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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -2.163484807190271 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -2.163484807190271 \cdot 10^{-7}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \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 r11839 = 2.0;
        double r11840 = x;
        double r11841 = r11839 * r11840;
        double r11842 = exp(r11841);
        double r11843 = 1.0;
        double r11844 = r11842 - r11843;
        double r11845 = exp(r11840);
        double r11846 = r11845 - r11843;
        double r11847 = r11844 / r11846;
        double r11848 = sqrt(r11847);
        return r11848;
}

↓

double f(double x) {
        double r11849 = x;
        double r11850 = -2.163484807190271e-07;
        bool r11851 = r11849 <= r11850;
        double r11852 = 2.0;
        double r11853 = r11852 * r11849;
        double r11854 = exp(r11853);
        double r11855 = 1.0;
        double r11856 = r11854 - r11855;
        double r11857 = -r11855;
        double r11858 = r11849 + r11849;
        double r11859 = exp(r11858);
        double r11860 = fma(r11857, r11855, r11859);
        double r11861 = r11856 / r11860;
        double r11862 = exp(r11849);
        double r11863 = sqrt(r11862);
        double r11864 = fma(r11863, r11863, r11855);
        double r11865 = r11861 * r11864;
        double r11866 = sqrt(r11865);
        double r11867 = 0.5;
        double r11868 = 2.0;
        double r11869 = pow(r11849, r11868);
        double r11870 = fma(r11855, r11849, r11852);
        double r11871 = fma(r11867, r11869, r11870);
        double r11872 = sqrt(r11871);
        double r11873 = r11851 ? r11866 : r11872;
        return r11873;
}

Derivation

Split input into 2 regimes
if x < -2.163484807190271e-07
1. Initial program 0.2
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied flip--0.1
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\color{blue}{\frac{e^{x} \cdot e^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]
4. Applied associate-/r/0.1
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1} \cdot \left(e^{x} + 1\right)}}\]
5. Simplified0.0
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}} \cdot \left(e^{x} + 1\right)}\]
6. Using strategy rm
7. Applied add-sqr-sqrt0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \left(\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}} + 1\right)}\]
8. Applied fma-def0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \color{blue}{\mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}}\]
if -2.163484807190271e-07 < x
1. Initial program 35.2
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.7
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.7
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.163484807190271 \cdot 10^{-7}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

Error

Derivation

`if x < -2.163484807190271e-07`

`if -2.163484807190271e-07 < x`

Reproduce