4

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

   9. Applied div-inv 37.3

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

   10. Applied associate-*r* 37.3

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

   12. Applied flip3-+ 37.3

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

4. Final simplification 12.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 2.0760474421781283 \cdot 10^{+39}:\\ \;\;\;\;\frac{1.0 + \frac{\frac{\left(\beta + \alpha\right) - 2 \cdot i}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}}}{\sqrt{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}} \cdot \left(\frac{\beta + \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \frac{\beta - \alpha}{\left(\beta + \alpha\right) - 2 \cdot i}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 3.6345013768301078 \cdot 10^{+118}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \mathbf{elif}\;\alpha \le 1.480588010249348 \cdot 10^{+179}:\\ \;\;\;\;\frac{\frac{{1.0}^{3} + {\left(\left(\left(\frac{\beta + \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\beta + \alpha\right) - 2 \cdot i}\right) \cdot \frac{\left(\beta + \alpha\right) - 2 \cdot i}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}\right)}^{3}}{\left(\left(\left(\frac{\beta + \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\beta + \alpha\right) - 2 \cdot i}\right) \cdot \frac{\left(\beta + \alpha\right) - 2 \cdot i}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}\right) \cdot \left(\left(\left(\frac{\beta + \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\beta + \alpha\right) - 2 \cdot i}\right) \cdot \frac{\left(\beta + \alpha\right) - 2 \cdot i}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}\right) + \left(1.0 \cdot 1.0 - 1.0 \cdot \left(\left(\left(\frac{\beta + \alpha}{2 \cdot i + \left(\beta + \alpha\right)} \cdot \left(\beta - \alpha\right)\right) \cdot \frac{1}{\left(\beta + \alpha\right) - 2 \cdot i}\right) \cdot \frac{\left(\beta + \alpha\right) - 2 \cdot i}{\left(2 \cdot i + \left(\beta + \alpha\right)\right) + 2.0}\right)\right)}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2.0}{\alpha} + \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019155 
(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))