\[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.367867422063156052558161035470618571708 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \end{array}\]

\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}

↓

\begin{array}{l}
\mathbf{if}\;x \le -2.367867422063156052558161035470618571708 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\

\end{array}

double f(double x) {
        double r23268 = 2.0;
        double r23269 = x;
        double r23270 = r23268 * r23269;
        double r23271 = exp(r23270);
        double r23272 = 1.0;
        double r23273 = r23271 - r23272;
        double r23274 = exp(r23269);
        double r23275 = r23274 - r23272;
        double r23276 = r23273 / r23275;
        double r23277 = sqrt(r23276);
        return r23277;
}

↓

double f(double x) {
        double r23278 = x;
        double r23279 = -2.367867422063156e-05;
        bool r23280 = r23278 <= r23279;
        double r23281 = 2.0;
        double r23282 = r23281 * r23278;
        double r23283 = exp(r23282);
        double r23284 = 1.0;
        double r23285 = r23283 - r23284;
        double r23286 = r23278 + r23278;
        double r23287 = exp(r23286);
        double r23288 = r23284 * r23284;
        double r23289 = r23287 - r23288;
        double r23290 = r23285 / r23289;
        double r23291 = sqrt(r23290);
        double r23292 = exp(r23278);
        double r23293 = r23292 + r23284;
        double r23294 = sqrt(r23293);
        double r23295 = r23291 * r23294;
        double r23296 = 2.0;
        double r23297 = pow(r23278, r23296);
        double r23298 = sqrt(r23281);
        double r23299 = r23297 / r23298;
        double r23300 = 0.25;
        double r23301 = 0.125;
        double r23302 = r23301 / r23281;
        double r23303 = r23300 - r23302;
        double r23304 = r23299 * r23303;
        double r23305 = 0.5;
        double r23306 = r23278 / r23298;
        double r23307 = r23305 * r23306;
        double r23308 = r23298 + r23307;
        double r23309 = r23304 + r23308;
        double r23310 = r23280 ? r23295 : r23309;
        return r23310;
}

Derivation

Split input into 2 regimes
if x < -2.367867422063156e-05
1. Initial program 0.1
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Applied associate-/r/0.0
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]
5. Applied sqrt-prod0.0
  \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}}\]
6. Simplified0.0
  \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}}} \cdot \sqrt{e^{x} + 1}\]
if -2.367867422063156e-05 < x
1. Initial program 34.6
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 5.7
  \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]
3. Simplified5.6
  \[\leadsto \color{blue}{\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.7
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.367867422063156052558161035470618571708 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \end{array}\]

could not determine a ground truth for program body (more)

Error

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Derivation

`if x < -2.367867422063156e-05`

`if -2.367867422063156e-05 < x`

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