Average Error: 23.9 → 11.2
Time: 34.2s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 4.511384635113818230514458629083655171473 \cdot 10^{132}:\\ \;\;\;\;\frac{\frac{{1}^{3} + {\left(\frac{\frac{\frac{\alpha + \beta}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}}\right)}^{3}}{1 \cdot 1 + \frac{\frac{\frac{\alpha + \beta}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}} \cdot \left(\frac{\frac{\frac{\alpha + \beta}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}} - 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]
\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 4.511384635113818230514458629083655171473 \cdot 10^{132}:\\
\;\;\;\;\frac{\frac{{1}^{3} + {\left(\frac{\frac{\frac{\alpha + \beta}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}}\right)}^{3}}{1 \cdot 1 + \frac{\frac{\frac{\alpha + \beta}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}} \cdot \left(\frac{\frac{\frac{\alpha + \beta}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}} - 1\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r90758 = alpha;
        double r90759 = beta;
        double r90760 = r90758 + r90759;
        double r90761 = r90759 - r90758;
        double r90762 = r90760 * r90761;
        double r90763 = 2.0;
        double r90764 = i;
        double r90765 = r90763 * r90764;
        double r90766 = r90760 + r90765;
        double r90767 = r90762 / r90766;
        double r90768 = r90766 + r90763;
        double r90769 = r90767 / r90768;
        double r90770 = 1.0;
        double r90771 = r90769 + r90770;
        double r90772 = r90771 / r90763;
        return r90772;
}
↓
double f(double alpha, double beta, double i) {
        double r90773 = alpha;
        double r90774 = 4.511384635113818e+132;
        bool r90775 = r90773 <= r90774;
        double r90776 = 1.0;
        double r90777 = 3.0;
        double r90778 = pow(r90776, r90777);
        double r90779 = beta;
        double r90780 = r90773 + r90779;
        double r90781 = 2.0;
        double r90782 = i;
        double r90783 = r90781 * r90782;
        double r90784 = r90780 + r90783;
        double r90785 = cbrt(r90784);
        double r90786 = r90779 - r90773;
        double r90787 = cbrt(r90786);
        double r90788 = r90785 / r90787;
        double r90789 = r90780 / r90788;
        double r90790 = r90784 + r90781;
        double r90791 = r90789 / r90790;
        double r90792 = r90785 * r90785;
        double r90793 = r90787 * r90787;
        double r90794 = r90792 / r90793;
        double r90795 = r90791 / r90794;
        double r90796 = pow(r90795, r90777);
        double r90797 = r90778 + r90796;
        double r90798 = r90776 * r90776;
        double r90799 = r90795 - r90776;
        double r90800 = r90795 * r90799;
        double r90801 = r90798 + r90800;
        double r90802 = r90797 / r90801;
        double r90803 = r90802 / r90781;
        double r90804 = 8.0;
        double r90805 = pow(r90773, r90777);
        double r90806 = r90804 / r90805;
        double r90807 = r90781 / r90773;
        double r90808 = 4.0;
        double r90809 = r90773 * r90773;
        double r90810 = r90808 / r90809;
        double r90811 = r90807 - r90810;
        double r90812 = r90806 + r90811;
        double r90813 = r90812 / r90781;
        double r90814 = r90775 ? r90803 : r90813;
        return r90814;
}

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 2 regimes

  2. if alpha < 4.511384635113818e+132

    1. Initial program 15.2

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

    3. Applied associate-/l*4.3

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

    5. Applied *-un-lft-identity4.3

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

    6. Applied add-cube-cbrt4.5

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

    7. Applied add-cube-cbrt4.3

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

    8. Applied times-frac4.3

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

    9. Applied *-un-lft-identity4.3

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

    10. Applied times-frac4.3

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

    11. Applied times-frac4.3

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

    12. Simplified4.3

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

    14. Applied flip3-+4.3

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

    15. Simplified4.3

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

    16. Simplified4.3

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

    if 4.511384635113818e+132 < alpha

    1. Initial program 61.8

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

    2. Taylor expanded around inf 41.0

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

    3. Simplified41.0

      \[\leadsto \frac{\color{blue}{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 4.511384635113818230514458629083655171473 \cdot 10^{132}:\\ \;\;\;\;\frac{\frac{{1}^{3} + {\left(\frac{\frac{\frac{\alpha + \beta}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}}\right)}^{3}}{1 \cdot 1 + \frac{\frac{\frac{\alpha + \beta}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}} \cdot \left(\frac{\frac{\frac{\alpha + \beta}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2}}{\frac{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}} - 1\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{{\alpha}^{3}} + \left(\frac{2}{\alpha} - \frac{4}{\alpha \cdot \alpha}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019208 
(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))