\[\frac{e^{x}}{e^{x} - 1}\]

↓

\[\begin{array}{l} \mathbf{if}\;x \le -0.0018711434362285975:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}} \cdot \frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}\\ \mathbf{elif}\;x \le 9.641487078301664 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{{\left(e^{2}\right)}^{\left(\log 1 - x\right)}}{1}} \cdot \left(1 + e^{\log 1 - x}\right)\\ \end{array}\]

\frac{e^{x}}{e^{x} - 1}

↓

\begin{array}{l}
\mathbf{if}\;x \le -0.0018711434362285975:\\
\;\;\;\;\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}} \cdot \frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}\\

\mathbf{elif}\;x \le 9.641487078301664 \cdot 10^{-4}:\\
\;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 - \frac{{\left(e^{2}\right)}^{\left(\log 1 - x\right)}}{1}} \cdot \left(1 + e^{\log 1 - x}\right)\\

\end{array}

double f(double x) {
        double r401 = x;
        double r402 = exp(r401);
        double r403 = 1.0;
        double r404 = r402 - r403;
        double r405 = r402 / r404;
        return r405;
}

↓

double f(double x) {
        double r406 = x;
        double r407 = -0.0018711434362285975;
        bool r408 = r406 <= r407;
        double r409 = 1.0;
        double r410 = 1.0;
        double r411 = log(r410);
        double r412 = r411 - r406;
        double r413 = exp(r412);
        double r414 = r409 - r413;
        double r415 = cbrt(r414);
        double r416 = r409 / r415;
        double r417 = r416 / r415;
        double r418 = r417 * r416;
        double r419 = 0.0009641487078301664;
        bool r420 = r406 <= r419;
        double r421 = 0.5;
        double r422 = 0.08333333333333333;
        double r423 = r422 * r406;
        double r424 = r409 / r406;
        double r425 = r423 + r424;
        double r426 = r421 + r425;
        double r427 = 2.0;
        double r428 = exp(r427);
        double r429 = pow(r428, r412);
        double r430 = r429 / r409;
        double r431 = r409 - r430;
        double r432 = r409 / r431;
        double r433 = r409 + r413;
        double r434 = r432 * r433;
        double r435 = r420 ? r426 : r434;
        double r436 = r408 ? r418 : r435;
        return r436;
}

Original	40.3
Target	39.8
Herbie	0.0

Derivation

Split input into 3 regimes
if x < -0.0018711434362285975
1. Initial program 0.0
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Using strategy rm
3. Applied clear-num0.0
  \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]
4. Simplified0.0
  \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
5. Using strategy rm
6. Applied add-exp-log0.0
  \[\leadsto \frac{1}{1 - \frac{\color{blue}{e^{\log 1}}}{e^{x}}}\]
7. Applied div-exp0.0
  \[\leadsto \frac{1}{1 - \color{blue}{e^{\log 1 - x}}}\]
8. Using strategy rm
9. Applied add-cube-cbrt0.0
  \[\leadsto \frac{1}{\color{blue}{\left(\sqrt[3]{1 - e^{\log 1 - x}} \cdot \sqrt[3]{1 - e^{\log 1 - x}}\right) \cdot \sqrt[3]{1 - e^{\log 1 - x}}}}\]
10. Applied add-cube-cbrt0.0
  \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{1} \cdot \sqrt[3]{1}\right) \cdot \sqrt[3]{1}}}{\left(\sqrt[3]{1 - e^{\log 1 - x}} \cdot \sqrt[3]{1 - e^{\log 1 - x}}\right) \cdot \sqrt[3]{1 - e^{\log 1 - x}}}\]
11. Applied times-frac0.0
  \[\leadsto \color{blue}{\frac{\sqrt[3]{1} \cdot \sqrt[3]{1}}{\sqrt[3]{1 - e^{\log 1 - x}} \cdot \sqrt[3]{1 - e^{\log 1 - x}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{1 - e^{\log 1 - x}}}}\]
12. Simplified0.0
  \[\leadsto \color{blue}{\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}}} \cdot \frac{\sqrt[3]{1}}{\sqrt[3]{1 - e^{\log 1 - x}}}\]
13. Simplified0.0
  \[\leadsto \frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}} \cdot \color{blue}{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}\]
if -0.0018711434362285975 < x < 0.0009641487078301664
1. Initial program 62.4
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Taylor expanded around 0 0.0
  \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]
if 0.0009641487078301664 < x
1. Initial program 35.8
  \[\frac{e^{x}}{e^{x} - 1}\]
2. Using strategy rm
3. Applied clear-num35.8
  \[\leadsto \color{blue}{\frac{1}{\frac{e^{x} - 1}{e^{x}}}}\]
4. Simplified1.1
  \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
5. Using strategy rm
6. Applied add-exp-log1.1
  \[\leadsto \frac{1}{1 - \frac{\color{blue}{e^{\log 1}}}{e^{x}}}\]
7. Applied div-exp0.7
  \[\leadsto \frac{1}{1 - \color{blue}{e^{\log 1 - x}}}\]
8. Using strategy rm
9. Applied flip--0.7
  \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot 1 - e^{\log 1 - x} \cdot e^{\log 1 - x}}{1 + e^{\log 1 - x}}}}\]
10. Applied associate-/r/0.7
  \[\leadsto \color{blue}{\frac{1}{1 \cdot 1 - e^{\log 1 - x} \cdot e^{\log 1 - x}} \cdot \left(1 + e^{\log 1 - x}\right)}\]
11. Simplified0.6
  \[\leadsto \color{blue}{\frac{1}{1 - \frac{{\left(e^{2}\right)}^{\left(\log 1 - x\right)}}{1}}} \cdot \left(1 + e^{\log 1 - x}\right)\]
Recombined 3 regimes into one program.
Final simplification0.0
\[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0018711434362285975:\\ \;\;\;\;\frac{\frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}}{\sqrt[3]{1 - e^{\log 1 - x}}} \cdot \frac{1}{\sqrt[3]{1 - e^{\log 1 - x}}}\\ \mathbf{elif}\;x \le 9.641487078301664 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{{\left(e^{2}\right)}^{\left(\log 1 - x\right)}}{1}} \cdot \left(1 + e^{\log 1 - x}\right)\\ \end{array}\]

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

Error

Try it out

Target

Derivation

`if x < -0.0018711434362285975`

`if -0.0018711434362285975 < x < 0.0009641487078301664`

`if 0.0009641487078301664 < x`

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