Average Error: 40.8 → 0.8
Time: 2.9s
Precision: 64
\[\frac{e^{x}}{e^{x} - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{2}}{x} + \left(\frac{7}{720} \cdot {x}^{3} + \frac{-1}{12} \cdot x\right)\right) \cdot \left(e^{x} + 1\right)\\ \end{array}\]
\frac{e^{x}}{e^{x} - 1}
↓
\begin{array}{l}
\mathbf{if}\;e^{x} \le 0.0:\\
\;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\

\mathbf{else}:\\
\;\;\;\;\left(\frac{\frac{1}{2}}{x} + \left(\frac{7}{720} \cdot {x}^{3} + \frac{-1}{12} \cdot x\right)\right) \cdot \left(e^{x} + 1\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)) <= 0.0)) {
		VAR = ((double) (1.0 / ((double) (1.0 - ((double) (1.0 / ((double) exp(x))))))));
	} else {
		VAR = ((double) (((double) (((double) (0.5 / x)) + ((double) (((double) (0.009722222222222222 * ((double) pow(x, 3.0)))) + ((double) (-0.08333333333333333 * x)))))) * ((double) (((double) exp(x)) + 1.0))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original40.8
Target40.4
Herbie0.8
\[\frac{1}{1 - e^{-x}}\]

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 clear-num0

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

    4. Simplified0

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

    if 0.0 < (exp x)

    1. Initial program 61.5

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

    3. Applied flip--61.5

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

    4. Applied associate-/r/61.5

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

    5. Taylor expanded around 0 1.2

      \[\leadsto \color{blue}{\left(\left(\frac{7}{720} \cdot {x}^{3} + \frac{1}{2} \cdot \frac{1}{x}\right) - \frac{1}{12} \cdot x\right)} \cdot \left(e^{x} + 1\right)\]

    6. Simplified1.2

      \[\leadsto \color{blue}{\left(\frac{\frac{1}{2}}{x} + \left(\frac{7}{720} \cdot {x}^{3} + \frac{-1}{12} \cdot x\right)\right)} \cdot \left(e^{x} + 1\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.8

    \[\leadsto \begin{array}{l} \mathbf{if}\;e^{x} \le 0.0:\\ \;\;\;\;\frac{1}{1 - \frac{1}{e^{x}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{\frac{1}{2}}{x} + \left(\frac{7}{720} \cdot {x}^{3} + \frac{-1}{12} \cdot x\right)\right) \cdot \left(e^{x} + 1\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020126 
(FPCore (x)
  :name "expq2 (section 3.11)"
  :precision binary64

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

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