Derivation

Split input into 2 regimes
if (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))) < 4.178112254527817e-10
1. Initial program 59.2
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Taylor expanded around inf 0.6
  \[\leadsto \color{blue}{-\left(3 \cdot \frac{1}{{x}^{3}} + \left(3 \cdot \frac{1}{x} + \frac{1}{{x}^{2}}\right)\right)}\]
3. Applied simplify0.3
  \[\leadsto \color{blue}{\frac{-3}{x} - \frac{1 + \frac{3}{x}}{x \cdot x}}\]
if 4.178112254527817e-10 < (- (/ x (+ x 1)) (/ (+ x 1) (- x 1)))
1. Initial program 0.2
  \[\frac{x}{x + 1} - \frac{x + 1}{x - 1}\]
2. Using strategy rm
3. Applied flip3--0.2
  \[\leadsto \frac{x}{x + 1} - \frac{x + 1}{\color{blue}{\frac{{x}^{3} - {1}^{3}}{x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)}}}\]
4. Applied associate-/r/0.2
  \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{x + 1}{{x}^{3} - {1}^{3}} \cdot \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right)}\]
5. Applied simplify0.2
  \[\leadsto \frac{x}{x + 1} - \color{blue}{\frac{1 + x}{{x}^{3} - 1}} \cdot \left(x \cdot x + \left(1 \cdot 1 + x \cdot 1\right)\right)\]
Recombined 2 regimes into one program.
Applied simplify0.3
\[\leadsto \color{blue}{\begin{array}{l} \mathbf{if}\;\frac{x}{1 + x} - \frac{1 + x}{x - 1} \le 4.178112254527817 \cdot 10^{-10}:\\ \;\;\;\;\frac{-3}{x} - \frac{\frac{3}{x} + 1}{x \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{1 + x} - \frac{1 + x}{{x}^{3} - 1} \cdot \left(\left(1 + x\right) + x \cdot x\right)\\ \end{array}}\]

Runtime

herbie shell --seed 2018170 
(FPCore (x)
  :name "Asymptote C"
  (- (/ x (+ x 1)) (/ (+ x 1) (- x 1))))

In
Out