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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.066294159780490556383492913420906617468 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \left(0.2499999999999998889776975374843459576368 - \frac{0.125}{2}\right) \cdot \frac{x}{\frac{\sqrt{2}}{x}}\right)\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.066294159780490556383492913420906617468 \cdot 10^{-14}:\\
\;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2} + \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \left(0.2499999999999998889776975374843459576368 - \frac{0.125}{2}\right) \cdot \frac{x}{\frac{\sqrt{2}}{x}}\right)\\

\end{array}

double f(double x) {
        double r1111090 = 2.0;
        double r1111091 = x;
        double r1111092 = r1111090 * r1111091;
        double r1111093 = exp(r1111092);
        double r1111094 = 1.0;
        double r1111095 = r1111093 - r1111094;
        double r1111096 = exp(r1111091);
        double r1111097 = r1111096 - r1111094;
        double r1111098 = r1111095 / r1111097;
        double r1111099 = sqrt(r1111098);
        return r1111099;
}

↓

double f(double x) {
        double r1111100 = x;
        double r1111101 = -1.0662941597804906e-14;
        bool r1111102 = r1111100 <= r1111101;
        double r1111103 = 2.0;
        double r1111104 = r1111103 * r1111100;
        double r1111105 = exp(r1111104);
        double r1111106 = sqrt(r1111105);
        double r1111107 = 1.0;
        double r1111108 = -r1111107;
        double r1111109 = fma(r1111106, r1111106, r1111108);
        double r1111110 = exp(r1111100);
        double r1111111 = r1111110 - r1111107;
        double r1111112 = r1111109 / r1111111;
        double r1111113 = sqrt(r1111112);
        double r1111114 = sqrt(r1111103);
        double r1111115 = 0.5;
        double r1111116 = r1111100 / r1111114;
        double r1111117 = 0.2499999999999999;
        double r1111118 = 0.125;
        double r1111119 = r1111118 / r1111103;
        double r1111120 = r1111117 - r1111119;
        double r1111121 = r1111114 / r1111100;
        double r1111122 = r1111100 / r1111121;
        double r1111123 = r1111120 * r1111122;
        double r1111124 = fma(r1111115, r1111116, r1111123);
        double r1111125 = r1111114 + r1111124;
        double r1111126 = r1111102 ? r1111113 : r1111125;
        return r1111126;
}

Derivation

Split input into 2 regimes
if x < -1.0662941597804906e-14
1. Initial program 0.7
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied add-sqr-sqrt0.5
  \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}} - 1}{e^{x} - 1}}\]
4. Applied fma-neg0.2
  \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}}{e^{x} - 1}}\]
if -1.0662941597804906e-14 < x
1. Initial program 36.6
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Using strategy rm
3. Applied add-sqr-sqrt34.7
  \[\leadsto \sqrt{\frac{\color{blue}{\sqrt{e^{2 \cdot x}} \cdot \sqrt{e^{2 \cdot x}}} - 1}{e^{x} - 1}}\]
4. Applied fma-neg31.0
  \[\leadsto \sqrt{\frac{\color{blue}{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}}{e^{x} - 1}}\]
5. Taylor expanded around 0 8.1
  \[\leadsto \color{blue}{\left(\sqrt{2} + \left(0.5 \cdot \frac{x}{\sqrt{2}} + 0.2499999999999998889776975374843459576368 \cdot \frac{{x}^{2}}{\sqrt{2}}\right)\right) - 0.125 \cdot \frac{{x}^{2}}{{\left(\sqrt{2}\right)}^{3}}}\]
6. Simplified8.1
  \[\leadsto \color{blue}{\sqrt{2} + \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \frac{x}{\frac{\sqrt{2}}{x}} \cdot \left(0.2499999999999998889776975374843459576368 - \frac{0.125}{2}\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification1.1
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.066294159780490556383492913420906617468 \cdot 10^{-14}:\\ \;\;\;\;\sqrt{\frac{\mathsf{fma}\left(\sqrt{e^{2 \cdot x}}, \sqrt{e^{2 \cdot x}}, -1\right)}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} + \mathsf{fma}\left(0.5, \frac{x}{\sqrt{2}}, \left(0.2499999999999998889776975374843459576368 - \frac{0.125}{2}\right) \cdot \frac{x}{\frac{\sqrt{2}}{x}}\right)\\ \end{array}\]

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

Error

Derivation

`if x < -1.0662941597804906e-14`

`if -1.0662941597804906e-14 < x`

Reproduce