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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.288683916673524238328662344654773619368 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.288683916673524238328662344654773619368 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\

\end{array}

double f(double x) {
        double r18852 = 2.0;
        double r18853 = x;
        double r18854 = r18852 * r18853;
        double r18855 = exp(r18854);
        double r18856 = 1.0;
        double r18857 = r18855 - r18856;
        double r18858 = exp(r18853);
        double r18859 = r18858 - r18856;
        double r18860 = r18857 / r18859;
        double r18861 = sqrt(r18860);
        return r18861;
}

↓

double f(double x) {
        double r18862 = x;
        double r18863 = -1.2886839166735242e-05;
        bool r18864 = r18862 <= r18863;
        double r18865 = 2.0;
        double r18866 = r18865 * r18862;
        double r18867 = exp(r18866);
        double r18868 = 1.0;
        double r18869 = r18867 - r18868;
        double r18870 = r18862 + r18862;
        double r18871 = exp(r18870);
        double r18872 = r18868 * r18868;
        double r18873 = r18871 - r18872;
        double r18874 = r18869 / r18873;
        double r18875 = sqrt(r18874);
        double r18876 = exp(r18862);
        double r18877 = r18876 + r18868;
        double r18878 = sqrt(r18877);
        double r18879 = r18875 * r18878;
        double r18880 = 0.5;
        double r18881 = r18880 * r18862;
        double r18882 = r18881 + r18868;
        double r18883 = r18862 * r18882;
        double r18884 = r18865 + r18883;
        double r18885 = sqrt(r18884);
        double r18886 = r18864 ? r18879 : r18885;
        return r18886;
}

Derivation

Split input into 2 regimes
if x < -1.2886839166735242e-05
1. Initial program 0.1
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.1
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Applied associate-/r/0.1
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]
5. Applied sqrt-prod0.1
  \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}}\]
6. Simplified0.0
  \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}}} \cdot \sqrt{e^{x} + 1}\]
if -1.2886839166735242e-05 < x
1. Initial program 33.9
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.5
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.5
  \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(0.5 \cdot x + 1\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.9
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.288683916673524238328662344654773619368 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]

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

Error

Try it out

Derivation

`if x < -1.2886839166735242e-05`

`if -1.2886839166735242e-05 < x`

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