Average Error: 39.8 → 0.3
Time: 3.2s
Precision: binary64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -2.06732437852957151 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \sqrt[3]{{\left(\sqrt{e^{x}} - \sqrt{1}\right)}^{3}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -2.06732437852957151 \cdot 10^{-4}:\\
\;\;\;\;\frac{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \sqrt[3]{{\left(\sqrt{e^{x}} - \sqrt{1}\right)}^{3}}}{x}\\

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

\end{array}
double code(double x) {
	return ((double) (((double) (((double) exp(x)) - 1.0)) / x));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -0.00020673243785295715)) {
		VAR = ((double) (((double) (((double) (((double) sqrt(((double) exp(x)))) + ((double) sqrt(1.0)))) * ((double) cbrt(((double) pow(((double) (((double) sqrt(((double) exp(x)))) - ((double) sqrt(1.0)))), 3.0)))))) / x));
	} else {
		VAR = ((double) (((double) (0.16666666666666666 * ((double) pow(x, 2.0)))) + ((double) (((double) (0.5 * x)) + 1.0))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original39.8
Target40.3
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 < -2.06732437852957151e-4

    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-cbrt-cube0.1

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

    8. Simplified0.1

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

    if -2.06732437852957151e-4 < x

    1. Initial program 60.2

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

    2. Taylor expanded around 0 0.4

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

  4. Final simplification0.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -2.06732437852957151 \cdot 10^{-4}:\\ \;\;\;\;\frac{\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \sqrt[3]{{\left(\sqrt{e^{x}} - \sqrt{1}\right)}^{3}}}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{6} \cdot {x}^{2} + \left(\frac{1}{2} \cdot x + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(FPCore (x)
  :name "Kahan's exp quotient"
  :precision binary64

  :herbie-target
  (if (and (< x 1.0) (> x -1.0)) (/ (- (exp x) 1.0) (log (exp x))) (/ (- (exp x) 1.0) x))

  (/ (- (exp x) 1.0) x))