Average Error: 24.5 → 12.0
Time: 16.9s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.5412307314962603 \cdot 10^{80}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\beta - \alpha}} \cdot \left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot i}\right) + 1}{2}\\ \mathbf{elif}\;\alpha \le 2.2620742697831166 \cdot 10^{119}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \mathbf{elif}\;\alpha \le 2.135954177821433 \cdot 10^{197}:\\ \;\;\;\;\frac{\left(\frac{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}\right) \cdot \left(\frac{\sqrt[3]{\alpha + \beta}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\beta - \alpha}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]
\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} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.5412307314962603 \cdot 10^{80}:\\
\;\;\;\;\frac{\frac{1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\beta - \alpha}} \cdot \left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot i}\right) + 1}{2}\\

\mathbf{elif}\;\alpha \le 2.2620742697831166 \cdot 10^{119}:\\
\;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\

\mathbf{elif}\;\alpha \le 2.135954177821433 \cdot 10^{197}:\\
\;\;\;\;\frac{\left(\frac{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}\right) \cdot \left(\frac{\sqrt[3]{\alpha + \beta}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\beta - \alpha}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}\right) + 1}{2}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r171650 = alpha;
        double r171651 = beta;
        double r171652 = r171650 + r171651;
        double r171653 = r171651 - r171650;
        double r171654 = r171652 * r171653;
        double r171655 = 2.0;
        double r171656 = i;
        double r171657 = r171655 * r171656;
        double r171658 = r171652 + r171657;
        double r171659 = r171654 / r171658;
        double r171660 = r171658 + r171655;
        double r171661 = r171659 / r171660;
        double r171662 = 1.0;
        double r171663 = r171661 + r171662;
        double r171664 = r171663 / r171655;
        return r171664;
}
↓
double f(double alpha, double beta, double i) {
        double r171665 = alpha;
        double r171666 = 1.5412307314962603e+80;
        bool r171667 = r171665 <= r171666;
        double r171668 = 1.0;
        double r171669 = beta;
        double r171670 = r171665 + r171669;
        double r171671 = 2.0;
        double r171672 = i;
        double r171673 = r171671 * r171672;
        double r171674 = r171670 + r171673;
        double r171675 = r171674 + r171671;
        double r171676 = sqrt(r171675);
        double r171677 = r171669 - r171665;
        double r171678 = r171676 / r171677;
        double r171679 = r171668 / r171678;
        double r171680 = r171670 / r171676;
        double r171681 = r171668 / r171674;
        double r171682 = r171680 * r171681;
        double r171683 = r171679 * r171682;
        double r171684 = 1.0;
        double r171685 = r171683 + r171684;
        double r171686 = r171685 / r171671;
        double r171687 = 2.2620742697831166e+119;
        bool r171688 = r171665 <= r171687;
        double r171689 = r171668 / r171665;
        double r171690 = r171671 * r171689;
        double r171691 = 8.0;
        double r171692 = 3.0;
        double r171693 = pow(r171665, r171692);
        double r171694 = r171668 / r171693;
        double r171695 = r171691 * r171694;
        double r171696 = r171690 + r171695;
        double r171697 = 4.0;
        double r171698 = 2.0;
        double r171699 = pow(r171665, r171698);
        double r171700 = r171668 / r171699;
        double r171701 = r171697 * r171700;
        double r171702 = r171696 - r171701;
        double r171703 = r171702 / r171671;
        double r171704 = 2.135954177821433e+197;
        bool r171705 = r171665 <= r171704;
        double r171706 = cbrt(r171670);
        double r171707 = r171706 * r171706;
        double r171708 = r171707 / r171676;
        double r171709 = cbrt(r171674);
        double r171710 = r171709 * r171709;
        double r171711 = r171668 / r171710;
        double r171712 = r171708 * r171711;
        double r171713 = r171706 / r171676;
        double r171714 = r171677 / r171709;
        double r171715 = r171713 * r171714;
        double r171716 = r171712 * r171715;
        double r171717 = r171716 + r171684;
        double r171718 = r171717 / r171671;
        double r171719 = r171705 ? r171718 : r171703;
        double r171720 = r171688 ? r171703 : r171719;
        double r171721 = r171667 ? r171686 : r171720;
        return r171721;
}

Error

Bits error versus alpha

Bits error versus beta

Bits error versus i

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 3 regimes

  2. if alpha < 1.5412307314962603e+80

    1. Initial program 14.1

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

    3. Applied *-un-lft-identity14.1

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

    4. Applied times-frac2.6

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

    5. Applied associate-/l*2.6

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

    7. Applied div-inv2.6

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

    8. Applied add-sqr-sqrt2.6

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

    9. Applied times-frac8.6

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

    10. Applied *-un-lft-identity8.6

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

    11. Applied *-un-lft-identity8.6

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

    12. Applied times-frac8.6

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

    13. Applied times-frac8.6

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

    14. Simplified8.6

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

    15. Simplified2.7

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

    if 1.5412307314962603e+80 < alpha < 2.2620742697831166e+119 or 2.135954177821433e+197 < alpha

    1. Initial program 57.7

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

    2. Taylor expanded around inf 42.0

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]

    if 2.2620742697831166e+119 < alpha < 2.135954177821433e+197

    1. Initial program 55.1

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

    3. Applied *-un-lft-identity55.1

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

    4. Applied times-frac38.2

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

    5. Applied associate-/l*38.2

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

    7. Applied add-cube-cbrt38.3

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

    8. Applied *-un-lft-identity38.3

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

    9. Applied times-frac38.2

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

    10. Applied add-sqr-sqrt38.3

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

    11. Applied times-frac40.0

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

    12. Applied add-cube-cbrt40.1

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

    13. Applied times-frac40.0

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

    14. Simplified38.3

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

    15. Simplified38.3

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

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.5412307314962603 \cdot 10^{80}:\\ \;\;\;\;\frac{\frac{1}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\beta - \alpha}} \cdot \left(\frac{\alpha + \beta}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{1}{\left(\alpha + \beta\right) + 2 \cdot i}\right) + 1}{2}\\ \mathbf{elif}\;\alpha \le 2.2620742697831166 \cdot 10^{119}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \mathbf{elif}\;\alpha \le 2.135954177821433 \cdot 10^{197}:\\ \;\;\;\;\frac{\left(\frac{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}\right) \cdot \left(\frac{\sqrt[3]{\alpha + \beta}}{\sqrt{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}} \cdot \frac{\beta - \alpha}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}\right) + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020056 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2 i))) (+ (+ (+ alpha beta) (* 2 i)) 2)) 1) 2))