Average Error: 39.9 → 0.4
Time: 3.1s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.636611049659256037804022643200596576207 \cdot 10^{-4}:\\ \;\;\;\;\frac{\sqrt{{\left(e^{x}\right)}^{3}} + \sqrt{{1}^{3}}}{\frac{\left(1 \cdot \left(1 + e^{x}\right) + e^{x + x}\right) \cdot x}{\sqrt{{\left(e^{x}\right)}^{3}} - \sqrt{{1}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1}{6} \cdot {x}^{2}}\right) + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.636611049659256037804022643200596576207 \cdot 10^{-4}:\\
\;\;\;\;\frac{\sqrt{{\left(e^{x}\right)}^{3}} + \sqrt{{1}^{3}}}{\frac{\left(1 \cdot \left(1 + e^{x}\right) + e^{x + x}\right) \cdot x}{\sqrt{{\left(e^{x}\right)}^{3}} - \sqrt{{1}^{3}}}}\\

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

\end{array}
double f(double x) {
        double r105029 = x;
        double r105030 = exp(r105029);
        double r105031 = 1.0;
        double r105032 = r105030 - r105031;
        double r105033 = r105032 / r105029;
        return r105033;
}
↓
double f(double x) {
        double r105034 = x;
        double r105035 = -0.0001636611049659256;
        bool r105036 = r105034 <= r105035;
        double r105037 = exp(r105034);
        double r105038 = 3.0;
        double r105039 = pow(r105037, r105038);
        double r105040 = sqrt(r105039);
        double r105041 = 1.0;
        double r105042 = pow(r105041, r105038);
        double r105043 = sqrt(r105042);
        double r105044 = r105040 + r105043;
        double r105045 = r105041 + r105037;
        double r105046 = r105041 * r105045;
        double r105047 = r105034 + r105034;
        double r105048 = exp(r105047);
        double r105049 = r105046 + r105048;
        double r105050 = r105049 * r105034;
        double r105051 = r105040 - r105043;
        double r105052 = r105050 / r105051;
        double r105053 = r105044 / r105052;
        double r105054 = 0.16666666666666666;
        double r105055 = 2.0;
        double r105056 = pow(r105034, r105055);
        double r105057 = r105054 * r105056;
        double r105058 = exp(r105057);
        double r105059 = log(r105058);
        double r105060 = 0.5;
        double r105061 = r105060 * r105034;
        double r105062 = 1.0;
        double r105063 = r105061 + r105062;
        double r105064 = r105059 + r105063;
        double r105065 = r105036 ? r105053 : r105064;
        return r105065;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.9
Target40.4
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.0001636611049659256

    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. Applied associate-/l/0.0

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

    5. Simplified0.0

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

    7. Applied add-sqr-sqrt0.0

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

    8. Applied add-sqr-sqrt0.0

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

    9. Applied difference-of-squares0.0

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

    10. Applied associate-/l*0.0

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

    if -0.0001636611049659256 < x

    1. Initial program 60.2

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

    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. Using strategy rm

    4. Applied add-log-exp0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.636611049659256037804022643200596576207 \cdot 10^{-4}:\\ \;\;\;\;\frac{\sqrt{{\left(e^{x}\right)}^{3}} + \sqrt{{1}^{3}}}{\frac{\left(1 \cdot \left(1 + e^{x}\right) + e^{x + x}\right) \cdot x}{\sqrt{{\left(e^{x}\right)}^{3}} - \sqrt{{1}^{3}}}}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1}{6} \cdot {x}^{2}}\right) + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020001 
(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))