Average Error: 41.4 → 0.5
Time: 16.3s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\frac{1}{e^{x}}} + 1}}{1 - \sqrt{\frac{1}{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{12}, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.9661998298952729768984681868460029363632:\\
\;\;\;\;\frac{\frac{1}{\sqrt{\frac{1}{e^{x}}} + 1}}{1 - \sqrt{\frac{1}{e^{x}}}}\\

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

\end{array}
double f(double x) {
        double r4661259 = x;
        double r4661260 = exp(r4661259);
        double r4661261 = 1.0;
        double r4661262 = r4661260 - r4661261;
        double r4661263 = r4661260 / r4661262;
        return r4661263;
}
↓
double f(double x) {
        double r4661264 = x;
        double r4661265 = exp(r4661264);
        double r4661266 = 0.966199829895273;
        bool r4661267 = r4661265 <= r4661266;
        double r4661268 = 1.0;
        double r4661269 = 1.0;
        double r4661270 = r4661269 / r4661265;
        double r4661271 = sqrt(r4661270);
        double r4661272 = r4661271 + r4661268;
        double r4661273 = r4661268 / r4661272;
        double r4661274 = r4661268 - r4661271;
        double r4661275 = r4661273 / r4661274;
        double r4661276 = 0.08333333333333333;
        double r4661277 = r4661268 / r4661264;
        double r4661278 = fma(r4661264, r4661276, r4661277);
        double r4661279 = 0.5;
        double r4661280 = r4661278 + r4661279;
        double r4661281 = r4661267 ? r4661275 : r4661280;
        return r4661281;
}

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

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

    5. Applied div-sub63.2

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

    6. Simplified0.0

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

    8. Applied add-sqr-sqrt0.0

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

    9. Applied *-un-lft-identity0.0

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

    10. Applied difference-of-squares0.0

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

    11. Applied associate-/r*0.0

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

    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

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

  4. Final simplification0.5

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

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)))