Average Error: 40.1 → 0.3
Time: 10.9s
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 r56055 = x;
        double r56056 = exp(r56055);
        double r56057 = 1.0;
        double r56058 = r56056 - r56057;
        double r56059 = r56058 / r56055;
        return r56059;
}
↓
double f(double x) {
        double r56060 = x;
        double r56061 = -0.00022388012385775216;
        bool r56062 = r56060 <= r56061;
        double r56063 = exp(r56060);
        double r56064 = sqrt(r56063);
        double r56065 = 1.0;
        double r56066 = sqrt(r56065);
        double r56067 = r56064 + r56066;
        double r56068 = cbrt(r56067);
        double r56069 = r56068 * r56068;
        double r56070 = r56064 - r56066;
        double r56071 = r56068 * r56070;
        double r56072 = r56069 * r56071;
        double r56073 = r56072 / r56060;
        double r56074 = 0.5;
        double r56075 = 0.16666666666666666;
        double r56076 = r56060 * r56060;
        double r56077 = 1.0;
        double r56078 = fma(r56075, r56076, r56077);
        double r56079 = fma(r56060, r56074, r56078);
        double r56080 = r56062 ? r56073 : r56079;
        return r56080;
}

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))