Average Error: 24.2 → 11.7
Time: 20.5s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1 \land i \gt 0.0\]
\[\frac{\frac{\frac{\left(\alpha + \beta\right) \cdot \left(\beta - \alpha\right)}{\left(\alpha + \beta\right) + 2 \cdot i}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 5.849296461317042084954782375689309523104 \cdot 10^{141}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}} \cdot \frac{\frac{\sqrt[3]{\alpha + \beta}}{\frac{\sqrt[3]{1}}{\frac{\beta - \alpha}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]
\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 5.849296461317042084954782375689309523104 \cdot 10^{141}:\\
\;\;\;\;\frac{\frac{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}} \cdot \frac{\frac{\sqrt[3]{\alpha + \beta}}{\frac{\sqrt[3]{1}}{\frac{\beta - \alpha}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 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 r133672 = alpha;
        double r133673 = beta;
        double r133674 = r133672 + r133673;
        double r133675 = r133673 - r133672;
        double r133676 = r133674 * r133675;
        double r133677 = 2.0;
        double r133678 = i;
        double r133679 = r133677 * r133678;
        double r133680 = r133674 + r133679;
        double r133681 = r133676 / r133680;
        double r133682 = r133680 + r133677;
        double r133683 = r133681 / r133682;
        double r133684 = 1.0;
        double r133685 = r133683 + r133684;
        double r133686 = r133685 / r133677;
        return r133686;
}
↓
double f(double alpha, double beta, double i) {
        double r133687 = alpha;
        double r133688 = 5.849296461317042e+141;
        bool r133689 = r133687 <= r133688;
        double r133690 = beta;
        double r133691 = r133687 + r133690;
        double r133692 = cbrt(r133691);
        double r133693 = r133692 * r133692;
        double r133694 = 1.0;
        double r133695 = cbrt(r133694);
        double r133696 = r133695 * r133695;
        double r133697 = 2.0;
        double r133698 = i;
        double r133699 = r133697 * r133698;
        double r133700 = r133691 + r133699;
        double r133701 = cbrt(r133700);
        double r133702 = r133701 * r133701;
        double r133703 = r133694 / r133702;
        double r133704 = r133696 / r133703;
        double r133705 = r133693 / r133704;
        double r133706 = r133690 - r133687;
        double r133707 = r133706 / r133701;
        double r133708 = r133695 / r133707;
        double r133709 = r133692 / r133708;
        double r133710 = r133700 + r133697;
        double r133711 = r133709 / r133710;
        double r133712 = r133705 * r133711;
        double r133713 = 1.0;
        double r133714 = r133712 + r133713;
        double r133715 = r133714 / r133697;
        double r133716 = r133694 / r133687;
        double r133717 = r133697 * r133716;
        double r133718 = 8.0;
        double r133719 = 3.0;
        double r133720 = pow(r133687, r133719);
        double r133721 = r133694 / r133720;
        double r133722 = r133718 * r133721;
        double r133723 = r133717 + r133722;
        double r133724 = 4.0;
        double r133725 = 2.0;
        double r133726 = pow(r133687, r133725);
        double r133727 = r133694 / r133726;
        double r133728 = r133724 * r133727;
        double r133729 = r133723 - r133728;
        double r133730 = r133729 / r133697;
        double r133731 = r133689 ? r133715 : r133730;
        return r133731;
}

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 < 5.849296461317042e+141

    1. Initial program 15.4

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

    3. Applied associate-/l*4.8

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

    5. Applied clear-num4.8

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

    7. Applied *-un-lft-identity4.8

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

    8. Applied add-cube-cbrt5.0

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

    9. Applied *-un-lft-identity5.0

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

    10. Applied times-frac5.0

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

    11. Applied add-cube-cbrt5.0

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

    12. Applied times-frac5.0

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

    13. Applied add-cube-cbrt4.8

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

    14. Applied times-frac4.8

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

    15. Applied times-frac4.8

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

    16. Simplified4.8

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

    if 5.849296461317042e+141 < alpha

    1. Initial program 62.6

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

    2. Taylor expanded around inf 41.8

      \[\leadsto \frac{\color{blue}{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}}{2}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification11.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 5.849296461317042084954782375689309523104 \cdot 10^{141}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\alpha + \beta} \cdot \sqrt[3]{\alpha + \beta}}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\frac{1}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}} \cdot \frac{\frac{\sqrt[3]{\alpha + \beta}}{\frac{\sqrt[3]{1}}{\frac{\beta - \alpha}{\sqrt[3]{\left(\alpha + \beta\right) + 2 \cdot i}}}}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) + 2} + 1}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right) - 4 \cdot \frac{1}{{\alpha}^{2}}}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019352 
(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))