Average Error: 40.0 → 0.3
Time: 1.7m
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00014435066433665848:\\ \;\;\;\;\frac{\frac{e^{x + \left(x + x\right)} - 1}{\frac{1 - e^{x} \cdot e^{x}}{1 - e^{x}} + e^{x} \cdot e^{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)}{x}\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.00014435066433665848:\\
\;\;\;\;\frac{\frac{e^{x + \left(x + x\right)} - 1}{\frac{1 - e^{x} \cdot e^{x}}{1 - e^{x}} + e^{x} \cdot e^{x}}}{x}\\

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

\end{array}
double f(double x) {
        double r12262806 = x;
        double r12262807 = exp(r12262806);
        double r12262808 = 1.0;
        double r12262809 = r12262807 - r12262808;
        double r12262810 = r12262809 / r12262806;
        return r12262810;
}
↓
double f(double x) {
        double r12262811 = x;
        double r12262812 = -0.00014435066433665848;
        bool r12262813 = r12262811 <= r12262812;
        double r12262814 = r12262811 + r12262811;
        double r12262815 = r12262811 + r12262814;
        double r12262816 = exp(r12262815);
        double r12262817 = 1.0;
        double r12262818 = r12262816 - r12262817;
        double r12262819 = exp(r12262811);
        double r12262820 = r12262819 * r12262819;
        double r12262821 = r12262817 - r12262820;
        double r12262822 = r12262817 - r12262819;
        double r12262823 = r12262821 / r12262822;
        double r12262824 = r12262823 + r12262820;
        double r12262825 = r12262818 / r12262824;
        double r12262826 = r12262825 / r12262811;
        double r12262827 = 0.5;
        double r12262828 = 0.16666666666666666;
        double r12262829 = r12262811 * r12262828;
        double r12262830 = r12262827 + r12262829;
        double r12262831 = r12262811 * r12262811;
        double r12262832 = r12262830 * r12262831;
        double r12262833 = r12262811 + r12262832;
        double r12262834 = r12262833 / r12262811;
        double r12262835 = r12262813 ? r12262826 : r12262834;
        return r12262835;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.0
Target39.2
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 < -0.00014435066433665848

    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. Using strategy rm

    5. Applied cube-mult0.0

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

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

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

    7. Applied distribute-lft-out--0.0

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

    8. Simplified0.0

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

    10. Applied flip-+0.0

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

    if -0.00014435066433665848 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00014435066433665848:\\ \;\;\;\;\frac{\frac{e^{x + \left(x + x\right)} - 1}{\frac{1 - e^{x} \cdot e^{x}}{1 - e^{x}} + e^{x} \cdot e^{x}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot \left(x \cdot x\right)}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2019119 
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

  (/ (- (exp x) 1) x))