Average Error: 41.4 → 0.7
Time: 13.3s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]
\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}:\\
\;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2} + \frac{1}{x}\right)\\

\end{array}
double f(double x) {
        double r41387 = x;
        double r41388 = exp(r41387);
        double r41389 = 1.0;
        double r41390 = r41388 - r41389;
        double r41391 = r41388 / r41390;
        return r41391;
}
↓
double f(double x) {
        double r41392 = x;
        double r41393 = exp(r41392);
        double r41394 = 0.0;
        bool r41395 = r41393 <= r41394;
        double r41396 = 1.0;
        double r41397 = 1.0;
        double r41398 = r41397 / r41393;
        double r41399 = r41396 - r41398;
        double r41400 = r41396 / r41399;
        double r41401 = 0.08333333333333333;
        double r41402 = 0.5;
        double r41403 = r41396 / r41392;
        double r41404 = r41402 + r41403;
        double r41405 = fma(r41401, r41392, r41404);
        double r41406 = r41395 ? r41400 : r41405;
        return r41406;
}

Error

Bits error versus x

Target

Original41.4
Target41.1
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

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 61.7

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

    2. Taylor expanded around 0 1.3

      \[\leadsto \frac{e^{x}}{\color{blue}{x + \left(\frac{1}{6} \cdot {x}^{3} + \frac{1}{2} \cdot {x}^{2}\right)}}\]

    3. Simplified1.3

      \[\leadsto \frac{e^{x}}{\color{blue}{\mathsf{fma}\left({x}^{3}, \frac{1}{6}, \mathsf{fma}\left(x \cdot x, \frac{1}{2}, x\right)\right)}}\]

    4. Taylor expanded around 0 1.1

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

    5. Simplified1.1

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

  4. Final simplification0.7

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

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

  (/ (exp x) (- (exp x) 1.0)))