Average Error: 41.4 → 0.5
Time: 15.6s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\ \;\;\;\;\frac{1}{\frac{e^{x} - 1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\log \left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right)\right)\right)} + \frac{1}{x}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\
\;\;\;\;\frac{1}{\frac{e^{x} - 1}{e^{x}}}\\

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

\end{array}
double f(double x) {
        double r4070377 = x;
        double r4070378 = exp(r4070377);
        double r4070379 = 1.0;
        double r4070380 = r4070378 - r4070379;
        double r4070381 = r4070378 / r4070380;
        return r4070381;
}
↓
double f(double x) {
        double r4070382 = x;
        double r4070383 = exp(r4070382);
        double r4070384 = 0.966199829895273;
        bool r4070385 = r4070383 <= r4070384;
        double r4070386 = 1.0;
        double r4070387 = 1.0;
        double r4070388 = r4070383 - r4070387;
        double r4070389 = r4070388 / r4070383;
        double r4070390 = r4070386 / r4070389;
        double r4070391 = exp(1.0);
        double r4070392 = 0.08333333333333333;
        double r4070393 = 0.5;
        double r4070394 = fma(r4070392, r4070382, r4070393);
        double r4070395 = log(r4070394);
        double r4070396 = pow(r4070391, r4070395);
        double r4070397 = r4070386 / r4070382;
        double r4070398 = r4070396 + r4070397;
        double r4070399 = r4070385 ? r4070390 : r4070398;
        return r4070399;
}

Error

Bits error versus x

Target

Original41.4
Target41.1
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.966199829895273

    1. Initial program 0.0

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

    3. Applied clear-num0.0

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

    if 0.966199829895273 < (exp x)

    1. Initial program 61.9

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

    2. Taylor expanded around 0 0.8

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

    3. Simplified0.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right) + \frac{1}{x}}\]
    4. Using strategy rm

    5. Applied add-exp-log0.8

      \[\leadsto \color{blue}{e^{\log \left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right)\right)}} + \frac{1}{x}\]
    6. Using strategy rm

    7. Applied pow10.8

      \[\leadsto e^{\log \color{blue}{\left({\left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right)\right)}^{1}\right)}} + \frac{1}{x}\]

    8. Applied log-pow0.8

      \[\leadsto e^{\color{blue}{1 \cdot \log \left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right)\right)}} + \frac{1}{x}\]

    9. Applied exp-prod0.8

      \[\leadsto \color{blue}{{\left(e^{1}\right)}^{\left(\log \left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right)\right)\right)}} + \frac{1}{x}\]

    10. Simplified0.8

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

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\ \;\;\;\;\frac{1}{\frac{e^{x} - 1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;{e}^{\left(\log \left(\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{2}\right)\right)\right)} + \frac{1}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019172 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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