Average Error: 40.1 → 0.3
Time: 10.1s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.238801238577521569503292475644684600411 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt[3]{\sqrt{e^{x}} + \sqrt{1}} \cdot \sqrt[3]{\sqrt{e^{x}} + \sqrt{1}}\right) \cdot \left(\sqrt[3]{\sqrt{e^{x}} + \sqrt{1}} \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\\ \end{array}\]
\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}:\\
\;\;\;\;x \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\\

\end{array}
double f(double x) {
        double r74950 = x;
        double r74951 = exp(r74950);
        double r74952 = 1.0;
        double r74953 = r74951 - r74952;
        double r74954 = r74953 / r74950;
        return r74954;
}
↓
double f(double x) {
        double r74955 = x;
        double r74956 = -0.00022388012385775216;
        bool r74957 = r74955 <= r74956;
        double r74958 = exp(r74955);
        double r74959 = sqrt(r74958);
        double r74960 = 1.0;
        double r74961 = sqrt(r74960);
        double r74962 = r74959 + r74961;
        double r74963 = cbrt(r74962);
        double r74964 = r74963 * r74963;
        double r74965 = r74959 - r74961;
        double r74966 = r74963 * r74965;
        double r74967 = r74964 * r74966;
        double r74968 = r74967 / r74955;
        double r74969 = 0.16666666666666666;
        double r74970 = r74955 * r74969;
        double r74971 = 0.5;
        double r74972 = r74970 + r74971;
        double r74973 = r74955 * r74972;
        double r74974 = 1.0;
        double r74975 = r74973 + r74974;
        double r74976 = r74957 ? r74968 : r74975;
        return r74976;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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

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

  4. Final simplification0.3

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

Reproduce

herbie shell --seed 2019194 
(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))