Average Error: 39.8 → 0.3
Time: 5.5s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.7806619917958195 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{{\left(e^{x}\right)}^{3}} \cdot \sqrt{{\left(e^{x}\right)}^{3}} - {1}^{3}} \cdot \sqrt[3]{\sqrt{{\left(e^{x}\right)}^{3}} \cdot \sqrt{{\left(e^{x}\right)}^{3}} - {1}^{3}}\right) \cdot \frac{\frac{\sqrt[3]{\sqrt{{\left(e^{x}\right)}^{3}} \cdot \sqrt{{\left(e^{x}\right)}^{3}} - {1}^{3}}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.7806619917958195 \cdot 10^{-4}:\\
\;\;\;\;\left(\sqrt[3]{\sqrt{{\left(e^{x}\right)}^{3}} \cdot \sqrt{{\left(e^{x}\right)}^{3}} - {1}^{3}} \cdot \sqrt[3]{\sqrt{{\left(e^{x}\right)}^{3}} \cdot \sqrt{{\left(e^{x}\right)}^{3}} - {1}^{3}}\right) \cdot \frac{\frac{\sqrt[3]{\sqrt{{\left(e^{x}\right)}^{3}} \cdot \sqrt{{\left(e^{x}\right)}^{3}} - {1}^{3}}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}}{x}\\

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

\end{array}
double f(double x) {
        double r87899 = x;
        double r87900 = exp(r87899);
        double r87901 = 1.0;
        double r87902 = r87900 - r87901;
        double r87903 = r87902 / r87899;
        return r87903;
}
↓
double f(double x) {
        double r87904 = x;
        double r87905 = -0.00017806619917958195;
        bool r87906 = r87904 <= r87905;
        double r87907 = exp(r87904);
        double r87908 = 3.0;
        double r87909 = pow(r87907, r87908);
        double r87910 = sqrt(r87909);
        double r87911 = r87910 * r87910;
        double r87912 = 1.0;
        double r87913 = pow(r87912, r87908);
        double r87914 = r87911 - r87913;
        double r87915 = cbrt(r87914);
        double r87916 = r87915 * r87915;
        double r87917 = r87912 + r87907;
        double r87918 = r87912 * r87917;
        double r87919 = r87904 + r87904;
        double r87920 = exp(r87919);
        double r87921 = r87918 + r87920;
        double r87922 = r87915 / r87921;
        double r87923 = r87922 / r87904;
        double r87924 = r87916 * r87923;
        double r87925 = 0.16666666666666666;
        double r87926 = 2.0;
        double r87927 = pow(r87904, r87926);
        double r87928 = r87925 * r87927;
        double r87929 = 0.5;
        double r87930 = r87929 * r87904;
        double r87931 = 1.0;
        double r87932 = r87930 + r87931;
        double r87933 = r87928 + r87932;
        double r87934 = r87906 ? r87924 : r87933;
        return r87934;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target40.3
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 < -0.00017806619917958195

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Simplified0.0

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

    6. Applied add-sqr-sqrt0.0

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

    8. Applied *-un-lft-identity0.0

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

    9. Applied *-un-lft-identity0.0

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

    10. Applied add-cube-cbrt0.0

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

    11. Applied times-frac0.0

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

    12. Applied times-frac0.0

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

    13. Simplified0.0

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

    if -0.00017806619917958195 < x

    1. Initial program 60.1

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

    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. Recombined 2 regimes into one program.

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.7806619917958195 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt[3]{\sqrt{{\left(e^{x}\right)}^{3}} \cdot \sqrt{{\left(e^{x}\right)}^{3}} - {1}^{3}} \cdot \sqrt[3]{\sqrt{{\left(e^{x}\right)}^{3}} \cdot \sqrt{{\left(e^{x}\right)}^{3}} - {1}^{3}}\right) \cdot \frac{\frac{\sqrt[3]{\sqrt{{\left(e^{x}\right)}^{3}} \cdot \sqrt{{\left(e^{x}\right)}^{3}} - {1}^{3}}}{1 \cdot \left(1 + e^{x}\right) + e^{x + x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020036 
(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))