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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.029656646791675586301382377962809755445 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.029656646791675586301382377962809755445 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(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 r10850 = 2.0;
        double r10851 = x;
        double r10852 = r10850 * r10851;
        double r10853 = exp(r10852);
        double r10854 = 1.0;
        double r10855 = r10853 - r10854;
        double r10856 = exp(r10851);
        double r10857 = r10856 - r10854;
        double r10858 = r10855 / r10857;
        double r10859 = sqrt(r10858);
        return r10859;
}

↓

double f(double x) {
        double r10860 = x;
        double r10861 = -1.0296566467916756e-05;
        bool r10862 = r10860 <= r10861;
        double r10863 = 2.0;
        double r10864 = r10863 * r10860;
        double r10865 = exp(r10864);
        double r10866 = 1.0;
        double r10867 = r10865 - r10866;
        double r10868 = -r10866;
        double r10869 = r10860 + r10860;
        double r10870 = exp(r10869);
        double r10871 = fma(r10868, r10866, r10870);
        double r10872 = r10867 / r10871;
        double r10873 = exp(r10860);
        double r10874 = r10873 + r10866;
        double r10875 = r10872 * r10874;
        double r10876 = sqrt(r10875);
        double r10877 = 0.5;
        double r10878 = 2.0;
        double r10879 = pow(r10860, r10878);
        double r10880 = fma(r10866, r10860, r10863);
        double r10881 = fma(r10877, r10879, r10880);
        double r10882 = sqrt(r10881);
        double r10883 = r10862 ? r10876 : r10882;
        return r10883;
}

Derivation

Split input into 2 regimes
if x < -1.0296566467916756e-05
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
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}} \cdot \left(e^{x} + 1\right)}\]
if -1.0296566467916756e-05 < x
1. Initial program 33.2
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.1
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.1
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.029656646791675586301382377962809755445 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

Error

Derivation

`if x < -1.0296566467916756e-05`

`if -1.0296566467916756e-05 < x`

Reproduce