Average Error: 39.8 → 0.4
Time: 15.0s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.6704963527402913 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{{\left({\left(e^{x}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{{1}^{3} \cdot \left({\left(e^{x}\right)}^{3} + {1}^{3}\right) + {\left(e^{x}\right)}^{6}}}{\left(x \cdot \sqrt{1 \cdot \left(e^{x} + 1\right) + e^{x + x}}\right) \cdot \sqrt{1 \cdot \left(e^{x} + 1\right) + e^{x + x}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.6704963527402913 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{{\left({\left(e^{x}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{{1}^{3} \cdot \left({\left(e^{x}\right)}^{3} + {1}^{3}\right) + {\left(e^{x}\right)}^{6}}}{\left(x \cdot \sqrt{1 \cdot \left(e^{x} + 1\right) + e^{x + x}}\right) \cdot \sqrt{1 \cdot \left(e^{x} + 1\right) + e^{x + x}}}\\

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

\end{array}
double f(double x) {
        double r87634 = x;
        double r87635 = exp(r87634);
        double r87636 = 1.0;
        double r87637 = r87635 - r87636;
        double r87638 = r87637 / r87634;
        return r87638;
}
↓
double f(double x) {
        double r87639 = x;
        double r87640 = -0.00016704963527402913;
        bool r87641 = r87639 <= r87640;
        double r87642 = exp(r87639);
        double r87643 = 3.0;
        double r87644 = pow(r87642, r87643);
        double r87645 = pow(r87644, r87643);
        double r87646 = 1.0;
        double r87647 = pow(r87646, r87643);
        double r87648 = pow(r87647, r87643);
        double r87649 = r87645 - r87648;
        double r87650 = r87644 + r87647;
        double r87651 = r87647 * r87650;
        double r87652 = 6.0;
        double r87653 = pow(r87642, r87652);
        double r87654 = r87651 + r87653;
        double r87655 = r87649 / r87654;
        double r87656 = r87642 + r87646;
        double r87657 = r87646 * r87656;
        double r87658 = r87639 + r87639;
        double r87659 = exp(r87658);
        double r87660 = r87657 + r87659;
        double r87661 = sqrt(r87660);
        double r87662 = r87639 * r87661;
        double r87663 = r87662 * r87661;
        double r87664 = r87655 / r87663;
        double r87665 = 0.16666666666666666;
        double r87666 = r87639 * r87665;
        double r87667 = 0.5;
        double r87668 = r87666 + r87667;
        double r87669 = r87639 * r87668;
        double r87670 = 1.0;
        double r87671 = r87669 + r87670;
        double r87672 = r87641 ? r87664 : r87671;
        return r87672;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target40.2
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 < -0.00016704963527402913

    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. Applied associate-/l/0.1

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

    5. Simplified0.1

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

    7. Applied add-sqr-sqrt0.1

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

    8. Applied associate-*r*0.1

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

    10. Applied flip3--0.1

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

    11. Simplified0.1

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

    if -0.00016704963527402913 < x

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.6704963527402913 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{{\left({\left(e^{x}\right)}^{3}\right)}^{3} - {\left({1}^{3}\right)}^{3}}{{1}^{3} \cdot \left({\left(e^{x}\right)}^{3} + {1}^{3}\right) + {\left(e^{x}\right)}^{6}}}{\left(x \cdot \sqrt{1 \cdot \left(e^{x} + 1\right) + e^{x + x}}\right) \cdot \sqrt{1 \cdot \left(e^{x} + 1\right) + e^{x + x}}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019195 
(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))