Average Error: 24.4 → 11.6
Time: 26.4s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 3.851388677432038317167398913439949496155 \cdot 10^{108}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \left(\frac{\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 2}} \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + i \cdot 2\right) + \beta}} \cdot \sqrt[3]{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 2}}{\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}}}\right)\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \le 8.09065536801492928071076414340505776935 \cdot 10^{157} \lor \neg \left(\alpha \le 3.039415483981533193944183363030540056882 \cdot 10^{202}\right):\\ \;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \left(\frac{\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 2}}{\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}}} \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 2}}\right)\right)}^{3}}}{2}\\ \end{array}\]
\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 3.851388677432038317167398913439949496155 \cdot 10^{108}:\\
\;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \left(\frac{\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 2}} \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + i \cdot 2\right) + \beta}} \cdot \sqrt[3]{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 2}}{\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}}}\right)\right)}^{3}}}{2}\\

\mathbf{elif}\;\alpha \le 8.09065536801492928071076414340505776935 \cdot 10^{157} \lor \neg \left(\alpha \le 3.039415483981533193944183363030540056882 \cdot 10^{202}\right):\\
\;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\

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

\end{array}
double f(double alpha, double beta, double i) {
        double r93503 = alpha;
        double r93504 = beta;
        double r93505 = r93503 + r93504;
        double r93506 = r93504 - r93503;
        double r93507 = r93505 * r93506;
        double r93508 = 2.0;
        double r93509 = i;
        double r93510 = r93508 * r93509;
        double r93511 = r93505 + r93510;
        double r93512 = r93507 / r93511;
        double r93513 = r93511 + r93508;
        double r93514 = r93512 / r93513;
        double r93515 = 1.0;
        double r93516 = r93514 + r93515;
        double r93517 = r93516 / r93508;
        return r93517;
}
↓
double f(double alpha, double beta, double i) {
        double r93518 = alpha;
        double r93519 = 3.8513886774320383e+108;
        bool r93520 = r93518 <= r93519;
        double r93521 = 1.0;
        double r93522 = beta;
        double r93523 = r93518 + r93522;
        double r93524 = r93522 - r93518;
        double r93525 = i;
        double r93526 = 2.0;
        double r93527 = r93525 * r93526;
        double r93528 = r93522 + r93527;
        double r93529 = r93518 + r93528;
        double r93530 = r93524 / r93529;
        double r93531 = cbrt(r93530);
        double r93532 = r93523 + r93527;
        double r93533 = r93532 + r93526;
        double r93534 = sqrt(r93533);
        double r93535 = r93531 / r93534;
        double r93536 = cbrt(r93524);
        double r93537 = r93518 + r93527;
        double r93538 = r93537 + r93522;
        double r93539 = r93536 / r93538;
        double r93540 = cbrt(r93539);
        double r93541 = r93536 * r93536;
        double r93542 = cbrt(r93541);
        double r93543 = r93540 * r93542;
        double r93544 = r93534 / r93531;
        double r93545 = r93543 / r93544;
        double r93546 = r93535 * r93545;
        double r93547 = r93523 * r93546;
        double r93548 = r93521 + r93547;
        double r93549 = 3.0;
        double r93550 = pow(r93548, r93549);
        double r93551 = cbrt(r93550);
        double r93552 = r93551 / r93526;
        double r93553 = 8.090655368014929e+157;
        bool r93554 = r93518 <= r93553;
        double r93555 = 3.039415483981533e+202;
        bool r93556 = r93518 <= r93555;
        double r93557 = !r93556;
        bool r93558 = r93554 || r93557;
        double r93559 = 8.0;
        double r93560 = pow(r93518, r93549);
        double r93561 = r93559 / r93560;
        double r93562 = r93526 / r93518;
        double r93563 = r93561 + r93562;
        double r93564 = 4.0;
        double r93565 = r93518 * r93518;
        double r93566 = r93564 / r93565;
        double r93567 = r93563 - r93566;
        double r93568 = r93567 / r93526;
        double r93569 = r93531 / r93544;
        double r93570 = r93569 * r93535;
        double r93571 = r93523 * r93570;
        double r93572 = r93521 + r93571;
        double r93573 = pow(r93572, r93549);
        double r93574 = cbrt(r93573);
        double r93575 = r93574 / r93526;
        double r93576 = r93558 ? r93568 : r93575;
        double r93577 = r93520 ? r93552 : r93576;
        return r93577;
}

