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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.15639988716910344 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.15639988716910344 \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 r13634 = 2.0;
        double r13635 = x;
        double r13636 = r13634 * r13635;
        double r13637 = exp(r13636);
        double r13638 = 1.0;
        double r13639 = r13637 - r13638;
        double r13640 = exp(r13635);
        double r13641 = r13640 - r13638;
        double r13642 = r13639 / r13641;
        double r13643 = sqrt(r13642);
        return r13643;
}

↓

double f(double x) {
        double r13644 = x;
        double r13645 = -1.1563998871691034e-05;
        bool r13646 = r13644 <= r13645;
        double r13647 = 2.0;
        double r13648 = r13647 * r13644;
        double r13649 = exp(r13648);
        double r13650 = sqrt(r13649);
        double r13651 = 1.0;
        double r13652 = sqrt(r13651);
        double r13653 = r13650 + r13652;
        double r13654 = r13650 - r13652;
        double r13655 = exp(r13644);
        double r13656 = r13655 - r13651;
        double r13657 = r13654 / r13656;
        double r13658 = r13653 * r13657;
        double r13659 = sqrt(r13658);
        double r13660 = 0.5;
        double r13661 = r13660 * r13644;
        double r13662 = r13651 + r13661;
        double r13663 = r13644 * r13662;
        double r13664 = r13663 + r13647;
        double r13665 = sqrt(r13664);
        double r13666 = r13646 ? r13659 : r13665;
        return r13666;
}

Derivation

Split input into 2 regimes
if x < -1.1563998871691034e-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 -1.1563998871691034e-05 < x
1. Initial program 34.9
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.4
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.4
  \[\leadsto \sqrt{\color{blue}{x \cdot \left(1 + 0.5 \cdot x\right) + 2}}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.15639988716910344 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{2 \cdot x}} - \sqrt{1}}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{x \cdot \left(1 + 0.5 \cdot x\right) + 2}\\ \end{array}\]

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

Error

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Derivation

`if x < -1.1563998871691034e-05`

`if -1.1563998871691034e-05 < x`

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