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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -6.87791414643454749 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -6.87791414643454749 \cdot 10^{-6}:\\
\;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\

\end{array}

double f(double x) {
        double r11906 = 2.0;
        double r11907 = x;
        double r11908 = r11906 * r11907;
        double r11909 = exp(r11908);
        double r11910 = 1.0;
        double r11911 = r11909 - r11910;
        double r11912 = exp(r11907);
        double r11913 = r11912 - r11910;
        double r11914 = r11911 / r11913;
        double r11915 = sqrt(r11914);
        return r11915;
}

↓

double f(double x) {
        double r11916 = x;
        double r11917 = -6.8779141464345475e-06;
        bool r11918 = r11916 <= r11917;
        double r11919 = exp(r11916);
        double r11920 = 1.0;
        double r11921 = r11919 + r11920;
        double r11922 = 2.0;
        double r11923 = r11922 * r11916;
        double r11924 = exp(r11923);
        double r11925 = r11924 - r11920;
        double r11926 = r11921 * r11925;
        double r11927 = 2.0;
        double r11928 = r11927 * r11916;
        double r11929 = exp(r11928);
        double r11930 = r11929 - r11920;
        double r11931 = r11926 / r11930;
        double r11932 = sqrt(r11931);
        double r11933 = 0.5;
        double r11934 = pow(r11916, r11927);
        double r11935 = fma(r11920, r11916, r11922);
        double r11936 = fma(r11933, r11934, r11935);
        double r11937 = sqrt(r11936);
        double r11938 = r11918 ? r11932 : r11937;
        return r11938;
}

Derivation

Split input into 2 regimes
if x < -6.8779141464345475e-06
1. Initial program 0.1
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Applied associate-/r/0.0
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]
5. Simplified0.0
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}} \cdot \left(e^{x} + 1\right)}\]
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 -6.8779141464345475e-06 < x
1. Initial program 34.1
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.5
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.5
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -6.87791414643454749 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\left(e^{x} + 1\right) \cdot \left(e^{2 \cdot x} - 1\right)}{e^{2 \cdot x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

Error

Derivation

`if x < -6.8779141464345475e-06`

`if -6.8779141464345475e-06 < x`

Reproduce