Average Error: 41.4 → 0.9
Time: 11.5s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\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}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\

\end{array}
double f(double x) {
        double r42893 = x;
        double r42894 = exp(r42893);
        double r42895 = 1.0;
        double r42896 = r42894 - r42895;
        double r42897 = r42894 / r42896;
        return r42897;
}
↓
double f(double x) {
        double r42898 = x;
        double r42899 = exp(r42898);
        double r42900 = 0.0;
        bool r42901 = r42899 <= r42900;
        double r42902 = 1.0;
        double r42903 = 1.0;
        double r42904 = r42903 / r42899;
        double r42905 = r42902 - r42904;
        double r42906 = r42902 / r42905;
        double r42907 = 0.08333333333333333;
        double r42908 = r42902 / r42898;
        double r42909 = fma(r42907, r42898, r42908);
        double r42910 = 0.5;
        double r42911 = r42909 + r42910;
        double r42912 = r42901 ? r42906 : r42911;
        return r42912;
}

Error

Bits error versus x

Target

Original41.4
Target41.0
Herbie0.9
\[\frac{1}{1 - e^{-x}}\]

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 61.3

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

    2. Taylor expanded around 0 1.3

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

    3. Simplified1.3

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

  4. Final simplification0.9

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

Reproduce

herbie shell --seed 2020046 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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