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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.138987560482873366194365993830928118768 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.138987560482873366194365993830928118768 \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 r10861 = 2.0;
        double r10862 = x;
        double r10863 = r10861 * r10862;
        double r10864 = exp(r10863);
        double r10865 = 1.0;
        double r10866 = r10864 - r10865;
        double r10867 = exp(r10862);
        double r10868 = r10867 - r10865;
        double r10869 = r10866 / r10868;
        double r10870 = sqrt(r10869);
        return r10870;
}

↓

double f(double x) {
        double r10871 = x;
        double r10872 = -1.1389875604828734e-05;
        bool r10873 = r10871 <= r10872;
        double r10874 = 2.0;
        double r10875 = r10874 * r10871;
        double r10876 = exp(r10875);
        double r10877 = 1.0;
        double r10878 = r10876 - r10877;
        double r10879 = -r10877;
        double r10880 = r10871 + r10871;
        double r10881 = exp(r10880);
        double r10882 = fma(r10879, r10877, r10881);
        double r10883 = r10878 / r10882;
        double r10884 = exp(r10871);
        double r10885 = sqrt(r10884);
        double r10886 = fma(r10885, r10885, r10877);
        double r10887 = r10883 * r10886;
        double r10888 = sqrt(r10887);
        double r10889 = 0.5;
        double r10890 = 2.0;
        double r10891 = pow(r10871, r10890);
        double r10892 = fma(r10877, r10871, r10874);
        double r10893 = fma(r10889, r10891, r10892);
        double r10894 = sqrt(r10893);
        double r10895 = r10873 ? r10888 : r10894;
        return r10895;
}

Derivation

Split input into 2 regimes
if x < -1.1389875604828734e-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 -1.1389875604828734e-05 < x
1. Initial program 33.2
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.1
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.1
  \[\leadsto \sqrt{\color{blue}{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.8
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.138987560482873366194365993830928118768 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)} \cdot \mathsf{fma}\left(\sqrt{e^{x}}, \sqrt{e^{x}}, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

Error

Derivation

`if x < -1.1389875604828734e-05`

`if -1.1389875604828734e-05 < x`

Reproduce