Average Error: 41.0 → 0.6
Time: 16.0s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.8605720028923194986347766644030343741179:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x} \cdot \left(\sqrt[3]{e^{x} \cdot e^{x}} \cdot \left(\sqrt[3]{e^{x} \cdot e^{x}} \cdot \sqrt[3]{e^{x} \cdot e^{x}}\right)\right) - 1}{e^{x} \cdot e^{x} + \left(e^{x} \cdot 1 + 1 \cdot 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.8605720028923194986347766644030343741179:\\
\;\;\;\;\frac{e^{x}}{\frac{e^{x} \cdot \left(\sqrt[3]{e^{x} \cdot e^{x}} \cdot \left(\sqrt[3]{e^{x} \cdot e^{x}} \cdot \sqrt[3]{e^{x} \cdot e^{x}}\right)\right) - 1}{e^{x} \cdot e^{x} + \left(e^{x} \cdot 1 + 1 \cdot 1\right)}}\\

\mathbf{else}:\\
\;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{x} + \frac{1}{2}\right)\\

\end{array}
double f(double x) {
        double r4860087 = x;
        double r4860088 = exp(r4860087);
        double r4860089 = 1.0;
        double r4860090 = r4860088 - r4860089;
        double r4860091 = r4860088 / r4860090;
        return r4860091;
}
↓
double f(double x) {
        double r4860092 = x;
        double r4860093 = exp(r4860092);
        double r4860094 = 0.8605720028923195;
        bool r4860095 = r4860093 <= r4860094;
        double r4860096 = r4860093 * r4860093;
        double r4860097 = cbrt(r4860096);
        double r4860098 = r4860097 * r4860097;
        double r4860099 = r4860097 * r4860098;
        double r4860100 = r4860093 * r4860099;
        double r4860101 = 1.0;
        double r4860102 = r4860100 - r4860101;
        double r4860103 = r4860093 * r4860101;
        double r4860104 = r4860101 * r4860101;
        double r4860105 = r4860103 + r4860104;
        double r4860106 = r4860096 + r4860105;
        double r4860107 = r4860102 / r4860106;
        double r4860108 = r4860093 / r4860107;
        double r4860109 = 0.08333333333333333;
        double r4860110 = r4860092 * r4860109;
        double r4860111 = 1.0;
        double r4860112 = r4860111 / r4860092;
        double r4860113 = 0.5;
        double r4860114 = r4860112 + r4860113;
        double r4860115 = r4860110 + r4860114;
        double r4860116 = r4860095 ? r4860108 : r4860115;
        return r4860116;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.0
Target40.6
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.8605720028923195

    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. Simplified0.0

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

    5. Taylor expanded around inf 0.0

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

    6. Simplified0.0

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

    8. Applied add-cube-cbrt0.0

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

    if 0.8605720028923195 < (exp x)

    1. Initial program 61.7

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

    2. Taylor expanded around 0 0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.8605720028923194986347766644030343741179:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x} \cdot \left(\sqrt[3]{e^{x} \cdot e^{x}} \cdot \left(\sqrt[3]{e^{x} \cdot e^{x}} \cdot \sqrt[3]{e^{x} \cdot e^{x}}\right)\right) - 1}{e^{x} \cdot e^{x} + \left(e^{x} \cdot 1 + 1 \cdot 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{12} + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019170 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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