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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -4.7552481292438652 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -4.7552481292438652 \cdot 10^{-11}:\\
\;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}\\

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

\end{array}

double f(double x) {
        double r65 = 2.0;
        double r66 = x;
        double r67 = r65 * r66;
        double r68 = exp(r67);
        double r69 = 1.0;
        double r70 = r68 - r69;
        double r71 = exp(r66);
        double r72 = r71 - r69;
        double r73 = r70 / r72;
        double r74 = sqrt(r73);
        return r74;
}

↓

double f(double x) {
        double r75 = x;
        double r76 = -4.755248129243865e-11;
        bool r77 = r75 <= r76;
        double r78 = 2.0;
        double r79 = r78 * r75;
        double r80 = exp(r79);
        double r81 = sqrt(r80);
        double r82 = 1.0;
        double r83 = sqrt(r82);
        double r84 = r81 + r83;
        double r85 = exp(r75);
        double r86 = r85 - r82;
        double r87 = r81 - r83;
        double r88 = r86 / r87;
        double r89 = r84 / r88;
        double r90 = sqrt(r89);
        double r91 = 0.5;
        double r92 = sqrt(r78);
        double r93 = r75 / r92;
        double r94 = r91 * r93;
        double r95 = 2.0;
        double r96 = pow(r75, r95);
        double r97 = r96 / r92;
        double r98 = 0.25;
        double r99 = 0.125;
        double r100 = r99 / r78;
        double r101 = r98 - r100;
        double r102 = r97 * r101;
        double r103 = r92 + r102;
        double r104 = r94 + r103;
        double r105 = r77 ? r90 : r104;
        return r105;
}

Derivation

Split input into 2 regimes
if x < -4.755248129243865e-11
1. Initial program 0.5
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.5
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{e^{x} - 1}}\]
4. Applied add-sqr-sqrt0.4
  \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}} - \sqrt{1} \cdot \sqrt{1}}{e^{x} - 1}}\]
5. Applied difference-of-squares0.1
  \[\leadsto \sqrt{\frac{\color{blue}{\left(\sqrt{e^{2 \cdot x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{2 \cdot x}} - \sqrt{1}\right)}}{e^{x} - 1}}\]
6. Applied associate-/l*0.1
  \[\leadsto \sqrt{\color{blue}{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}}\]
if -4.755248129243865e-11 < x
1. Initial program 36.3
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 7.0
  \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]
3. Simplified6.9
  \[\leadsto \color{blue}{0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification0.9
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -4.7552481292438652 \cdot 10^{-11}:\\ \;\;\;\;\sqrt{\frac{\sqrt{e^{2 \cdot x}} + \sqrt{1}}{\frac{e^{x} - 1}{\sqrt{e^{2 \cdot x}} - \sqrt{1}}}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\sqrt{2}} + \left(\sqrt{2} + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\right)\\ \end{array}\]

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

Error

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Derivation

`if x < -4.755248129243865e-11`

`if -4.755248129243865e-11 < x`

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