\[e^{a \cdot x} - 1\]

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.008022852700594357:\\ \;\;\;\;\frac{\log \left(e^{{\left(e^{a \cdot x}\right)}^{3} - 1}\right)}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right) + \left(x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\right)\\ \end{array}\]

double f(double a, double x) {
        double r18188844 = a;
        double r18188845 = x;
        double r18188846 = r18188844 * r18188845;
        double r18188847 = exp(r18188846);
        double r18188848 = 1.0;
        double r18188849 = r18188847 - r18188848;
        return r18188849;
}

↓

double f(double a, double x) {
        double r18188850 = a;
        double r18188851 = x;
        double r18188852 = r18188850 * r18188851;
        double r18188853 = -0.008022852700594357;
        bool r18188854 = r18188852 <= r18188853;
        double r18188855 = exp(r18188852);
        double r18188856 = 3.0;
        double r18188857 = pow(r18188855, r18188856);
        double r18188858 = 1.0;
        double r18188859 = r18188857 - r18188858;
        double r18188860 = exp(r18188859);
        double r18188861 = log(r18188860);
        double r18188862 = r18188855 * r18188855;
        double r18188863 = r18188858 + r18188855;
        double r18188864 = r18188862 + r18188863;
        double r18188865 = r18188861 / r18188864;
        double r18188866 = 0.5;
        double r18188867 = r18188866 * r18188852;
        double r18188868 = r18188867 * r18188852;
        double r18188869 = 0.16666666666666666;
        double r18188870 = r18188850 * r18188869;
        double r18188871 = r18188852 * r18188852;
        double r18188872 = r18188870 * r18188871;
        double r18188873 = r18188851 * r18188872;
        double r18188874 = r18188873 + r18188852;
        double r18188875 = r18188868 + r18188874;
        double r18188876 = r18188854 ? r18188865 : r18188875;
        return r18188876;
}

e^{a \cdot x} - 1

↓

\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.008022852700594357:\\
\;\;\;\;\frac{\log \left(e^{{\left(e^{a \cdot x}\right)}^{3} - 1}\right)}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right) + \left(x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\right)\\

\end{array}

Original	29.9
Target	0.2
Herbie	0.3

Derivation

Split input into 2 regimes
if (* a x) < -0.008022852700594357
1. Initial program 0.0
  \[e^{a \cdot x} - 1\]
2. Using strategy rm
3. Applied flip3--0.0
  \[\leadsto \color{blue}{\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}\]
4. Using strategy rm
5. Applied add-log-exp0.0
  \[\leadsto \frac{\color{blue}{\log \left(e^{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}\right)}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}\]
if -0.008022852700594357 < (* a x)
1. Initial program 44.6
  \[e^{a \cdot x} - 1\]
2. Taylor expanded around 0 14.3
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]
3. Simplified0.5
  \[\leadsto \color{blue}{\left(a \cdot x + x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\right) + \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)}\]
Recombined 2 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.008022852700594357:\\ \;\;\;\;\frac{\log \left(e^{{\left(e^{a \cdot x}\right)}^{3} - 1}\right)}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right) + \left(x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + a \cdot x\right)\\ \end{array}\]

Error

Target

Derivation

`if (* a x) < -0.008022852700594357`

`if -0.008022852700594357 < (* a x)`

Reproduce