Average Error: 52.4 → 10.8
Time: 28.7s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 1\]
\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
↓
\[\begin{array}{l} \mathbf{if}\;i \le 1.6205126458778218 \cdot 10^{+133}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2} - \sqrt{\sqrt{1.0}}} \cdot \frac{\sqrt{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2} + \sqrt{\sqrt{1.0}}}\right) \cdot \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\alpha + \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}} \cdot \left(\sqrt[3]{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \left(\sqrt[3]{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \sqrt[3]{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)\right)\\ \end{array}\]
\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}
↓
\begin{array}{l}
\mathbf{if}\;i \le 1.6205126458778218 \cdot 10^{+133}:\\
\;\;\;\;\left(\frac{\sqrt{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2} - \sqrt{\sqrt{1.0}}} \cdot \frac{\sqrt{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2} + \sqrt{\sqrt{1.0}}}\right) \cdot \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\alpha + \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}} \cdot \left(\sqrt[3]{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \left(\sqrt[3]{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \sqrt[3]{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)\right)\\

\end{array}
double f(double alpha, double beta, double i) {
        double r2123198 = i;
        double r2123199 = alpha;
        double r2123200 = beta;
        double r2123201 = r2123199 + r2123200;
        double r2123202 = r2123201 + r2123198;
        double r2123203 = r2123198 * r2123202;
        double r2123204 = r2123200 * r2123199;
        double r2123205 = r2123204 + r2123203;
        double r2123206 = r2123203 * r2123205;
        double r2123207 = 2.0;
        double r2123208 = r2123207 * r2123198;
        double r2123209 = r2123201 + r2123208;
        double r2123210 = r2123209 * r2123209;
        double r2123211 = r2123206 / r2123210;
        double r2123212 = 1.0;
        double r2123213 = r2123210 - r2123212;
        double r2123214 = r2123211 / r2123213;
        return r2123214;
}
↓
double f(double alpha, double beta, double i) {
        double r2123215 = i;
        double r2123216 = 1.6205126458778218e+133;
        bool r2123217 = r2123215 <= r2123216;
        double r2123218 = alpha;
        double r2123219 = beta;
        double r2123220 = r2123218 + r2123219;
        double r2123221 = r2123220 + r2123215;
        double r2123222 = r2123215 * r2123221;
        double r2123223 = r2123218 * r2123219;
        double r2123224 = r2123222 + r2123223;
        double r2123225 = 2.0;
        double r2123226 = r2123215 * r2123225;
        double r2123227 = r2123220 + r2123226;
        double r2123228 = r2123224 / r2123227;
        double r2123229 = sqrt(r2123228);
        double r2123230 = sqrt(r2123227);
        double r2123231 = 1.0;
        double r2123232 = sqrt(r2123231);
        double r2123233 = sqrt(r2123232);
        double r2123234 = r2123230 - r2123233;
        double r2123235 = r2123229 / r2123234;
        double r2123236 = r2123230 + r2123233;
        double r2123237 = r2123229 / r2123236;
        double r2123238 = r2123235 * r2123237;
        double r2123239 = r2123222 / r2123227;
        double r2123240 = r2123232 + r2123227;
        double r2123241 = r2123239 / r2123240;
        double r2123242 = r2123238 * r2123241;
        double r2123243 = 0.5;
        double r2123244 = r2123243 * r2123215;
        double r2123245 = 0.25;
        double r2123246 = r2123245 * r2123220;
        double r2123247 = r2123244 + r2123246;
        double r2123248 = r2123227 - r2123232;
        double r2123249 = r2123247 / r2123248;
        double r2123250 = r2123221 / r2123227;
        double r2123251 = r2123215 * r2123250;
        double r2123252 = r2123251 / r2123240;
        double r2123253 = cbrt(r2123252);
        double r2123254 = r2123253 * r2123253;
        double r2123255 = r2123253 * r2123254;
        double r2123256 = r2123249 * r2123255;
        double r2123257 = r2123217 ? r2123242 : r2123256;
        return r2123257;
}

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 i < 1.6205126458778218e+133

    1. Initial program 39.5

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

    3. Applied add-sqr-sqrt39.5

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

    4. Applied difference-of-squares39.5

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

    5. Applied times-frac15.1

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

    6. Applied times-frac10.2

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

    8. Applied add-sqr-sqrt10.2

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

    9. Applied sqrt-prod10.2

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

    10. Applied add-sqr-sqrt10.5

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

    11. Applied difference-of-squares10.5

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

    12. Applied add-sqr-sqrt10.3

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

    13. Applied times-frac10.3

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

    if 1.6205126458778218e+133 < i

    1. Initial program 62.1

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

    3. Applied add-sqr-sqrt62.1

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

    4. Applied difference-of-squares62.1

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

    5. Applied times-frac57.0

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

    6. Applied times-frac56.9

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

    8. Applied *-un-lft-identity56.9

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

    9. Applied times-frac56.8

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

    10. Simplified56.8

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

    11. Taylor expanded around 0 11.1

      \[\leadsto \frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\color{blue}{\frac{1}{2} \cdot i + \left(\frac{1}{4} \cdot \beta + \frac{1}{4} \cdot \alpha\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]

    12. Simplified11.1

      \[\leadsto \frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}} \cdot \frac{\color{blue}{\frac{1}{4} \cdot \left(\alpha + \beta\right) + i \cdot \frac{1}{2}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]
    13. Using strategy rm

    14. Applied add-cube-cbrt11.1

      \[\leadsto \color{blue}{\left(\left(\sqrt[3]{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}} \cdot \sqrt[3]{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}}\right) \cdot \sqrt[3]{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + \sqrt{1.0}}}\right)} \cdot \frac{\frac{1}{4} \cdot \left(\alpha + \beta\right) + i \cdot \frac{1}{2}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) - \sqrt{1.0}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification10.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;i \le 1.6205126458778218 \cdot 10^{+133}:\\ \;\;\;\;\left(\frac{\sqrt{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2} - \sqrt{\sqrt{1.0}}} \cdot \frac{\sqrt{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right) + \alpha \cdot \beta}{\left(\alpha + \beta\right) + i \cdot 2}}}{\sqrt{\left(\alpha + \beta\right) + i \cdot 2} + \sqrt{\sqrt{1.0}}}\right) \cdot \frac{\frac{i \cdot \left(\left(\alpha + \beta\right) + i\right)}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{2} \cdot i + \frac{1}{4} \cdot \left(\alpha + \beta\right)}{\left(\left(\alpha + \beta\right) + i \cdot 2\right) - \sqrt{1.0}} \cdot \left(\sqrt[3]{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \left(\sqrt[3]{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}} \cdot \sqrt[3]{\frac{i \cdot \frac{\left(\alpha + \beta\right) + i}{\left(\alpha + \beta\right) + i \cdot 2}}{\sqrt{1.0} + \left(\left(\alpha + \beta\right) + i \cdot 2\right)}}\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019156 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/4"
  :pre (and (> alpha -1) (> beta -1) (> i 1))
  (/ (/ (* (* i (+ (+ alpha beta) i)) (+ (* beta alpha) (* i (+ (+ alpha beta) i)))) (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i)))) (- (* (+ (+ alpha beta) (* 2 i)) (+ (+ alpha beta) (* 2 i))) 1.0)))