Average Error: 16.3 → 5.9
Time: 17.3s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 307451316.10967922210693359375:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 307451316.10967922210693359375:\\
\;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r77615 = beta;
        double r77616 = alpha;
        double r77617 = r77615 - r77616;
        double r77618 = r77616 + r77615;
        double r77619 = 2.0;
        double r77620 = r77618 + r77619;
        double r77621 = r77617 / r77620;
        double r77622 = 1.0;
        double r77623 = r77621 + r77622;
        double r77624 = r77623 / r77619;
        return r77624;
}
↓
double f(double alpha, double beta) {
        double r77625 = alpha;
        double r77626 = 307451316.1096792;
        bool r77627 = r77625 <= r77626;
        double r77628 = beta;
        double r77629 = r77625 + r77628;
        double r77630 = 2.0;
        double r77631 = r77629 + r77630;
        double r77632 = r77628 / r77631;
        double r77633 = r77625 / r77631;
        double r77634 = 1.0;
        double r77635 = r77633 - r77634;
        double r77636 = r77632 - r77635;
        double r77637 = log(r77636);
        double r77638 = exp(r77637);
        double r77639 = r77638 / r77630;
        double r77640 = cbrt(r77628);
        double r77641 = r77640 * r77640;
        double r77642 = cbrt(r77631);
        double r77643 = r77642 * r77642;
        double r77644 = r77641 / r77643;
        double r77645 = r77640 / r77642;
        double r77646 = r77644 * r77645;
        double r77647 = 4.0;
        double r77648 = 2.0;
        double r77649 = pow(r77625, r77648);
        double r77650 = r77647 / r77649;
        double r77651 = r77630 / r77625;
        double r77652 = r77650 - r77651;
        double r77653 = 8.0;
        double r77654 = 3.0;
        double r77655 = pow(r77625, r77654);
        double r77656 = r77653 / r77655;
        double r77657 = r77652 - r77656;
        double r77658 = r77646 - r77657;
        double r77659 = r77658 / r77630;
        double r77660 = r77627 ? r77639 : r77659;
        return r77660;
}

Error

Bits error versus alpha

Bits error versus beta

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if alpha < 307451316.1096792

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    4. Applied associate-+l-0.1

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

    6. Applied add-exp-log0.1

      \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}}{2}\]

    if 307451316.1096792 < alpha

    1. Initial program 49.9

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

    3. Applied div-sub49.9

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

    4. Applied associate-+l-48.3

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

    6. Applied add-log-exp48.3

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - \color{blue}{\log \left(e^{1}\right)}\right)}{2}\]

    7. Applied add-log-exp48.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\color{blue}{\log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}\right)} - \log \left(e^{1}\right)\right)}{2}\]

    8. Applied diff-log48.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\log \left(\frac{e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}}{e^{1}}\right)}}{2}\]

    9. Simplified48.4

      \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \color{blue}{\left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}}{2}\]
    10. Using strategy rm

    11. Applied add-cube-cbrt48.5

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

    12. Applied add-cube-cbrt48.4

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

    13. Applied times-frac48.4

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

    14. Taylor expanded around inf 17.8

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

    15. Simplified17.8

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

  4. Final simplification5.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 307451316.10967922210693359375:\\ \;\;\;\;\frac{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sqrt[3]{\beta} \cdot \sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2} \cdot \sqrt[3]{\left(\alpha + \beta\right) + 2}} \cdot \frac{\sqrt[3]{\beta}}{\sqrt[3]{\left(\alpha + \beta\right) + 2}} - \left(\left(\frac{4}{{\alpha}^{2}} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))