Average Error: 39.5 → 0.3
Time: 16.1s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.011815026060015150141668804906203149585 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.011815026060015150141668804906203149585 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}}{x}\\

\mathbf{else}:\\
\;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\\

\end{array}
double f(double x) {
        double r98875 = x;
        double r98876 = exp(r98875);
        double r98877 = 1.0;
        double r98878 = r98876 - r98877;
        double r98879 = r98878 / r98875;
        return r98879;
}
↓
double f(double x) {
        double r98880 = x;
        double r98881 = -0.00010118150260600152;
        bool r98882 = r98880 <= r98881;
        double r98883 = r98880 + r98880;
        double r98884 = exp(r98883);
        double r98885 = 1.0;
        double r98886 = r98885 * r98885;
        double r98887 = r98884 - r98886;
        double r98888 = exp(r98880);
        double r98889 = r98885 + r98888;
        double r98890 = r98887 / r98889;
        double r98891 = r98890 / r98880;
        double r98892 = 0.16666666666666666;
        double r98893 = 0.5;
        double r98894 = fma(r98892, r98880, r98893);
        double r98895 = 1.0;
        double r98896 = fma(r98880, r98894, r98895);
        double r98897 = r98882 ? r98891 : r98896;
        return r98897;
}

Error

Bits error versus x

Target

Original39.5
Target39.9
Herbie0.3
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.00010118150260600152

    1. Initial program 0.1

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

    3. Applied flip--0.1

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

    4. Simplified0.1

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

    5. Simplified0.1

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

    if -0.00010118150260600152 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.011815026060015150141668804906203149585 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019325 +o rules:numerics
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

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