\[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]

↓

\[\begin{array}{l} \mathbf{if}\;\beta \le 5.714299342882349 \cdot 10^{+204}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{(\left(i + \left(\alpha + \beta\right)\right) \cdot i + \left(\alpha \cdot \beta\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}} \cdot \sqrt{\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\right) \cdot \left(\sqrt{\frac{\frac{(\left(i + \left(\alpha + \beta\right)\right) \cdot i + \left(\alpha \cdot \beta\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}} \cdot \sqrt{\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

double f(double alpha, double beta, double i) {
        double r17291202 = i;
        double r17291203 = alpha;
        double r17291204 = beta;
        double r17291205 = r17291203 + r17291204;
        double r17291206 = r17291205 + r17291202;
        double r17291207 = r17291202 * r17291206;
        double r17291208 = r17291204 * r17291203;
        double r17291209 = r17291208 + r17291207;
        double r17291210 = r17291207 * r17291209;
        double r17291211 = 2.0;
        double r17291212 = r17291211 * r17291202;
        double r17291213 = r17291205 + r17291212;
        double r17291214 = r17291213 * r17291213;
        double r17291215 = r17291210 / r17291214;
        double r17291216 = 1.0;
        double r17291217 = r17291214 - r17291216;
        double r17291218 = r17291215 / r17291217;
        return r17291218;
}

↓

double f(double alpha, double beta, double i) {
        double r17291219 = beta;
        double r17291220 = 5.714299342882349e+204;
        bool r17291221 = r17291219 <= r17291220;
        double r17291222 = i;
        double r17291223 = alpha;
        double r17291224 = r17291223 + r17291219;
        double r17291225 = r17291222 + r17291224;
        double r17291226 = r17291223 * r17291219;
        double r17291227 = fma(r17291225, r17291222, r17291226);
        double r17291228 = 2.0;
        double r17291229 = fma(r17291228, r17291222, r17291224);
        double r17291230 = r17291227 / r17291229;
        double r17291231 = 1.0;
        double r17291232 = sqrt(r17291231);
        double r17291233 = r17291232 + r17291229;
        double r17291234 = r17291230 / r17291233;
        double r17291235 = sqrt(r17291234);
        double r17291236 = r17291222 * r17291225;
        double r17291237 = r17291236 / r17291229;
        double r17291238 = r17291229 - r17291232;
        double r17291239 = r17291237 / r17291238;
        double r17291240 = sqrt(r17291239);
        double r17291241 = r17291235 * r17291240;
        double r17291242 = r17291241 * r17291241;
        double r17291243 = 0.0;
        double r17291244 = r17291221 ? r17291242 : r17291243;
        return r17291244;
}

\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}

↓

\begin{array}{l}
\mathbf{if}\;\beta \le 5.714299342882349 \cdot 10^{+204}:\\
\;\;\;\;\left(\sqrt{\frac{\frac{(\left(i + \left(\alpha + \beta\right)\right) \cdot i + \left(\alpha \cdot \beta\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}} \cdot \sqrt{\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\right) \cdot \left(\sqrt{\frac{\frac{(\left(i + \left(\alpha + \beta\right)\right) \cdot i + \left(\alpha \cdot \beta\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}} \cdot \sqrt{\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\right)\\

