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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.275446733100294322569082106078042215813 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.275446733100294322569082106078042215813 \cdot 10^{-5}:\\
\;\;\;\;\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 r14133 = 2.0;
        double r14134 = x;
        double r14135 = r14133 * r14134;
        double r14136 = exp(r14135);
        double r14137 = 1.0;
        double r14138 = r14136 - r14137;
        double r14139 = exp(r14134);
        double r14140 = r14139 - r14137;
        double r14141 = r14138 / r14140;
        double r14142 = sqrt(r14141);
        return r14142;
}

↓

double f(double x) {
        double r14143 = x;
        double r14144 = -1.2754467331002943e-05;
        bool r14145 = r14143 <= r14144;
        double r14146 = 2.0;
        double r14147 = r14146 * r14143;
        double r14148 = exp(r14147);
        double r14149 = 1.0;
        double r14150 = r14148 - r14149;
        double r14151 = -r14149;
        double r14152 = r14143 + r14143;
        double r14153 = exp(r14152);
        double r14154 = fma(r14151, r14149, r14153);
        double r14155 = r14150 / r14154;
        double r14156 = exp(r14143);
        double r14157 = sqrt(r14156);
        double r14158 = fma(r14157, r14157, r14149);
        double r14159 = r14155 * r14158;
        double r14160 = sqrt(r14159);
        double r14161 = 0.5;
        double r14162 = 2.0;
        double r14163 = pow(r14143, r14162);
        double r14164 = fma(r14149, r14143, r14146);
        double r14165 = fma(r14161, r14163, r14164);
        double r14166 = sqrt(r14165);
        double r14167 = r14145 ? r14160 : r14166;
        return r14167;
}

Derivation

Split input into 2 regimes
if x < -1.2754467331002943e-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
  \[\leadsto \sqrt{\color{blue}{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}} \cdot \left(e^{x} + 1\right)}\]
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 -1.2754467331002943e-05 < x
1. Initial program 33.8
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.8
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.8
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.9
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.275446733100294322569082106078042215813 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

Error

Derivation

`if x < -1.2754467331002943e-05`

`if -1.2754467331002943e-05 < x`

Reproduce