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

↓

\[\begin{array}{l} \mathbf{if}\;x \le -1.34331473476043902905803159586461958952 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(e^{2 \cdot x} - 1\right) \cdot \frac{1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + 0.5 \cdot x\right)}\\ \end{array}\]

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

↓

\begin{array}{l}
\mathbf{if}\;x \le -1.34331473476043902905803159586461958952 \cdot 10^{-5}:\\
\;\;\;\;\sqrt{\left(e^{2 \cdot x} - 1\right) \cdot \frac{1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\

\mathbf{else}:\\
\;\;\;\;\sqrt{2 + x \cdot \left(1 + 0.5 \cdot x\right)}\\

\end{array}

double f(double x) {
        double r23068 = 2.0;
        double r23069 = x;
        double r23070 = r23068 * r23069;
        double r23071 = exp(r23070);
        double r23072 = 1.0;
        double r23073 = r23071 - r23072;
        double r23074 = exp(r23069);
        double r23075 = r23074 - r23072;
        double r23076 = r23073 / r23075;
        double r23077 = sqrt(r23076);
        return r23077;
}

↓

double f(double x) {
        double r23078 = x;
        double r23079 = -1.343314734760439e-05;
        bool r23080 = r23078 <= r23079;
        double r23081 = 2.0;
        double r23082 = r23081 * r23078;
        double r23083 = exp(r23082);
        double r23084 = 1.0;
        double r23085 = r23083 - r23084;
        double r23086 = 1.0;
        double r23087 = r23078 + r23078;
        double r23088 = exp(r23087);
        double r23089 = r23084 * r23084;
        double r23090 = r23088 - r23089;
        double r23091 = r23086 / r23090;
        double r23092 = r23085 * r23091;
        double r23093 = sqrt(r23092);
        double r23094 = exp(r23078);
        double r23095 = r23094 + r23084;
        double r23096 = sqrt(r23095);
        double r23097 = r23093 * r23096;
        double r23098 = 0.5;
        double r23099 = r23098 * r23078;
        double r23100 = r23084 + r23099;
        double r23101 = r23078 * r23100;
        double r23102 = r23081 + r23101;
        double r23103 = sqrt(r23102);
        double r23104 = r23080 ? r23097 : r23103;
        return r23104;
}

Derivation

Split input into 2 regimes
if x < -1.343314734760439e-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 div-inv0.0
  \[\leadsto \sqrt{\color{blue}{\left(e^{2 \cdot x} - 1\right) \cdot \frac{1}{e^{x + x} - 1 \cdot 1}}} \cdot \sqrt{e^{x} + 1}\]
if -1.343314734760439e-05 < x
1. Initial program 34.5
  \[\sqrt{\frac{e^{2 \cdot x} - 1}{e^{x} - 1}}\]
2. Taylor expanded around 0 6.4
  \[\leadsto \sqrt{\color{blue}{0.5 \cdot {x}^{2} + \left(1 \cdot x + 2\right)}}\]
3. Simplified6.4
  \[\leadsto \sqrt{\color{blue}{2 + x \cdot \left(1 + 0.5 \cdot x\right)}}\]
Recombined 2 regimes into one program.
Final simplification0.9
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.34331473476043902905803159586461958952 \cdot 10^{-5}:\\ \;\;\;\;\sqrt{\left(e^{2 \cdot x} - 1\right) \cdot \frac{1}{e^{x + x} - 1 \cdot 1}} \cdot \sqrt{e^{x} + 1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 + x \cdot \left(1 + 0.5 \cdot x\right)}\\ \end{array}\]

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

Error

Try it out

Derivation

`if x < -1.343314734760439e-05`

`if -1.343314734760439e-05 < x`

Reproduce

In
Out