Average Error: 40.0 → 0.4
Time: 3.0s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.67759019414094238 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}}\right) \cdot \left(\frac{1}{36} \cdot {x}^{2} + \left(\frac{1}{6} \cdot x + 1\right)\right)\right) \cdot \sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.67759019414094238 \cdot 10^{-4}:\\
\;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\

\mathbf{else}:\\
\;\;\;\;\left(\left(\left(\sqrt[3]{\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}}\right) \cdot \left(\frac{1}{36} \cdot {x}^{2} + \left(\frac{1}{6} \cdot x + 1\right)\right)\right) \cdot \sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\\

\end{array}
double f(double x) {
        double r97175 = x;
        double r97176 = exp(r97175);
        double r97177 = 1.0;
        double r97178 = r97176 - r97177;
        double r97179 = r97178 / r97175;
        return r97179;
}
↓
double f(double x) {
        double r97180 = x;
        double r97181 = -0.00016775901941409424;
        bool r97182 = r97180 <= r97181;
        double r97183 = exp(r97180);
        double r97184 = r97183 / r97180;
        double r97185 = 1.0;
        double r97186 = r97185 / r97180;
        double r97187 = r97184 - r97186;
        double r97188 = 0.16666666666666666;
        double r97189 = 2.0;
        double r97190 = pow(r97180, r97189);
        double r97191 = r97188 * r97190;
        double r97192 = 0.5;
        double r97193 = r97192 * r97180;
        double r97194 = 1.0;
        double r97195 = r97193 + r97194;
        double r97196 = r97191 + r97195;
        double r97197 = cbrt(r97196);
        double r97198 = cbrt(r97197);
        double r97199 = r97198 * r97198;
        double r97200 = r97199 * r97198;
        double r97201 = 0.027777777777777776;
        double r97202 = r97201 * r97190;
        double r97203 = r97188 * r97180;
        double r97204 = r97203 + r97194;
        double r97205 = r97202 + r97204;
        double r97206 = r97200 * r97205;
        double r97207 = r97206 * r97197;
        double r97208 = r97182 ? r97187 : r97207;
        return r97208;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.0
Target40.5
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.00016775901941409424

    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 -0.00016775901941409424 < x

    1. Initial program 60.0

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

    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. Using strategy rm

    4. Applied add-cube-cbrt0.5

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

    6. Applied add-cube-cbrt0.5

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

    7. Taylor expanded around 0 0.5

      \[\leadsto \left(\left(\left(\sqrt[3]{\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}}\right) \cdot \color{blue}{\left(\frac{1}{36} \cdot {x}^{2} + \left(\frac{1}{6} \cdot x + 1\right)\right)}\right) \cdot \sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.67759019414094238 \cdot 10^{-4}:\\ \;\;\;\;\frac{e^{x}}{x} - \frac{1}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\left(\sqrt[3]{\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}} \cdot \sqrt[3]{\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}}\right) \cdot \sqrt[3]{\sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}}\right) \cdot \left(\frac{1}{36} \cdot {x}^{2} + \left(\frac{1}{6} \cdot x + 1\right)\right)\right) \cdot \sqrt[3]{\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020046 
(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))