Average Error: 41.7 → 0.6
Time: 2.8s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.99740808223063815:\\ \;\;\;\;\frac{e^{x}}{\left(\sqrt[3]{\frac{{\left(e^{2}\right)}^{x} - 1 \cdot 1}{e^{x} + 1}} \cdot \sqrt[3]{\frac{{\left(e^{2}\right)}^{x} - 1 \cdot 1}{e^{x} + 1}}\right) \cdot \sqrt[3]{\frac{{\left(e^{2}\right)}^{x} - 1 \cdot 1}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.99740808223063815:\\
\;\;\;\;\frac{e^{x}}{\left(\sqrt[3]{\frac{{\left(e^{2}\right)}^{x} - 1 \cdot 1}{e^{x} + 1}} \cdot \sqrt[3]{\frac{{\left(e^{2}\right)}^{x} - 1 \cdot 1}{e^{x} + 1}}\right) \cdot \sqrt[3]{\frac{{\left(e^{2}\right)}^{x} - 1 \cdot 1}{e^{x} + 1}}}\\

\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) <= 0.9974080822306381)) {
		VAR = (exp(x) / ((cbrt(((pow(exp(2.0), x) - (1.0 * 1.0)) / (exp(x) + 1.0))) * cbrt(((pow(exp(2.0), x) - (1.0 * 1.0)) / (exp(x) + 1.0)))) * cbrt(((pow(exp(2.0), 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

Original41.7
Target41.2
Herbie0.6
\[\frac{1}{1 - e^{-x}}\]

Derivation

  1. Split input into 2 regimes

  2. if (exp x) < 0.9974080822306381

    1. Initial program 0.0

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

    3. Applied flip--0.0

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

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

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

    6. Applied exp-prod0.0

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

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

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

    8. Applied exp-prod0.0

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

    9. Applied pow-prod-down0.0

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

    10. Simplified0.0

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

    12. Applied add-cube-cbrt0.0

      \[\leadsto \frac{e^{x}}{\color{blue}{\left(\sqrt[3]{\frac{{\left(e^{2}\right)}^{x} - 1 \cdot 1}{e^{x} + 1}} \cdot \sqrt[3]{\frac{{\left(e^{2}\right)}^{x} - 1 \cdot 1}{e^{x} + 1}}\right) \cdot \sqrt[3]{\frac{{\left(e^{2}\right)}^{x} - 1 \cdot 1}{e^{x} + 1}}}}\]

    if 0.9974080822306381 < (exp x)

    1. Initial program 62.0

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

    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

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.99740808223063815:\\ \;\;\;\;\frac{e^{x}}{\left(\sqrt[3]{\frac{{\left(e^{2}\right)}^{x} - 1 \cdot 1}{e^{x} + 1}} \cdot \sqrt[3]{\frac{{\left(e^{2}\right)}^{x} - 1 \cdot 1}{e^{x} + 1}}\right) \cdot \sqrt[3]{\frac{{\left(e^{2}\right)}^{x} - 1 \cdot 1}{e^{x} + 1}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{2} + \left(\frac{1}{12} \cdot x + \frac{1}{x}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020091 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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