Average Error: 40.2 → 0.6
Time: 12.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0017346667534861506:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0017346667534861506:\\
\;\;\;\;\frac{e^{x}}{e^{x} - 1}\\

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

\end{array}
double f(double x) {
        double r3223808 = x;
        double r3223809 = exp(r3223808);
        double r3223810 = 1.0;
        double r3223811 = r3223809 - r3223810;
        double r3223812 = r3223809 / r3223811;
        return r3223812;
}
↓
double f(double x) {
        double r3223813 = x;
        double r3223814 = -0.0017346667534861506;
        bool r3223815 = r3223813 <= r3223814;
        double r3223816 = exp(r3223813);
        double r3223817 = 1.0;
        double r3223818 = r3223816 - r3223817;
        double r3223819 = r3223816 / r3223818;
        double r3223820 = 0.08333333333333333;
        double r3223821 = r3223820 * r3223813;
        double r3223822 = 0.5;
        double r3223823 = r3223817 / r3223813;
        double r3223824 = r3223822 + r3223823;
        double r3223825 = r3223821 + r3223824;
        double r3223826 = r3223815 ? r3223819 : r3223825;
        return r3223826;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.2
Target39.8
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0017346667534861506

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

      \[\leadsto \color{blue}{\frac{e^{x}}{e^{x} - 1}}\]

    if -0.0017346667534861506 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.8

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

    3. Taylor expanded around 0 0.8

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0017346667534861506:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019121 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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