Average Error: 39.5 → 0.3
Time: 3.1s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.6731406755706916 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\sqrt{e^{x}} + \sqrt{1}}{\frac{x}{{\left(\sqrt{e^{x}}\right)}^{3} - {\left(\sqrt{1}\right)}^{3}}}}{\sqrt{e^{x}} \cdot \sqrt{e^{x}} + \left(\sqrt{1} \cdot \sqrt{1} + \sqrt{e^{x}} \cdot \sqrt{1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.6731406755706916 \cdot 10^{-4}:\\
\;\;\;\;\frac{\frac{\sqrt{e^{x}} + \sqrt{1}}{\frac{x}{{\left(\sqrt{e^{x}}\right)}^{3} - {\left(\sqrt{1}\right)}^{3}}}}{\sqrt{e^{x}} \cdot \sqrt{e^{x}} + \left(\sqrt{1} \cdot \sqrt{1} + \sqrt{e^{x}} \cdot \sqrt{1}\right)}\\

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

\end{array}
double f(double x) {
        double r74293 = x;
        double r74294 = exp(r74293);
        double r74295 = 1.0;
        double r74296 = r74294 - r74295;
        double r74297 = r74296 / r74293;
        return r74297;
}
↓
double f(double x) {
        double r74298 = x;
        double r74299 = -0.00016731406755706916;
        bool r74300 = r74298 <= r74299;
        double r74301 = exp(r74298);
        double r74302 = sqrt(r74301);
        double r74303 = 1.0;
        double r74304 = sqrt(r74303);
        double r74305 = r74302 + r74304;
        double r74306 = 3.0;
        double r74307 = pow(r74302, r74306);
        double r74308 = pow(r74304, r74306);
        double r74309 = r74307 - r74308;
        double r74310 = r74298 / r74309;
        double r74311 = r74305 / r74310;
        double r74312 = r74302 * r74302;
        double r74313 = r74304 * r74304;
        double r74314 = r74302 * r74304;
        double r74315 = r74313 + r74314;
        double r74316 = r74312 + r74315;
        double r74317 = r74311 / r74316;
        double r74318 = 2.0;
        double r74319 = pow(r74298, r74318);
        double r74320 = 0.16666666666666666;
        double r74321 = r74298 * r74320;
        double r74322 = 0.5;
        double r74323 = r74321 + r74322;
        double r74324 = r74319 * r74323;
        double r74325 = r74324 + r74298;
        double r74326 = r74325 / r74298;
        double r74327 = r74300 ? r74317 : r74326;
        return r74327;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.5
Target39.9
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 < -0.00016731406755706916

    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

      \[\leadsto \color{blue}{\frac{\sqrt{e^{x}} + \sqrt{1}}{\frac{x}{\sqrt{e^{x}} - \sqrt{1}}}}\]
    7. Using strategy rm

    8. Applied flip3--0.1

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

    9. Applied associate-/r/0.1

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

    10. Applied associate-/r*0.1

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

    if -0.00016731406755706916 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.6731406755706916 \cdot 10^{-4}:\\ \;\;\;\;\frac{\frac{\sqrt{e^{x}} + \sqrt{1}}{\frac{x}{{\left(\sqrt{e^{x}}\right)}^{3} - {\left(\sqrt{1}\right)}^{3}}}}{\sqrt{e^{x}} \cdot \sqrt{e^{x}} + \left(\sqrt{1} \cdot \sqrt{1} + \sqrt{e^{x}} \cdot \sqrt{1}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(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))