Average Error: 16.1 → 6.4
Time: 1.0m
Precision: 64
\[\alpha \gt -1 \land \beta \gt -1\]
\[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
↓
\[\begin{array}{l} \mathbf{if}\;\alpha \le 1.30567877700455932 \cdot 10^{40}:\\ \;\;\;\;\frac{\log \left(e^{\sqrt[3]{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]
\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}
↓
\begin{array}{l}
\mathbf{if}\;\alpha \le 1.30567877700455932 \cdot 10^{40}:\\
\;\;\;\;\frac{\log \left(e^{\sqrt[3]{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}}\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 r136737 = beta;
        double r136738 = alpha;
        double r136739 = r136737 - r136738;
        double r136740 = r136738 + r136737;
        double r136741 = 2.0;
        double r136742 = r136740 + r136741;
        double r136743 = r136739 / r136742;
        double r136744 = 1.0;
        double r136745 = r136743 + r136744;
        double r136746 = r136745 / r136741;
        return r136746;
}
↓
double f(double alpha, double beta) {
        double r136747 = alpha;
        double r136748 = 1.3056787770045593e+40;
        bool r136749 = r136747 <= r136748;
        double r136750 = beta;
        double r136751 = r136747 + r136750;
        double r136752 = 2.0;
        double r136753 = r136751 + r136752;
        double r136754 = r136750 / r136753;
        double r136755 = cbrt(r136754);
        double r136756 = 6.0;
        double r136757 = pow(r136755, r136756);
        double r136758 = cbrt(r136757);
        double r136759 = exp(r136758);
        double r136760 = log(r136759);
        double r136761 = r136760 * r136755;
        double r136762 = r136747 / r136753;
        double r136763 = 1.0;
        double r136764 = r136762 - r136763;
        double r136765 = r136761 - r136764;
        double r136766 = r136765 / r136752;
        double r136767 = 4.0;
        double r136768 = r136747 * r136747;
        double r136769 = r136767 / r136768;
        double r136770 = r136752 / r136747;
        double r136771 = r136769 - r136770;
        double r136772 = 8.0;
        double r136773 = 3.0;
        double r136774 = pow(r136747, r136773);
        double r136775 = r136772 / r136774;
        double r136776 = r136771 - r136775;
        double r136777 = r136754 - r136776;
        double r136778 = r136777 / r136752;
        double r136779 = r136749 ? r136766 : r136778;
        return r136779;
}

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 < 1.3056787770045593e+40

    1. Initial program 1.9

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

    3. Applied div-sub1.9

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

    4. Applied associate-+l-1.9

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

    6. Applied add-cube-cbrt1.9

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

    8. Applied pow1/321.7

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

    9. Applied pow1/321.7

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

    10. Applied pow-prod-down1.9

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

    11. Simplified1.9

      \[\leadsto \frac{{\color{blue}{\left({\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}\right)}}^{\frac{1}{3}} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]
    12. Using strategy rm

    13. Applied add-log-exp1.9

      \[\leadsto \frac{\color{blue}{\log \left(e^{{\left({\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}\right)}^{\frac{1}{3}}}\right)} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]

    14. Simplified1.9

      \[\leadsto \frac{\log \color{blue}{\left(e^{\sqrt[3]{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}}\right)} \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\]

    if 1.3056787770045593e+40 < alpha

    1. Initial program 51.1

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

    3. Applied div-sub51.1

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

    4. Applied associate-+l-49.5

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

    5. Taylor expanded around inf 17.5

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

    6. Simplified17.5

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

  4. Final simplification6.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 1.30567877700455932 \cdot 10^{40}:\\ \;\;\;\;\frac{\log \left(e^{\sqrt[3]{{\left(\sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)}^{6}}}\right) \cdot \sqrt[3]{\frac{\beta}{\left(\alpha + \beta\right) + 2}} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/1"
  :precision binary64
  :pre (and (> alpha -1) (> beta -1))
  (/ (+ (/ (- beta alpha) (+ (+ alpha beta) 2)) 1) 2))