Average Error: 39.5 → 0.2
Time: 2.8s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.338894298495746 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} \cdot e^{x} - 1 \cdot 1}}{\frac{x}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.338894298495746 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} \cdot e^{x} - 1 \cdot 1}}{\frac{x}{e^{x} - 1}}\\

\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 r79819 = x;
        double r79820 = exp(r79819);
        double r79821 = 1.0;
        double r79822 = r79820 - r79821;
        double r79823 = r79822 / r79819;
        return r79823;
}
↓
double f(double x) {
        double r79824 = x;
        double r79825 = -0.0001338894298495746;
        bool r79826 = r79824 <= r79825;
        double r79827 = 1.0;
        double r79828 = -r79827;
        double r79829 = r79824 + r79824;
        double r79830 = exp(r79829);
        double r79831 = fma(r79828, r79827, r79830);
        double r79832 = exp(r79824);
        double r79833 = r79832 * r79832;
        double r79834 = r79827 * r79827;
        double r79835 = r79833 - r79834;
        double r79836 = r79831 / r79835;
        double r79837 = r79832 - r79827;
        double r79838 = r79824 / r79837;
        double r79839 = r79836 / r79838;
        double r79840 = 0.16666666666666666;
        double r79841 = 2.0;
        double r79842 = pow(r79824, r79841);
        double r79843 = 0.5;
        double r79844 = 1.0;
        double r79845 = fma(r79843, r79824, r79844);
        double r79846 = fma(r79840, r79842, r79845);
        double r79847 = r79826 ? r79839 : r79846;
        return r79847;
}

Error

Bits error versus x

Target

Original39.5
Target39.9
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0001338894298495746

    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

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

    6. Applied flip-+0.1

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

    7. Applied associate-/r/0.1

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

    8. Applied associate-/l*0.1

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

    if -0.0001338894298495746 < 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

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.338894298495746 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} \cdot e^{x} - 1 \cdot 1}}{\frac{x}{e^{x} - 1}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 +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))