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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -5.5556517639115665 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{\sqrt{\sqrt{e^{2 \cdot x}}} \cdot \sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -5.5556517639115665 \cdot 10^{-17}:\\
\;\;\;\;\sqrt{\frac{\sqrt{\sqrt{e^{2 \cdot x}}} \cdot \sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\

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

\end{array}

double f(double x) {
        double r11825 = 2.0;
        double r11826 = x;
        double r11827 = r11825 * r11826;
        double r11828 = exp(r11827);
        double r11829 = 1.0;
        double r11830 = r11828 - r11829;
        double r11831 = exp(r11826);
        double r11832 = r11831 - r11829;
        double r11833 = r11830 / r11832;
        double r11834 = sqrt(r11833);
        return r11834;
}

↓

double f(double x) {
        double r11835 = x;
        double r11836 = -5.555651763911567e-17;
        bool r11837 = r11835 <= r11836;
        double r11838 = 2.0;
        double r11839 = r11838 * r11835;
        double r11840 = exp(r11839);
        double r11841 = sqrt(r11840);
        double r11842 = sqrt(r11841);
        double r11843 = r11842 * r11842;
        double r11844 = 1.0;
        double r11845 = sqrt(r11844);
        double r11846 = r11843 + r11845;
        double r11847 = 1.0;
        double r11848 = r11846 / r11847;
        double r11849 = sqrt(r11848);
        double r11850 = exp(r11838);
        double r11851 = 0.5;
        double r11852 = r11851 * r11835;
        double r11853 = pow(r11850, r11852);
        double r11854 = r11853 - r11845;
        double r11855 = exp(r11835);
        double r11856 = r11855 - r11844;
        double r11857 = r11854 / r11856;
        double r11858 = sqrt(r11857);
        double r11859 = r11849 * r11858;
        double r11860 = 0.5;
        double r11861 = r11860 * r11835;
        double r11862 = r11844 + r11861;
        double r11863 = r11835 * r11862;
        double r11864 = r11863 + r11838;
        double r11865 = sqrt(r11864);
        double r11866 = r11837 ? r11859 : r11865;
        return r11866;
}

Derivation

Split input into 2 regimes
if x < -5.555651763911567e-17
1. Initial program 0.8
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied *-un-lft-identity0.8
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{1 \cdot \left(e^{x} - 1\right)}}}\]
4. Applied add-sqr-sqrt0.8
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot \left(e^{x} - 1\right)}}\]
5. Applied add-sqr-sqrt0.7
  \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}} - \sqrt{1} \cdot \sqrt{1}}{1 \cdot \left(e^{x} - 1\right)}}\]
6. Applied difference-of-squares0.3
  \[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}}{1 \cdot \left(e^{x} - 1\right)}}\]
7. Applied times-frac0.3
  \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1} \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}}\]
8. Applied sqrt-prod0.3
  \[\leadsto \color{blue}{\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}}\]
9. Using strategy rm
10. Applied add-log-exp0.3
  \[\leadsto \sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{e^{\color{blue}{\log \left(e^{2}\right)} \cdot x}} - \sqrt{1}}{e^{x} - 1}}\]
11. Applied exp-to-pow0.3
  \[\leadsto \sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\sqrt{\color{blue}{{\left(e^{2}\right)}^{x}}} - \sqrt{1}}{e^{x} - 1}}\]
12. Applied sqrt-pow10.0
  \[\leadsto \sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{\color{blue}{{\left(e^{2}\right)}^{\left(\frac{x}{2}\right)}} - \sqrt{1}}{e^{x} - 1}}\]
13. Simplified0.0
  \[\leadsto \sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\color{blue}{\left(\frac{1}{2} \cdot x\right)}} - \sqrt{1}}{e^{x} - 1}}\]
14. Using strategy rm
15. Applied add-sqr-sqrt0.0
  \[\leadsto \sqrt{\frac{\sqrt{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\]
16. Applied sqrt-prod0.0
  \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{\sqrt{e^{2 \cdot x}}} \cdot \sqrt{\sqrt{e^{2 \cdot x}}}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\]
if -5.555651763911567e-17 < x
1. Initial program 37.7
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 9.1
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified9.1
  \[\leadsto \sqrt{\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right) + 2}}\]
Recombined 2 regimes into one program.
Final simplification0.9
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -5.5556517639115665 \cdot 10^{-17}:\\ \;\;\;\;\sqrt{\frac{\sqrt{\sqrt{e^{2 \cdot x}}} \cdot \sqrt{\sqrt{e^{2 \cdot x}}} + \sqrt{1}}{1}} \cdot \sqrt{\frac{{\left(e^{2}\right)}^{\left(\frac{1}{2} \cdot x\right)} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

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

Error

Try it out

Derivation

`if x < -5.555651763911567e-17`

`if -5.555651763911567e-17 < x`

Reproduce

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