Average Error: 40.1 → 0.3
Time: 11.2s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.238801238577521569503292475644684600411 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{e^{x}} + \sqrt{1}} \cdot \sqrt[3]{\sqrt{e^{x}} + \sqrt{1}}\right) \cdot \left(\sqrt[3]{\sqrt{e^{x}} + \sqrt{1}} \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{2}, \mathsf{fma}\left(\frac{1}{6}, x \cdot x, 1\right)\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.238801238577521569503292475644684600411 \cdot 10^{-4}:\\
\;\;\;\;\frac{\left(\sqrt[3]{\sqrt{e^{x}} + \sqrt{1}} \cdot \sqrt[3]{\sqrt{e^{x}} + \sqrt{1}}\right) \cdot \left(\sqrt[3]{\sqrt{e^{x}} + \sqrt{1}} \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)\right)}{x}\\

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

\end{array}
double f(double x) {
        double r55300 = x;
        double r55301 = exp(r55300);
        double r55302 = 1.0;
        double r55303 = r55301 - r55302;
        double r55304 = r55303 / r55300;
        return r55304;
}
↓
double f(double x) {
        double r55305 = x;
        double r55306 = -0.00022388012385775216;
        bool r55307 = r55305 <= r55306;
        double r55308 = exp(r55305);
        double r55309 = sqrt(r55308);
        double r55310 = 1.0;
        double r55311 = sqrt(r55310);
        double r55312 = r55309 + r55311;
        double r55313 = cbrt(r55312);
        double r55314 = r55313 * r55313;
        double r55315 = r55309 - r55311;
        double r55316 = r55313 * r55315;
        double r55317 = r55314 * r55316;
        double r55318 = r55317 / r55305;
        double r55319 = 0.5;
        double r55320 = 0.16666666666666666;
        double r55321 = r55305 * r55305;
        double r55322 = 1.0;
        double r55323 = fma(r55320, r55321, r55322);
        double r55324 = fma(r55305, r55319, r55323);
        double r55325 = r55307 ? r55318 : r55324;
        return r55325;
}

Error

Bits error versus x

Target

Original40.1
Target40.6
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 < -0.00022388012385775216

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied difference-of-squares0.1

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

    6. Simplified0.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt{1} + \sqrt{e^{x}}\right)} \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}{x}\]
    7. Using strategy rm

    8. Applied add-cube-cbrt0.1

      \[\leadsto \frac{\color{blue}{\left(\left(\sqrt[3]{\sqrt{1} + \sqrt{e^{x}}} \cdot \sqrt[3]{\sqrt{1} + \sqrt{e^{x}}}\right) \cdot \sqrt[3]{\sqrt{1} + \sqrt{e^{x}}}\right)} \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}{x}\]

    9. Applied associate-*l*0.1

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{1} + \sqrt{e^{x}}} \cdot \sqrt[3]{\sqrt{1} + \sqrt{e^{x}}}\right) \cdot \left(\sqrt[3]{\sqrt{1} + \sqrt{e^{x}}} \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)\right)}}{x}\]

    10. Simplified0.1

      \[\leadsto \frac{\left(\sqrt[3]{\sqrt{1} + \sqrt{e^{x}}} \cdot \sqrt[3]{\sqrt{1} + \sqrt{e^{x}}}\right) \cdot \color{blue}{\left(\sqrt[3]{\sqrt{e^{x}} + \sqrt{1}} \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)\right)}}{x}\]

    if -0.00022388012385775216 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.4

      \[\leadsto \color{blue}{\frac{1}{2} \cdot x + \left(\frac{1}{6} \cdot {x}^{2} + 1\right)}\]

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.238801238577521569503292475644684600411 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{e^{x}} + \sqrt{1}} \cdot \sqrt[3]{\sqrt{e^{x}} + \sqrt{1}}\right) \cdot \left(\sqrt[3]{\sqrt{e^{x}} + \sqrt{1}} \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{1}{2}, \mathsf{fma}\left(\frac{1}{6}, x \cdot x, 1\right)\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019194 +o rules:numerics
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1.0) (> x -1.0)) (/ (- (exp x) 1.0) (log (exp x))) (/ (- (exp x) 1.0) x))

  (/ (- (exp x) 1.0) x))