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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.3921498547628525 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.3921498547628525 \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 r65 = 2.0;
        double r66 = x;
        double r67 = r65 * r66;
        double r68 = exp(r67);
        double r69 = 1.0;
        double r70 = r68 - r69;
        double r71 = exp(r66);
        double r72 = r71 - r69;
        double r73 = r70 / r72;
        double r74 = sqrt(r73);
        return r74;
}

↓

double f(double x) {
        double r75 = x;
        double r76 = -1.3921498547628525e-05;
        bool r77 = r75 <= r76;
        double r78 = 2.0;
        double r79 = r78 * r75;
        double r80 = exp(r79);
        double r81 = 1.0;
        double r82 = r80 - r81;
        double r83 = -r81;
        double r84 = r75 + r75;
        double r85 = exp(r84);
        double r86 = fma(r83, r81, r85);
        double r87 = r82 / r86;
        double r88 = exp(r75);
        double r89 = r88 + r81;
        double r90 = r87 * r89;
        double r91 = sqrt(r90);
        double r92 = 0.5;
        double r93 = 2.0;
        double r94 = pow(r75, r93);
        double r95 = fma(r81, r75, r78);
        double r96 = fma(r92, r94, r95);
        double r97 = sqrt(r96);
        double r98 = r77 ? r91 : r97;
        return r98;
}

Derivation

Split input into 2 regimes
if x < -1.3921498547628525e-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 -1.3921498547628525e-05 < x
1. Initial program 34.0
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.0
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.0
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.3921498547628525 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(e^{x} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

Error

Derivation

`if x < -1.3921498547628525e-05`

`if -1.3921498547628525e-05 < x`

Reproduce