Average Error: 39.2 → 0.4
Time: 5.6s
Precision: 64
\[\frac{1 - e^{-x}}{x}\]
\[\begin{array}{l} \mathbf{if}\;x \le 1.619478639329793683010461080584718729369 \cdot 10^{-4}:\\ \;\;\;\;1 + x \cdot \left(\frac{1}{6} \cdot x + \frac{-1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{e^{-x}}{x}\\ \end{array}\]
\frac{1 - e^{-x}}{x}
\begin{array}{l}
\mathbf{if}\;x \le 1.619478639329793683010461080584718729369 \cdot 10^{-4}:\\
\;\;\;\;1 + x \cdot \left(\frac{1}{6} \cdot x + \frac{-1}{2}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{x} - \frac{e^{-x}}{x}\\

\end{array}
double f(double x) {
        double r685965 = 1.0;
        double r685966 = x;
        double r685967 = -r685966;
        double r685968 = exp(r685967);
        double r685969 = r685965 - r685968;
        double r685970 = r685969 / r685966;
        return r685970;
}

double f(double x) {
        double r685971 = x;
        double r685972 = 0.00016194786393297937;
        bool r685973 = r685971 <= r685972;
        double r685974 = 1.0;
        double r685975 = 0.16666666666666666;
        double r685976 = r685975 * r685971;
        double r685977 = -0.5;
        double r685978 = r685976 + r685977;
        double r685979 = r685971 * r685978;
        double r685980 = r685974 + r685979;
        double r685981 = 1.0;
        double r685982 = r685981 / r685971;
        double r685983 = -r685971;
        double r685984 = exp(r685983);
        double r685985 = r685984 / r685971;
        double r685986 = r685982 - r685985;
        double r685987 = r685973 ? r685980 : r685986;
        return r685987;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes
  2. if x < 0.00016194786393297937

    1. Initial program 59.9

      \[\frac{1 - e^{-x}}{x}\]
    2. Taylor expanded around 0 0.5

      \[\leadsto \color{blue}{\left(\frac{1}{6} \cdot {x}^{2} + 1\right) - \frac{1}{2} \cdot x}\]
    3. Simplified0.5

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{6} \cdot x + \frac{-1}{2}\right)}\]

    if 0.00016194786393297937 < x

    1. Initial program 0.1

      \[\frac{1 - e^{-x}}{x}\]
    2. Using strategy rm
    3. Applied div-sub0.1

      \[\leadsto \color{blue}{\frac{1}{x} - \frac{e^{-x}}{x}}\]
  3. Recombined 2 regimes into one program.
  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le 1.619478639329793683010461080584718729369 \cdot 10^{-4}:\\ \;\;\;\;1 + x \cdot \left(\frac{1}{6} \cdot x + \frac{-1}{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} - \frac{e^{-x}}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 1 
(FPCore (x)
  :name "(1-exp(-x))/x"
  :precision binary64
  (/ (- 1 (exp (- x))) x))