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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -7.26854291605878706 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -7.26854291605878706 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\

\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 r14797 = 2.0;
        double r14798 = x;
        double r14799 = r14797 * r14798;
        double r14800 = exp(r14799);
        double r14801 = 1.0;
        double r14802 = r14800 - r14801;
        double r14803 = exp(r14798);
        double r14804 = r14803 - r14801;
        double r14805 = r14802 / r14804;
        double r14806 = sqrt(r14805);
        return r14806;
}

↓

double f(double x) {
        double r14807 = x;
        double r14808 = -7.268542916058787e-16;
        bool r14809 = r14807 <= r14808;
        double r14810 = 2.0;
        double r14811 = r14810 * r14807;
        double r14812 = exp(r14811);
        double r14813 = 1.0;
        double r14814 = r14812 - r14813;
        double r14815 = r14807 + r14807;
        double r14816 = exp(r14815);
        double r14817 = r14813 * r14813;
        double r14818 = r14816 - r14817;
        double r14819 = r14814 / r14818;
        double r14820 = sqrt(r14819);
        double r14821 = exp(r14807);
        double r14822 = r14821 + r14813;
        double r14823 = sqrt(r14822);
        double r14824 = r14820 * r14823;
        double r14825 = 2.0;
        double r14826 = pow(r14807, r14825);
        double r14827 = sqrt(r14810);
        double r14828 = r14826 / r14827;
        double r14829 = 0.25;
        double r14830 = 0.125;
        double r14831 = r14830 / r14810;
        double r14832 = r14829 - r14831;
        double r14833 = r14828 * r14832;
        double r14834 = 0.5;
        double r14835 = r14807 / r14827;
        double r14836 = r14834 * r14835;
        double r14837 = r14827 + r14836;
        double r14838 = r14833 + r14837;
        double r14839 = r14809 ? r14824 : r14838;
        return r14839;
}

Derivation

Split input into 2 regimes
if x < -7.268542916058787e-16
1. Initial program 0.7
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.5
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Applied associate-/r/0.5
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]
5. Applied sqrt-prod0.5
  \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}}\]
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 -7.268542916058787e-16 < x
1. Initial program 37.3
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 7.3
  \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]
3. Simplified7.3
  \[\leadsto \color{blue}{\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.7
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.26854291605878706 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right) + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\\ \end{array}\]

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 < -7.268542916058787e-16`

`if -7.268542916058787e-16 < x`

Reproduce

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