Average Error: 40.0 → 0.7
Time: 10.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.0016474661445984768:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.0016474661445984768:\\
\;\;\;\;\frac{e^{x}}{e^{x} - 1}\\

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

\end{array}
double f(double x) {
        double r2780301 = x;
        double r2780302 = exp(r2780301);
        double r2780303 = 1.0;
        double r2780304 = r2780302 - r2780303;
        double r2780305 = r2780302 / r2780304;
        return r2780305;
}
↓
double f(double x) {
        double r2780306 = x;
        double r2780307 = -0.0016474661445984768;
        bool r2780308 = r2780306 <= r2780307;
        double r2780309 = exp(r2780306);
        double r2780310 = 1.0;
        double r2780311 = r2780309 - r2780310;
        double r2780312 = r2780309 / r2780311;
        double r2780313 = 0.08333333333333333;
        double r2780314 = r2780313 * r2780306;
        double r2780315 = 0.5;
        double r2780316 = r2780310 / r2780306;
        double r2780317 = r2780315 + r2780316;
        double r2780318 = r2780314 + r2780317;
        double r2780319 = r2780308 ? r2780312 : r2780318;
        return r2780319;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.0
Target39.5
Herbie0.7
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0016474661445984768

    1. Initial program 0.0

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

    2. Taylor expanded around inf 0.0

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

    if -0.0016474661445984768 < x

    1. Initial program 60.1

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

    2. Taylor expanded around 0 1.1

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

    3. Taylor expanded around inf 1.1

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

  4. Final simplification0.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.0016474661445984768:\\ \;\;\;\;\frac{e^{x}}{e^{x} - 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{12} \cdot x + \left(\frac{1}{2} + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019133 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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