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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.42926588964696801061903669305408470791 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -2.42926588964696801061903669305408470791 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \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 r15988 = 2.0;
        double r15989 = x;
        double r15990 = r15988 * r15989;
        double r15991 = exp(r15990);
        double r15992 = 1.0;
        double r15993 = r15991 - r15992;
        double r15994 = exp(r15989);
        double r15995 = r15994 - r15992;
        double r15996 = r15993 / r15995;
        double r15997 = sqrt(r15996);
        return r15997;
}

↓

double f(double x) {
        double r15998 = x;
        double r15999 = -2.429265889646968e-05;
        bool r16000 = r15998 <= r15999;
        double r16001 = 2.0;
        double r16002 = r16001 * r15998;
        double r16003 = exp(r16002);
        double r16004 = sqrt(r16003);
        double r16005 = 1.0;
        double r16006 = sqrt(r16005);
        double r16007 = r16004 + r16006;
        double r16008 = r16004 - r16006;
        double r16009 = exp(r15998);
        double r16010 = r16009 - r16005;
        double r16011 = r16008 / r16010;
        double r16012 = r16007 * r16011;
        double r16013 = sqrt(r16012);
        double r16014 = 0.5;
        double r16015 = r16014 * r15998;
        double r16016 = r16005 + r16015;
        double r16017 = r15998 * r16016;
        double r16018 = r16017 + r16001;
        double r16019 = sqrt(r16018);
        double r16020 = r16000 ? r16013 : r16019;
        return r16020;
}

Derivation

Split input into 2 regimes
if x < -2.429265889646968e-05
1. Initial program 0.1
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied *-un-lft-identity0.1
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{1 \cdot \left(e^{x} - 1\right)}}}\]
4. Applied add-sqr-sqrt0.1
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot \left(e^{x} - 1\right)}}\]
5. Applied add-sqr-sqrt0.1
  \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}} - \sqrt{1} \cdot \sqrt{1}}{1 \cdot \left(e^{x} - 1\right)}}\]
6. Applied difference-of-squares0.0
  \[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}}{1 \cdot \left(e^{x} - 1\right)}}\]
7. Applied times-frac0.0
  \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{1} \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}}\]
8. Simplified0.0
  \[\leadsto \sqrt{\color{blue}{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right)} \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\]
if -2.429265889646968e-05 < x
1. Initial program 33.7
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.8
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.8
  \[\leadsto \sqrt{\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right) + 2}}\]
Recombined 2 regimes into one program.
Final simplification0.9
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.42926588964696801061903669305408470791 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

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

Error

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Derivation

`if x < -2.429265889646968e-05`

`if -2.429265889646968e-05 < x`

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