Average Error: 40.1 → 0.3
Time: 3.9s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -8.83342028199871082 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{e^{x + x} \cdot e^{x} - {1}^{3}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -8.83342028199871082 \cdot 10^{-5}:\\
\;\;\;\;\frac{\frac{e^{x + x} \cdot e^{x} - {1}^{3}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}}{x}\\

\mathbf{else}:\\
\;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\

\end{array}
double f(double x) {
        double r80930 = x;
        double r80931 = exp(r80930);
        double r80932 = 1.0;
        double r80933 = r80931 - r80932;
        double r80934 = r80933 / r80930;
        return r80934;
}
↓
double f(double x) {
        double r80935 = x;
        double r80936 = -8.833420281998711e-05;
        bool r80937 = r80935 <= r80936;
        double r80938 = r80935 + r80935;
        double r80939 = exp(r80938);
        double r80940 = exp(r80935);
        double r80941 = r80939 * r80940;
        double r80942 = 1.0;
        double r80943 = 3.0;
        double r80944 = pow(r80942, r80943);
        double r80945 = r80941 - r80944;
        double r80946 = r80942 + r80940;
        double r80947 = r80942 * r80946;
        double r80948 = r80947 + r80939;
        double r80949 = r80945 / r80948;
        double r80950 = r80949 / r80935;
        double r80951 = 2.0;
        double r80952 = pow(r80935, r80951);
        double r80953 = 0.16666666666666666;
        double r80954 = r80935 * r80953;
        double r80955 = 0.5;
        double r80956 = r80954 + r80955;
        double r80957 = r80952 * r80956;
        double r80958 = r80957 + r80935;
        double r80959 = r80958 / r80935;
        double r80960 = r80937 ? r80950 : r80959;
        return r80960;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.1
Target40.5
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 < -8.833420281998711e-05

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.1

      \[\leadsto \frac{\frac{{\left(e^{x}\right)}^{3} - {1}^{3}}{\color{blue}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}}}{x}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt0.1

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

    7. Applied unpow-prod-down0.1

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

    8. Simplified0.1

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

    9. Simplified0.1

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

    if -8.833420281998711e-05 < x

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.5

      \[\leadsto \frac{\color{blue}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}}{x}\]

    3. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -8.83342028199871082 \cdot 10^{-5}:\\ \;\;\;\;\frac{\frac{e^{x + x} \cdot e^{x} - {1}^{3}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020039 
(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))