Average Error: 29.3 → 0.2
Time: 17.0s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -15417.9220111292761 \lor \neg \left(x \le 14949.242297652785\right):\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x}{x + 1} + \frac{x + 1}{x - 1}, \frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}\right)}^{3}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -15417.9220111292761 \lor \neg \left(x \le 14949.242297652785\right):\\
\;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\

\mathbf{else}:\\
\;\;\;\;\sqrt[3]{{\left(\frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x}{x + 1} + \frac{x + 1}{x - 1}, \frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}\right)}^{3}}\\

\end{array}
double code(double x) {
	return ((double) (((double) (x / ((double) (x + 1.0)))) - ((double) (((double) (x + 1.0)) / ((double) (x - 1.0))))));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -15417.922011129276) || !(x <= 14949.242297652785))) {
		VAR = ((double) (((double) (((double) -(1.0)) / ((double) pow(x, 2.0)))) - ((double) fma(3.0, ((double) (1.0 / x)), ((double) (3.0 * ((double) (1.0 / ((double) pow(x, 3.0))))))))));
	} else {
		VAR = ((double) cbrt(((double) pow(((double) (((double) (((double) pow(((double) (x / ((double) (x + 1.0)))), 3.0)) - ((double) pow(((double) (((double) (x + 1.0)) / ((double) (x - 1.0)))), 3.0)))) / ((double) fma(((double) (((double) (x + 1.0)) / ((double) (x - 1.0)))), ((double) (((double) (x / ((double) (x + 1.0)))) + ((double) (((double) (x + 1.0)) / ((double) (x - 1.0)))))), ((double) (((double) (x / ((double) (x + 1.0)))) * ((double) (x / ((double) (x + 1.0)))))))))), 3.0))));
	}
	return VAR;
}

Error

Bits error versus x

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if x < -15417.922011129276 or 14949.242297652785 < x

    1. Initial program 59.5

      \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]

    2. Taylor expanded around inf 0.3

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

    3. Simplified0.3

      \[\leadsto \color{blue}{\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)}\]

    if -15417.922011129276 < x < 14949.242297652785

    1. Initial program 0.1

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

    3. Applied flip3--0.1

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

    4. Simplified0.1

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

    6. Applied add-cbrt-cube0.1

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

    7. Applied add-cbrt-cube0.1

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

    8. Applied cbrt-undiv0.1

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

    9. Simplified0.1

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

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -15417.9220111292761 \lor \neg \left(x \le 14949.242297652785\right):\\ \;\;\;\;\frac{-1}{{x}^{2}} - \mathsf{fma}\left(3, \frac{1}{x}, 3 \cdot \frac{1}{{x}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt[3]{{\left(\frac{{\left(\frac{x}{x + 1}\right)}^{3} - {\left(\frac{x + 1}{x - 1}\right)}^{3}}{\mathsf{fma}\left(\frac{x + 1}{x - 1}, \frac{x}{x + 1} + \frac{x + 1}{x - 1}, \frac{x}{x + 1} \cdot \frac{x}{x + 1}\right)}\right)}^{3}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020121 +o rules:numerics
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))