Average Error: 39.8 → 0.3
Time: 3.2s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.2721971246653334 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.2721971246653334 \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 r103131 = x;
        double r103132 = exp(r103131);
        double r103133 = 1.0;
        double r103134 = r103132 - r103133;
        double r103135 = r103134 / r103131;
        return r103135;
}
↓
double f(double x) {
        double r103136 = x;
        double r103137 = -0.00012721971246653334;
        bool r103138 = r103136 <= r103137;
        double r103139 = 1.0;
        double r103140 = -r103139;
        double r103141 = r103136 + r103136;
        double r103142 = exp(r103141);
        double r103143 = fma(r103140, r103139, r103142);
        double r103144 = exp(r103136);
        double r103145 = r103144 + r103139;
        double r103146 = r103143 / r103145;
        double r103147 = r103146 / r103136;
        double r103148 = 0.16666666666666666;
        double r103149 = 2.0;
        double r103150 = pow(r103136, r103149);
        double r103151 = 0.5;
        double r103152 = 1.0;
        double r103153 = fma(r103151, r103136, r103152);
        double r103154 = fma(r103148, r103150, r103153);
        double r103155 = r103138 ? r103147 : r103154;
        return r103155;
}

Error

Bits error versus x

Target

Original39.8
Target40.3
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 < -0.00012721971246653334

    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

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

    if -0.00012721971246653334 < x

    1. Initial program 60.3

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

    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 simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.2721971246653334 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\mathsf{fma}\left(-1, 1, e^{x + x}\right)}{e^{x} + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, \mathsf{fma}\left(\frac{1}{2}, x, 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020081 +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))