Average Error: 16.6 → 6.1
Time: 16.6s
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 14037838382120007700:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}}\right)\right)\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 14037838382120007700:\\
\;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}}\right)\right)\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\

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

\end{array}
double f(double alpha, double beta) {
        double r85634 = beta;
        double r85635 = alpha;
        double r85636 = r85634 - r85635;
        double r85637 = r85635 + r85634;
        double r85638 = 2.0;
        double r85639 = r85637 + r85638;
        double r85640 = r85636 / r85639;
        double r85641 = 1.0;
        double r85642 = r85640 + r85641;
        double r85643 = r85642 / r85638;
        return r85643;
}
↓
double f(double alpha, double beta) {
        double r85644 = alpha;
        double r85645 = 1.4037838382120008e+19;
        bool r85646 = r85644 <= r85645;
        double r85647 = beta;
        double r85648 = r85644 + r85647;
        double r85649 = 2.0;
        double r85650 = r85648 + r85649;
        double r85651 = r85647 / r85650;
        double r85652 = cbrt(r85651);
        double r85653 = exp(r85652);
        double r85654 = sqrt(r85653);
        double r85655 = log(r85654);
        double r85656 = r85655 + r85655;
        double r85657 = r85652 * r85656;
        double r85658 = r85657 * r85652;
        double r85659 = r85644 / r85650;
        double r85660 = 1.0;
        double r85661 = r85659 - r85660;
        double r85662 = r85658 - r85661;
        double r85663 = r85662 / r85649;
        double r85664 = 4.0;
        double r85665 = r85644 * r85644;
        double r85666 = r85664 / r85665;
        double r85667 = r85649 / r85644;
        double r85668 = r85666 - r85667;
        double r85669 = 8.0;
        double r85670 = 3.0;
        double r85671 = pow(r85644, r85670);
        double r85672 = r85669 / r85671;
        double r85673 = r85668 - r85672;
        double r85674 = r85651 - r85673;
        double r85675 = r85674 / r85649;
        double r85676 = r85646 ? r85663 : r85675;
        return r85676;
}

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 < 1.4037838382120008e+19

    1. Initial program 0.6

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

    3. Applied div-sub0.6

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

    4. Applied associate-+l-0.6

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

    6. Applied add-cube-cbrt0.6

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

    8. Applied add-log-exp0.6

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

    10. Applied add-sqr-sqrt0.6

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

    11. Applied log-prod0.6

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

    if 1.4037838382120008e+19 < alpha

    1. Initial program 50.6

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

    3. Applied div-sub50.6

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

    4. Applied associate-+l-49.1

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

    5. Taylor expanded around inf 17.9

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

    6. Simplified17.9

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

  4. Final simplification6.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 14037838382120007700:\\ \;\;\;\;\frac{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} \cdot \left(\log \left(\sqrt{e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}}\right) + \log \left(\sqrt{e^{\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}}}\right)\right)\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 +o rules:numerics
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :pre (and (> alpha -1.0) (> beta -1.0))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2.0)) 1.0) 2.0))