Average Error: 39.8 → 0.4
Time: 2.6s
Precision: 64
\[\frac{e^{x} - 1}{x}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -1.8101971030998605 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{x}} - \sqrt{1}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + 1\\ \end{array}\]
\frac{e^{x} - 1}{x}
↓
\begin{array}{l}
\mathbf{if}\;x \le -1.8101971030998605 \cdot 10^{-4}:\\
\;\;\;\;\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{x}} - \sqrt{1}}{x}\\

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

\end{array}
double code(double x) {
	return ((double) (((double) (((double) exp(x)) - 1.0)) / x));
}
↓
double code(double x) {
	double VAR;
	if ((x <= -0.00018101971030998605)) {
		VAR = ((double) (((double) (((double) sqrt(((double) exp(x)))) + ((double) sqrt(1.0)))) * ((double) (((double) (((double) sqrt(((double) exp(x)))) - ((double) sqrt(1.0)))) / x))));
	} else {
		VAR = ((double) (((double) (x * ((double) (0.5 + ((double) (x * 0.16666666666666666)))))) + 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
Target40.2
Herbie0.4
\[\begin{array}{l} \mathbf{if}\;x \lt 1 \land x \gt -1:\\ \;\;\;\;\frac{e^{x} - 1}{\log \left(e^{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} - 1}{x}\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if x < -0.00018101971030998605

    1. Initial program 0.0

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

    3. Applied *-un-lft-identity0.0

      \[\leadsto \frac{e^{x} - 1}{\color{blue}{1 \cdot x}}\]

    4. Applied add-sqr-sqrt0.0

      \[\leadsto \frac{e^{x} - \color{blue}{\sqrt{1} \cdot \sqrt{1}}}{1 \cdot x}\]

    5. Applied add-sqr-sqrt0.1

      \[\leadsto \frac{\color{blue}{\sqrt{e^{x}} \cdot \sqrt{e^{x}}} - \sqrt{1} \cdot \sqrt{1}}{1 \cdot x}\]

    6. Applied difference-of-squares0.0

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

    7. Applied times-frac0.1

      \[\leadsto \color{blue}{\frac{\sqrt{e^{x}} + \sqrt{1}}{1} \cdot \frac{\sqrt{e^{x}} - \sqrt{1}}{x}}\]

    8. Simplified0.1

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

    if -0.00018101971030998605 < x

    1. Initial program 60.0

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

    2. Taylor expanded around 0 0.5

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

    4. Applied associate-+r+0.5

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

    5. Simplified0.5

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

  4. Final simplification0.4

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -1.8101971030998605 \cdot 10^{-4}:\\ \;\;\;\;\left(\sqrt{e^{x}} + \sqrt{1}\right) \cdot \frac{\sqrt{e^{x}} - \sqrt{1}}{x}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{1}{2} + x \cdot \frac{1}{6}\right) + 1\\ \end{array}\]

Reproduce

herbie shell --seed 2020123 
(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))