Average Error: 40.0 → 0.3
Time: 2.8s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.012327742145682 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{x} \cdot \left(e^{x} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.012327742145682 \cdot 10^{-4}:\\
\;\;\;\;\frac{1}{x} \cdot \left(e^{x} - 1\right)\\

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

\end{array}
double f(double x) {
        double r73773 = x;
        double r73774 = exp(r73773);
        double r73775 = 1.0;
        double r73776 = r73774 - r73775;
        double r73777 = r73776 / r73773;
        return r73777;
}
↓
double f(double x) {
        double r73778 = x;
        double r73779 = -0.0001012327742145682;
        bool r73780 = r73778 <= r73779;
        double r73781 = 1.0;
        double r73782 = r73781 / r73778;
        double r73783 = exp(r73778);
        double r73784 = 1.0;
        double r73785 = r73783 - r73784;
        double r73786 = r73782 * r73785;
        double r73787 = 0.16666666666666666;
        double r73788 = 2.0;
        double r73789 = pow(r73778, r73788);
        double r73790 = 0.5;
        double r73791 = fma(r73790, r73778, r73781);
        double r73792 = fma(r73787, r73789, r73791);
        double r73793 = r73780 ? r73786 : r73792;
        return r73793;
}

Error

Bits error versus x

Target

Original40.0
Target40.5
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 < -0.0001012327742145682

    1. Initial program 0.1

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

    3. Applied div-sub0.1

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

    5. Applied div-inv0.1

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

    6. Applied div-inv0.1

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

    7. Applied distribute-rgt-out--0.1

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

    if -0.0001012327742145682 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.012327742145682 \cdot 10^{-4}:\\ \;\;\;\;\frac{1}{x} \cdot \left(e^{x} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020035 +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))