Average Error: 40.4 → 0.5
Time: 11.4s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.001819987733615175:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x} \cdot \left(e^{x} \cdot e^{x}\right) + -1}{e^{x} \cdot e^{x} + \left(e^{x} + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.001819987733615175:\\
\;\;\;\;\frac{e^{x}}{\frac{e^{x} \cdot \left(e^{x} \cdot e^{x}\right) + -1}{e^{x} \cdot e^{x} + \left(e^{x} + 1\right)}}\\

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

\end{array}
double f(double x) {
        double r3021757 = x;
        double r3021758 = exp(r3021757);
        double r3021759 = 1.0;
        double r3021760 = r3021758 - r3021759;
        double r3021761 = r3021758 / r3021760;
        return r3021761;
}
↓
double f(double x) {
        double r3021762 = x;
        double r3021763 = -0.001819987733615175;
        bool r3021764 = r3021762 <= r3021763;
        double r3021765 = exp(r3021762);
        double r3021766 = r3021765 * r3021765;
        double r3021767 = r3021765 * r3021766;
        double r3021768 = -1.0;
        double r3021769 = r3021767 + r3021768;
        double r3021770 = 1.0;
        double r3021771 = r3021765 + r3021770;
        double r3021772 = r3021766 + r3021771;
        double r3021773 = r3021769 / r3021772;
        double r3021774 = r3021765 / r3021773;
        double r3021775 = 0.08333333333333333;
        double r3021776 = r3021775 * r3021762;
        double r3021777 = r3021770 / r3021762;
        double r3021778 = 0.5;
        double r3021779 = r3021777 + r3021778;
        double r3021780 = r3021776 + r3021779;
        double r3021781 = r3021764 ? r3021774 : r3021780;
        return r3021781;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.4
Target40.1
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.001819987733615175

    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

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

    5. Simplified0.0

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

    if -0.001819987733615175 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.7

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.001819987733615175:\\ \;\;\;\;\frac{e^{x}}{\frac{e^{x} \cdot \left(e^{x} \cdot e^{x}\right) + -1}{e^{x} \cdot e^{x} + \left(e^{x} + 1\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019132 
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))