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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.627556263318037368709352563644898737039 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left({\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}, {\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 2\right)\right)}^{3}}}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -9.627556263318037368709352563644898737039 \cdot 10^{-6}:\\
\;\;\;\;\sqrt{\frac{\mathsf{fma}\left({\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}, {\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}, -1\right)}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 2\right)\right)}^{3}}}\\

\end{array}

double f(double x) {
        double r24161 = 2.0;
        double r24162 = x;
        double r24163 = r24161 * r24162;
        double r24164 = exp(r24163);
        double r24165 = 1.0;
        double r24166 = r24164 - r24165;
        double r24167 = exp(r24162);
        double r24168 = r24167 - r24165;
        double r24169 = r24166 / r24168;
        double r24170 = sqrt(r24169);
        return r24170;
}

↓

double f(double x) {
        double r24171 = x;
        double r24172 = -9.627556263318037e-06;
        bool r24173 = r24171 <= r24172;
        double r24174 = 2.0;
        double r24175 = exp(r24174);
        double r24176 = 2.0;
        double r24177 = r24171 / r24176;
        double r24178 = pow(r24175, r24177);
        double r24179 = 1.0;
        double r24180 = -r24179;
        double r24181 = fma(r24178, r24178, r24180);
        double r24182 = exp(r24171);
        double r24183 = r24182 - r24179;
        double r24184 = r24181 / r24183;
        double r24185 = sqrt(r24184);
        double r24186 = 0.5;
        double r24187 = fma(r24171, r24186, r24179);
        double r24188 = fma(r24171, r24187, r24174);
        double r24189 = 3.0;
        double r24190 = pow(r24188, r24189);
        double r24191 = cbrt(r24190);
        double r24192 = sqrt(r24191);
        double r24193 = r24173 ? r24185 : r24192;
        return r24193;
}

Derivation

Split input into 2 regimes
if x < -9.627556263318037e-06
1. Initial program 0.1
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Simplified0.1
  \[\leadsto \color{blue}{\sqrt{\frac{{\left(e^{2}\right)}^{x} - 1}{e^{x} - 1}}}\]
3. Using strategy rm
4. Applied sqr-pow0.1
  \[\leadsto \sqrt{\frac{\color{blue}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}} - 1}{e^{x} - 1}}\]
5. Applied fma-neg0.0
  \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left({\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}, {\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}, -1\right)}}{e^{x} - 1}}\]
if -9.627556263318037e-06 < x
1. Initial program 34.4
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Simplified34.5
  \[\leadsto \color{blue}{\sqrt{\frac{{\left(e^{2}\right)}^{x} - 1}{e^{x} - 1}}}\]
3. Taylor expanded around 0 6.2
  \[\leadsto \sqrt{\color{blue}{1 \cdot x + \left(0.5 \cdot {x}^{2} + 2\right)}}\]
4. Simplified6.2
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(1, x, \mathsf{fma}\left(x \cdot x, 0.5, 2\right)\right)}}\]
5. Using strategy rm
6. Applied add-cbrt-cube6.3
  \[\leadsto \sqrt{\color{blue}{\sqrt[3]{\left(\mathsf{fma}\left(1, x, \mathsf{fma}\left(x \cdot x, 0.5, 2\right)\right) \cdot \mathsf{fma}\left(1, x, \mathsf{fma}\left(x \cdot x, 0.5, 2\right)\right)\right) \cdot \mathsf{fma}\left(1, x, \mathsf{fma}\left(x \cdot x, 0.5, 2\right)\right)}}}\]
7. Simplified6.3
  \[\leadsto \sqrt{\sqrt[3]{\color{blue}{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 2\right)\right)}^{3}}}}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.627556263318037368709352563644898737039 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left({\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}, {\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\sqrt[3]{{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 2\right)\right)}^{3}}}\\ \end{array}\]

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

Error

Derivation

`if x < -9.627556263318037e-06`

`if -9.627556263318037e-06 < x`

Reproduce