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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.31818012348102291 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{x} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.31818012348102291 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{x} + 1\right)\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 r14090 = 2.0;
        double r14091 = x;
        double r14092 = r14090 * r14091;
        double r14093 = exp(r14092);
        double r14094 = 1.0;
        double r14095 = r14093 - r14094;
        double r14096 = exp(r14091);
        double r14097 = r14096 - r14094;
        double r14098 = r14095 / r14097;
        double r14099 = sqrt(r14098);
        return r14099;
}

↓

double f(double x) {
        double r14100 = x;
        double r14101 = -1.3181801234810229e-05;
        bool r14102 = r14100 <= r14101;
        double r14103 = 2.0;
        double r14104 = r14103 * r14100;
        double r14105 = exp(r14104);
        double r14106 = 1.0;
        double r14107 = r14105 - r14106;
        double r14108 = r14100 + r14100;
        double r14109 = exp(r14108);
        double r14110 = r14106 * r14106;
        double r14111 = r14109 - r14110;
        double r14112 = r14107 / r14111;
        double r14113 = sqrt(r14112);
        double r14114 = exp(r14100);
        double r14115 = r14114 + r14106;
        double r14116 = log1p(r14115);
        double r14117 = expm1(r14116);
        double r14118 = sqrt(r14117);
        double r14119 = r14113 * r14118;
        double r14120 = 0.5;
        double r14121 = 2.0;
        double r14122 = pow(r14100, r14121);
        double r14123 = fma(r14106, r14100, r14103);
        double r14124 = fma(r14120, r14122, r14123);
        double r14125 = sqrt(r14124);
        double r14126 = r14102 ? r14119 : r14125;
        return r14126;
}

Derivation

Split input into 2 regimes
if x < -1.3181801234810229e-05
1. Initial program 0.1
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
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. Applied sqrt-prod0.1
  \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} \cdot e^{x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}}\]
6. Simplified0.0
  \[\leadsto \color{blue}{\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}}} \cdot \sqrt{e^{x} + 1}\]
7. Using strategy rm
8. Applied expm1-log1p-u0.0
  \[\leadsto \sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{x} + 1\right)\right)}}\]
if -1.3181801234810229e-05 < x
1. Initial program 34.5
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
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 simplification0.9
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.31818012348102291 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{x} + 1\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\mathsf{fma}\left(0.5, {x}^{2}, \mathsf{fma}\left(1, x, 2\right)\right)}\\ \end{array}\]

Falling back on racket egraph because egg-herbie package not installed (more)

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

Error

Derivation

`if x < -1.3181801234810229e-05`

`if -1.3181801234810229e-05 < x`

Reproduce