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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.2907203718339361 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt{2}}, 0.5, \sqrt{2}\right) + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.2907203718339361 \cdot 10^{-15}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\

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

\end{array}

double f(double x) {
        double r17370 = 2.0;
        double r17371 = x;
        double r17372 = r17370 * r17371;
        double r17373 = exp(r17372);
        double r17374 = 1.0;
        double r17375 = r17373 - r17374;
        double r17376 = exp(r17371);
        double r17377 = r17376 - r17374;
        double r17378 = r17375 / r17377;
        double r17379 = sqrt(r17378);
        return r17379;
}

↓

double f(double x) {
        double r17380 = x;
        double r17381 = -1.290720371833936e-15;
        bool r17382 = r17380 <= r17381;
        double r17383 = 2.0;
        double r17384 = r17383 * r17380;
        double r17385 = exp(r17384);
        double r17386 = 1.0;
        double r17387 = r17385 - r17386;
        double r17388 = -r17386;
        double r17389 = r17380 + r17380;
        double r17390 = exp(r17389);
        double r17391 = fma(r17388, r17386, r17390);
        double r17392 = exp(r17380);
        double r17393 = r17392 + r17386;
        double r17394 = r17391 / r17393;
        double r17395 = r17387 / r17394;
        double r17396 = sqrt(r17395);
        double r17397 = sqrt(r17383);
        double r17398 = r17380 / r17397;
        double r17399 = 0.5;
        double r17400 = fma(r17398, r17399, r17397);
        double r17401 = 2.0;
        double r17402 = pow(r17380, r17401);
        double r17403 = r17402 / r17397;
        double r17404 = 0.25;
        double r17405 = 0.125;
        double r17406 = r17405 / r17383;
        double r17407 = r17404 - r17406;
        double r17408 = r17403 * r17407;
        double r17409 = r17400 + r17408;
        double r17410 = r17382 ? r17396 : r17409;
        return r17410;
}

Derivation

Split input into 2 regimes
if x < -1.290720371833936e-15
1. Initial program 0.8
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
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. Simplified0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\color{blue}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{e^{x} + 1}}}\]
if -1.290720371833936e-15 < x
1. Initial program 38.3
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 7.9
  \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]
3. Simplified7.9
  \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x}{\sqrt{2}}, 0.5, \sqrt{2}\right) + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.2907203718339361 \cdot 10^{-15}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{\sqrt{2}}, 0.5, \sqrt{2}\right) + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\\ \end{array}\]

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

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

Error

Derivation

`if x < -1.290720371833936e-15`

`if -1.290720371833936e-15 < x`

Reproduce