Average Error: 39.1 → 0.3
Time: 12.7s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -7.91242022931417 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\frac{\frac{e^{\left(9 \cdot x + 9 \cdot x\right) + 9 \cdot x} + -1}{\left(1 + e^{9 \cdot x} \cdot e^{9 \cdot x}\right) + e^{9 \cdot x}}}{\left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right) + 1\right) + \left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right) \cdot \left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right)}}{e^{x} \cdot \left(e^{x} + 1\right) + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x + 1\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -7.91242022931417 \cdot 10^{-05}:\\
\;\;\;\;\frac{\frac{\frac{\frac{e^{\left(9 \cdot x + 9 \cdot x\right) + 9 \cdot x} + -1}{\left(1 + e^{9 \cdot x} \cdot e^{9 \cdot x}\right) + e^{9 \cdot x}}}{\left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right) + 1\right) + \left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right) \cdot \left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right)}}{e^{x} \cdot \left(e^{x} + 1\right) + 1}}{x}\\

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

\end{array}
double f(double x) {
        double r3768286 = x;
        double r3768287 = exp(r3768286);
        double r3768288 = 1.0;
        double r3768289 = r3768287 - r3768288;
        double r3768290 = r3768289 / r3768286;
        return r3768290;
}
↓
double f(double x) {
        double r3768291 = x;
        double r3768292 = -7.91242022931417e-05;
        bool r3768293 = r3768291 <= r3768292;
        double r3768294 = 9.0;
        double r3768295 = r3768294 * r3768291;
        double r3768296 = r3768295 + r3768295;
        double r3768297 = r3768296 + r3768295;
        double r3768298 = exp(r3768297);
        double r3768299 = -1.0;
        double r3768300 = r3768298 + r3768299;
        double r3768301 = 1.0;
        double r3768302 = exp(r3768295);
        double r3768303 = r3768302 * r3768302;
        double r3768304 = r3768301 + r3768303;
        double r3768305 = r3768304 + r3768302;
        double r3768306 = r3768300 / r3768305;
        double r3768307 = exp(r3768291);
        double r3768308 = r3768307 * r3768307;
        double r3768309 = r3768307 * r3768308;
        double r3768310 = r3768309 + r3768301;
        double r3768311 = r3768309 * r3768309;
        double r3768312 = r3768310 + r3768311;
        double r3768313 = r3768306 / r3768312;
        double r3768314 = r3768307 + r3768301;
        double r3768315 = r3768307 * r3768314;
        double r3768316 = r3768315 + r3768301;
        double r3768317 = r3768313 / r3768316;
        double r3768318 = r3768317 / r3768291;
        double r3768319 = 0.5;
        double r3768320 = 0.16666666666666666;
        double r3768321 = r3768320 * r3768291;
        double r3768322 = r3768319 + r3768321;
        double r3768323 = r3768322 * r3768291;
        double r3768324 = r3768323 + r3768301;
        double r3768325 = r3768293 ? r3768318 : r3768324;
        return r3768325;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.1
Target38.2
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 < -7.91242022931417e-05

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.1

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

    5. Simplified0.1

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

    7. Applied flip3-+0.1

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

    8. Simplified0.0

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

    9. Simplified0.0

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

    11. Applied flip3-+0.1

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

    12. Simplified0.0

      \[\leadsto \frac{\frac{\frac{\frac{\color{blue}{e^{\left(9 \cdot x + 9 \cdot x\right) + 9 \cdot x} + -1}}{e^{3 \cdot x + \left(3 \cdot x + 3 \cdot x\right)} \cdot e^{3 \cdot x + \left(3 \cdot x + 3 \cdot x\right)} + \left(-1 \cdot -1 - e^{3 \cdot x + \left(3 \cdot x + 3 \cdot x\right)} \cdot -1\right)}}{\left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right) \cdot \left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right) + \left(1 + e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right)}}{1 + e^{x} \cdot \left(1 + e^{x}\right)}}{x}\]

    13. Simplified0.0

      \[\leadsto \frac{\frac{\frac{\frac{e^{\left(9 \cdot x + 9 \cdot x\right) + 9 \cdot x} + -1}{\color{blue}{\left(e^{9 \cdot x} \cdot e^{9 \cdot x} + 1\right) + e^{9 \cdot x}}}}{\left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right) \cdot \left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right) + \left(1 + e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right)}}{1 + e^{x} \cdot \left(1 + e^{x}\right)}}{x}\]

    if -7.91242022931417e-05 < x

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -7.91242022931417 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\frac{\frac{e^{\left(9 \cdot x + 9 \cdot x\right) + 9 \cdot x} + -1}{\left(1 + e^{9 \cdot x} \cdot e^{9 \cdot x}\right) + e^{9 \cdot x}}}{\left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right) + 1\right) + \left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right) \cdot \left(e^{x} \cdot \left(e^{x} \cdot e^{x}\right)\right)}}{e^{x} \cdot \left(e^{x} + 1\right) + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{1}{2} + \frac{1}{6} \cdot x\right) \cdot x + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019165 
(FPCore (x)
  :name "Kahan's exp quotient"

  :herbie-target
  (if (and (< x 1) (> x -1)) (/ (- (exp x) 1) (log (exp x))) (/ (- (exp x) 1) x))

  (/ (- (exp x) 1) x))