Average Error: 40.2 → 0.3
Time: 2.3s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.55293744862148258 \cdot 10^{-4}:\\ \;\;\;\;-1 \cdot \frac{1 - e^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.55293744862148258 \cdot 10^{-4}:\\
\;\;\;\;-1 \cdot \frac{1 - e^{x}}{x}\\

\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 r72655 = x;
        double r72656 = exp(r72655);
        double r72657 = 1.0;
        double r72658 = r72656 - r72657;
        double r72659 = r72658 / r72655;
        return r72659;
}
↓
double f(double x) {
        double r72660 = x;
        double r72661 = -0.00015529374486214826;
        bool r72662 = r72660 <= r72661;
        double r72663 = -1.0;
        double r72664 = 1.0;
        double r72665 = exp(r72660);
        double r72666 = r72664 - r72665;
        double r72667 = r72666 / r72660;
        double r72668 = r72663 * r72667;
        double r72669 = 0.16666666666666666;
        double r72670 = 2.0;
        double r72671 = pow(r72660, r72670);
        double r72672 = 0.5;
        double r72673 = 1.0;
        double r72674 = fma(r72672, r72660, r72673);
        double r72675 = fma(r72669, r72671, r72674);
        double r72676 = r72662 ? r72668 : r72675;
        return r72676;
}

Error

Bits error versus x

Target

Original40.2
Target40.7
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 < -0.00015529374486214826

    1. Initial program 0.1

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

    2. Taylor expanded around -inf 0.1

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

    if -0.00015529374486214826 < 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

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.55293744862148258 \cdot 10^{-4}:\\ \;\;\;\;-1 \cdot \frac{1 - e^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020003 +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))