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

↓

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

\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 r12878 = 2.0;
        double r12879 = x;
        double r12880 = r12878 * r12879;
        double r12881 = exp(r12880);
        double r12882 = 1.0;
        double r12883 = r12881 - r12882;
        double r12884 = exp(r12879);
        double r12885 = r12884 - r12882;
        double r12886 = r12883 / r12885;
        double r12887 = sqrt(r12886);
        return r12887;
}

↓

double f(double x) {
        double r12888 = x;
        double r12889 = -7.268542916058787e-16;
        bool r12890 = r12888 <= r12889;
        double r12891 = 2.0;
        double r12892 = r12891 * r12888;
        double r12893 = exp(r12892);
        double r12894 = 1.0;
        double r12895 = r12893 - r12894;
        double r12896 = r12888 + r12888;
        double r12897 = exp(r12896);
        double r12898 = r12894 * r12894;
        double r12899 = r12897 - r12898;
        double r12900 = r12895 / r12899;
        double r12901 = exp(r12888);
        double r12902 = sqrt(r12901);
        double r12903 = fma(r12902, r12902, r12894);
        double r12904 = r12900 * r12903;
        double r12905 = sqrt(r12904);
        double r12906 = 0.5;
        double r12907 = 2.0;
        double r12908 = pow(r12888, r12907);
        double r12909 = fma(r12894, r12888, r12891);
        double r12910 = fma(r12906, r12908, r12909);
        double r12911 = sqrt(r12910);
        double r12912 = r12890 ? r12905 : r12911;
        return r12912;
}

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