Average Error: 41.9 → 1.0
Time: 3.2s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\sqrt[3]{{\left(\frac{e^{x}}{e^{x} - 1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.0:\\
\;\;\;\;\sqrt[3]{{\left(\frac{e^{x}}{e^{x} - 1}\right)}^{3}}\\

\mathbf{else}:\\
\;\;\;\;\frac{e^{x}}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\\

\end{array}
double code(double x) {
	return (exp(x) / (exp(x) - 1.0));
}
↓
double code(double x) {
	double temp;
	if ((exp(x) <= 0.0)) {
		temp = cbrt(pow((exp(x) / (exp(x) - 1.0)), 3.0));
	} else {
		temp = (exp(x) / ((0.5 * pow(x, 2.0)) + ((0.16666666666666666 * pow(x, 3.0)) + x)));
	}
	return temp;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.9
Target41.5
Herbie1.0
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.0

    1. Initial program 0

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

    3. Applied add-cbrt-cube0

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

    4. Applied add-cbrt-cube0

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

    5. Applied cbrt-undiv0

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

    6. Simplified0

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

    if 0.0 < (exp x)

    1. Initial program 61.4

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

    2. Taylor expanded around 0 1.4

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

  4. Final simplification1.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\sqrt[3]{{\left(\frac{e^{x}}{e^{x} - 1}\right)}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x}}{\frac{1}{2} \cdot {x}^{2} + \left(\frac{1}{6} \cdot {x}^{3} + x\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020066 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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