Average Error: 39.7 → 0.3
Time: 8.5s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le \frac{-1858094157389103}{9223372036854775808}:\\ \;\;\;\;\frac{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le \frac{-1858094157389103}{9223372036854775808}:\\
\;\;\;\;\frac{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}{x}\\

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

\end{array}
double f(double x) {
        double r95079 = x;
        double r95080 = exp(r95079);
        double r95081 = 1.0;
        double r95082 = r95080 - r95081;
        double r95083 = r95082 / r95079;
        return r95083;
}
↓
double f(double x) {
        double r95084 = x;
        double r95085 = -1858094157389103.0;
        double r95086 = 9.223372036854776e+18;
        double r95087 = r95085 / r95086;
        bool r95088 = r95084 <= r95087;
        double r95089 = exp(r95084);
        double r95090 = sqrt(r95089);
        double r95091 = 1.0;
        double r95092 = sqrt(r95091);
        double r95093 = r95090 + r95092;
        double r95094 = r95090 - r95092;
        double r95095 = r95093 * r95094;
        double r95096 = r95095 / r95084;
        double r95097 = 1.0;
        double r95098 = 6.0;
        double r95099 = r95097 / r95098;
        double r95100 = 2.0;
        double r95101 = pow(r95084, r95100);
        double r95102 = r95099 * r95101;
        double r95103 = r95097 / r95100;
        double r95104 = r95103 * r95084;
        double r95105 = r95104 + r95097;
        double r95106 = r95102 + r95105;
        double r95107 = r95088 ? r95096 : r95106;
        return r95107;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.7
Target40.1
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 < -0.00020145497221238883

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    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

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

    if -0.00020145497221238883 < x

    1. Initial program 60.1

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le \frac{-1858094157389103}{9223372036854775808}:\\ \;\;\;\;\frac{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \left(\sqrt{e^{x}} - \sqrt{1}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019303 
(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))