Average Error: 39.8 → 0.3
Time: 4.7s
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 r76974 = x;
        double r76975 = exp(r76974);
        double r76976 = 1.0;
        double r76977 = r76975 - r76976;
        double r76978 = r76977 / r76974;
        return r76978;
}
↓
double f(double x) {
        double r76979 = x;
        double r76980 = -0.00017806619917958195;
        bool r76981 = r76979 <= r76980;
        double r76982 = exp(r76979);
        double r76983 = 3.0;
        double r76984 = pow(r76982, r76983);
        double r76985 = sqrt(r76984);
        double r76986 = r76985 * r76985;
        double r76987 = 1.0;
        double r76988 = pow(r76987, r76983);
        double r76989 = r76986 - r76988;
        double r76990 = cbrt(r76989);
        double r76991 = r76990 * r76990;
        double r76992 = r76987 + r76982;
        double r76993 = r76987 * r76992;
        double r76994 = r76979 + r76979;
        double r76995 = exp(r76994);
        double r76996 = r76993 + r76995;
        double r76997 = r76990 / r76996;
        double r76998 = r76997 / r76979;
        double r76999 = r76991 * r76998;
        double r77000 = 0.16666666666666666;
        double r77001 = 2.0;
        double r77002 = pow(r76979, r77001);
        double r77003 = r77000 * r77002;
        double r77004 = 0.5;
        double r77005 = r77004 * r76979;
        double r77006 = 1.0;
        double r77007 = r77005 + r77006;
        double r77008 = r77003 + r77007;
        double r77009 = r76981 ? r76999 : r77008;
        return r77009;
}

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))