Average Error: 39.3 → 0.3
Time: 18.0s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -0.00014547784294540906:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(e^{\left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot x}\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -0.00014547784294540906:\\
\;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\

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

\end{array}
double f(double x) {
        double r3280707 = x;
        double r3280708 = exp(r3280707);
        double r3280709 = 1.0;
        double r3280710 = r3280708 - r3280709;
        double r3280711 = r3280710 / r3280707;
        return r3280711;
}
↓
double f(double x) {
        double r3280712 = x;
        double r3280713 = -0.00014547784294540906;
        bool r3280714 = r3280712 <= r3280713;
        double r3280715 = exp(r3280712);
        double r3280716 = r3280715 / r3280712;
        double r3280717 = 1.0;
        double r3280718 = r3280717 / r3280712;
        double r3280719 = r3280716 - r3280718;
        double r3280720 = 0.5;
        double r3280721 = 0.16666666666666666;
        double r3280722 = r3280712 * r3280721;
        double r3280723 = r3280720 + r3280722;
        double r3280724 = r3280723 * r3280712;
        double r3280725 = exp(r3280724);
        double r3280726 = log(r3280725);
        double r3280727 = r3280717 + r3280726;
        double r3280728 = r3280714 ? r3280719 : r3280727;
        return r3280728;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.3
Target38.5
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 < -0.00014547784294540906

    1. Initial program 0.1

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

    3. Applied div-sub0.1

      \[\leadsto \color{blue}{\frac{e^{x}}{x} - \frac{1}{x}}\]

    if -0.00014547784294540906 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

      \[\leadsto \color{blue}{1 + x \cdot \left(\frac{1}{6} \cdot x + \frac{1}{2}\right)}\]
    4. Using strategy rm

    5. Applied add-log-exp0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -0.00014547784294540906:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;1 + \log \left(e^{\left(\frac{1}{2} + x \cdot \frac{1}{6}\right) \cdot x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019139 
(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))