Average Error: 24.0 → 11.9
Time: 1.4m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2.0} + 1.0}{2.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.2994643925028505 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{\frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}} \cdot \left(\left(\frac{\sqrt[3]{\beta + \alpha}}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}} \cdot \frac{\sqrt[3]{\beta + \alpha} \cdot \sqrt[3]{\beta + \alpha}}{\frac{1}{\frac{1}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)} \cdot \sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}}\right) \cdot \frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}}\right) + \left(1.0 \cdot 1.0\right) \cdot 1.0}{\left(1.0 - \frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}}\right) \cdot 1.0 + \frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}} \cdot \frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]
\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) + 2.0} + 1.0}{2.0}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.2994643925028505 \cdot 10^{+247}:\\
\;\;\;\;\frac{\frac{\frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}} \cdot \left(\left(\frac{\sqrt[3]{\beta + \alpha}}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}} \cdot \frac{\sqrt[3]{\beta + \alpha} \cdot \sqrt[3]{\beta + \alpha}}{\frac{1}{\frac{1}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)} \cdot \sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}}\right) \cdot \frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}}\right) + \left(1.0 \cdot 1.0\right) \cdot 1.0}{\left(1.0 - \frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}}\right) \cdot 1.0 + \frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}} \cdot \frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}}}}{2.0}\\

\mathbf{else}:\\
\;\;\;\;\frac{\left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\

\end{array}
double f(double alpha, double beta, double i) {
        double r5221097 = alpha;
        double r5221098 = beta;
        double r5221099 = r5221097 + r5221098;
        double r5221100 = r5221098 - r5221097;
        double r5221101 = r5221099 * r5221100;
        double r5221102 = 2.0;
        double r5221103 = i;
        double r5221104 = r5221102 * r5221103;
        double r5221105 = r5221099 + r5221104;
        double r5221106 = r5221101 / r5221105;
        double r5221107 = 2.0;
        double r5221108 = r5221105 + r5221107;
        double r5221109 = r5221106 / r5221108;
        double r5221110 = 1.0;
        double r5221111 = r5221109 + r5221110;
        double r5221112 = r5221111 / r5221107;
        return r5221112;
}
↓
double f(double alpha, double beta, double i) {
        double r5221113 = alpha;
        double r5221114 = 1.2994643925028505e+247;
        bool r5221115 = r5221113 <= r5221114;
        double r5221116 = beta;
        double r5221117 = r5221116 + r5221113;
        double r5221118 = 2.0;
        double r5221119 = 2.0;
        double r5221120 = i;
        double r5221121 = r5221119 * r5221120;
        double r5221122 = r5221121 + r5221117;
        double r5221123 = r5221118 + r5221122;
        double r5221124 = r5221116 - r5221113;
        double r5221125 = r5221124 / r5221122;
        double r5221126 = r5221123 / r5221125;
        double r5221127 = r5221117 / r5221126;
        double r5221128 = cbrt(r5221117);
        double r5221129 = cbrt(r5221122);
        double r5221130 = r5221124 / r5221129;
        double r5221131 = r5221123 / r5221130;
        double r5221132 = r5221128 / r5221131;
        double r5221133 = r5221128 * r5221128;
        double r5221134 = 1.0;
        double r5221135 = r5221129 * r5221129;
        double r5221136 = r5221134 / r5221135;
        double r5221137 = r5221134 / r5221136;
        double r5221138 = r5221133 / r5221137;
        double r5221139 = r5221132 * r5221138;
        double r5221140 = r5221139 * r5221127;
        double r5221141 = r5221127 * r5221140;
        double r5221142 = 1.0;
        double r5221143 = r5221142 * r5221142;
        double r5221144 = r5221143 * r5221142;
        double r5221145 = r5221141 + r5221144;
        double r5221146 = r5221142 - r5221127;
        double r5221147 = r5221146 * r5221142;
        double r5221148 = r5221127 * r5221127;
        double r5221149 = r5221147 + r5221148;
        double r5221150 = r5221145 / r5221149;
        double r5221151 = r5221150 / r5221118;
        double r5221152 = 8.0;
        double r5221153 = r5221113 * r5221113;
        double r5221154 = r5221113 * r5221153;
        double r5221155 = r5221152 / r5221154;
        double r5221156 = 4.0;
        double r5221157 = r5221156 / r5221153;
        double r5221158 = r5221155 - r5221157;
        double r5221159 = r5221118 / r5221113;
        double r5221160 = r5221158 + r5221159;
        double r5221161 = r5221160 / r5221118;
        double r5221162 = r5221115 ? r5221151 : r5221161;
        return r5221162;
}

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 < 1.2994643925028505e+247

    1. Initial program 21.2

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

    3. Applied *-un-lft-identity21.2

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

    4. Applied times-frac9.9

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

    5. Applied associate-/l*9.9

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

    7. Applied flip3-+9.9

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

    8. Simplified9.9

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

    9. Simplified9.9

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

    11. Applied add-cube-cbrt10.1

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

    12. Applied *-un-lft-identity10.1

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

    13. Applied times-frac10.1

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

    14. Applied *-un-lft-identity10.1

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

    15. Applied times-frac10.1

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

    16. Applied add-cube-cbrt9.9

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

    17. Applied times-frac9.9

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

    if 1.2994643925028505e+247 < alpha

    1. Initial program 63.2

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

    2. Taylor expanded around inf 40.0

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

    3. Simplified40.0

      \[\leadsto \frac{\color{blue}{\frac{2.0}{\alpha} + \left(\frac{8.0}{\left(\alpha \cdot \alpha\right) \cdot \alpha} - \frac{4.0}{\alpha \cdot \alpha}\right)}}{2.0}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.2994643925028505 \cdot 10^{+247}:\\ \;\;\;\;\frac{\frac{\frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}} \cdot \left(\left(\frac{\sqrt[3]{\beta + \alpha}}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}} \cdot \frac{\sqrt[3]{\beta + \alpha} \cdot \sqrt[3]{\beta + \alpha}}{\frac{1}{\frac{1}{\sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)} \cdot \sqrt[3]{2 \cdot i + \left(\beta + \alpha\right)}}}}\right) \cdot \frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}}\right) + \left(1.0 \cdot 1.0\right) \cdot 1.0}{\left(1.0 - \frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}}\right) \cdot 1.0 + \frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}} \cdot \frac{\beta + \alpha}{\frac{2.0 + \left(2 \cdot i + \left(\beta + \alpha\right)\right)}{\frac{\beta - \alpha}{2 \cdot i + \left(\beta + \alpha\right)}}}}}{2.0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\frac{8.0}{\alpha \cdot \left(\alpha \cdot \alpha\right)} - \frac{4.0}{\alpha \cdot \alpha}\right) + \frac{2.0}{\alpha}}{2.0}\\ \end{array}\]

Reproduce

herbie shell --seed 2019152 
(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))