Average Error: 41.2 → 0.6
Time: 1.2m
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.001574667532423496430632470222121810365934:\\ \;\;\;\;\left(e^{x} \cdot e^{x} + \left(1 \cdot e^{x} + 1 \cdot 1\right)\right) \cdot \frac{e^{x}}{e^{\left(x + x\right) + x} - \left(1 \cdot 1\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \left(\frac{1}{12} \cdot {\left({\left(x \cdot x\right)}^{\left(\sqrt[3]{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{3}}\right)}\right) \cdot \sqrt[3]{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.001574667532423496430632470222121810365934:\\
\;\;\;\;\left(e^{x} \cdot e^{x} + \left(1 \cdot e^{x} + 1 \cdot 1\right)\right) \cdot \frac{e^{x}}{e^{\left(x + x\right) + x} - \left(1 \cdot 1\right) \cdot 1}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \left(\frac{1}{12} \cdot {\left({\left(x \cdot x\right)}^{\left(\sqrt[3]{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{3}}\right)}\right) \cdot \sqrt[3]{x}\right)\\

\end{array}
double f(double x) {
        double r3090253 = x;
        double r3090254 = exp(r3090253);
        double r3090255 = 1.0;
        double r3090256 = r3090254 - r3090255;
        double r3090257 = r3090254 / r3090256;
        return r3090257;
}
↓
double f(double x) {
        double r3090258 = x;
        double r3090259 = -0.0015746675324234964;
        bool r3090260 = r3090258 <= r3090259;
        double r3090261 = exp(r3090258);
        double r3090262 = r3090261 * r3090261;
        double r3090263 = 1.0;
        double r3090264 = r3090263 * r3090261;
        double r3090265 = r3090263 * r3090263;
        double r3090266 = r3090264 + r3090265;
        double r3090267 = r3090262 + r3090266;
        double r3090268 = r3090258 + r3090258;
        double r3090269 = r3090268 + r3090258;
        double r3090270 = exp(r3090269);
        double r3090271 = r3090265 * r3090263;
        double r3090272 = r3090270 - r3090271;
        double r3090273 = r3090261 / r3090272;
        double r3090274 = r3090267 * r3090273;
        double r3090275 = 0.5;
        double r3090276 = 1.0;
        double r3090277 = r3090276 / r3090258;
        double r3090278 = 0.08333333333333333;
        double r3090279 = r3090258 * r3090258;
        double r3090280 = 0.3333333333333333;
        double r3090281 = cbrt(r3090280);
        double r3090282 = r3090281 * r3090281;
        double r3090283 = pow(r3090279, r3090282);
        double r3090284 = pow(r3090283, r3090281);
        double r3090285 = r3090278 * r3090284;
        double r3090286 = cbrt(r3090258);
        double r3090287 = r3090285 * r3090286;
        double r3090288 = r3090277 + r3090287;
        double r3090289 = r3090275 + r3090288;
        double r3090290 = r3090260 ? r3090274 : r3090289;
        return r3090290;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.2
Target40.8
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0015746675324234964

    1. Initial program 0.0

      \[\frac{e^{x}}{e^{x} - 1}\]
    2. Using strategy rm

    3. Applied flip3--0.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)}}}\]

    4. Applied associate-/r/0.0

      \[\leadsto \color{blue}{\frac{e^{x}}{{\left(e^{x}\right)}^{3} - {1}^{3}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)}\]

    5. Simplified0.0

      \[\leadsto \color{blue}{\frac{e^{x}}{e^{\left(x + x\right) + x} - 1 \cdot \left(1 \cdot 1\right)}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)\]

    if -0.0015746675324234964 < x

    1. Initial program 61.7

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

    2. Taylor expanded around 0 0.8

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]
    3. Using strategy rm

    4. Applied add-cube-cbrt0.8

      \[\leadsto \frac{1}{2} + \left(\frac{1}{12} \cdot \color{blue}{\left(\left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right) \cdot \sqrt[3]{x}\right)} + \frac{1}{x}\right)\]

    5. Applied associate-*r*0.8

      \[\leadsto \frac{1}{2} + \left(\color{blue}{\left(\frac{1}{12} \cdot \left(\sqrt[3]{x} \cdot \sqrt[3]{x}\right)\right) \cdot \sqrt[3]{x}} + \frac{1}{x}\right)\]
    6. Using strategy rm

    7. Applied pow1/331.9

      \[\leadsto \frac{1}{2} + \left(\left(\frac{1}{12} \cdot \left(\sqrt[3]{x} \cdot \color{blue}{{x}^{\frac{1}{3}}}\right)\right) \cdot \sqrt[3]{x} + \frac{1}{x}\right)\]

    8. Applied pow1/331.9

      \[\leadsto \frac{1}{2} + \left(\left(\frac{1}{12} \cdot \left(\color{blue}{{x}^{\frac{1}{3}}} \cdot {x}^{\frac{1}{3}}\right)\right) \cdot \sqrt[3]{x} + \frac{1}{x}\right)\]

    9. Applied pow-prod-down0.8

      \[\leadsto \frac{1}{2} + \left(\left(\frac{1}{12} \cdot \color{blue}{{\left(x \cdot x\right)}^{\frac{1}{3}}}\right) \cdot \sqrt[3]{x} + \frac{1}{x}\right)\]
    10. Using strategy rm

    11. Applied add-cube-cbrt0.8

      \[\leadsto \frac{1}{2} + \left(\left(\frac{1}{12} \cdot {\left(x \cdot x\right)}^{\color{blue}{\left(\left(\sqrt[3]{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{3}}\right) \cdot \sqrt[3]{\frac{1}{3}}\right)}}\right) \cdot \sqrt[3]{x} + \frac{1}{x}\right)\]

    12. Applied pow-unpow0.8

      \[\leadsto \frac{1}{2} + \left(\left(\frac{1}{12} \cdot \color{blue}{{\left({\left(x \cdot x\right)}^{\left(\sqrt[3]{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{3}}\right)}}\right) \cdot \sqrt[3]{x} + \frac{1}{x}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.001574667532423496430632470222121810365934:\\ \;\;\;\;\left(e^{x} \cdot e^{x} + \left(1 \cdot e^{x} + 1 \cdot 1\right)\right) \cdot \frac{e^{x}}{e^{\left(x + x\right) + x} - \left(1 \cdot 1\right) \cdot 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{x} + \left(\frac{1}{12} \cdot {\left({\left(x \cdot x\right)}^{\left(\sqrt[3]{\frac{1}{3}} \cdot \sqrt[3]{\frac{1}{3}}\right)}\right)}^{\left(\sqrt[3]{\frac{1}{3}}\right)}\right) \cdot \sqrt[3]{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019200 
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

  (/ (exp x) (- (exp x) 1.0)))