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

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

↓

\[\begin{array}{l} \mathbf{if}\;\alpha \le 6566215071237633024:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;\alpha \le 6566215071237633024:\\
\;\;\;\;\frac{\log \left(e^{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\

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

\end{array}

double f(double alpha, double beta) {
        double r159536 = beta;
        double r159537 = alpha;
        double r159538 = r159536 - r159537;
        double r159539 = r159537 + r159536;
        double r159540 = 2.0;
        double r159541 = r159539 + r159540;
        double r159542 = r159538 / r159541;
        double r159543 = 1.0;
        double r159544 = r159542 + r159543;
        double r159545 = r159544 / r159540;
        return r159545;
}

↓

double f(double alpha, double beta) {
        double r159546 = alpha;
        double r159547 = 6.566215071237633e+18;
        bool r159548 = r159546 <= r159547;
        double r159549 = beta;
        double r159550 = r159546 + r159549;
        double r159551 = 2.0;
        double r159552 = r159550 + r159551;
        double r159553 = r159549 / r159552;
        double r159554 = exp(r159553);
        double r159555 = log(r159554);
        double r159556 = r159546 / r159552;
        double r159557 = 1.0;
        double r159558 = r159556 - r159557;
        double r159559 = exp(r159558);
        double r159560 = log(r159559);
        double r159561 = r159555 - r159560;
        double r159562 = r159561 / r159551;
        double r159563 = 4.0;
        double r159564 = r159563 / r159546;
        double r159565 = r159564 / r159546;
        double r159566 = 8.0;
        double r159567 = -r159566;
        double r159568 = 3.0;
        double r159569 = pow(r159546, r159568);
        double r159570 = r159567 / r159569;
        double r159571 = r159565 + r159570;
        double r159572 = -r159551;
        double r159573 = r159572 / r159546;
        double r159574 = r159571 + r159573;
        double r159575 = r159553 - r159574;
        double r159576 = r159575 / r159551;
        double r159577 = r159548 ? r159562 : r159576;
        return r159577;
}

Derivation

Split input into 2 regimes
if alpha < 6.566215071237633e+18
1. Initial program 0.5
  \[\frac{\frac{\beta - \alpha}{\left(\alpha + \beta\right) + 2} + 1}{2}\]
2. Using strategy rm
3. Applied div-sub0.5
  \[\leadsto \frac{\color{blue}{\left(\frac{\beta}{\left(\alpha + \beta\right) + 2} - \frac{\alpha}{\left(\alpha + \beta\right) + 2}\right)} + 1}{2}\]
4. Applied associate-+l-0.5
  \[\leadsto \frac{\color{blue}{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1\right)}}{2}\]
5. Using strategy rm
6. Applied add-log-exp0.5
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\frac{\alpha}{\left(\alpha + \beta\right) + 2} - \color{blue}{\log \left(e^{1}\right)}\right)}{2}\]
7. Applied add-log-exp0.6
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\color{blue}{\log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}\right)} - \log \left(e^{1}\right)\right)}{2}\]
8. Applied diff-log0.6
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\log \left(\frac{e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2}}}{e^{1}}\right)}}{2}\]
9. Simplified0.5
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \log \color{blue}{\left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}}{2}\]
10. Using strategy rm
11. Applied add-log-exp0.5
  \[\leadsto \frac{\color{blue}{\log \left(e^{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right)} - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\]
if 6.566215071237633e+18 < 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
  \[\leadsto \frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \color{blue}{\left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}}{2}\]
Recombined 2 regimes into one program.
Final simplification6.1
\[\leadsto \begin{array}{l} \mathbf{if}\;\alpha \le 6566215071237633024:\\ \;\;\;\;\frac{\log \left(e^{\frac{\beta}{\left(\alpha + \beta\right) + 2}}\right) - \log \left(e^{\frac{\alpha}{\left(\alpha + \beta\right) + 2} - 1}\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\beta}{\left(\alpha + \beta\right) + 2} - \left(\left(\frac{\frac{4}{\alpha}}{\alpha} + \frac{-8}{{\alpha}^{3}}\right) + \frac{-2}{\alpha}\right)}{2}\\ \end{array}\]

Error

Try it out

Derivation

`if alpha < 6.566215071237633e+18`

`if 6.566215071237633e+18 < alpha`

Reproduce

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