Average Error: 39.9 → 0.6
Time: 16.3s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.002288391212180564:\\ \;\;\;\;\frac{e^{x}}{\log \left(\frac{e^{e^{x}}}{e}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.002288391212180564:\\
\;\;\;\;\frac{e^{x}}{\log \left(\frac{e^{e^{x}}}{e}\right)}\\

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

\end{array}
double f(double x) {
        double r2493233 = x;
        double r2493234 = exp(r2493233);
        double r2493235 = 1.0;
        double r2493236 = r2493234 - r2493235;
        double r2493237 = r2493234 / r2493236;
        return r2493237;
}
↓
double f(double x) {
        double r2493238 = x;
        double r2493239 = -0.002288391212180564;
        bool r2493240 = r2493238 <= r2493239;
        double r2493241 = exp(r2493238);
        double r2493242 = exp(r2493241);
        double r2493243 = exp(1.0);
        double r2493244 = r2493242 / r2493243;
        double r2493245 = log(r2493244);
        double r2493246 = r2493241 / r2493245;
        double r2493247 = 0.08333333333333333;
        double r2493248 = r2493247 * r2493238;
        double r2493249 = 1.0;
        double r2493250 = r2493249 / r2493238;
        double r2493251 = 0.5;
        double r2493252 = r2493250 + r2493251;
        double r2493253 = r2493248 + r2493252;
        double r2493254 = r2493240 ? r2493246 : r2493253;
        return r2493254;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.9
Target39.6
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.002288391212180564

    1. Initial program 0.0

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

    3. Applied add-log-exp0.0

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

    4. Applied add-log-exp0.0

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

    5. Applied diff-log0.0

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

    6. Simplified0.0

      \[\leadsto \frac{e^{x}}{\log \color{blue}{\left(\frac{e^{e^{x}}}{e}\right)}}\]

    if -0.002288391212180564 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.8

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.002288391212180564:\\ \;\;\;\;\frac{e^{x}}{\log \left(\frac{e^{e^{x}}}{e}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{x} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019138 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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