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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -3.0314812521352058 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -3.0314812521352058 \cdot 10^{-11}:\\
\;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\

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

\end{array}

double f(double x) {
        double r21106 = 2.0;
        double r21107 = x;
        double r21108 = r21106 * r21107;
        double r21109 = exp(r21108);
        double r21110 = 1.0;
        double r21111 = r21109 - r21110;
        double r21112 = exp(r21107);
        double r21113 = r21112 - r21110;
        double r21114 = r21111 / r21113;
        double r21115 = sqrt(r21114);
        return r21115;
}

↓

double f(double x) {
        double r21116 = x;
        double r21117 = -3.031481252135206e-11;
        bool r21118 = r21116 <= r21117;
        double r21119 = exp(r21116);
        double r21120 = 1.0;
        double r21121 = r21119 + r21120;
        double r21122 = 2.0;
        double r21123 = r21122 * r21116;
        double r21124 = exp(r21123);
        double r21125 = r21124 - r21120;
        double r21126 = r21121 * r21125;
        double r21127 = 2.0;
        double r21128 = r21127 * r21116;
        double r21129 = exp(r21128);
        double r21130 = r21129 - r21120;
        double r21131 = r21126 / r21130;
        double r21132 = sqrt(r21131);
        double r21133 = 0.5;
        double r21134 = r21133 * r21116;
        double r21135 = r21134 + r21120;
        double r21136 = r21116 * r21135;
        double r21137 = r21122 + r21136;
        double r21138 = sqrt(r21137);
        double r21139 = r21118 ? r21132 : r21138;
        return r21139;
}

Derivation

Split input into 2 regimes
if x < -3.031481252135206e-11
1. Initial program 0.5
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.3
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Applied associate-/r/0.3
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]
5. Simplified0.0
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \left(e^{x} + 1\right)}\]
6. Taylor expanded around inf 0.0
  \[\leadsto \color{blue}{\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}}\]
if -3.031481252135206e-11 < x
1. Initial program 37.1
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.6
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.6
  \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(0.5 \cdot x + 1\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.7
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -3.0314812521352058 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(0.5 \cdot x + 1\right)}\\ \end{array}\]

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

Error

Try it out

Derivation

`if x < -3.031481252135206e-11`

`if -3.031481252135206e-11 < x`

Reproduce

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