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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.9528378137991099 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.9528378137991099 \cdot 10^{-15}:\\
\;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 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 r19359 = 2.0;
        double r19360 = x;
        double r19361 = r19359 * r19360;
        double r19362 = exp(r19361);
        double r19363 = 1.0;
        double r19364 = r19362 - r19363;
        double r19365 = exp(r19360);
        double r19366 = r19365 - r19363;
        double r19367 = r19364 / r19366;
        double r19368 = sqrt(r19367);
        return r19368;
}

↓

double f(double x) {
        double r19369 = x;
        double r19370 = -1.95283781379911e-15;
        bool r19371 = r19369 <= r19370;
        double r19372 = exp(r19369);
        double r19373 = 1.0;
        double r19374 = r19372 + r19373;
        double r19375 = 2.0;
        double r19376 = r19375 * r19369;
        double r19377 = exp(r19376);
        double r19378 = r19377 - r19373;
        double r19379 = r19374 * r19378;
        double r19380 = 2.0;
        double r19381 = r19380 * r19369;
        double r19382 = exp(r19381);
        double r19383 = r19382 - r19373;
        double r19384 = r19379 / r19383;
        double r19385 = sqrt(r19384);
        double r19386 = 0.5;
        double r19387 = sqrt(r19375);
        double r19388 = r19369 / r19387;
        double r19389 = r19386 * r19388;
        double r19390 = pow(r19369, r19380);
        double r19391 = r19390 / r19387;
        double r19392 = 0.25;
        double r19393 = 0.125;
        double r19394 = r19393 / r19375;
        double r19395 = r19392 - r19394;
        double r19396 = r19391 * r19395;
        double r19397 = r19387 + r19396;
        double r19398 = r19389 + r19397;
        double r19399 = r19371 ? r19385 : r19398;
        return r19399;
}

Derivation

Split input into 2 regimes
if x < -1.95283781379911e-15
1. Initial program 0.7
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.5
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Simplified0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\color{blue}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{e^{x} + 1}}}\]
5. Taylor expanded around inf 0.0
  \[\leadsto \sqrt{\color{blue}{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}}\]
if -1.95283781379911e-15 < x
1. Initial program 37.6
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 8.2
  \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]
3. Simplified8.2
  \[\leadsto \color{blue}{0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.9528378137991099 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

Error

Try it out

Derivation

`if x < -1.95283781379911e-15`

`if -1.95283781379911e-15 < x`

Reproduce

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