Average Error: 40.0 → 0.3
Time: 7.4s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.02234544586829302 \cdot 10^{-4}:\\ \;\;\;\;\frac{\sqrt{e^{x}} + \sqrt{1}}{\frac{x}{\sqrt{e^{x}} - \sqrt{1}}}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.02234544586829302 \cdot 10^{-4}:\\
\;\;\;\;\frac{\sqrt{e^{x}} + \sqrt{1}}{\frac{x}{\sqrt{e^{x}} - \sqrt{1}}}\\

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

\end{array}
double f(double x) {
        double r98165 = x;
        double r98166 = exp(r98165);
        double r98167 = 1.0;
        double r98168 = r98166 - r98167;
        double r98169 = r98168 / r98165;
        return r98169;
}
↓
double f(double x) {
        double r98170 = x;
        double r98171 = -0.0002022345445868293;
        bool r98172 = r98170 <= r98171;
        double r98173 = exp(r98170);
        double r98174 = sqrt(r98173);
        double r98175 = 1.0;
        double r98176 = sqrt(r98175);
        double r98177 = r98174 + r98176;
        double r98178 = r98174 - r98176;
        double r98179 = r98170 / r98178;
        double r98180 = r98177 / r98179;
        double r98181 = 1.0;
        double r98182 = 0.16666666666666666;
        double r98183 = r98170 * r98182;
        double r98184 = 0.5;
        double r98185 = r98183 + r98184;
        double r98186 = r98170 * r98185;
        double r98187 = r98181 + r98186;
        double r98188 = r98172 ? r98180 : r98187;
        return r98188;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.0
Target40.5
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 < -0.0002022345445868293

    1. Initial program 0.1

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

    3. Applied add-sqr-sqrt0.1

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

    4. Applied add-sqr-sqrt0.1

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

    5. Applied difference-of-squares0.1

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

    6. Applied associate-/l*0.1

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

    if -0.0002022345445868293 < 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. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.02234544586829302 \cdot 10^{-4}:\\ \;\;\;\;\frac{\sqrt{e^{x}} + \sqrt{1}}{\frac{x}{\sqrt{e^{x}} - \sqrt{1}}}\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020047 
(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))