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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.998472655500366441157916730375632141659 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{1} + \sqrt{e^{2 \cdot x}}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\sqrt{2}} + \left(\sqrt{2} + \left(0.25 - \frac{0.125}{2}\right) \cdot \frac{x \cdot x}{\sqrt{2}}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.998472655500366441157916730375632141659 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{\left(\sqrt{1} + \sqrt{e^{2 \cdot x}}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\frac{x \cdot 0.5}{\sqrt{2}} + \left(\sqrt{2} + \left(0.25 - \frac{0.125}{2}\right) \cdot \frac{x \cdot x}{\sqrt{2}}\right)\\

\end{array}

double f(double x) {
        double r1541562 = 2.0;
        double r1541563 = x;
        double r1541564 = r1541562 * r1541563;
        double r1541565 = exp(r1541564);
        double r1541566 = 1.0;
        double r1541567 = r1541565 - r1541566;
        double r1541568 = exp(r1541563);
        double r1541569 = r1541568 - r1541566;
        double r1541570 = r1541567 / r1541569;
        double r1541571 = sqrt(r1541570);
        return r1541571;
}

↓

double f(double x) {
        double r1541572 = x;
        double r1541573 = -1.9984726555003664e-05;
        bool r1541574 = r1541572 <= r1541573;
        double r1541575 = 1.0;
        double r1541576 = sqrt(r1541575);
        double r1541577 = 2.0;
        double r1541578 = r1541577 * r1541572;
        double r1541579 = exp(r1541578);
        double r1541580 = sqrt(r1541579);
        double r1541581 = r1541576 + r1541580;
        double r1541582 = r1541580 - r1541576;
        double r1541583 = r1541581 * r1541582;
        double r1541584 = exp(r1541572);
        double r1541585 = r1541584 - r1541575;
        double r1541586 = r1541583 / r1541585;
        double r1541587 = sqrt(r1541586);
        double r1541588 = 0.5;
        double r1541589 = r1541572 * r1541588;
        double r1541590 = sqrt(r1541577);
        double r1541591 = r1541589 / r1541590;
        double r1541592 = 0.25;
        double r1541593 = 0.125;
        double r1541594 = r1541593 / r1541577;
        double r1541595 = r1541592 - r1541594;
        double r1541596 = r1541572 * r1541572;
        double r1541597 = r1541596 / r1541590;
        double r1541598 = r1541595 * r1541597;
        double r1541599 = r1541590 + r1541598;
        double r1541600 = r1541591 + r1541599;
        double r1541601 = r1541574 ? r1541587 : r1541600;
        return r1541601;
}

Derivation

Split input into 2 regimes
if x < -1.9984726555003664e-05
1. Initial program 0.1
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.1
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{e^{x} - 1}}\]
4. Applied add-sqr-sqrt0.1
  \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}} - \sqrt{1} \cdot \sqrt{1}}{e^{x} - 1}}\]
5. Applied difference-of-squares0.0
  \[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}}{e^{x} - 1}}\]
if -1.9984726555003664e-05 < x
1. Initial program 34.0
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.0
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.0
  \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(0.5 \cdot x + 1\right)}}\]
4. Taylor expanded around 0 6.0
  \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]
5. Simplified6.0
  \[\leadsto \color{blue}{\frac{0.5 \cdot x}{\sqrt{2}} + \left(\sqrt{2} + \frac{x \cdot x}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.998472655500366441157916730375632141659 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\left(\sqrt{1} + \sqrt{e^{2 \cdot x}}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 0.5}{\sqrt{2}} + \left(\sqrt{2} + \left(0.25 - \frac{0.125}{2}\right) \cdot \frac{x \cdot x}{\sqrt{2}}\right)\\ \end{array}\]

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

Error

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Derivation

`if x < -1.9984726555003664e-05`

`if -1.9984726555003664e-05 < x`

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