Average Error: 39.8 → 0.3
Time: 22.8s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9.238900741085779 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1 - e^{\left(x + \left(x + x\right)\right) \cdot 3} \cdot e^{\left(x + \left(x + x\right)\right) \cdot 3}}{-1 - e^{\left(x + \left(x + x\right)\right) \cdot 3}}}{e^{3 \cdot x} \cdot \left(e^{3 \cdot x} + 1\right) + 1}}{e^{x} \cdot \left(e^{x} + 1\right) + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9.238900741085779 \cdot 10^{-05}:\\
\;\;\;\;\frac{\frac{\frac{\frac{1 - e^{\left(x + \left(x + x\right)\right) \cdot 3} \cdot e^{\left(x + \left(x + x\right)\right) \cdot 3}}{-1 - e^{\left(x + \left(x + x\right)\right) \cdot 3}}}{e^{3 \cdot x} \cdot \left(e^{3 \cdot x} + 1\right) + 1}}{e^{x} \cdot \left(e^{x} + 1\right) + 1}}{x}\\

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

\end{array}
double f(double x) {
        double r3490171 = x;
        double r3490172 = exp(r3490171);
        double r3490173 = 1.0;
        double r3490174 = r3490172 - r3490173;
        double r3490175 = r3490174 / r3490171;
        return r3490175;
}
↓
double f(double x) {
        double r3490176 = x;
        double r3490177 = -9.238900741085779e-05;
        bool r3490178 = r3490176 <= r3490177;
        double r3490179 = 1.0;
        double r3490180 = r3490176 + r3490176;
        double r3490181 = r3490176 + r3490180;
        double r3490182 = 3.0;
        double r3490183 = r3490181 * r3490182;
        double r3490184 = exp(r3490183);
        double r3490185 = r3490184 * r3490184;
        double r3490186 = r3490179 - r3490185;
        double r3490187 = -1.0;
        double r3490188 = r3490187 - r3490184;
        double r3490189 = r3490186 / r3490188;
        double r3490190 = r3490182 * r3490176;
        double r3490191 = exp(r3490190);
        double r3490192 = r3490191 + r3490179;
        double r3490193 = r3490191 * r3490192;
        double r3490194 = r3490193 + r3490179;
        double r3490195 = r3490189 / r3490194;
        double r3490196 = exp(r3490176);
        double r3490197 = r3490196 + r3490179;
        double r3490198 = r3490196 * r3490197;
        double r3490199 = r3490198 + r3490179;
        double r3490200 = r3490195 / r3490199;
        double r3490201 = r3490200 / r3490176;
        double r3490202 = 0.5;
        double r3490203 = 0.16666666666666666;
        double r3490204 = r3490203 * r3490176;
        double r3490205 = r3490202 + r3490204;
        double r3490206 = r3490176 * r3490205;
        double r3490207 = r3490206 + r3490179;
        double r3490208 = r3490178 ? r3490201 : r3490207;
        return r3490208;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target39.0
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 < -9.238900741085779e-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

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

    5. Simplified0.1

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

    7. Applied flip3-+0.1

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

    8. Simplified0.0

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

    9. Simplified0.0

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

    11. Applied flip-+0.1

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

    if -9.238900741085779e-05 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.4

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

    3. Simplified0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9.238900741085779 \cdot 10^{-05}:\\ \;\;\;\;\frac{\frac{\frac{\frac{1 - e^{\left(x + \left(x + x\right)\right) \cdot 3} \cdot e^{\left(x + \left(x + x\right)\right) \cdot 3}}{-1 - e^{\left(x + \left(x + x\right)\right) \cdot 3}}}{e^{3 \cdot x} \cdot \left(e^{3 \cdot x} + 1\right) + 1}}{e^{x} \cdot \left(e^{x} + 1\right) + 1}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot x\right) + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2019146 
(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))