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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.293819911268271 \cdot 10^{-5} \lor \neg \left(x \le 1.1457866946809401 \cdot 10^{-14}\right):\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.293819911268271 \cdot 10^{-5} \lor \neg \left(x \le 1.1457866946809401 \cdot 10^{-14}\right):\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}\\

\end{array}

double f(double x) {
        double r13282 = 2.0;
        double r13283 = x;
        double r13284 = r13282 * r13283;
        double r13285 = exp(r13284);
        double r13286 = 1.0;
        double r13287 = r13285 - r13286;
        double r13288 = exp(r13283);
        double r13289 = r13288 - r13286;
        double r13290 = r13287 / r13289;
        double r13291 = sqrt(r13290);
        return r13291;
}

↓

double f(double x) {
        double r13292 = x;
        double r13293 = -1.2938199112682712e-05;
        bool r13294 = r13292 <= r13293;
        double r13295 = 1.1457866946809401e-14;
        bool r13296 = r13292 <= r13295;
        double r13297 = !r13296;
        bool r13298 = r13294 || r13297;
        double r13299 = 2.0;
        double r13300 = r13299 * r13292;
        double r13301 = exp(r13300);
        double r13302 = 1.0;
        double r13303 = r13301 - r13302;
        double r13304 = r13292 + r13292;
        double r13305 = exp(r13304);
        double r13306 = r13302 * r13302;
        double r13307 = r13305 - r13306;
        double r13308 = exp(r13292);
        double r13309 = r13308 + r13302;
        double r13310 = r13307 / r13309;
        double r13311 = r13303 / r13310;
        double r13312 = sqrt(r13311);
        double r13313 = 0.5;
        double r13314 = 2.0;
        double r13315 = pow(r13292, r13314);
        double r13316 = r13313 * r13315;
        double r13317 = r13302 * r13292;
        double r13318 = r13317 + r13299;
        double r13319 = r13316 + r13318;
        double r13320 = sqrt(r13319);
        double r13321 = r13298 ? r13312 : r13320;
        return r13321;
}

Derivation

Split input into 2 regimes
if x < -1.2938199112682712e-05 or 1.1457866946809401e-14 < x
1. Initial program 0.9
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.6
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Simplified0.1
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\color{blue}{e^{x + x} - 1 \cdot 1}}{e^{x} + 1}}}\]
if -1.2938199112682712e-05 < x < 1.1457866946809401e-14
1. Initial program 45.9
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--43.6
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Simplified34.5
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\color{blue}{e^{x + x} - 1 \cdot 1}}{e^{x} + 1}}}\]
5. Taylor expanded around 0 0.1
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.293819911268271 \cdot 10^{-5} \lor \neg \left(x \le 1.1457866946809401 \cdot 10^{-14}\right):\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{e^{x + x} - 1 \cdot 1}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}\\ \end{array}\]

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

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

Error

Try it out

Derivation

`if x < -1.2938199112682712e-05 or 1.1457866946809401e-14 < x`

`if -1.2938199112682712e-05 < x < 1.1457866946809401e-14`

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