\[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le 64.8190525609748:\\ \;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{2}{3}, -1\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(e^{\left(\varepsilon - 1\right) \cdot x} + \frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon}\right) - \frac{\frac{1}{\varepsilon} - 1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)\\ \end{array}\]

\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}

↓

\begin{array}{l}
\mathbf{if}\;x \le 64.8190525609748:\\
\;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{2}{3}, -1\right), 2\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} \cdot \left(\left(e^{\left(\varepsilon - 1\right) \cdot x} + \frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon}\right) - \frac{\frac{1}{\varepsilon} - 1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)\\

\end{array}

double f(double x, double eps) {
        double r1853233 = 1.0;
        double r1853234 = eps;
        double r1853235 = r1853233 / r1853234;
        double r1853236 = r1853233 + r1853235;
        double r1853237 = r1853233 - r1853234;
        double r1853238 = x;
        double r1853239 = r1853237 * r1853238;
        double r1853240 = -r1853239;
        double r1853241 = exp(r1853240);
        double r1853242 = r1853236 * r1853241;
        double r1853243 = r1853235 - r1853233;
        double r1853244 = r1853233 + r1853234;
        double r1853245 = r1853244 * r1853238;
        double r1853246 = -r1853245;
        double r1853247 = exp(r1853246);
        double r1853248 = r1853243 * r1853247;
        double r1853249 = r1853242 - r1853248;
        double r1853250 = 2.0;
        double r1853251 = r1853249 / r1853250;
        return r1853251;
}

↓

double f(double x, double eps) {
        double r1853252 = x;
        double r1853253 = 64.8190525609748;
        bool r1853254 = r1853252 <= r1853253;
        double r1853255 = 0.5;
        double r1853256 = r1853252 * r1853252;
        double r1853257 = 0.6666666666666666;
        double r1853258 = -1.0;
        double r1853259 = fma(r1853252, r1853257, r1853258);
        double r1853260 = 2.0;
        double r1853261 = fma(r1853256, r1853259, r1853260);
        double r1853262 = r1853255 * r1853261;
        double r1853263 = eps;
        double r1853264 = 1.0;
        double r1853265 = r1853263 - r1853264;
        double r1853266 = r1853265 * r1853252;
        double r1853267 = exp(r1853266);
        double r1853268 = r1853267 / r1853263;
        double r1853269 = r1853267 + r1853268;
        double r1853270 = r1853264 / r1853263;
        double r1853271 = r1853270 - r1853264;
        double r1853272 = fma(r1853263, r1853252, r1853252);
        double r1853273 = exp(r1853272);
        double r1853274 = r1853271 / r1853273;
        double r1853275 = r1853269 - r1853274;
        double r1853276 = r1853255 * r1853275;
        double r1853277 = r1853254 ? r1853262 : r1853276;
        return r1853277;
}

Derivation

Split input into 2 regimes
if x < 64.8190525609748
1. Initial program 39.0
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Simplified39.0
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\mathsf{fma}\left(e^{x \cdot \left(-1 + \varepsilon\right)}, \frac{1}{\varepsilon}, e^{x \cdot \left(-1 + \varepsilon\right)}\right) - \frac{\frac{1}{\varepsilon} - 1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}\]
3. Taylor expanded around 0 1.4
  \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right)}\]
4. Simplified1.4
  \[\leadsto \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{2}{3}, 2 - x \cdot x\right)}\]
5. Taylor expanded around 0 1.4
  \[\leadsto \frac{1}{2} \cdot \color{blue}{\left(\left(\frac{2}{3} \cdot {x}^{3} + 2\right) - {x}^{2}\right)}\]
6. Simplified1.4
  \[\leadsto \frac{1}{2} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{2}{3}, -1\right), 2\right)}\]
if 64.8190525609748 < x
1. Initial program 0.1
  \[\frac{\left(1 + \frac{1}{\varepsilon}\right) \cdot e^{-\left(1 - \varepsilon\right) \cdot x} - \left(\frac{1}{\varepsilon} - 1\right) \cdot e^{-\left(1 + \varepsilon\right) \cdot x}}{2}\]
2. Simplified0.2
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \left(\mathsf{fma}\left(e^{x \cdot \left(-1 + \varepsilon\right)}, \frac{1}{\varepsilon}, e^{x \cdot \left(-1 + \varepsilon\right)}\right) - \frac{\frac{1}{\varepsilon} - 1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)}\]
3. Taylor expanded around inf 0.2
  \[\leadsto \frac{1}{2} \cdot \left(\color{blue}{\left(e^{\left(\varepsilon - 1\right) \cdot x} + \frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon}\right)} - \frac{\frac{1}{\varepsilon} - 1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)\]
Recombined 2 regimes into one program.
Final simplification1.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le 64.8190525609748:\\ \;\;\;\;\frac{1}{2} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \frac{2}{3}, -1\right), 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} \cdot \left(\left(e^{\left(\varepsilon - 1\right) \cdot x} + \frac{e^{\left(\varepsilon - 1\right) \cdot x}}{\varepsilon}\right) - \frac{\frac{1}{\varepsilon} - 1}{e^{\mathsf{fma}\left(\varepsilon, x, x\right)}}\right)\\ \end{array}\]

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

Error

Derivation

`if x < 64.8190525609748`

`if 64.8190525609748 < x`

Reproduce