Average Error: 41.6 → 0.6
Time: 2.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.99622329995855252:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{1 - \frac{1}{e^{x}}}}}{\sqrt[3]{1 - \frac{1}{e^{x}}}}}{\sqrt[3]{1 - \frac{1}{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.99622329995855252:\\
\;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{1 - \frac{1}{e^{x}}}}}{\sqrt[3]{1 - \frac{1}{e^{x}}}}}{\sqrt[3]{1 - \frac{1}{e^{x}}}}\\

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

\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)) <= 0.9962232999585525)) {
		VAR = ((double) (((double) (((double) (1.0 / ((double) cbrt(((double) (1.0 - ((double) (1.0 / ((double) exp(x)))))))))) / ((double) cbrt(((double) (1.0 - ((double) (1.0 / ((double) exp(x)))))))))) / ((double) cbrt(((double) (1.0 - ((double) (1.0 / ((double) exp(x))))))))));
	} else {
		VAR = ((double) (((double) fma(0.08333333333333333, x, ((double) (1.0 / x)))) + 0.5));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original41.6
Target41.2
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9962232999585525

    1. Initial program 0.0

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

    3. Applied clear-num0.0

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

    4. Simplified0.0

      \[\leadsto \frac{1}{\color{blue}{1 - \frac{1}{e^{x}}}}\]
    5. Using strategy rm

    6. Applied add-cube-cbrt0.0

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

    7. Applied associate-/r*0.0

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

    8. Simplified0.0

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

    if 0.9962232999585525 < (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. Simplified0.9

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

  4. Final simplification0.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.99622329995855252:\\ \;\;\;\;\frac{\frac{\frac{1}{\sqrt[3]{1 - \frac{1}{e^{x}}}}}{\sqrt[3]{1 - \frac{1}{e^{x}}}}}{\sqrt[3]{1 - \frac{1}{e^{x}}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{12}, x, \frac{1}{x}\right) + \frac{1}{2}\\ \end{array}\]

Reproduce

herbie shell --seed 2020113 +o rules:numerics
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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