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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.367867422063156052558161035470618571708 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -2.367867422063156052558161035470618571708 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\

\end{array}

double f(double x) {
        double r24071 = 2.0;
        double r24072 = x;
        double r24073 = r24071 * r24072;
        double r24074 = exp(r24073);
        double r24075 = 1.0;
        double r24076 = r24074 - r24075;
        double r24077 = exp(r24072);
        double r24078 = r24077 - r24075;
        double r24079 = r24076 / r24078;
        double r24080 = sqrt(r24079);
        return r24080;
}

↓

double f(double x) {
        double r24081 = x;
        double r24082 = -2.367867422063156e-05;
        bool r24083 = r24081 <= r24082;
        double r24084 = 2.0;
        double r24085 = r24084 * r24081;
        double r24086 = exp(r24085);
        double r24087 = 1.0;
        double r24088 = r24086 - r24087;
        double r24089 = r24081 + r24081;
        double r24090 = exp(r24089);
        double r24091 = r24087 * r24087;
        double r24092 = r24090 - r24091;
        double r24093 = r24088 / r24092;
        double r24094 = exp(r24081);
        double r24095 = r24094 + r24087;
        double r24096 = r24093 * r24095;
        double r24097 = sqrt(r24096);
        double r24098 = 2.0;
        double r24099 = pow(r24081, r24098);
        double r24100 = sqrt(r24084);
        double r24101 = r24099 / r24100;
        double r24102 = 0.25;
        double r24103 = 0.125;
        double r24104 = r24103 / r24084;
        double r24105 = r24102 - r24104;
        double r24106 = r24101 * r24105;
        double r24107 = 0.5;
        double r24108 = r24081 / r24100;
        double r24109 = r24107 * r24108;
        double r24110 = r24100 + r24109;
        double r24111 = r24106 + r24110;
        double r24112 = r24083 ? r24097 : r24111;
        return r24112;
}

Derivation

Split input into 2 regimes
if x < -2.367867422063156e-05
1. Initial program 0.1
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Applied associate-/r/0.0
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]
5. Simplified0.0
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \left(e^{x} + 1\right)}\]
if -2.367867422063156e-05 < x
1. Initial program 34.6
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 5.7
  \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]
3. Simplified5.6
  \[\leadsto \color{blue}{\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.7
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.367867422063156052558161035470618571708 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \end{array}\]

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

Error

Try it out

Derivation

`if x < -2.367867422063156e-05`

`if -2.367867422063156e-05 < x`

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