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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.1247925004240297 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^{3}}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.1247925004240297 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^{3}}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\

\end{array}

double f(double x) {
        double r11476 = 2.0;
        double r11477 = x;
        double r11478 = r11476 * r11477;
        double r11479 = exp(r11478);
        double r11480 = 1.0;
        double r11481 = r11479 - r11480;
        double r11482 = exp(r11477);
        double r11483 = r11482 - r11480;
        double r11484 = r11481 / r11483;
        double r11485 = sqrt(r11484);
        return r11485;
}

↓

double f(double x) {
        double r11486 = x;
        double r11487 = -1.1247925004240297e-05;
        bool r11488 = r11486 <= r11487;
        double r11489 = 2.0;
        double r11490 = r11489 * r11486;
        double r11491 = exp(r11490);
        double r11492 = 1.0;
        double r11493 = r11491 - r11492;
        double r11494 = 3.0;
        double r11495 = pow(r11493, r11494);
        double r11496 = cbrt(r11495);
        double r11497 = -r11492;
        double r11498 = r11486 + r11486;
        double r11499 = exp(r11498);
        double r11500 = fma(r11497, r11492, r11499);
        double r11501 = r11496 / r11500;
        double r11502 = exp(r11486);
        double r11503 = r11502 + r11492;
        double r11504 = r11501 * r11503;
        double r11505 = sqrt(r11504);
        double r11506 = 0.5;
        double r11507 = 2.0;
        double r11508 = pow(r11486, r11507);
        double r11509 = fma(r11492, r11486, r11489);
        double r11510 = fma(r11506, r11508, r11509);
        double r11511 = sqrt(r11510);
        double r11512 = r11488 ? r11505 : r11511;
        return r11512;
}

Derivation

Split input into 2 regimes
if x < -1.1247925004240297e-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. Simplified0.0
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}} \cdot \left(e^{x} + 1\right)}\]
6. Using strategy rm
7. Applied add-cbrt-cube0.0
  \[\leadsto \sqrt{\frac{\color{blue}{\sqrt[3]{\left(\left(e^{2 \cdot x} - 1\right) \cdot \left(e^{2 \cdot x} - 1\right)\right) \cdot \left(e^{2 \cdot x} - 1\right)}}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\]
8. Simplified0.0
  \[\leadsto \sqrt{\frac{\sqrt[3]{\color{blue}{{\left(e^{2 \cdot x} - 1\right)}^{3}}}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\]
if -1.1247925004240297e-05 < x
1. Initial program 33.8
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 5.9
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified5.9
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.1247925004240297 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\sqrt[3]{{\left(e^{2 \cdot x} - 1\right)}^{3}}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

Error

Derivation

`if x < -1.1247925004240297e-05`

`if -1.1247925004240297e-05 < x`

Reproduce