Average Error: 39.9 → 0.4
Time: 4.1s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.656199158469729814779219934095522148709 \cdot 10^{-4}:\\ \;\;\;\;-1 \cdot \frac{1 - e^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1}{6} \cdot {x}^{2}}\right) + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.656199158469729814779219934095522148709 \cdot 10^{-4}:\\
\;\;\;\;-1 \cdot \frac{1 - e^{x}}{x}\\

\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 r96130 = x;
        double r96131 = exp(r96130);
        double r96132 = 1.0;
        double r96133 = r96131 - r96132;
        double r96134 = r96133 / r96130;
        return r96134;
}
↓
double f(double x) {
        double r96135 = x;
        double r96136 = -0.00016561991584697298;
        bool r96137 = r96135 <= r96136;
        double r96138 = -1.0;
        double r96139 = 1.0;
        double r96140 = exp(r96135);
        double r96141 = r96139 - r96140;
        double r96142 = r96141 / r96135;
        double r96143 = r96138 * r96142;
        double r96144 = 0.16666666666666666;
        double r96145 = 2.0;
        double r96146 = pow(r96135, r96145);
        double r96147 = r96144 * r96146;
        double r96148 = exp(r96147);
        double r96149 = log(r96148);
        double r96150 = 0.5;
        double r96151 = r96150 * r96135;
        double r96152 = 1.0;
        double r96153 = r96151 + r96152;
        double r96154 = r96149 + r96153;
        double r96155 = r96137 ? r96143 : r96154;
        return r96155;
}

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 < -0.00016561991584697298

    1. Initial program 0.1

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

    2. Taylor expanded around -inf 0.1

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

    if -0.00016561991584697298 < x

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.6

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

    4. Applied add-log-exp0.6

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.656199158469729814779219934095522148709 \cdot 10^{-4}:\\ \;\;\;\;-1 \cdot \frac{1 - e^{x}}{x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{1}{6} \cdot {x}^{2}}\right) + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019318 
(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))