Average Error: 41.2 → 0.5
Time: 2.4s
Precision: binary64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.99533251375977216:\\ \;\;\;\;\frac{\sqrt{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt{e^{x}}}{\sqrt[3]{{\left(\sqrt{e^{x}} - \sqrt{1}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.99533251375977216:\\
\;\;\;\;\frac{\sqrt{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt{e^{x}}}{\sqrt[3]{{\left(\sqrt{e^{x}} - \sqrt{1}\right)}^{3}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\

\end{array}
double code(double x) {
	return ((double) (((double) exp(x)) / ((double) (((double) exp(x)) - 1.0))));
}
↓
double code(double x) {
	double VAR;
	if ((((double) exp(x)) <= 0.9953325137597722)) {
		VAR = ((double) (((double) (((double) sqrt(((double) exp(x)))) / ((double) (((double) sqrt(((double) exp(x)))) + ((double) sqrt(1.0)))))) * ((double) (((double) sqrt(((double) exp(x)))) / ((double) cbrt(((double) pow(((double) (((double) sqrt(((double) exp(x)))) - ((double) sqrt(1.0)))), 3.0))))))));
	} else {
		VAR = ((double) (0.5 + ((double) (((double) (0.08333333333333333 * x)) + ((double) (1.0 / x))))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.2
Target40.9
Herbie0.5
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.99533251375977216

    1. Initial program 0.0

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

    3. Applied add-sqr-sqrt0.0

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

    4. Applied add-sqr-sqrt0.0

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

    5. Applied difference-of-squares0.0

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

    6. Applied add-sqr-sqrt0.0

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

    7. Applied times-frac0.0

      \[\leadsto \color{blue}{\frac{\sqrt{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt{e^{x}}}{\sqrt{e^{x}} - \sqrt{1}}}\]
    8. Using strategy rm

    9. Applied add-cbrt-cube0.0

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

    10. Simplified0.0

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

    if 0.99533251375977216 < (exp x)

    1. Initial program 62.0

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

    2. Taylor expanded around 0 0.7

      \[\leadsto \color{blue}{\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.99533251375977216:\\ \;\;\;\;\frac{\sqrt{e^{x}}}{\sqrt{e^{x}} + \sqrt{1}} \cdot \frac{\sqrt{e^{x}}}{\sqrt[3]{{\left(\sqrt{e^{x}} - \sqrt{1}\right)}^{3}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020149 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1.0 (- 1.0 (exp (neg x))))

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