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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.635655884635412785585475300345031302932 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right) + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.635655884635412785585475300345031302932 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\

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

\end{array}

double f(double x) {
        double r24085 = 2.0;
        double r24086 = x;
        double r24087 = r24085 * r24086;
        double r24088 = exp(r24087);
        double r24089 = 1.0;
        double r24090 = r24088 - r24089;
        double r24091 = exp(r24086);
        double r24092 = r24091 - r24089;
        double r24093 = r24090 / r24092;
        double r24094 = sqrt(r24093);
        return r24094;
}

↓

double f(double x) {
        double r24095 = x;
        double r24096 = -1.6356558846354128e-16;
        bool r24097 = r24095 <= r24096;
        double r24098 = 2.0;
        double r24099 = r24098 * r24095;
        double r24100 = exp(r24099);
        double r24101 = 1.0;
        double r24102 = r24100 - r24101;
        double r24103 = r24095 + r24095;
        double r24104 = exp(r24103);
        double r24105 = r24101 * r24101;
        double r24106 = r24104 - r24105;
        double r24107 = r24102 / r24106;
        double r24108 = sqrt(r24107);
        double r24109 = exp(r24095);
        double r24110 = r24109 + r24101;
        double r24111 = sqrt(r24110);
        double r24112 = r24108 * r24111;
        double r24113 = sqrt(r24098);
        double r24114 = 0.5;
        double r24115 = r24095 / r24113;
        double r24116 = r24114 * r24115;
        double r24117 = r24113 + r24116;
        double r24118 = 2.0;
        double r24119 = pow(r24095, r24118);
        double r24120 = r24119 / r24113;
        double r24121 = 0.25;
        double r24122 = 0.125;
        double r24123 = r24122 / r24098;
        double r24124 = r24121 - r24123;
        double r24125 = r24120 * r24124;
        double r24126 = r24117 + r24125;
        double r24127 = r24097 ? r24112 : r24126;
        return r24127;
}

Derivation

Split input into 2 regimes
if x < -1.6356558846354128e-16
1. Initial program 0.8
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.6
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Applied associate-/r/0.6
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]
5. Applied sqrt-prod0.6
  \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}}\]
6. Simplified0.0
  \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}}} \cdot \sqrt{e^{x} + 1}\]
if -1.6356558846354128e-16 < x
1. Initial program 38.4
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 9.1
  \[\leadsto \color{blue}{\left(0.25 \cdot \frac{{x}^{2}}{\sqrt{2}} + \left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]
3. Simplified9.1
  \[\leadsto \color{blue}{\left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right) + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)}\]
Recombined 2 regimes into one program.
Final simplification0.9
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.635655884635412785585475300345031302932 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{2} + 0.5 \cdot \frac{x}{\sqrt{2}}\right) + \frac{{x}^{2}}{\sqrt{2}} \cdot \left(0.25 - \frac{0.125}{2}\right)\\ \end{array}\]

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

Error

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Derivation

`if x < -1.6356558846354128e-16`

`if -1.6356558846354128e-16 < x`

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