Average Error: 39.8 → 0.3
Time: 6.8s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.2153975327080830811602579766272924644 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.2153975327080830811602579766272924644 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}}{x}\\

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

\end{array}
double f(double x) {
        double r89162 = x;
        double r89163 = exp(r89162);
        double r89164 = 1.0;
        double r89165 = r89163 - r89164;
        double r89166 = r89165 / r89162;
        return r89166;
}
↓
double f(double x) {
        double r89167 = x;
        double r89168 = -0.00012153975327080831;
        bool r89169 = r89167 <= r89168;
        double r89170 = r89167 + r89167;
        double r89171 = exp(r89170);
        double r89172 = 1.0;
        double r89173 = r89172 * r89172;
        double r89174 = r89171 - r89173;
        double r89175 = exp(r89167);
        double r89176 = r89172 + r89175;
        double r89177 = r89174 / r89176;
        double r89178 = r89177 / r89167;
        double r89179 = 0.16666666666666666;
        double r89180 = 0.5;
        double r89181 = fma(r89179, r89167, r89180);
        double r89182 = 1.0;
        double r89183 = fma(r89167, r89181, r89182);
        double r89184 = r89169 ? r89178 : r89183;
        return r89184;
}

Error

Bits error versus x

Target

Original39.8
Target40.2
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 < -0.00012153975327080831

    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. Simplified0.0

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

    5. Simplified0.0

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

    if -0.00012153975327080831 < x

    1. Initial program 60.0

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

    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

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.2153975327080830811602579766272924644 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{e^{x + x} - 1 \cdot 1}{1 + e^{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \mathsf{fma}\left(\frac{1}{6}, x, \frac{1}{2}\right), 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019351 +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))