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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.2887874422510308 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + 0.5 \cdot x\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.2887874422510308 \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(1 + 0.5 \cdot x\right)}\\

\end{array}

double f(double x) {
        double r10912 = 2.0;
        double r10913 = x;
        double r10914 = r10912 * r10913;
        double r10915 = exp(r10914);
        double r10916 = 1.0;
        double r10917 = r10915 - r10916;
        double r10918 = exp(r10913);
        double r10919 = r10918 - r10916;
        double r10920 = r10917 / r10919;
        double r10921 = sqrt(r10920);
        return r10921;
}

↓

double f(double x) {
        double r10922 = x;
        double r10923 = -1.2887874422510308e-05;
        bool r10924 = r10922 <= r10923;
        double r10925 = 2.0;
        double r10926 = r10925 * r10922;
        double r10927 = exp(r10926);
        double r10928 = 1.0;
        double r10929 = r10927 - r10928;
        double r10930 = r10922 + r10922;
        double r10931 = exp(r10930);
        double r10932 = r10928 * r10928;
        double r10933 = r10931 - r10932;
        double r10934 = r10929 / r10933;
        double r10935 = sqrt(r10934);
        double r10936 = exp(r10922);
        double r10937 = r10936 + r10928;
        double r10938 = sqrt(r10937);
        double r10939 = r10935 * r10938;
        double r10940 = 0.5;
        double r10941 = r10940 * r10922;
        double r10942 = r10928 + r10941;
        double r10943 = r10922 * r10942;
        double r10944 = r10925 + r10943;
        double r10945 = sqrt(r10944);
        double r10946 = r10924 ? r10939 : r10945;
        return r10946;
}

Derivation

Split input into 2 regimes
if x < -1.2887874422510308e-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 -1.2887874422510308e-05 < x
1. Initial program 34.6
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.7
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.7
  \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(1 + 0.5 \cdot x\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.9
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.2887874422510308 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + 0.5 \cdot x\right)}\\ \end{array}\]

Falling back on racket egraph because egg-herbie package not installed (more)

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

Error

Try it out

Derivation

`if x < -1.2887874422510308e-05`

`if -1.2887874422510308e-05 < x`

Reproduce

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