\mathbf{else}:\\
\;\;\;\;0\\

\end{array}

Derivation

Split input into 2 regimes
if beta < 5.714299342882349e+204
1. Initial program 51.1
  \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
2. Simplified51.1
  \[\leadsto \color{blue}{\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_* \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}{(2 \cdot i + \left(\alpha + \beta\right))_* \cdot (2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* \cdot (2 \cdot i + \left(\alpha + \beta\right))_* - 1.0}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt51.1
  \[\leadsto \frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_* \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}{(2 \cdot i + \left(\alpha + \beta\right))_* \cdot (2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* \cdot (2 \cdot i + \left(\alpha + \beta\right))_* - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]
5. Applied difference-of-squares51.1
  \[\leadsto \frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_* \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}{(2 \cdot i + \left(\alpha + \beta\right))_* \cdot (2 \cdot i + \left(\alpha + \beta\right))_*}}{\color{blue}{\left((2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}\right) \cdot \left((2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}\right)}}\]
6. Applied times-frac36.1
  \[\leadsto \frac{\color{blue}{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}}{\left((2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}\right) \cdot \left((2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}\right)}\]
7. Applied times-frac34.0
  \[\leadsto \color{blue}{\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}} \cdot \frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\]
8. Using strategy rm
9. Applied add-sqr-sqrt34.0
  \[\leadsto \frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}} \cdot \color{blue}{\left(\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}} \cdot \sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\right)}\]
10. Applied add-sqr-sqrt34.0
  \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}}} \cdot \sqrt{\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}}}\right)} \cdot \left(\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}} \cdot \sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\right)\]
11. Applied unswap-sqr34.0
  \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}}} \cdot \sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\right) \cdot \left(\sqrt{\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}}} \cdot \sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\right)}\]
if 5.714299342882349e+204 < beta
1. Initial program 62.6
  \[\frac{\frac{\left(i \cdot \left(\left(\alpha + \beta\right) + i\right)\right) \cdot \left(\beta \cdot \alpha + i \cdot \left(\left(\alpha + \beta\right) + i\right)\right)}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right)}}{\left(\left(\alpha + \beta\right) + 2 \cdot i\right) \cdot \left(\left(\alpha + \beta\right) + 2 \cdot i\right) - 1.0}\]
2. Simplified62.6
  \[\leadsto \color{blue}{\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_* \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}{(2 \cdot i + \left(\alpha + \beta\right))_* \cdot (2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* \cdot (2 \cdot i + \left(\alpha + \beta\right))_* - 1.0}}\]
3. Using strategy rm
4. Applied add-sqr-sqrt62.6
  \[\leadsto \frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_* \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}{(2 \cdot i + \left(\alpha + \beta\right))_* \cdot (2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* \cdot (2 \cdot i + \left(\alpha + \beta\right))_* - \color{blue}{\sqrt{1.0} \cdot \sqrt{1.0}}}\]
5. Applied difference-of-squares62.6
  \[\leadsto \frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_* \cdot \left(\left(\left(\alpha + \beta\right) + i\right) \cdot i\right)}{(2 \cdot i + \left(\alpha + \beta\right))_* \cdot (2 \cdot i + \left(\alpha + \beta\right))_*}}{\color{blue}{\left((2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}\right) \cdot \left((2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}\right)}}\]
6. Applied times-frac56.1
  \[\leadsto \frac{\color{blue}{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*} \cdot \frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}}{\left((2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}\right) \cdot \left((2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}\right)}\]
7. Applied times-frac54.5
  \[\leadsto \color{blue}{\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}} \cdot \frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\]
8. Using strategy rm
9. Applied add-sqr-sqrt54.5
  \[\leadsto \frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}} \cdot \color{blue}{\left(\sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}} \cdot \sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\right)}\]
10. Applied associate-*r*54.5
  \[\leadsto \color{blue}{\left(\frac{\frac{(\left(\left(\alpha + \beta\right) + i\right) \cdot i + \left(\beta \cdot \alpha\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* + \sqrt{1.0}} \cdot \sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\right) \cdot \sqrt{\frac{\frac{\left(\left(\alpha + \beta\right) + i\right) \cdot i}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}}\]
11. Taylor expanded around inf 45.0
  \[\leadsto \color{blue}{0}\]
Recombined 2 regimes into one program.
Final simplification35.2
\[\leadsto \begin{array}{l} \mathbf{if}\;\beta \le 5.714299342882349 \cdot 10^{+204}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{(\left(i + \left(\alpha + \beta\right)\right) \cdot i + \left(\alpha \cdot \beta\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}} \cdot \sqrt{\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\right) \cdot \left(\sqrt{\frac{\frac{(\left(i + \left(\alpha + \beta\right)\right) \cdot i + \left(\alpha \cdot \beta\right))_*}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{\sqrt{1.0} + (2 \cdot i + \left(\alpha + \beta\right))_*}} \cdot \sqrt{\frac{\frac{i \cdot \left(i + \left(\alpha + \beta\right)\right)}{(2 \cdot i + \left(\alpha + \beta\right))_*}}{(2 \cdot i + \left(\alpha + \beta\right))_* - \sqrt{1.0}}}\right)\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array}\]

Error

Derivation

`if beta < 5.714299342882349e+204`

`if 5.714299342882349e+204 < beta`

Reproduce