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

↓

\[\begin{array}{l} \mathbf{if}\;a \cdot x \le -0.05945617636539035:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(e^{\left(a \cdot x + a \cdot x\right) + a \cdot x} + -1\right) \cdot \left(e^{\left(a \cdot x + a \cdot x\right) + a \cdot x} + -1\right)\right) \cdot \left(e^{a \cdot x} - 1\right)}}{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

double f(double a, double x) {
        double r12052779 = a;
        double r12052780 = x;
        double r12052781 = r12052779 * r12052780;
        double r12052782 = exp(r12052781);
        double r12052783 = 1.0;
        double r12052784 = r12052782 - r12052783;
        return r12052784;
}

↓

double f(double a, double x) {
        double r12052785 = a;
        double r12052786 = x;
        double r12052787 = r12052785 * r12052786;
        double r12052788 = -0.05945617636539035;
        bool r12052789 = r12052787 <= r12052788;
        double r12052790 = r12052787 + r12052787;
        double r12052791 = r12052790 + r12052787;
        double r12052792 = exp(r12052791);
        double r12052793 = -1.0;
        double r12052794 = r12052792 + r12052793;
        double r12052795 = r12052794 * r12052794;
        double r12052796 = exp(r12052787);
        double r12052797 = 1.0;
        double r12052798 = r12052796 - r12052797;
        double r12052799 = r12052795 * r12052798;
        double r12052800 = cbrt(r12052799);
        double r12052801 = r12052796 * r12052796;
        double r12052802 = r12052797 + r12052796;
        double r12052803 = r12052801 + r12052802;
        double r12052804 = r12052803 * r12052803;
        double r12052805 = cbrt(r12052804);
        double r12052806 = r12052800 / r12052805;
        double r12052807 = 0.16666666666666666;
        double r12052808 = r12052785 * r12052807;
        double r12052809 = r12052787 * r12052787;
        double r12052810 = r12052808 * r12052809;
        double r12052811 = r12052786 * r12052810;
        double r12052812 = 0.5;
        double r12052813 = r12052812 * r12052787;
        double r12052814 = r12052787 * r12052813;
        double r12052815 = r12052811 + r12052814;
        double r12052816 = r12052787 + r12052815;
        double r12052817 = r12052789 ? r12052806 : r12052816;
        return r12052817;
}

e^{a \cdot x} - 1

↓

\begin{array}{l}
\mathbf{if}\;a \cdot x \le -0.05945617636539035:\\
\;\;\;\;\frac{\sqrt[3]{\left(\left(e^{\left(a \cdot x + a \cdot x\right) + a \cdot x} + -1\right) \cdot \left(e^{\left(a \cdot x + a \cdot x\right) + a \cdot x} + -1\right)\right) \cdot \left(e^{a \cdot x} - 1\right)}}{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)\right)}}\\

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

\end{array}

Original	29.3
Target	0.2
Herbie	0.3

Derivation

Split input into 2 regimes
if (* a x) < -0.05945617636539035
1. Initial program 0.0
  \[e^{a \cdot x} - 1\]
2. Using strategy rm
3. Applied add-cbrt-cube0.0
  \[\leadsto \color{blue}{\sqrt[3]{\left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right) \cdot \left(e^{a \cdot x} - 1\right)}}\]
4. Using strategy rm
5. Applied flip3--0.0
  \[\leadsto \sqrt[3]{\left(\left(e^{a \cdot x} - 1\right) \cdot \left(e^{a \cdot x} - 1\right)\right) \cdot \color{blue}{\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}}\]
6. Applied flip3--0.0
  \[\leadsto \sqrt[3]{\left(\left(e^{a \cdot x} - 1\right) \cdot \color{blue}{\frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}\right) \cdot \frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}\]
7. Applied associate-*r/0.0
  \[\leadsto \sqrt[3]{\color{blue}{\frac{\left(e^{a \cdot x} - 1\right) \cdot \left({\left(e^{a \cdot x}\right)}^{3} - {1}^{3}\right)}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}} \cdot \frac{{\left(e^{a \cdot x}\right)}^{3} - {1}^{3}}{e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)}}\]
8. Applied frac-times0.0
  \[\leadsto \sqrt[3]{\color{blue}{\frac{\left(\left(e^{a \cdot x} - 1\right) \cdot \left({\left(e^{a \cdot x}\right)}^{3} - {1}^{3}\right)\right) \cdot \left({\left(e^{a \cdot x}\right)}^{3} - {1}^{3}\right)}{\left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)\right)}}}\]
9. Applied cbrt-div0.0
  \[\leadsto \color{blue}{\frac{\sqrt[3]{\left(\left(e^{a \cdot x} - 1\right) \cdot \left({\left(e^{a \cdot x}\right)}^{3} - {1}^{3}\right)\right) \cdot \left({\left(e^{a \cdot x}\right)}^{3} - {1}^{3}\right)}}{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)\right)}}}\]
10. Simplified0.0
  \[\leadsto \frac{\color{blue}{\sqrt[3]{\left(\left(e^{\left(a \cdot x + a \cdot x\right) + a \cdot x} + -1\right) \cdot \left(e^{\left(a \cdot x + a \cdot x\right) + a \cdot x} + -1\right)\right) \cdot \left(e^{a \cdot x} - 1\right)}}}{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 \cdot 1 + e^{a \cdot x} \cdot 1\right)\right)}}\]
if -0.05945617636539035 < (* a x)
1. Initial program 44.4
  \[e^{a \cdot x} - 1\]
2. Taylor expanded around 0 13.6
  \[\leadsto \color{blue}{\frac{1}{2} \cdot \left({a}^{2} \cdot {x}^{2}\right) + \left(a \cdot x + \frac{1}{6} \cdot \left({a}^{3} \cdot {x}^{3}\right)\right)}\]
3. Simplified0.4
  \[\leadsto \color{blue}{\left(a \cdot x + x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right)\right) + \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)}\]
4. Using strategy rm
5. Applied associate-+l+0.4
  \[\leadsto \color{blue}{a \cdot x + \left(x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right) \cdot \left(a \cdot x\right)\right)}\]
Recombined 2 regimes into one program.
Final simplification0.3
\[\leadsto \begin{array}{l} \mathbf{if}\;a \cdot x \le -0.05945617636539035:\\ \;\;\;\;\frac{\sqrt[3]{\left(\left(e^{\left(a \cdot x + a \cdot x\right) + a \cdot x} + -1\right) \cdot \left(e^{\left(a \cdot x + a \cdot x\right) + a \cdot x} + -1\right)\right) \cdot \left(e^{a \cdot x} - 1\right)}}{\sqrt[3]{\left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)\right) \cdot \left(e^{a \cdot x} \cdot e^{a \cdot x} + \left(1 + e^{a \cdot x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;a \cdot x + \left(x \cdot \left(\left(a \cdot \frac{1}{6}\right) \cdot \left(\left(a \cdot x\right) \cdot \left(a \cdot x\right)\right)\right) + \left(a \cdot x\right) \cdot \left(\frac{1}{2} \cdot \left(a \cdot x\right)\right)\right)\\ \end{array}\]

Error

Target

Derivation

`if (* a x) < -0.05945617636539035`

`if -0.05945617636539035 < (* a x)`

Reproduce