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

↓

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

\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 \left(e^{x} + 1\right)}\\

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

\end{array}

double f(double x) {
        double r11918 = 2.0;
        double r11919 = x;
        double r11920 = r11918 * r11919;
        double r11921 = exp(r11920);
        double r11922 = 1.0;
        double r11923 = r11921 - r11922;
        double r11924 = exp(r11919);
        double r11925 = r11924 - r11922;
        double r11926 = r11923 / r11925;
        double r11927 = sqrt(r11926);
        return r11927;
}

↓

double f(double x) {
        double r11928 = x;
        double r11929 = -1.2887874422510308e-05;
        bool r11930 = r11928 <= r11929;
        double r11931 = 2.0;
        double r11932 = r11931 * r11928;
        double r11933 = exp(r11932);
        double r11934 = 1.0;
        double r11935 = r11933 - r11934;
        double r11936 = r11928 + r11928;
        double r11937 = exp(r11936);
        double r11938 = r11934 * r11934;
        double r11939 = r11937 - r11938;
        double r11940 = r11935 / r11939;
        double r11941 = exp(r11928);
        double r11942 = r11941 + r11934;
        double r11943 = r11940 * r11942;
        double r11944 = sqrt(r11943);
        double r11945 = 0.5;
        double r11946 = r11945 * r11928;
        double r11947 = r11934 + r11946;
        double r11948 = r11928 * r11947;
        double r11949 = r11931 + r11948;
        double r11950 = sqrt(r11949);
        double r11951 = r11930 ? r11944 : r11950;
        return r11951;
}

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

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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