Average Error: 40.0 → 0.6
Time: 49.4s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.8767957581638243:\\ \;\;\;\;\left(e^{x} + 1\right) \cdot \frac{e^{x}}{e^{x} \cdot e^{x} + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + \frac{1}{12} \cdot x\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.8767957581638243:\\
\;\;\;\;\left(e^{x} + 1\right) \cdot \frac{e^{x}}{e^{x} \cdot e^{x} + -1}\\

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

\end{array}
double f(double x) {
        double r3509280 = x;
        double r3509281 = exp(r3509280);
        double r3509282 = 1.0;
        double r3509283 = r3509281 - r3509282;
        double r3509284 = r3509281 / r3509283;
        return r3509284;
}
↓
double f(double x) {
        double r3509285 = x;
        double r3509286 = exp(r3509285);
        double r3509287 = 0.8767957581638243;
        bool r3509288 = r3509286 <= r3509287;
        double r3509289 = 1.0;
        double r3509290 = r3509286 + r3509289;
        double r3509291 = r3509286 * r3509286;
        double r3509292 = -1.0;
        double r3509293 = r3509291 + r3509292;
        double r3509294 = r3509286 / r3509293;
        double r3509295 = r3509290 * r3509294;
        double r3509296 = r3509289 / r3509285;
        double r3509297 = 0.5;
        double r3509298 = r3509296 + r3509297;
        double r3509299 = 0.08333333333333333;
        double r3509300 = r3509299 * r3509285;
        double r3509301 = r3509298 + r3509300;
        double r3509302 = r3509288 ? r3509295 : r3509301;
        return r3509302;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.0
Target39.6
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.8767957581638243

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Applied associate-/r/0.0

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

    5. Simplified0.0

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

    if 0.8767957581638243 < (exp x)

    1. Initial program 59.9

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

    2. Taylor expanded around 0 1.0

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

    3. Taylor expanded around -inf 1.0

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.8767957581638243:\\ \;\;\;\;\left(e^{x} + 1\right) \cdot \frac{e^{x}}{e^{x} \cdot e^{x} + -1}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{x} + \frac{1}{2}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019141 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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