Average Error: 41.0 → 0.9
Time: 12.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{e^{x}}{e^{x} \cdot \left(e^{x} \cdot e^{x}\right) - \left(1 \cdot 1\right) \cdot 1} \cdot \left(\left(1 \cdot e^{x} + 1 \cdot 1\right) + e^{x} \cdot e^{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \frac{1}{12} \cdot x\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.0:\\
\;\;\;\;\frac{e^{x}}{e^{x} \cdot \left(e^{x} \cdot e^{x}\right) - \left(1 \cdot 1\right) \cdot 1} \cdot \left(\left(1 \cdot e^{x} + 1 \cdot 1\right) + e^{x} \cdot e^{x}\right)\\

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

\end{array}
double f(double x) {
        double r5256981 = x;
        double r5256982 = exp(r5256981);
        double r5256983 = 1.0;
        double r5256984 = r5256982 - r5256983;
        double r5256985 = r5256982 / r5256984;
        return r5256985;
}
↓
double f(double x) {
        double r5256986 = x;
        double r5256987 = exp(r5256986);
        double r5256988 = 0.0;
        bool r5256989 = r5256987 <= r5256988;
        double r5256990 = r5256987 * r5256987;
        double r5256991 = r5256987 * r5256990;
        double r5256992 = 1.0;
        double r5256993 = r5256992 * r5256992;
        double r5256994 = r5256993 * r5256992;
        double r5256995 = r5256991 - r5256994;
        double r5256996 = r5256987 / r5256995;
        double r5256997 = r5256992 * r5256987;
        double r5256998 = r5256997 + r5256993;
        double r5256999 = r5256998 + r5256990;
        double r5257000 = r5256996 * r5256999;
        double r5257001 = 0.5;
        double r5257002 = 1.0;
        double r5257003 = r5257002 / r5256986;
        double r5257004 = r5257001 + r5257003;
        double r5257005 = 0.08333333333333333;
        double r5257006 = r5257005 * r5256986;
        double r5257007 = r5257004 + r5257006;
        double r5257008 = r5256989 ? r5257000 : r5257007;
        return r5257008;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.0
Target40.6
Herbie0.9
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.0

    1. Initial program 0

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

    3. Applied flip3--0

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

    4. Applied associate-/r/0

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

    5. Simplified0

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

    if 0.0 < (exp x)

    1. Initial program 61.4

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

    2. Taylor expanded around 0 1.3

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

  4. Final simplification0.9

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{e^{x}}{e^{x} \cdot \left(e^{x} \cdot e^{x}\right) - \left(1 \cdot 1\right) \cdot 1} \cdot \left(\left(1 \cdot e^{x} + 1 \cdot 1\right) + e^{x} \cdot e^{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019169 
(FPCore (x)
  :name "expq2 (section 3.11)"

  :herbie-target
  (/ 1.0 (- 1.0 (exp (- x))))

  (/ (exp x) (- (exp x) 1.0)))