Average Error: 29.4 → 0.1
Time: 3.7s
Precision: 64
\[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
↓
\[\begin{array}{l} \mathbf{if}\;x \le -9397.35870742111 \lor \neg \left(x \le 8391.56628072750573\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{x}{\frac{x + 1}{x} \cdot {\left(\sqrt[3]{x + 1}\right)}^{3}} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}{\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} + \frac{x + 1}{x - 1}}\\ \end{array}\]
\frac{x}{x + 1} - \frac{x + 1}{x - 1}
↓
\begin{array}{l}
\mathbf{if}\;x \le -9397.35870742111 \lor \neg \left(x \le 8391.56628072750573\right):\\
\;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\

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

\end{array}
double code(double x) {
	return ((x / (x + 1.0)) - ((x + 1.0) / (x - 1.0)));
}
↓
double code(double x) {
	double VAR;
	if (((x <= -9397.35870742111) || !(x <= 8391.566280727506))) {
		VAR = (((-1.0 / pow(x, 2.0)) - (3.0 / x)) - (3.0 / pow(x, 3.0)));
	} else {
		VAR = ((1.0 * ((x / (((x + 1.0) / x) * pow(cbrt((x + 1.0)), 3.0))) - (((x + 1.0) / (x - 1.0)) * ((x + 1.0) / (x - 1.0))))) / (((x / (cbrt((x + 1.0)) * cbrt((x + 1.0)))) / cbrt((x + 1.0))) + ((x + 1.0) / (x - 1.0))));
	}
	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 < -9397.35870742111 or 8391.566280727506 < x

    1. Initial program 59.3

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

    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. Simplified0.0

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

    if -9397.35870742111 < x < 8391.566280727506

    1. Initial program 0.1

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

    3. Applied add-cube-cbrt0.1

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

    4. Applied associate-/r*0.1

      \[\leadsto \color{blue}{\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}}} - \frac{x + 1}{x - 1}\]
    5. Using strategy rm

    6. Applied flip--0.1

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

    7. Simplified0.1

      \[\leadsto \frac{\color{blue}{\left(-\frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right) + \frac{\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} \cdot x}{x + 1}}}{\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} + \frac{x + 1}{x - 1}}\]
    8. Using strategy rm

    9. Applied *-un-lft-identity0.1

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

    10. Applied *-un-lft-identity0.1

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

    11. Applied distribute-lft-out0.1

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

    12. Simplified0.1

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

  4. Final simplification0.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;x \le -9397.35870742111 \lor \neg \left(x \le 8391.56628072750573\right):\\ \;\;\;\;\left(\frac{-1}{{x}^{2}} - \frac{3}{x}\right) - \frac{3}{{x}^{3}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 \cdot \left(\frac{x}{\frac{x + 1}{x} \cdot {\left(\sqrt[3]{x + 1}\right)}^{3}} - \frac{x + 1}{x - 1} \cdot \frac{x + 1}{x - 1}\right)}{\frac{\frac{x}{\sqrt[3]{x + 1} \cdot \sqrt[3]{x + 1}}}{\sqrt[3]{x + 1}} + \frac{x + 1}{x - 1}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020079 
(FPCore (x)
  :name "Asymptote C"
  :precision binary64
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))