Average Error: 41.2 → 0.9
Time: 2.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 1.6351282551502975 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{1}{12}, \mathsf{fma}\left({x}^{3}, \frac{7}{720}, \frac{\frac{1}{2}}{x}\right)\right) \cdot \left(e^{x} + 1\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 1.6351282551502975 \cdot 10^{-4}:\\
\;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(-x, \frac{1}{12}, \mathsf{fma}\left({x}^{3}, \frac{7}{720}, \frac{\frac{1}{2}}{x}\right)\right) \cdot \left(e^{x} + 1\right)\\

\end{array}
double f(double x) {
        double r88592 = x;
        double r88593 = exp(r88592);
        double r88594 = 1.0;
        double r88595 = r88593 - r88594;
        double r88596 = r88593 / r88595;
        return r88596;
}
↓
double f(double x) {
        double r88597 = x;
        double r88598 = exp(r88597);
        double r88599 = 0.00016351282551502975;
        bool r88600 = r88598 <= r88599;
        double r88601 = 1.0;
        double r88602 = 1.0;
        double r88603 = r88602 / r88598;
        double r88604 = r88601 - r88603;
        double r88605 = r88601 / r88604;
        double r88606 = -r88597;
        double r88607 = 0.08333333333333333;
        double r88608 = 3.0;
        double r88609 = pow(r88597, r88608);
        double r88610 = 0.009722222222222222;
        double r88611 = 0.5;
        double r88612 = r88611 / r88597;
        double r88613 = fma(r88609, r88610, r88612);
        double r88614 = fma(r88606, r88607, r88613);
        double r88615 = r88598 + r88602;
        double r88616 = r88614 * r88615;
        double r88617 = r88600 ? r88605 : r88616;
        return r88617;
}

Error

Bits error versus x

Target

Original41.2
Target40.7
Herbie0.9
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.00016351282551502975

    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

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

    if 0.00016351282551502975 < (exp x)

    1. Initial program 61.6

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

    3. Applied flip--61.6

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

    4. Applied associate-/r/61.6

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

    5. Simplified61.6

      \[\leadsto \color{blue}{\frac{e^{x}}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}} \cdot \left(e^{x} + 1\right)\]

    6. Taylor expanded around 0 1.3

      \[\leadsto \color{blue}{\left(\left(\frac{7}{720} \cdot {x}^{3} + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{12} \cdot x\right)} \cdot \left(e^{x} + 1\right)\]

    7. Simplified1.3

      \[\leadsto \color{blue}{\mathsf{fma}\left(-x, \frac{1}{12}, \mathsf{fma}\left({x}^{3}, \frac{7}{720}, \frac{\frac{1}{2}}{x}\right)\right)} \cdot \left(e^{x} + 1\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 1.6351282551502975 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{1}{12}, \mathsf{fma}\left({x}^{3}, \frac{7}{720}, \frac{\frac{1}{2}}{x}\right)\right) \cdot \left(e^{x} + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020062 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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