Average Error: 41.3 → 0.7
Time: 3.2s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9576068429522089919814220593252684921026:\\ \;\;\;\;\sqrt[3]{\frac{1}{{\left(1 - \frac{1}{e^{x}}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9576068429522089919814220593252684921026:\\
\;\;\;\;\sqrt[3]{\frac{1}{{\left(1 - \frac{1}{e^{x}}\right)}^{3}}}\\

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

\end{array}
double f(double x) {
        double r84747 = x;
        double r84748 = exp(r84747);
        double r84749 = 1.0;
        double r84750 = r84748 - r84749;
        double r84751 = r84748 / r84750;
        return r84751;
}
↓
double f(double x) {
        double r84752 = x;
        double r84753 = exp(r84752);
        double r84754 = 0.957606842952209;
        bool r84755 = r84753 <= r84754;
        double r84756 = 1.0;
        double r84757 = 1.0;
        double r84758 = r84757 / r84753;
        double r84759 = r84756 - r84758;
        double r84760 = 3.0;
        double r84761 = pow(r84759, r84760);
        double r84762 = r84756 / r84761;
        double r84763 = cbrt(r84762);
        double r84764 = 0.08333333333333333;
        double r84765 = r84756 / r84752;
        double r84766 = fma(r84764, r84752, r84765);
        double r84767 = 0.5;
        double r84768 = r84766 + r84767;
        double r84769 = r84755 ? r84763 : r84768;
        return r84769;
}

Error

Bits error versus x

Target

Original41.3
Target40.8
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.957606842952209

    1. Initial program 0.0

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

    3. Applied clear-num0.0

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

    4. Simplified0.0

      \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
    5. Using strategy rm

    6. Applied add-cbrt-cube0.1

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

    7. Applied add-cbrt-cube0.1

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

    8. Applied cbrt-undiv0.1

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

    9. Simplified0.1

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

    if 0.957606842952209 < (exp x)

    1. Initial program 62.0

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

    2. Taylor expanded around 0 1.1

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

    3. Simplified1.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9576068429522089919814220593252684921026:\\ \;\;\;\;\sqrt[3]{\frac{1}{{\left(1 - \frac{1}{e^{x}}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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