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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.998472655500366441157916730375632141659 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.4999999999999997779553950749686919152737, x, 1\right), 2\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.998472655500366441157916730375632141659 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.4999999999999997779553950749686919152737, x, 1\right), 2\right)}\\

\end{array}

double f(double x) {
        double r1489875 = 2.0;
        double r1489876 = x;
        double r1489877 = r1489875 * r1489876;
        double r1489878 = exp(r1489877);
        double r1489879 = 1.0;
        double r1489880 = r1489878 - r1489879;
        double r1489881 = exp(r1489876);
        double r1489882 = r1489881 - r1489879;
        double r1489883 = r1489880 / r1489882;
        double r1489884 = sqrt(r1489883);
        return r1489884;
}

↓

double f(double x) {
        double r1489885 = x;
        double r1489886 = -1.9984726555003664e-05;
        bool r1489887 = r1489885 <= r1489886;
        double r1489888 = 2.0;
        double r1489889 = r1489888 * r1489885;
        double r1489890 = exp(r1489889);
        double r1489891 = sqrt(r1489890);
        double r1489892 = 1.0;
        double r1489893 = -r1489892;
        double r1489894 = fma(r1489891, r1489891, r1489893);
        double r1489895 = exp(r1489885);
        double r1489896 = r1489895 - r1489892;
        double r1489897 = r1489894 / r1489896;
        double r1489898 = sqrt(r1489897);
        double r1489899 = 0.4999999999999998;
        double r1489900 = fma(r1489899, r1489885, r1489892);
        double r1489901 = fma(r1489885, r1489900, r1489888);
        double r1489902 = sqrt(r1489901);
        double r1489903 = r1489887 ? r1489898 : r1489902;
        return r1489903;
}

Derivation

Split input into 2 regimes
if x < -1.9984726555003664e-05
1. Initial program 0.1
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.1
  \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}} - 1}{e^{x} - 1}}\]
4. Applied fma-neg0.0
  \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}}{e^{x} - 1}}\]
if -1.9984726555003664e-05 < x
1. Initial program 34.0
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied add-sqr-sqrt31.6
  \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}} - 1}{e^{x} - 1}}\]
4. Applied fma-neg26.2
  \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}}{e^{x} - 1}}\]
5. Taylor expanded around 0 6.0
  \[\leadsto \sqrt{\color{blue}{0.4999999999999997779553950749686919152737 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
6. Simplified6.0
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(0.4999999999999997779553950749686919152737, x, 1\right), 2\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.998472655500366441157916730375632141659 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(0.4999999999999997779553950749686919152737, x, 1\right), 2\right)}\\ \end{array}\]

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

Error

Derivation

`if x < -1.9984726555003664e-05`

`if -1.9984726555003664e-05 < x`

Reproduce