Average Error: 41.4 → 0.4
Time: 11.1s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 1.000081178772427970002922847925219684839:\\ \;\;\;\;\frac{e^{x}}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 1.000081178772427970002922847925219684839:\\
\;\;\;\;\frac{e^{x}}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + x}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\

\end{array}
double f(double x) {
        double r239266 = x;
        double r239267 = exp(r239266);
        double r239268 = 1.0;
        double r239269 = r239267 - r239268;
        double r239270 = r239267 / r239269;
        return r239270;
}
↓
double f(double x) {
        double r239271 = x;
        double r239272 = exp(r239271);
        double r239273 = 1.000081178772428;
        bool r239274 = r239272 <= r239273;
        double r239275 = 2.0;
        double r239276 = pow(r239271, r239275);
        double r239277 = 0.5;
        double r239278 = 0.16666666666666666;
        double r239279 = r239278 * r239271;
        double r239280 = r239277 + r239279;
        double r239281 = r239276 * r239280;
        double r239282 = r239281 + r239271;
        double r239283 = r239272 / r239282;
        double r239284 = 1.0;
        double r239285 = 1.0;
        double r239286 = r239285 / r239272;
        double r239287 = r239284 - r239286;
        double r239288 = r239284 / r239287;
        double r239289 = r239274 ? r239283 : r239288;
        return r239289;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.4
Target41.0
Herbie0.4
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 1.000081178772428

    1. Initial program 41.5

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

    2. Taylor expanded around 0 11.3

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

    3. Simplified0.4

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

    if 1.000081178772428 < (exp x)

    1. Initial program 31.5

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

    3. Applied clear-num31.6

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

    4. Simplified1.5

      \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 1.000081178772427970002922847925219684839:\\ \;\;\;\;\frac{e^{x}}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019291 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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