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 r79697 = x;
        double r79698 = exp(r79697);
        double r79699 = 1.0;
        double r79700 = r79698 - r79699;
        double r79701 = r79698 / r79700;
        return r79701;
}
↓
double f(double x) {
        double r79702 = x;
        double r79703 = exp(r79702);
        double r79704 = 0.957606842952209;
        bool r79705 = r79703 <= r79704;
        double r79706 = 1.0;
        double r79707 = 1.0;
        double r79708 = r79707 / r79703;
        double r79709 = r79706 - r79708;
        double r79710 = 3.0;
        double r79711 = pow(r79709, r79710);
        double r79712 = r79706 / r79711;
        double r79713 = cbrt(r79712);
        double r79714 = 0.08333333333333333;
        double r79715 = r79706 / r79702;
        double r79716 = fma(r79714, r79702, r79715);
        double r79717 = 0.5;
        double r79718 = r79716 + r79717;
        double r79719 = r79705 ? r79713 : r79718;
        return r79719;
}

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)))