\[-1 \lt \varepsilon \land \varepsilon \lt 1\]

\[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;\varepsilon \le -7.739597797624075 \cdot 10^{-75}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \le 5.71803006796694 \cdot 10^{-72}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \end{array}\]

\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}

↓

\begin{array}{l}
\mathbf{if}\;\varepsilon \le -7.739597797624075 \cdot 10^{-75}:\\
\;\;\;\;\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\

\mathbf{elif}\;\varepsilon \le 5.71803006796694 \cdot 10^{-72}:\\
\;\;\;\;\frac{1}{b} + \frac{1}{a}\\

\mathbf{else}:\\
\;\;\;\;\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\

\end{array}

double f(double a, double b, double eps) {
        double r2788901 = eps;
        double r2788902 = a;
        double r2788903 = b;
        double r2788904 = r2788902 + r2788903;
        double r2788905 = r2788904 * r2788901;
        double r2788906 = exp(r2788905);
        double r2788907 = 1.0;
        double r2788908 = r2788906 - r2788907;
        double r2788909 = r2788901 * r2788908;
        double r2788910 = r2788902 * r2788901;
        double r2788911 = exp(r2788910);
        double r2788912 = r2788911 - r2788907;
        double r2788913 = r2788903 * r2788901;
        double r2788914 = exp(r2788913);
        double r2788915 = r2788914 - r2788907;
        double r2788916 = r2788912 * r2788915;
        double r2788917 = r2788909 / r2788916;
        return r2788917;
}

↓

double f(double a, double b, double eps) {
        double r2788918 = eps;
        double r2788919 = -7.739597797624075e-75;
        bool r2788920 = r2788918 <= r2788919;
        double r2788921 = a;
        double r2788922 = b;
        double r2788923 = r2788921 + r2788922;
        double r2788924 = r2788923 * r2788918;
        double r2788925 = expm1(r2788924);
        double r2788926 = r2788921 * r2788918;
        double r2788927 = expm1(r2788926);
        double r2788928 = r2788918 / r2788927;
        double r2788929 = r2788925 * r2788928;
        double r2788930 = r2788922 * r2788918;
        double r2788931 = expm1(r2788930);
        double r2788932 = r2788929 / r2788931;
        double r2788933 = 5.71803006796694e-72;
        bool r2788934 = r2788918 <= r2788933;
        double r2788935 = 1.0;
        double r2788936 = r2788935 / r2788922;
        double r2788937 = r2788935 / r2788921;
        double r2788938 = r2788936 + r2788937;
        double r2788939 = r2788934 ? r2788938 : r2788932;
        double r2788940 = r2788920 ? r2788932 : r2788939;
        return r2788940;
}

Original	58.6
Target	14.4
Herbie	2.8

Derivation

Split input into 2 regimes
if eps < -7.739597797624075e-75 or 5.71803006796694e-72 < eps
1. Initial program 52.0
  \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
2. Simplified7.0
  \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]
3. Using strategy rm
4. Applied associate-*l/6.6
  \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}{\mathsf{expm1}\left(\varepsilon \cdot b\right)}}\]
if -7.739597797624075e-75 < eps < 5.71803006796694e-72
1. Initial program 60.2
  \[\frac{\varepsilon \cdot \left(e^{\left(a + b\right) \cdot \varepsilon} - 1\right)}{\left(e^{a \cdot \varepsilon} - 1\right) \cdot \left(e^{b \cdot \varepsilon} - 1\right)}\]
2. Simplified32.7
  \[\leadsto \color{blue}{\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right)}{\mathsf{expm1}\left(\varepsilon \cdot b\right)} \cdot \frac{\varepsilon}{\mathsf{expm1}\left(\varepsilon \cdot a\right)}}\]
3. Taylor expanded around 0 1.9
  \[\leadsto \color{blue}{\frac{1}{a} + \frac{1}{b}}\]
Recombined 2 regimes into one program.
Final simplification2.8
\[\leadsto \begin{array}{l} \mathbf{if}\;\varepsilon \le -7.739597797624075 \cdot 10^{-75}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \mathbf{elif}\;\varepsilon \le 5.71803006796694 \cdot 10^{-72}:\\ \;\;\;\;\frac{1}{b} + \frac{1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{expm1}\left(\left(a + b\right) \cdot \varepsilon\right) \cdot \frac{\varepsilon}{\mathsf{expm1}\left(a \cdot \varepsilon\right)}}{\mathsf{expm1}\left(b \cdot \varepsilon\right)}\\ \end{array}\]

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

Error

Try it out

Target

Derivation

`if eps < -7.739597797624075e-75 or 5.71803006796694e-72 < eps`

`if -7.739597797624075e-75 < eps < 5.71803006796694e-72`

Reproduce

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