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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.693221537821368658447131741849752371853 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.693221537821368658447131741849752371853 \cdot 10^{-10}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\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 r12311 = 2.0;
        double r12312 = x;
        double r12313 = r12311 * r12312;
        double r12314 = exp(r12313);
        double r12315 = 1.0;
        double r12316 = r12314 - r12315;
        double r12317 = exp(r12312);
        double r12318 = r12317 - r12315;
        double r12319 = r12316 / r12318;
        double r12320 = sqrt(r12319);
        return r12320;
}

↓

double f(double x) {
        double r12321 = x;
        double r12322 = -1.6932215378213687e-10;
        bool r12323 = r12321 <= r12322;
        double r12324 = 2.0;
        double r12325 = r12324 * r12321;
        double r12326 = exp(r12325);
        double r12327 = 1.0;
        double r12328 = r12326 - r12327;
        double r12329 = -r12327;
        double r12330 = r12321 + r12321;
        double r12331 = exp(r12330);
        double r12332 = fma(r12329, r12327, r12331);
        double r12333 = r12328 / r12332;
        double r12334 = exp(r12321);
        double r12335 = r12334 + r12327;
        double r12336 = r12333 * r12335;
        double r12337 = sqrt(r12336);
        double r12338 = 0.5;
        double r12339 = 2.0;
        double r12340 = pow(r12321, r12339);
        double r12341 = fma(r12327, r12321, r12324);
        double r12342 = fma(r12338, r12340, r12341);
        double r12343 = sqrt(r12342);
        double r12344 = r12323 ? r12337 : r12343;
        return r12344;
}

Derivation

Split input into 2 regimes
if x < -1.6932215378213687e-10
1. Initial program 0.4
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.2
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Applied associate-/r/0.2
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]
5. Simplified0.0
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}} \cdot \left(e^{x} + 1\right)}\]
if -1.6932215378213687e-10 < x
1. Initial program 36.0
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.9
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.9
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.693221537821368658447131741849752371853 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

Error

Derivation

`if x < -1.6932215378213687e-10`

`if -1.6932215378213687e-10 < x`

Reproduce