Average Error: 39.9 → 0.4
Time: 2.8s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.383162727126432478899631384194890415529 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.383162727126432478899631384194890415529 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}{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 r93763 = x;
        double r93764 = exp(r93763);
        double r93765 = 1.0;
        double r93766 = r93764 - r93765;
        double r93767 = r93766 / r93763;
        return r93767;
}
↓
double f(double x) {
        double r93768 = x;
        double r93769 = -0.00013831627271264325;
        bool r93770 = r93768 <= r93769;
        double r93771 = 1.0;
        double r93772 = -r93771;
        double r93773 = r93768 + r93768;
        double r93774 = exp(r93773);
        double r93775 = fma(r93772, r93771, r93774);
        double r93776 = exp(r93768);
        double r93777 = r93776 + r93771;
        double r93778 = r93775 / r93777;
        double r93779 = r93778 / r93768;
        double r93780 = 0.16666666666666666;
        double r93781 = 2.0;
        double r93782 = pow(r93768, r93781);
        double r93783 = 0.5;
        double r93784 = 1.0;
        double r93785 = fma(r93783, r93768, r93784);
        double r93786 = fma(r93780, r93782, r93785);
        double r93787 = r93770 ? r93779 : r93786;
        return r93787;
}

Error

Bits error versus x

Target

Original39.9
Target40.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.00013831627271264325

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

    4. Simplified0.0

      \[\leadsto \frac{\frac{\color{blue}{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}}{e^{x} + 1}}{x}\]

    if -0.00013831627271264325 < 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 simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.383162727126432478899631384194890415529 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 +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))