Average Error: 40.3 → 0.5
Time: 3.8m
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.002055367404917252:\\ \;\;\;\;\frac{\left(1 + e^{x} \cdot e^{x}\right) + e^{x}}{\frac{e^{x} \cdot e^{x}}{\frac{e^{x} \cdot \left(e^{x} - 1\right) - 1}{e^{x}}} - \frac{1}{e^{x} \cdot \left(e^{x} - 1\right) - 1}} \cdot \frac{1}{e^{x} - \left(\frac{1}{e^{x}} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \frac{1}{12} \cdot x\\ \end{array}\]
double f(double x) {
        double r23082834 = x;
        double r23082835 = exp(r23082834);
        double r23082836 = 1.0;
        double r23082837 = r23082835 - r23082836;
        double r23082838 = r23082835 / r23082837;
        return r23082838;
}
↓
double f(double x) {
        double r23082839 = x;
        double r23082840 = -0.002055367404917252;
        bool r23082841 = r23082839 <= r23082840;
        double r23082842 = 1.0;
        double r23082843 = exp(r23082839);
        double r23082844 = r23082843 * r23082843;
        double r23082845 = r23082842 + r23082844;
        double r23082846 = r23082845 + r23082843;
        double r23082847 = r23082843 - r23082842;
        double r23082848 = r23082843 * r23082847;
        double r23082849 = r23082848 - r23082842;
        double r23082850 = r23082849 / r23082843;
        double r23082851 = r23082844 / r23082850;
        double r23082852 = r23082842 / r23082849;
        double r23082853 = r23082851 - r23082852;
        double r23082854 = r23082846 / r23082853;
        double r23082855 = r23082842 / r23082843;
        double r23082856 = r23082855 + r23082842;
        double r23082857 = r23082843 - r23082856;
        double r23082858 = r23082842 / r23082857;
        double r23082859 = r23082854 * r23082858;
        double r23082860 = 0.5;
        double r23082861 = r23082842 / r23082839;
        double r23082862 = r23082860 + r23082861;
        double r23082863 = 0.08333333333333333;
        double r23082864 = r23082863 * r23082839;
        double r23082865 = r23082862 + r23082864;
        double r23082866 = r23082841 ? r23082859 : r23082865;
        return r23082866;
}


\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.002055367404917252:\\
\;\;\;\;\frac{\left(1 + e^{x} \cdot e^{x}\right) + e^{x}}{\frac{e^{x} \cdot e^{x}}{\frac{e^{x} \cdot \left(e^{x} - 1\right) - 1}{e^{x}}} - \frac{1}{e^{x} \cdot \left(e^{x} - 1\right) - 1}} \cdot \frac{1}{e^{x} - \left(\frac{1}{e^{x}} + 1\right)}\\

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

\end{array}

Error

Bits error versus x

Target

Original40.3
Target40.0
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.002055367404917252

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    5. Applied flip-+0.0

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

    6. Applied associate-/r/0.0

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

    7. Applied *-un-lft-identity0.0

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

    8. Applied times-frac0.0

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

    9. Simplified0.0

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

    10. Simplified0.0

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

    if -0.002055367404917252 < x

    1. Initial program 60.1

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

    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 simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.002055367404917252:\\ \;\;\;\;\frac{\left(1 + e^{x} \cdot e^{x}\right) + e^{x}}{\frac{e^{x} \cdot e^{x}}{\frac{e^{x} \cdot \left(e^{x} - 1\right) - 1}{e^{x}}} - \frac{1}{e^{x} \cdot \left(e^{x} - 1\right) - 1}} \cdot \frac{1}{e^{x} - \left(\frac{1}{e^{x}} + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{x}\right) + \frac{1}{12} \cdot x\\ \end{array}\]

Reproduce

herbie shell --seed 2019101 
(FPCore (x)
  :name "expq2 (section 3.11)"

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

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