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

↓

\[\sqrt[3]{{\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}^{3}}\]

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

↓

\sqrt[3]{{\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}^{3}}

double code(double x) {
	return ((1.0 / (x - 1.0)) + (x / (x + 1.0)));
}

↓

double code(double x) {
	return cbrt(pow(((1.0 / (x - 1.0)) + (x / (x + 1.0))), 3.0));
}

Try it out

In
Out

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Derivation

Initial program 0.0
\[\frac{1}{x - 1} + \frac{x}{x + 1}\]
Using strategy rm
Applied add-cbrt-cube0.0
\[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)\right) \cdot \left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}}\]
Simplified0.0
\[\leadsto \sqrt[3]{\color{blue}{{\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}^{3}}}\]
Final simplification0.0
\[\leadsto \sqrt[3]{{\left(\frac{1}{x - 1} + \frac{x}{x + 1}\right)}^{3}}\]

herbie shell --seed 2020102 +o rules:numerics
(FPCore (x)
  :name "Asymptote B"
  :precision binary64
  (+ (/ 1 (- x 1)) (/ x (+ x 1))))