Average Error: 41.1 → 0.6
Time: 3.8s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.94930661802810024:\\ \;\;\;\;\frac{e^{x}}{\sqrt[3]{{\left(e^{x \cdot 3} - {1}^{3}\right)}^{3}}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.94930661802810024:\\
\;\;\;\;\frac{e^{x}}{\sqrt[3]{{\left(e^{x \cdot 3} - {1}^{3}\right)}^{3}}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)\\

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

\end{array}
double code(double x) {
	return (exp(x) / (exp(x) - 1.0));
}
↓
double code(double x) {
	double VAR;
	if ((exp(x) <= 0.9493066180281002)) {
		VAR = ((exp(x) / cbrt(pow((exp((x * 3.0)) - pow(1.0, 3.0)), 3.0))) * ((exp(x) * exp(x)) + ((1.0 * 1.0) + (exp(x) * 1.0))));
	} else {
		VAR = (0.5 + ((0.08333333333333333 * x) + (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

Original41.1
Target40.7
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9493066180281002

    1. Initial program 0.0

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

    3. Applied flip3--0.0

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

    4. Applied associate-/r/0.0

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

    6. Applied pow-exp0.0

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

    8. Applied add-cbrt-cube0.0

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

    9. Simplified0.0

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

    if 0.9493066180281002 < (exp x)

    1. Initial program 61.7

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

    2. Taylor expanded around 0 0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.94930661802810024:\\ \;\;\;\;\frac{e^{x}}{\sqrt[3]{{\left(e^{x \cdot 3} - {1}^{3}\right)}^{3}}} \cdot \left(e^{x} \cdot e^{x} + \left(1 \cdot 1 + e^{x} \cdot 1\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020106 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

  :herbie-target
  (/ 1 (- 1 (exp (- x))))

  (/ (exp x) (- (exp x) 1)))