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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -9.47195806647322485446235246220325620925 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 2\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -9.47195806647322485446235246220325620925 \cdot 10^{-6}:\\
\;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}, -1\right)}{e^{x} - 1}}\\

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

\end{array}

double f(double x) {
        double r1245065 = 2.0;
        double r1245066 = x;
        double r1245067 = r1245065 * r1245066;
        double r1245068 = exp(r1245067);
        double r1245069 = 1.0;
        double r1245070 = r1245068 - r1245069;
        double r1245071 = exp(r1245066);
        double r1245072 = r1245071 - r1245069;
        double r1245073 = r1245070 / r1245072;
        double r1245074 = sqrt(r1245073);
        return r1245074;
}

↓

double f(double x) {
        double r1245075 = x;
        double r1245076 = -9.471958066473225e-06;
        bool r1245077 = r1245075 <= r1245076;
        double r1245078 = 2.0;
        double r1245079 = r1245078 * r1245075;
        double r1245080 = exp(r1245079);
        double r1245081 = sqrt(r1245080);
        double r1245082 = log(r1245081);
        double r1245083 = exp(r1245082);
        double r1245084 = 1.0;
        double r1245085 = -r1245084;
        double r1245086 = fma(r1245081, r1245083, r1245085);
        double r1245087 = exp(r1245075);
        double r1245088 = r1245087 - r1245084;
        double r1245089 = r1245086 / r1245088;
        double r1245090 = sqrt(r1245089);
        double r1245091 = 0.5;
        double r1245092 = fma(r1245075, r1245091, r1245084);
        double r1245093 = fma(r1245075, r1245092, r1245078);
        double r1245094 = sqrt(r1245093);
        double r1245095 = r1245077 ? r1245090 : r1245094;
        return r1245095;
}

Derivation

Split input into 2 regimes
if x < -9.471958066473225e-06
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
  \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}}{e^{x} - 1}}\]
5. Using strategy rm
6. Applied add-exp-log0.0
  \[\leadsto \sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \color{blue}{e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}}, -1\right)}{e^{x} - 1}}\]
if -9.471958066473225e-06 < x
1. Initial program 34.5
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.5
  \[\leadsto \sqrt{\color{blue}{1 \cdot x + \left(0.5 \cdot {x}^{2} + 2\right)}}\]
3. Simplified6.5
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 2\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.47195806647322485446235246220325620925 \cdot 10^{-6}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, e^{\log \left(\sqrt{e^{2 \cdot x}}\right)}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 2\right)}\\ \end{array}\]

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

Error

Derivation

`if x < -9.471958066473225e-06`

`if -9.471958066473225e-06 < x`

Reproduce