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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -6.207265300769888841194911957148031651441 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{\left|\sqrt[3]{e^{2 \cdot x}}\right| \cdot \sqrt{\sqrt[3]{e^{2 \cdot x}}} + \sqrt{1}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -6.207265300769888841194911957148031651441 \cdot 10^{-12}:\\
\;\;\;\;\sqrt{\frac{\left|\sqrt[3]{e^{2 \cdot x}}\right| \cdot \sqrt{\sqrt[3]{e^{2 \cdot x}}} + \sqrt{1}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\\

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

\end{array}

double f(double x) {
        double r27180 = 2.0;
        double r27181 = x;
        double r27182 = r27180 * r27181;
        double r27183 = exp(r27182);
        double r27184 = 1.0;
        double r27185 = r27183 - r27184;
        double r27186 = exp(r27181);
        double r27187 = r27186 - r27184;
        double r27188 = r27185 / r27187;
        double r27189 = sqrt(r27188);
        return r27189;
}

↓

double f(double x) {
        double r27190 = x;
        double r27191 = -6.207265300769889e-12;
        bool r27192 = r27190 <= r27191;
        double r27193 = 2.0;
        double r27194 = r27193 * r27190;
        double r27195 = exp(r27194);
        double r27196 = cbrt(r27195);
        double r27197 = fabs(r27196);
        double r27198 = sqrt(r27196);
        double r27199 = r27197 * r27198;
        double r27200 = 1.0;
        double r27201 = sqrt(r27200);
        double r27202 = r27199 + r27201;
        double r27203 = exp(r27190);
        double r27204 = r27203 - r27200;
        double r27205 = exp(r27193);
        double r27206 = 2.0;
        double r27207 = r27190 / r27206;
        double r27208 = pow(r27205, r27207);
        double r27209 = r27208 - r27201;
        double r27210 = r27204 / r27209;
        double r27211 = r27202 / r27210;
        double r27212 = sqrt(r27211);
        double r27213 = 0.5;
        double r27214 = sqrt(r27193);
        double r27215 = r27190 / r27214;
        double r27216 = r27213 * r27215;
        double r27217 = pow(r27190, r27206);
        double r27218 = r27217 / r27214;
        double r27219 = 0.25;
        double r27220 = 0.125;
        double r27221 = r27220 / r27193;
        double r27222 = r27219 - r27221;
        double r27223 = r27218 * r27222;
        double r27224 = r27214 + r27223;
        double r27225 = r27216 + r27224;
        double r27226 = r27192 ? r27212 : r27225;
        return r27226;
}

Derivation

Split input into 2 regimes
if x < -6.207265300769889e-12
1. Initial program 0.5
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.5
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{e^{x} - 1}}\]
4. Applied add-sqr-sqrt0.4
  \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}} - \sqrt{1} \cdot \sqrt{1}}{e^{x} - 1}}\]
5. Applied difference-of-squares0.1
  \[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}}{e^{x} - 1}}\]
6. Applied associate-/l*0.1
  \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}}\]
7. Using strategy rm
8. Applied add-log-exp0.1
  \[\leadsto \sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{\sqrt{e^{\color{blue}{\log \left(e^{2}\right)} \cdot x}} - \sqrt{1}}}}\]
9. Applied exp-to-pow0.1
  \[\leadsto \sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{\sqrt{\color{blue}{{\left(e^{2}\right)}^{x}}} - \sqrt{1}}}}\]
10. Applied sqrt-pow10.0
  \[\leadsto \sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{\color{blue}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}} - \sqrt{1}}}}\]
11. Using strategy rm
12. Applied add-cube-cbrt0.0
  \[\leadsto \sqrt{\frac{\sqrt{\color{blue}{\left(\sqrt[3]{e^{2 \cdot x}} \cdot \sqrt[3]{e^{2 \cdot x}}\right) \cdot \sqrt[3]{e^{2 \cdot x}}}} + \sqrt{1}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\]
13. Applied sqrt-prod0.0
  \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{\sqrt[3]{e^{2 \cdot x}} \cdot \sqrt[3]{e^{2 \cdot x}}} \cdot \sqrt{\sqrt[3]{e^{2 \cdot x}}}} + \sqrt{1}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\]
14. Simplified0.0
  \[\leadsto \sqrt{\frac{\color{blue}{\left|\sqrt[3]{e^{2 \cdot x}}\right|} \cdot \sqrt{\sqrt[3]{e^{2 \cdot x}}} + \sqrt{1}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\]
if -6.207265300769889e-12 < x
1. Initial program 37.2
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 7.4
  \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]
3. Simplified7.4
  \[\leadsto \color{blue}{0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.207265300769888841194911957148031651441 \cdot 10^{-12}:\\ \;\;\;\;\sqrt{\frac{\left|\sqrt[3]{e^{2 \cdot x}}\right| \cdot \sqrt{\sqrt[3]{e^{2 \cdot x}}} + \sqrt{1}}{\frac{e^{x} - 1}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

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

Error

Try it out

Derivation

`if x < -6.207265300769889e-12`

`if -6.207265300769889e-12 < x`

Reproduce

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