Average Error: 39.6 → 0.3
Time: 3.0s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.9657375756151034 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \log \left(e^{\sqrt{e^{x}} - \sqrt{1}}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.9657375756151034 \cdot 10^{-4}:\\
\;\;\;\;\frac{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \log \left(e^{\sqrt{e^{x}} - \sqrt{1}}\right)}{x}\\

\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 r81443 = x;
        double r81444 = exp(r81443);
        double r81445 = 1.0;
        double r81446 = r81444 - r81445;
        double r81447 = r81446 / r81443;
        return r81447;
}
↓
double f(double x) {
        double r81448 = x;
        double r81449 = -0.00019657375756151034;
        bool r81450 = r81448 <= r81449;
        double r81451 = exp(r81448);
        double r81452 = sqrt(r81451);
        double r81453 = 1.0;
        double r81454 = sqrt(r81453);
        double r81455 = r81452 + r81454;
        double r81456 = r81452 - r81454;
        double r81457 = exp(r81456);
        double r81458 = log(r81457);
        double r81459 = r81455 * r81458;
        double r81460 = r81459 / r81448;
        double r81461 = 2.0;
        double r81462 = pow(r81448, r81461);
        double r81463 = 0.16666666666666666;
        double r81464 = r81448 * r81463;
        double r81465 = 0.5;
        double r81466 = r81464 + r81465;
        double r81467 = r81462 * r81466;
        double r81468 = r81467 + r81448;
        double r81469 = r81468 / r81448;
        double r81470 = r81450 ? r81460 : r81469;
        return r81470;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.6
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 < -0.00019657375756151034

    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. Using strategy rm

    7. Applied add-log-exp0.1

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

    8. Applied add-log-exp0.1

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

    9. Applied diff-log0.1

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

    10. Simplified0.1

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

    if -0.00019657375756151034 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.5

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

    3. Simplified0.5

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.9657375756151034 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \log \left(e^{\sqrt{e^{x}} - \sqrt{1}}\right)}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{{x}^{2} \cdot \left(x \cdot \frac{1}{6} + \frac{1}{2}\right) + x}{x}\\ \end{array}\]

Reproduce

herbie shell --seed 2020065 
(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))