Average Error: 40.8 → 0.8
Time: 10.4s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.0:\\
\;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\

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

\end{array}
double f(double x) {
        double r113546 = x;
        double r113547 = exp(r113546);
        double r113548 = 1.0;
        double r113549 = r113547 - r113548;
        double r113550 = r113547 / r113549;
        return r113550;
}
↓
double f(double x) {
        double r113551 = x;
        double r113552 = exp(r113551);
        double r113553 = 0.0;
        bool r113554 = r113552 <= r113553;
        double r113555 = 1.0;
        double r113556 = 1.0;
        double r113557 = r113556 / r113552;
        double r113558 = r113555 - r113557;
        double r113559 = r113555 / r113558;
        double r113560 = 0.5;
        double r113561 = 0.08333333333333333;
        double r113562 = r113561 * r113551;
        double r113563 = r113555 / r113551;
        double r113564 = r113562 + r113563;
        double r113565 = r113560 + r113564;
        double r113566 = r113554 ? r113559 : r113565;
        return r113566;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.8
Target40.4
Herbie0.8
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.0

    1. Initial program 0

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

    3. Applied clear-num0

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

    4. Simplified0

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

    if 0.0 < (exp x)

    1. Initial program 61.2

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

    2. Taylor expanded around 0 1.3

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

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020045 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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