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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -7.26854291605878706 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -7.26854291605878706 \cdot 10^{-16}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\

\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 r12827 = 2.0;
        double r12828 = x;
        double r12829 = r12827 * r12828;
        double r12830 = exp(r12829);
        double r12831 = 1.0;
        double r12832 = r12830 - r12831;
        double r12833 = exp(r12828);
        double r12834 = r12833 - r12831;
        double r12835 = r12832 / r12834;
        double r12836 = sqrt(r12835);
        return r12836;
}

↓

double f(double x) {
        double r12837 = x;
        double r12838 = -7.268542916058787e-16;
        bool r12839 = r12837 <= r12838;
        double r12840 = 2.0;
        double r12841 = r12840 * r12837;
        double r12842 = exp(r12841);
        double r12843 = 1.0;
        double r12844 = r12842 - r12843;
        double r12845 = r12837 + r12837;
        double r12846 = exp(r12845);
        double r12847 = r12843 * r12843;
        double r12848 = r12846 - r12847;
        double r12849 = r12844 / r12848;
        double r12850 = exp(r12837);
        double r12851 = sqrt(r12850);
        double r12852 = fma(r12851, r12851, r12843);
        double r12853 = r12849 * r12852;
        double r12854 = sqrt(r12853);
        double r12855 = 0.5;
        double r12856 = 2.0;
        double r12857 = pow(r12837, r12856);
        double r12858 = fma(r12843, r12837, r12840);
        double r12859 = fma(r12855, r12857, r12858);
        double r12860 = sqrt(r12859);
        double r12861 = r12839 ? r12854 : r12860;
        return r12861;
}

Derivation

Split input into 2 regimes
if x < -7.268542916058787e-16
1. Initial program 0.7
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.5
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Applied associate-/r/0.5
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]
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. Using strategy rm
7. Applied add-sqr-sqrt0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \left(\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}} + 1\right)}\]
8. Applied fma-def0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \color{blue}{\mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}}\]
if -7.268542916058787e-16 < x
1. Initial program 37.3
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 7.4
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified7.4
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.7
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.26854291605878706 \cdot 10^{-16}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

Falling back on racket egraph because egg-herbie package not installed (more)

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

Error

Derivation

`if x < -7.268542916058787e-16`

`if -7.268542916058787e-16 < x`

Reproduce