Average Error: 41.6 → 0.2
Time: 2.8s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.997692962517644166275943007349269464612:\\ \;\;\;\;\frac{e^{x}}{{\left(e^{x}\right)}^{3} - {1}^{3}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)\\ \mathbf{elif}\;e^{x} \le 1.00000000133302457960837728023761883378:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \log \left(e^{\frac{1}{e^{x}}}\right)}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.997692962517644166275943007349269464612:\\
\;\;\;\;\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)\\

\mathbf{elif}\;e^{x} \le 1.00000000133302457960837728023761883378:\\
\;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{1 - \log \left(e^{\frac{1}{e^{x}}}\right)}\\

\end{array}
double f(double x) {
        double r80854 = x;
        double r80855 = exp(r80854);
        double r80856 = 1.0;
        double r80857 = r80855 - r80856;
        double r80858 = r80855 / r80857;
        return r80858;
}
↓
double f(double x) {
        double r80859 = x;
        double r80860 = exp(r80859);
        double r80861 = 0.9976929625176442;
        bool r80862 = r80860 <= r80861;
        double r80863 = 3.0;
        double r80864 = pow(r80860, r80863);
        double r80865 = 1.0;
        double r80866 = pow(r80865, r80863);
        double r80867 = r80864 - r80866;
        double r80868 = r80860 / r80867;
        double r80869 = r80860 * r80860;
        double r80870 = r80865 * r80865;
        double r80871 = r80860 * r80865;
        double r80872 = r80870 + r80871;
        double r80873 = r80869 + r80872;
        double r80874 = r80868 * r80873;
        double r80875 = 1.0000000013330246;
        bool r80876 = r80860 <= r80875;
        double r80877 = 0.5;
        double r80878 = 0.08333333333333333;
        double r80879 = r80878 * r80859;
        double r80880 = 1.0;
        double r80881 = r80880 / r80859;
        double r80882 = r80879 + r80881;
        double r80883 = r80877 + r80882;
        double r80884 = r80865 / r80860;
        double r80885 = exp(r80884);
        double r80886 = log(r80885);
        double r80887 = r80880 - r80886;
        double r80888 = r80880 / r80887;
        double r80889 = r80876 ? r80883 : r80888;
        double r80890 = r80862 ? r80874 : r80889;
        return r80890;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.6
Target41.1
Herbie0.2
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 3 regimes

  2. if (exp x) < 0.9976929625176442

    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. Applied associate-/r/0.0

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

    if 0.9976929625176442 < (exp x) < 1.0000000013330246

    1. Initial program 62.8

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

    2. Taylor expanded around 0 0.0

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

    if 1.0000000013330246 < (exp x)

    1. Initial program 28.3

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

    3. Applied clear-num28.3

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

    4. Simplified6.8

      \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
    5. Using strategy rm

    6. Applied add-log-exp7.0

      \[\leadsto \frac{1}{1 - \color{blue}{\log \left(e^{\frac{1}{e^{x}}}\right)}}\]
  3. Recombined 3 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.997692962517644166275943007349269464612:\\ \;\;\;\;\frac{e^{x}}{{\left(e^{x}\right)}^{3} - {1}^{3}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)\\ \mathbf{elif}\;e^{x} \le 1.00000000133302457960837728023761883378:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \log \left(e^{\frac{1}{e^{x}}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019362 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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