\[\alpha \gt -1 \land \beta \gt -1\]

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 6.48438097382688997 \cdot 10^{29}:\\ \;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}^{3}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}

↓

\begin{array}{l}
\mathbf{if}\;\alpha \le 6.48438097382688997 \cdot 10^{29}:\\
\;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}^{3}}}}{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 r113710 = beta;
        double r113711 = alpha;
        double r113712 = r113710 - r113711;
        double r113713 = r113711 + r113710;
        double r113714 = 2.0;
        double r113715 = r113713 + r113714;
        double r113716 = r113712 / r113715;
        double r113717 = 1.0;
        double r113718 = r113716 + r113717;
        double r113719 = r113718 / r113714;
        return r113719;
}

↓

double f(double alpha, double beta) {
        double r113720 = alpha;
        double r113721 = 6.48438097382689e+29;
        bool r113722 = r113720 <= r113721;
        double r113723 = beta;
        double r113724 = r113720 + r113723;
        double r113725 = 2.0;
        double r113726 = r113724 + r113725;
        double r113727 = r113723 / r113726;
        double r113728 = r113720 / r113726;
        double r113729 = 1.0;
        double r113730 = r113728 - r113729;
        double r113731 = r113727 - r113730;
        double r113732 = log(r113731);
        double r113733 = 3.0;
        double r113734 = pow(r113732, r113733);
        double r113735 = cbrt(r113734);
        double r113736 = exp(r113735);
        double r113737 = r113736 / r113725;
        double r113738 = 4.0;
        double r113739 = r113720 * r113720;
        double r113740 = r113738 / r113739;
        double r113741 = r113725 / r113720;
        double r113742 = r113740 - r113741;
        double r113743 = 8.0;
        double r113744 = pow(r113720, r113733);
        double r113745 = r113743 / r113744;
        double r113746 = r113742 - r113745;
        double r113747 = r113727 - r113746;
        double r113748 = r113747 / r113725;
        double r113749 = r113722 ? r113737 : r113748;
        return r113749;
}

Derivation

Split input into 2 regimes
if alpha < 6.48438097382689e+29
1. Initial program 1.4
  \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
2. Using strategy rm
3. Applied div-sub1.4
  \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]
4. Applied associate-+l-1.4
  \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
5. Using strategy rm
6. Applied add-exp-log1.4
  \[\leadsto \frac{\color{blue}{e^{\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}}{2}\]
7. Using strategy rm
8. Applied add-cbrt-cube1.4
  \[\leadsto \frac{e^{\color{blue}{\sqrt[3]{\left(\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right) \cdot \log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right) \cdot \log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)}}}}{2}\]
9. Simplified1.4
  \[\leadsto \frac{e^{\sqrt[3]{\color{blue}{{\left(\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}^{3}}}}}{2}\]
if 6.48438097382689e+29 < alpha
1. Initial program 52.1
  \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
2. Using strategy rm
3. Applied div-sub52.1
  \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]
4. Applied associate-+l-50.4
  \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
5. Taylor expanded around inf 17.7
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(4 \cdot \frac{1}{{\alpha}^{2}} - \left(2 \cdot \frac{1}{\alpha} + 8 \cdot \frac{1}{{\alpha}^{3}}\right)\right)}}{2}\]
6. Simplified17.7
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}}{2}\]
Recombined 2 regimes into one program.
Final simplification6.2
\[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6.48438097382688997 \cdot 10^{29}:\\ \;\;\;\;\frac{e^{\sqrt[3]{{\left(\log \left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)\right)\right)}^{3}}}}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{4}{\alpha \cdot \alpha} - \frac{2}{\alpha}\right) - \frac{8}{{\alpha}^{3}}\right)}{2}\\ \end{array}\]

Falling back on racket egraph because egg-herbie package not installed (more)

Error

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Derivation

`if alpha < 6.48438097382689e+29`

`if 6.48438097382689e+29 < alpha`

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