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 < 3.8513886774320383e+108

    1. Initial program 14.4

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

    3. Applied *-un-lft-identity14.4

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

    4. Applied *-un-lft-identity14.4

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

    5. Applied times-frac3.7

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

    6. Applied times-frac3.7

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

    7. Simplified3.7

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

    8. Simplified3.7

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

    10. Applied add-sqr-sqrt3.8

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

    11. Applied add-cube-cbrt3.8

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

    12. Applied times-frac3.8

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

    13. Simplified3.8

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

    14. Simplified3.8

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

    16. Applied add-cbrt-cube3.7

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

    17. Simplified3.7

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

    19. Applied *-un-lft-identity3.7

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

    20. Applied add-cube-cbrt3.8

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

    21. Applied times-frac3.8

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

    22. Applied cbrt-prod3.8

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

    23. Simplified3.8

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

    24. Simplified3.8

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

    if 3.8513886774320383e+108 < alpha < 8.090655368014929e+157 or 3.039415483981533e+202 < alpha

    1. Initial program 59.1

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

    3. Applied *-un-lft-identity59.1

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

    4. Applied *-un-lft-identity59.1

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

    5. Applied times-frac46.2

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

    6. Applied times-frac46.3

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

    7. Simplified46.3

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

    8. Simplified46.3

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

    9. Taylor expanded around inf 41.0

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

    10. Simplified41.0

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

    if 8.090655368014929e+157 < alpha < 3.039415483981533e+202

    1. Initial program 64.0

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

    3. Applied *-un-lft-identity64.0

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

    4. Applied *-un-lft-identity64.0

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

    5. Applied times-frac35.4

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

    6. Applied times-frac35.3

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

    7. Simplified35.3

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

    8. Simplified35.3

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

    10. Applied add-sqr-sqrt35.4

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

    11. Applied add-cube-cbrt35.4

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

    12. Applied times-frac35.5

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

    13. Simplified35.4

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

    14. Simplified35.4

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

    16. Applied add-cbrt-cube35.4

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

    17. Simplified35.4

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

  4. Final simplification11.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 3.851388677432038317167398913439949496155 \cdot 10^{108}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \left(\frac{\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 2}} \cdot \frac{\sqrt[3]{\frac{\sqrt[3]{\beta - \alpha}}{\left(\alpha + i \cdot 2\right) + \beta}} \cdot \sqrt[3]{\sqrt[3]{\beta - \alpha} \cdot \sqrt[3]{\beta - \alpha}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 2}}{\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}}}\right)\right)}^{3}}}{2}\\ \mathbf{elif}\;\alpha \le 8.09065536801492928071076414340505776935 \cdot 10^{157} \lor \neg \left(\alpha \le 3.039415483981533193944183363030540056882 \cdot 10^{202}\right):\\ \;\;\;\;\frac{\left(\frac{8}{{\alpha}^{3}} + \frac{2}{\alpha}\right) - \frac{4}{\alpha \cdot \alpha}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{{\left(1 + \left(\alpha + \beta\right) \cdot \left(\frac{\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}}{\frac{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 2}}{\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}}} \cdot \frac{\sqrt[3]{\frac{\beta - \alpha}{\alpha + \left(\beta + i \cdot 2\right)}}}{\sqrt{\left(\left(\alpha + \beta\right) + i \cdot 2\right) + 2}}\right)\right)}^{3}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019196 
(FPCore (alpha beta i)
  :name "Octave 3.8, jcobi/2"
  :pre (and (> alpha -1.0) (> beta -1.0) (> i 0.0))
  (/ (+ (/ (/ (* (+ alpha beta) (- beta alpha)) (+ (+ alpha beta) (* 2.0 i))) (+ (+ (+ alpha beta) (* 2.0 i)) 2.0)) 1.0) 2.0))